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Closed-Loop Control of Flow-Induced Cavity Oscillations

Permanent Link: http://ufdc.ufl.edu/UFE0021633/00001

Material Information

Title: Closed-Loop Control of Flow-Induced Cavity Oscillations
Physical Description: 1 online resource (227 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: adaptive, cavity, control, flow
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Flow-induced cavity oscillations are a coupled flow-acoustic problem in which the inherent closed-loop system dynamics can lead to large unsteady pressure levels in and around the cavity, resulting in both broadband noise and discrete tones. This problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, and automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been working on this problem for more than fifty years. This is because not only is the physical nature of this problem rich and complex, but also it has become a standard test bed for controller deign and implementation in flow control. The ultimate goal of this research is to minimize the cavity acoustic tones and the broadband noise level over a range of freestream Mach numbers. Although open-loop and closed-loop control methodologies have been explored extensively in recent years, there are still some issues that need to be studied further. For example, a low-order theoretical model suitable for controller design does not exist. Most recent flow-induced cavity models are based either on Rossiter?s semi-expirical formula or a proper orthogonal decomposition (POD) based models. These models cannot be implemented in adaptive controller design directly. In addition, closed-loop control of high subsonic and supersonic flows remains an unexplored area. In order to achieve these objectives, an analytical system model is first developed in this research. This analytical model is a transfer function based model and it can be used as a potential model for controller design. Then, a MIMO system identification algorithm is derived and combined with the generalized prediction control (GPC) algorithm. The resultant integration of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics and reduce the flow-induced oscillations. A novel piezoelectric-driven synthetic jet actuator array is designed for this research. The resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. A validation vibration beam problem is used to demonstrate the combination of system ID and GPC algorithms. The result shows a ~20 dB reduction at the single resonance peak and a ~9 dB reduction of the integrated vibration levels. Both open-loop control and closed-loop control are applied to the flow-induced cavity oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced for both cases. The GPC controller can generate a series of control signals to drive the actuator array resulting in dB reduction for the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closed-loop controller (i.e., the OASPL reduction is 3 dB). The relevant flow physics and active flow control actuators are examined and explained in this research. The limitations of the present setup are discussed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cattafesta III, Louis N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021633:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021633/00001

Material Information

Title: Closed-Loop Control of Flow-Induced Cavity Oscillations
Physical Description: 1 online resource (227 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: adaptive, cavity, control, flow
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Flow-induced cavity oscillations are a coupled flow-acoustic problem in which the inherent closed-loop system dynamics can lead to large unsteady pressure levels in and around the cavity, resulting in both broadband noise and discrete tones. This problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, and automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been working on this problem for more than fifty years. This is because not only is the physical nature of this problem rich and complex, but also it has become a standard test bed for controller deign and implementation in flow control. The ultimate goal of this research is to minimize the cavity acoustic tones and the broadband noise level over a range of freestream Mach numbers. Although open-loop and closed-loop control methodologies have been explored extensively in recent years, there are still some issues that need to be studied further. For example, a low-order theoretical model suitable for controller design does not exist. Most recent flow-induced cavity models are based either on Rossiter?s semi-expirical formula or a proper orthogonal decomposition (POD) based models. These models cannot be implemented in adaptive controller design directly. In addition, closed-loop control of high subsonic and supersonic flows remains an unexplored area. In order to achieve these objectives, an analytical system model is first developed in this research. This analytical model is a transfer function based model and it can be used as a potential model for controller design. Then, a MIMO system identification algorithm is derived and combined with the generalized prediction control (GPC) algorithm. The resultant integration of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics and reduce the flow-induced oscillations. A novel piezoelectric-driven synthetic jet actuator array is designed for this research. The resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. A validation vibration beam problem is used to demonstrate the combination of system ID and GPC algorithms. The result shows a ~20 dB reduction at the single resonance peak and a ~9 dB reduction of the integrated vibration levels. Both open-loop control and closed-loop control are applied to the flow-induced cavity oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced for both cases. The GPC controller can generate a series of control signals to drive the actuator array resulting in dB reduction for the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closed-loop controller (i.e., the OASPL reduction is 3 dB). The relevant flow physics and active flow control actuators are examined and explained in this research. The limitations of the present setup are discussed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cattafesta III, Louis N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021633:00001


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CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS


By

QI SONG
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

































2008 Qi Song


































To my wife, Jingyan Wang; and my lovely son, Lawrence W. Song









ACKNOWLEDGMENTS

This study was performed while I was a member of the Interdisciplinary Microsystems

Group (IMG) in the Department of Mechanical and Aerospace at the University of Florida in

Gainesville, Florida, USA. First, I sincerely acknowledge my advisor, Dr. Lou Cattafesta, for

providing me with this opportunity and giving me so much precious advice during my course

time at UF. His guidance and encouragement always gave me sufficient confidence to conquer

any difficulty. I thank all of my colleagues in the IMG group for their invaluable assistance.

Finally, I appreciate my friends and my dear family for their tremendous consideration and

unselfish support during my journey.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IS T O F T A B L E S .............................................................................. ............... 8

LIST OF FIGURES .................................. .. ..... ..... ................. .9

LIST OF ABBREVIATION S ......... ............................................ .... .................. 14

A B S T R A C T ................................ ............................................................ 16

CHAPTER

1 INTRODUCTON ............................... ... .. .... ..... ................. 18

L iteratu re R ev iew .............................................................................2 0
Physical M odels................................................... 21
Physics-Based Models.................. .. ................................21
Num erical Simulations ......................... ........................ ......... 27
P O D -T ype M odels................ .... .... ...... .................... ............ ........ .. .......... ...... 28
On-Line System ID and Active Closed-Loop Control Methodologies...........................28
U resolved T technical Issues ........................................ ............................................33
Technical Objectives ............................... .. .. ... .... ................... 33
Approach and Outline ............... ................. .............................. ....... 34

2 SYSTEM IDENTIFICATION ALGORITHMS ....................................... ............... 38

O v erv iew ................... ...................3...................8..........
SISO IIR F ilter A lgorithm s .......................................................................... ....................39
IIR O E A lgorithm ........................................................................ .......... .. 40
IIR EE Algorithm ..................................... .............................41
IIR SM Algorithm ..................................... .. .... ...... .. ............41
IIR C E A lgorithm ............... .... .... ........ .. .......................................... ................. 4 1
Recursive IIR Filters Simulation Results and Analyses............................................41
Accuracy comparison for sufficient system.......................................................43
Accuracy comparison for insufficient system............................. ....................43
Convergence rate ............................................... .. ......................... 44
C om putational com plexity ............................................... ............................ 44
C o n c lu sio n s ..............................................................................4 5
M IM O IIR F ilter A lgorithm ......................................................................... ....................46

3 GENERALIZED PREDICTIVE CONTROL ALGORIHTM ............................................62

Introduction ................... .......... ...... ...........................................62
M IM O A adaptive G P C M odel ........................................................................ ...................63









MIMO Adaptive GPC Cost Function ........... .. ........ ............................ 66
M IM O A adaptive G PC Law ..................................................... ................................... 66
M IM O A adaptive GPC Optim um Solution ........................................... .....................67
MIMO Adaptive GPC Recursive Solution................................... ...............68

4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES ...........................................71

Schematic of the Vibration Beam Test Bed ............... .............. .....................71
System Identification Experimental Results........................................... .......................... 72
C om putational C om plexity ..................................................................... ..................72
Sy stem Id entification ............................................................................. .................... 7 3
Disturbance Effect ...................... .............. .... ................ 74
Closed-Loop Control Experiment Results.. .. ......................... .....................74
C om putational C om plexity ..................................................................... ..................74
Closed-Loop Results ......................................... .... .. ..... .............. .. 74
Estimated Order Effect ......... ... ........................ ....... 75
Predict Horizon Effect ......... ... .... ......... .. .. ......... ............. 76
Input W eight Effect ............. .... .... .. ................. ... .. .. ............. .. .............. 76
Disturbance Effect for Different SNR Levels During System ID..............................76
Su m m ary ......... .... ..... ....... ............ ....................................................77

5 WIND TUNNEL EXPERIMENTAL SETUP................................. 90

W ind T unnel F facility ................................................................................... ....................90
Test Section and C avity M odel.................................................... ............................... 91
Pressure/Temperature M easurement Systems ............................................. ............... 93
Facility Data Acquisition and Control Systems................................. ........................ 94
A ctu ato r S y ste m ................................................................................................................ 9 5

6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION .............................113

B a c k g ro u n d ........................................................................................................................... 1 1 3
Data Analysis M ethods..................................... .................................... ........ 115
Noise Floor of Unsteady Pressure Transducers................................................................116
Effects of Structural Vibrations on Unsteady Pressure Transducers ..................................116
Baseline Experimental Results and Analysis ............................................... ............... 117
Open-Loop Experim ental Results and Analysis............................................................... 118
Closed-Loop Experimental Results and Analysis ..................................... .................120

7 SUMMARY AND FUTURE WORK ....................................................... ...................140

Sum m ary of C contributions .................................................... ................................... 140
F utu re W ork ......................................................14 1

APPENDIX

A M A TR IX O PR A T IO N S ............................................................................ .....................143



6









Vector Derivatives ............... ................. ............ .................. ......... 143
D definition of V ectors ..................................... ........................ ...... ............... 143
Derivative of Scalar with Respect to Vector .....................................................143
Derivative of Vector with Respect to Vector ............................................................ 143
Second Derivative of Scalar With Respect to Vector (Hessian Matrix) .....................144
Table of Several Useful Vector Derivative Formulas ................................................144
Proof of the Formulas ....... .. ................................ ............ ................. 145
P roof (a) ...................................................................................................... ....... 145
P ro o f (b ) ...............................................................14 5
P ro o f (c ) ............................................................................................................ 1 4 6
P roof (d) ............... ......................146
The Chain Rule of the V ector Functions ............................................... .......... ..... 148
The Derivative of Scalar Functions Respect to a Matrix .................................................150

B CAVITY OSCILLATION MODELS .................................................155

R o ssiter M odel ...................................................................................................155
Linear Models of Cavity Flow Oscillations.................................. 156
Global Model for the Cavity Oscillations in Supersonic Flow ............................................159
Global Model for the Cavity Oscillations in Subsonic Flow.................... ..................163

C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS ...................... .......169

M IM O Sy stem Identification .......................................................................................... 169
Generalized Predictive Control Model ..................... ....... ...............171

D A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC
R E SO N A N C E S .............................................................................179

M ason's R ule ............................................. ..... ................................... 179
Global Model for a Cavity Oscillation in Supersonic Flow ................................................180
Global Model for a Cavity Oscillation in Subsonic Flow ...................................................183

E CENTER VELOCITY OF ACTUATOR ARRAY .............................................................196

F PARAMETRIC STUDY FOR OPEN-LOOP CONTROL ..................................... 207

L IST O F R E F E R E N C E S ....................................................................................................... 2 19

BIO GR A PH ICA L SK ETCH ...............................................................227









LIST OF TABLES


Table page

2-1 Summary of the IIR OE algorithm. .......................... ............. ...............................49

2-2 Sum m ary of the IIR EE algorithm ......... ................. .................................... ............... 50

2-3 Sum m ary of the IIR SM algorithm ........... ............... ............................. ............... 51

2-4 Sum m ary of the IIR CE algorithm ............................................................. .....................52

2-5 Simulation results of IIR algorithms for sufficient case .................................................53

2-6 Simulation results of IIR algorithms for insufficient case................................................54

2-7 Simulation conditions of IIR algorithms for sufficient case............................................55

2-8 Sum m ary of the IIR/LM S algorithm s.......................................................... ............... 56

4-1 Parameters selection of the vibration beam experiment ................ ......... .......................78

4-2 Summary of the results of the adaptive GPC algorithm.................... ................................ 79

5-1 Physical and piezoelectric properties of APC 850 device ...................................................100

5-2 Geometric properties and parameters for the actuator ....................................................... 101

5-3 Resonant frequencies with respective centerline velocities for each input voltage. ............102

A V sector derivative form ulas. ........................................................................ ...................154

D-1 Components of the Mason's formula for supersonic case...................................................189

D-2 Components of the Mason's formula for subsonic case.....................................................190

D-3 Components of the Mason's formula for subsonic case.....................................................192









LIST OF FIGURES


Figure page

1-1 Schematic illustrating flow-induced cavity resonance for an upstream turbulent
boundary layer. .......................................................................................................... ......36

1-2 Tam and Block (1978) model of acoustic wave field inside and outside the rectangular
cav ity ................... ............................................................ ................ 3 6

1-3 C classification of flow control. ...................................................................... ....................37

1-4 Block diagram of system ID and on-line control. ...................................... ............... 37

2-1 Linear time-invariant (LTI) IIR Filter Structure..................................................................57

2-2 Simulation structure of the adaptive IIR filter..................................................................... 57

2-3 z-plane of the test m odel ........... ... ................. ....... .. ...... .. .................. 58

2-4 3D plot of the MSOE performance surface of the insufficient order test system. .................58

2-5 Contour plot of the MSOE performance surface.................. ... ... ................ 59

2-6 Simulation results of weight track of the IIR algorithms for sufficient case..........................59

2-7 Simulation results of weight track of the IIR algorithms for insufficient case.....................60

2-8 Learning curve of IIR algorithms for sufficient case. ............. ........... .... ...........60

2-9 Computational complexity results from the experiment. ...................................................... 61

3-1 M odel predictive control strategy ................................................. .............................. 70

4-1 Schematic diagram of the vibration beam test bed ..................................................... 80

5-1 Schematic of the wind tunnel facility. ....................... ..................... ..........103

5-2 Schematic of the test section and the cavity model.......................................................... 103

5-3 Schematic of the control hardware setup ............. ................................ ... ............ 104

5-4 Bimorph bender disc actuator in parallel operation................................... ...............104

5-5 D designed ZN M F actuator array ......................... ....... ................................ ............... 105

5-6 Dimensions of the slot for designed actuator array. .................................. .................106

5-7 ZNMF actuator array mounted in wind tunnel.............................................................107









5-8 Bimorph 3 centerline rms velocities of the single unit piezoelectric based synthetic
actuator with different excitation sinusoid input signal...........................108

5-9 The comparison plot of the experiment and simulation result of the actuator design code
for bim orph 3 ............ ......... ................ ........................................ ........................ .... 109

5-10 Current saturation effects of the amplifier .............................. ..............110

5-11 Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic
treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25, K=0.7).........111

5-12 Schematic of a single periodic cell of the actuator jets and the proposed interaction
with the income ing boundary layer. ............................................................................112

6-1 Schematic of simplified wind tunnel and cavity regions acoustic resonances for
sub sonic flow .......................................................................... 123

6-2 Noise floor level comparison at different discrete Mach numbers with acoustic
treatment at trailing edge floor of the cavity with L/D=6 ............................ ..........123

6-3 x -acceleration unsteady power spectrum (dB ref. g) for case with acoustic treatment
and no cavity .............................................................................124

6-4 y -acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic treatment
and no cavity .............................................................................125

6-5 z -acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic treatment
and no cavity .............................................................................126

6-6 Spectrogram of pressure measurement (dB ref. 20e-6 Pa) on the trailing edge floor of
the cavity for the case with acoustic treatment and no cavity .............. .... ...............127

6-7 Spectrogram of pressure measurement (ref 20e-6 Pa) on the trailing edge cavity floor
without acoustic treatm ent at L/D=6. ........................................ .......................... 128

6-8 Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic
treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25, K=0.7).........129

6-9 Noise floor of the unsteady pressure level at the surface of the trailing edge of the cavity
with and without the actuator turned on. .............................................. ............... 130

6-10 Open-loop sinusoidal control results for flow-induced cavity oscillations at trailing
edge floor of the cavity. ................................................... ...... .. ........ .... 131

6-11 Running error variance plot for the system identification algorithm. .............................133

6-12 Closed-Loop active control result for flow-induced cavity oscillations at Mach 0.27 at
the trailing edge floor of the L/D =6 cavity. .......................................... ...............134









6-13 Input signal of the Closed-Loop active control result for flow-induced cavity
oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity........................135

6-14 Sensitivity function (Equation 4-1) of the closed-loop control for M=0.27 upstream
flow condition .. ................................................................................ 136

6-15 Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream flow
condition w ith varying estim ated order.. .......... .................... ................................ 137

6-16 Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream flow
condition with varying predictive horizon s. ....................................... ............... 138

6-17 Unsteady pressure level comparison between the open-loop control and closed-loop
control for M=0.27 upstream flow condition..... ..................... ............139

B Schem atic of R ossiter m odel. ...................... ....... ........ ...................................... 166

B-2 Block diagram of the linear model of the flow-induced cavity oscillations....................... 166

B-3 Block diagram of the reflection model. ........................................ ......................... 167

B-4 Global model for the cavity oscillations in supersonic flow.......................................... 167

B-5 Block diagram of the global model for a cavity oscillation in supersonic flow ................167

B-6 Global model for a cavity oscillation in subsonic flow. ................... ............................. 168

B-7 Block diagram of the global model for a cavity oscillation in subsonic flow. ....................168

D-1 Global model for a cavity oscillation in supersonic flow ..................................................193

D-2 Block diagram of the global model for a cavity oscillation in supersonic flow................193

D-3 Signal flow graph of the global model for a cavity oscillation in supersonic flow............. 194

D-4 Global model for a cavity oscillation in subsonic flow. ............................................. 194

D-5 Block diagram of the global model for a cavity oscillation in subsonic flow...................195

D-6 Signal flow graph of the global model for a cavity oscillation in subsonic flow..............195

E-1 Hot-wire measurement for actuator array slot la.................. .......................... 197

E-2 Hot-wire measurement for actuator array slot lb. .................................... .................198

E-3 Hot-wire measurement for actuator array slot 2a.............................................199

E-4 Hot-wire measurement for actuator array slot 2b.. ..................................... ............... 200









E-5 Hot-wire measurement for actuator array slot 3a..................................... .................201

E-6 Hot-wire measurement for actuator array slot 3b.. ..................................... ...............202

E-7 Hot-wire measurement for actuator array slot 4a ...... ......... ..............................203

E-8 Hot-wire measurement for actuator array slot 14b.. ............................... ...............204

E-9 Hot-wire measurement for actuator array slot 5a..................................... .................205

E-10 Hot-wire measurement for actuator array slot 5b.. ....................................................206

F-l Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
500 H z and 100 V pp voltage. ........................................ ............................................208

F-2 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
500 H z and 150 V pp voltage. ........................................ ............................................208

F-3 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
600 H z and 100 V pp voltage. ........................................ ............................................209

F-4 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
600 H z and 150 V pp voltage. ........................................ ............................................209

F-5 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
700 H z and 100 V pp voltage. ........................................ ............................................2 10

F-6 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
700 H z and 150 V pp voltage. ........................................ ............................................2 10

F-7 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
800 H z and 100 V pp voltage. ...................................................................... ..............2 11

F-8 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
800 H z and 150 V pp voltage. ....................................................................... .........2 11

F-9 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
900 H z and 100 V pp voltage. ........................................ ............................................2 12

F-10 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
900 H z and 150 V pp voltage. ........................................ ............................................2 12

F-11 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1000 H z and 100 V pp voltage. ............................................................. .....................2 13

F-12 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1000 H z and 150 V pp voltage. ............................................................. .....................2 13









F-13 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1100 H z and 100 V pp voltage. ...................................................................... ...........2 14

F-14 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1100 H z and 150 V pp voltage. ...................................................................... ...........2 14

F-15 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1200 H z and 100 V pp voltage. ............................................................. .....................2 15

F-16 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1200 H z and 150 V pp voltage. ............................................................. .....................2 15

F-17 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1300 H z and 100 V pp voltage. ............................................................. .....................2 16

F-18 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1300 H z and 150 V pp voltage. ............................................................. .....................2 16

F-19 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1400 H z and 100 V pp voltage. ............................................................. .....................217

F-20 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1400 H z and 150 V pp voltage. ............................................................. .....................2 17

F-21 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1500 H z and 100 V pp voltage. ............................................................. .....................2 18

F-22 Open-Loop control result for M=0.31 and excitation sinusoidal input with frequency
1500 H z and 150 V pp voltage. ............................................................. .....................2 18









LIST OF ABBREVIATIONS


D Cavity depth

L Cavity length

M Freestream flow Mach number

U. Freestream flow velocity

a Mean sound speed inside the cavity

m, Mode number (integer number 1,2...)

Wo Natural frequency of second order system

r Reflection coefficient

1 Damping ratio

a Phase lag factor

K Proportion of the vortices speed to the freestream speed

Y Ratio of the specific heats

2, Spacing of the vortices

ra Time delay inside the cavity

T, Time delay inside the shear layer

ADC Analog to digital converter

ARMA Autoregressive and moving-average

CARIMA Auto-regressive and integrated moving average

CE Composite error

DAC Digital to analog converter

DNS Direct Numerical Simulations

DSP Digital signal processing

EE Equation error









FFT Fast Fourier transform

FIR Finite impulse response

FRF Frequency response function

GPC Generalized predictive control

ID Identification

IIR Infinite impulse response

JTFA Joint-time frequency analysis

LES Large Eddy Simulations

LMS Least mean square

LQG Linear quadratic Gaussian

LTI Linear time-invariant

MIMO Multiple-input multiple-output

MPC Model predictive control

MSOE Mean square output error

OE Output error

PDF Probability density function

POD Proper orthogonal decomposition

RANS Reynolds Averaged Navier-stokes

RLS Recursive least square

SISO Single-input single-output

SM Steiglitz and McBride

SNR Signal to noise ratio

SPL Sound pressure level

STR Self-tuning regulator

TITO Two-input Two-output









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS

By

Qi Song

May 2008

Chair: Louis Cattafesta
Major: Aerospace Engineering

Flow-induced cavity oscillations are a coupled flow-acoustic problem in which the

inherent closed-loop system dynamics can lead to large unsteady pressure levels in and around

the cavity, resulting in both broadband noise and discrete tones. This problem exists in many

practical environments, such as landing gear bays and weapon delivery systems on aircraft, and

automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been

working on this problem for more than fifty years. This is because not only is the physical

nature of this problem rich and complex, but also it has become a standard test bed for controller

deign and implementation in flow control.

The ultimate goal of this research is to minimize the cavity acoustic tones and the

broadband noise level over a range of freestream Mach numbers. Although open-loop and

closed-loop control methodologies have been explored extensively in recent years, there are still

some issues that need to be studied further. For example, a low-order theoretical model suitable

for controller design does not exist. Most recent flow-induced cavity models are based either on

Rossiter's semi-expirical formula or a proper orthogonal decomposition (POD) based models.

These models cannot be implemented in adaptive controller design directly. In addition, closed-

loop control of high subsonic and supersonic flows remains an unexplored area.









In order to achieve these objectives, an analytical system model is first developed in this

research. This analytical model is a transfer function based model and it can be used as a

potential model for controller design. Then, a MIMO system identification algorithm is derived

and combined with the generalized prediction control (GPC) algorithm. The resultant integration

of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics

and reduce the flow-induced oscillations.

A novel piezoelectric-driven synthetic jet actuator array is designed for this research. The

resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well as a

large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of

interest at low subsonic Mach numbers.

A validation vibration beam problem is used to demonstrate the combination of system ID

and GPC algorithms. The result shows a -20 dB reduction at the single resonance peak and a -9

dB reduction of the integrated vibration levels.

Both open-loop control and closed-loop control are applied to the flow-induced cavity

oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the

trailing edge floor are reduced for both cases. The GPC controller can generate a series of

control signals to drive the actuator array resulting in dB reduction for the second, third, and

fouth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, the broadband

background noise is also reduced by this closed-loop controller (i.e., the OASPL reduction is 3

dB). The relevant flow physics and active flow control actuators are examined and explained in

this research. The limitations of the present setup are discussed.









CHAPTER 1
INTRODUCTION


Flow-induced cavity oscillations have been studied for more than fifty years, and the

problem has attracted researchers in both fluid dynamics and controls. First, this problem exists

in many practical environments, such as landing gear bays and weapon delivery systems on

aircraft, sunroofs and windows "buffeting" in automobiles, and junctions between structural and

aerodynamic components in both (Kook et al. 1997). The flow-acoustic coupling inherent in

cavity resonance can lead to high unsteady pressure levels (both broadband noise and discrete

tones), and can cause fatigue failure of the cavity and its contents. For example, the measured

sound pressure levels in and around a weapons bay can exceed 170 dB ref 20 [aPa. For this

reason, researchers are usually interested in suppression of flow-induced open cavity oscillations.

Furthermore, this problem has become a standard test problem for designing, testing, and

analyzing real-time feedback control systems. Although the standard rectangular cavity

geometry is relatively simple, the physical nature of this problem is both rich and complex.

Several good review articles on the flow-induced cavity oscillation problem are available in the

literature (Rockwell, 1978, Komerath 1987, Colonius 2001, Cattafesta 2003).

Figure 1-1 is a simplified schematic for two typical flow situations, corresponding to

external (a) supersonic and (b) subsonic flow over a rectangular cavity with length, L, depth, D,

and width, W. The cavity oscillation process can be summarized as follows. A (usually)

turbulent boundary layer with thickness, 3, and momentum thickness, 0, separates at the

upstream edge of the cavity. Both a turbulent boundary layer and laminar boundary layer

generate the discrete tones caused by the external flow. However, a laminar boundary layer has

been shown to produce louder tones, presumably because a turbulent flow generally results in a









thicker shear layer with broadband disturbances, which leads to overall lower levels of

oscillations (Tam and Block 1978; Colonius 2001).

Following the description of Kerschen and Tumin (2003) and Alvarez et al.(2004), when

the turbulent boundary layer separates at the upstream edge of the cavity, the resulting high

speed or "fast" acoustic wave, E,, and the low speed or "slow" acoustic wave, E,, propagate

downstream in the supersonic flow case. In the subsonic flow case, only the so-called

disturbance wave, E,, propagates downstream. In both cases, the shear layer instability, s,

develops based upon its initial conditions coupled by the upstream traveling acoustic feedback

wave, U, inside the cavity (and E, outside the cavity for subsonic flow). Kelvin-Helmholtz-type

(Tam and Block 1978) convective instability waves develop and amplify in the shear layer as

they propagate downstream and finally saturate due to nonlinearity. In particular, the instability

waves grow and form large-scale vortical structures that convect downstream at a fraction of the

freestream velocity. These structures then impinge near the trailing edge of "open" cavities

(LID <10). The reattachment region acts as the primary acoustic source, which has been

modeled as a monopole (Tam and Block 1978) or a dipole (Bilanin and Covert 1973) source.

As a result, an upstream traveling acoustic wave, U, is generated inside the cavity. In

subsonic flow, an additional acoustic wave, E,, propagates upstream outside the cavity. In this

description, the acoustic feedback is modeled via acoustic waves that travel in the -x direction.

Finally, the loop is closed by a receptivity process, in which the upstream traveling waves are

converted to downstream traveling instability and acoustic waves. The initial amplitude and

phase of these waves are set by the incident acoustic disturbances through this receptivity

process. Physically, some of the acoustic disturbance energy is converted to the instability

waves at the upstream separation edge. Since the wavelength and the velocity of the instability









waves and the acoustic disturbances differ, only those waves that are in-phase ensure

reinforcement of disturbances at that frequency. Therefore, this process is normally considered

an introduction of a disturbance into the system, ultimately resulting in large-amplitude discrete

tones inside and around the cavity. The measured broadband noise component is mainly due to

the turbulent shear layer.

The relevant dimensionless parameters are: L/D, L/W, L/O, and shape factor H = 6 /0

with the freestream flow parameters, Reynolds number Re, and Mach number Mn, all of which

lead to tones (with Strouhal number St = jL/U, ) characterized by their strength as unsteady

pressure normalized by the freestream dynamic pressure, p,,/q .

In this study, three dimensional effects are not considered, since the cavity tones are

generated by the interaction of the freestream flow and the longitudinal modes (coupled with

vertical depth modes). Width modes are not relevant in this feedback loop if the width is small

enough to prevent higher-order spanwise modes but large enough so that the mean flow over the

cavity length is approximately two dimensional. Note that the width of the cavity does affect the

amplitude of the cavity oscillations (Rossiter 1966) but is of secondary importance (Cain 1999).

Therefore, a two-dimensional model is reasonable from a physical perspective even though the

unsteady turbulent motion is inevitably three-dimensional (Bilanin and Covert 1973).

Literature Review

In this section, some published results related to the physics of flow-induced cavity

oscillations are discussed. Since the ultimate goal of this research is to minimize the cavity

acoustic tones and perhaps the broadband noise level, potential control methodologies and

algorithms are also reviewed. A recent review paper by Cattafesta et al. (2003) gives a summary

of the various passive and open-loop cavity suppression studies.









Physical Models

In order to suppress the discrete tones and the broadband acoustic level of flow-induced

open cavity resonance, an understanding of the physics is essential. From a control engineer's

point of view, a simplified and low-order model is desirable in order to predict the resonant

frequencies and amplitudes over a broad range of the governing dimensionless parameters.

Physics-Based Models

Rossiter (1964) performed an extensive experimental study on the measurement of the

unsteady pressure in and around a rectangular open cavity (2ft x 1.5ft) in a subsonic and

transonic freestream air flow (0.5
unsteady acoustic tones generated in the cavity. For the deeper cavities (LID < 4 ), there was

usually a single dominant tone, and the dominant frequency was observed to jump between

different cavity tones. For the shallower cavities (LID > 4), two or more peaks were often

observed and were approximately equal in magnitude. He proposed that the flow entering the

cavity caused the external stream to accelerate, and then the flow decelerated near the

reattachment region. As a result, pressure was lower near the separation region (leading edge)

and higher near the reattachment region (trailing edge). As a result, he suggested that large

eddies developed within the cavity due to this pressure gradient. He also used shadowgraphs to

illustrate that the shear layer separates from the cavity leading edge, and the instability waves

develop into discrete vortices that are shed at regular time internals from the front lip of the

cavity (at Mach number 0.6 and with two-dimensional cavity LID = 4 and a laminar boundary

layer). He postulated that there were some connections between the vortex shedding and the

acoustic feedback, and this phenomenon produced a series of periodic pressure fluctuations.









When the frequency of one of these components is close to the natural frequency of the cavity,

resonance occurs.

In his study, Rossiter gave a semi-empirical formula for predicting the resonant

frequencies of these peaks at a specific Mach number. The derivation of the Rossiter equation is

given in Appendix B, and the resulting formula for the dimensionless Strouhal number is

S L _(m, -a)
St fmL (m ) (1-1)
U 1

where f, is the resonance frequency for integer mode m,, L is the length of the cavity,

U. is the freestream velocity, a is the phase lag factor (in fractions of a wavelength), K is the

ratio of the vortex propagation speed to the freestream velocity, and MAJ is the freestream Mach

number. Empirical constant values of K = 0.57 and a = 0.25 are shown to best fit the measured

frequencies of resonances over a wide range of the Mach numbers for his experiment. These

experimental constants account for the phase shift associated with the coupling between the

shear layer and acoustic waves at the two ends of the cavity, and this phase shift is approximately

independent of frequency. The phase speed cKU of the vortices is a weak function of MA,, L/O

and D/O (Colonius 2001). Different integer values m, give different frequencies, commonly

referred to as "shear layer" or "Rossiter" modes. In conclusion, Rossiter's formula is based on

an integer number of 2r phase shifts, 2k r, around a resonant feedback loop consisting of a

downstream unstable shear layer disturbance and an upstream feedback acoustic wave inside the

cavity. This phase shift is a necessary condition for self-sustaining oscillations (Cattafesta et al.

1999a). However, Rossiter's expression does not account for the depth or width of the cavity

and only successfully predicts the longitudinal cavity resonant frequencies at moderate-to-high

Mach numbers. It also does not predict the amplitude of the oscillations.









Heller and Bliss (1975) corrected the Rossiter equation for the higher sound speed in the

cavity, in which the static temperature in the cavity was assumed to be the stagnation

temperature of the upstream. The modified Rossiter formula is

f,L (m a)
St f LU' ) (1-2)


1 2 1+ M 2


where y is defined as the ratio of specific heats. They gave a discussion on the physical

mechanisms of the oscillation process based on water table visualization experiments. They

suggested that the unsteady motion of the shear layer leads to a periodic mass addition and

removal at the cavity trailing edge, leading to subsequent modeling efforts that employ an

acoustic monopole source. In addition, the wave motion of the shear layer and the wave

structure within the cavity were strongly coupled.

Bilanin and Covert (1973) modeled the cavity problem by splitting the domain into two

parts outside and inside the cavity. These two flow fields were separated by a thin mixing layer,

which was approximated by a vortex sheet, and the flow was assumed to be inviscid. The

dominant pressure oscillations at the trailing edge were modeled by a single periodic acoustic

monopole. They also assumed that the pressure field from the trailing edge source had no effect

on the vortex sheet itself. Hence, the main disturbance was introduced at the leading edge of the

shear layer. Kegerise et al. (2004) illustrated the agreement between the disturbance sensitivity

function defined in control systems and the performance measurement of output disturbances.

Their analysis confirmed the notion that the disturbances were mainly introduced into the cavity

at the cavity leading edge.









Tam and Block (1978) carried out extensive experimental investigations at low subsonic

Mach numbers (M < 0.4) and postulated that vortex shedding was probably not the main factor

for cavity resonance over the entire Mach number range. They made two key assumptions,

namely that the rectangular cavity flow was two-dimensional, and the mean flow velocity inside

the cavity was zero. These two assumptions were based on experimental evidence of little

correlation between the mean flow and the acoustic feedback inside the cavity. Tam and Block

proposed a process of flow-induced cavity oscillations as follows. The shear layer oscillated up

and down at the trailing edge of the cavity. The upward movement was uncorrelated with the

generation of the acoustic waves, because if the shear layer covered the trailing edge, then the

external flow passes over the trailing edge without impingement. They argued that only the

downward motion of the shear layer into the cavity caused significant generation of pressure

waves and subsequent radiation of acoustic waves in all directions (Figure 1-2).

For example, some of the waves radiating into the external flow (e.g., wave A) were

argued to have minor effects on the oscillations inside the cavity. However, the effect of the

waves propagating inside of the cavity was deemed more significant. The resulting acoustic

waves included the upstream propagating waves (e.g., wave C) and the reflected waves from the

floor (e.g., wave F) and the upstream wall (e.g., wave E). Subsequent reflections of the acoustic

waves by the walls, the cavity, or the shear layer were deemed negligible. They concluded that

the directly radiated wave and the first reflected waves by the floor and upstream end wall of the

cavity provided the energy to excite the instability waves of the shear layer. These disturbances

within the shear layer were then amplified as the instability waves propagate downstream. When

the disturbances amplitudes became large, nonlinear effects were important and ultimately

established the amplitude of the discrete tones. A mathematical model of the cavity oscillation









and acoustic field were developed. In order to calculate the phases and waves generated at the

trailing edge, a periodic line source was simulated at the trailing edge of the cavity. In addition,

the reflections of the acoustic waves by the cavity walls were modeled by periodic line image

sources about the cavity walls. Their model accounts for the finite shear layer thickness effects

and produces a more accurate estimation of the resonance frequencies than Rossiter's model.

However, their resulting model is complicated and difficult to employ for control law design.

Rowley et al. (2002 b, 2003, 2006) provided an alternative viewpoint for understanding

flow-induced cavity oscillations. They showed that self-sustained oscillations existed only under

certain conditions. The resonant frequencies were due to the instabilities in the shear layer

interacting with the flow and acoustic fields. The amplitude of the oscillations was determined

by nonlinear saturation. However, at other conditions, the cavity oscillations could be

represented as a lightly damped but stable linear system. The oscillations were caused by the

amplification of external disturbances via the closed-loop dynamics of the cavity. The amplitude

of each mode was determined by the amplitude of the external forcing disturbances and some

frequency-dependent gain of the system. They modeled the dynamics of the shear layer as a

second-order system and the acoustic propagation process via a one-dimensional, standing-wave

model. The impingement and receptivity procedures were simply modeled as a constant unity

gain. Finally, the Rossiter formula was derived under some specific conditions. The derivation

of this model is provided in Appendix B.

They also used Gaussian white noise as input and examined the probability density

function (PDF) and the phase portrait of the output pressure signal at different Mach numbers.

Their results showed that under some conditions, the self-sustained regime of Rossiter modes

was valid. However, at other conditions, called the forced regime, open cavity oscillations may









be represented as lightly damped stable linear systems. External random forces drove the finite-

amplitude cavity oscillations, which implies they will disappear if the external forces were

removed. This physical linear model was also proposed as a potential model for controller

design.

Kerschen and Tumin. (2003) and Alvarez et al. (2004) provided a promising global model

to describe the flow-induced cavity oscillation problem for two different flow patterns (Figure 1-

1). Their model combined scattering analyses for the two ends of the cavity and a propagation

analysis of the cavity shear layer, internal region of the cavity, and acoustic near-field. They

solved a matrix eigenvalue problem to identify the resonant frequencies of the cavity oscillation.

From their resulting characteristic functions, four and twelve closed loops could be identified for

the supersonic flow and subsonic flow cases, respectively. One more feedback loop makes the

subsonic flow much more complex than the supersonic flow. For example, some of these closed

loops were major loops, such as closed loop s, U and U, D (Figure 1-1), while the other closed

loops were considered minor loops. The combined effects of these loops caused the cavity

resonances in the cavity flow. Besides these closed loops, the forward propagation paths, such as

S, D, Ed, E,, and E,, also have critical effects on the amplitude of the oscillations. This global

model provides more insights for controller design. A detailed derivation of this model is

provided in Appendix B.

Clearly, the physics-based models described above provide physical insight concerning

flow-induced cavity oscillations. However, the original Rossiter model and the global model

derived by Kerschen and Tumin. (2003) can only estimate the resonance frequencies of the

cavity flow. The linear model derived by Rowley et al. (2002b, 2003, 2006) is transfer function

based model but is not sufficiently accurate to design a control system. A transfer function based









model, which is an extension of Kerschen et al.'s model, of cavity acoustic resonances is derived

and given in Appendix D. For this approach, a signal flow graph is first constructed from the

block diagram of the Kerschen et al's physical model, and then Mason's rule (Nise 2004) is

applied to obtain the transfer function from the disturbance input to the selected system output.

This method can give predictions for both the resonant frequencies of the flow-induced cavity

oscillations and the amplitude of the cavity tones. In addition, this method also provides a linear

estimate for the system transfer function from the disturbance input at the leading edge and the

pressure sensor output within the cavity walls. Therefore, this model is a potential global model

for controller design in this research.

Numerical Simulations

Some computational fluid dynamic (CFD) methods, such as Direct Numerical Simulations

(DNS), Large Eddy Simulations (LES) and Reynolds Averaged Navier-stokes (RANS), provide

useful information for understanding the issues of physical modeling of cavity oscillations. A

review paper by Colonius (2001) gives a summary of issues related to each of these topics. More

recent research on these topics can be found by Rizzetta et al. (2002, 2003) and Gloerfelt (2004).

The Detached Eddy Simulation (DES) method, which is a involves a hybrid turbulence modeling

methodology, has also been used to calculate the flow and acoustic fields of the cavity (Allen

and Mendonca 2004; Hamed et al. 2003, 2004). Another hybrid RANS-LES turbulence

modeling approach is presented by Arunajatesan and Sinha (2001, 2003). They model the

upstream boundary layer flow field and the shear layer region via RANS and LES models,

respectively. All of these computational methods provide, at a minimum, good flow

visualization and physical insight, and, at a maximum, quantitative information on the details of

flow dynamics.









POD-Type Models

The previous analytical physical models are not accurate enough to design a control

system. Furthermore, CFD methods are far too computationally intensive at the present time to

provide a reasonable framework to design and test potential controllers. This translates into the

need for new methods to develop more accurate reduced-order models. Therefore, simulation

and experimental data based models were proposed and later used for the controller design.

Rowley et al. (2001) introduced a nonlinear dynamical model for flow-induced rectangular

cavity oscillations, which was based on the method of vector-valued proper orthogonal

decomposition (POD) and Galerkin projection. The POD method obtains low-dimensional

descriptions of a high-order system (Chatterjee 2000). For the cavity flow problem, data

resulting from the temporal-spatial evolution of the numerical simulations or experiments is used

to construct a low-order subspace system that captures the main features (coherent structures) of

the cavity flow. A more detailed explanation of POD methods for cavity flow are given by

Rowley et al. (2000, 2001, 2002c, 2003a). Some of the control methodologies discussed in next

section can be constructed based on the resultant model obtained by POD (Caraballo et al. 2003,

2004, 2005; Samimy et al. 2003, 2004; Yuan et al. 2005).

Instead, we turn our attention to an alternative experimental-based modeling approach that

employs system identification techniques. Here, the nonlinear infinite-dimensional governing

equations are modeled by a reduced set of differential (in continuous) or difference (in discrete

time) equations. This method is the focus of this study and is discussed in the following section.

On-Line System ID and Active Closed-Loop Control Methodologies

Previous studies aimed at suppression of the flow-induced cavity tones have employed

mainly passive or open-loop active flow control methodologies. The standard classification of

the flow control techniques is shown in Figure 1-3. The review paper by Cattafesta et al. (2003)









provides a detailed overview of various passive and open-loop control methodologies. However,

passive and open-loop approaches are only effective for a limited range of flow conditions.

Active feedback flow control has recently been applied to the flow-induced cavity oscillation

problems. The closed-loop control approaches have advantages of reduced energy consumption

(Cattafesta et al. 1997), no additional drag penalty, and robustness to parameter changes and

modeling uncertainties. In general, closed-loop flow control measures and feeds back pressure

fluctuations at the surface of the cavity walls (or floor) to an actuator at the cavity leading edge

to suppress the cavity oscillations in a closed-loop fashion.

In general, past active control strategies have taken one of two approaches for the purpose

of reducing cavity resonance. First, they can thicken the boundary layer in order to reduce the

growth of the instabilities in the shear layer. Alternatively, they can be used to break the internal

feedback loop of the cavity dynamics. Most closed-loop schemes exploit the latter approach.

Early closed-loop control applications used manual tuning of the gain and delay of simple

feedback loops to suppress resonance (Gharib et al. 1987; Williams et al. 2000a,b). Mongeau et

al. (1998) and Kook et al. (2002) used an active spoiler driven at the leading edge and a loop-

shaping algorithm to obtain significant attenuation with small actuation effort. Debiasi et al.

(2003, 2004) and Samimy et al. (2003) proposed a simple logic-based controller for closed-loop

cavity flow control. Low-order model-based controllers with different bandwidths, gains and

time delays have also been designed and implemented (Rowley et al. 2002, 2003, Williams et al.

2002, Micheau et al. 2004, Debiasi et al. 2004). Linear optimal controllers (Cattafesta et al.

1997, Cabell et al. 2002, Debiasi et al. 2004, Samimy et al. 2004, Caraballo et al. 2005) have

been successfully designed for operation at a single flow condition. These models are all based

on reduced-order system models, and most of these controller design methods are based on









model forms of the frequency response function, rational discrete/continuous transfer function,

or state-space form. However, the coefficients of these model forms are assumed to be constant,

and this assumption requires that the system is time invariant or at least a quasi-static system

with a fixed Mach number.

Although the physical models of flow-induced cavity oscillations have been explored

extensively, they are not convenient for control realization. This is because these models are

highly dependent on the accuracy of the estimated internal states of the cavity system. In

addition, cavity flow is known to be quite sensitive to slight changes in flow parameters. So a

small change in Mach number can deteriorate the performance of a single-point designed

controller (Rowley and Williams 2003). Therefore, adaptive control is certainly a reasonable

approach to consider for reducing oscillations in the flow past a cavity. Adaptive control

methodology combines a general control strategy and system identification (ID) algorithms.

This method is thus potentially able to adapt to the changes of the cavity dimension and flow

conditions. It updates the controller parameters for optimum performance automatically.

The structure of this method is illustrated in Figure 1-4. Two distinct loops can be

observed in the controller. The outer loop is a standard feedback control system comprised of

the process block and the controller block. The controller operates at a sample rate that is

suitable for the discrete process under control. The inner loop consists of a parameter estimator

block and a controller design block. An ID algorithm and a specified cost function are then used

to design a controller that will minimize the output. The steps for real-time flow control include:

(i) Use a broadband system ID input from the actuator(s) and the measured pressure fluctuation

output(s) on the walls of the cavity to estimate the system (plant and disturbance) parameters. (ii)









Design a controller based on the estimated system parameters. (iii) Control the whole system to

minimize the effects of the disturbance, measured noise, and the uncertainties in the plant.

Based on this adaptive control methodology, some adaptive algorithms adjust the

controller design parameters to track dynamic changes in the system. However, only a few

researchers have demonstrated the on-line adaptive closed-loop control of flow-induced cavity

oscillations. Cattafesta et al. (1999 a, b) applied an adaptive disturbance rejection algorithm,

which was based upon the ARMARKOV/Toeplitz models (Akers and Bernstein 1997;

Venugopal and Bernstein 2000, 2001), to identify and control a cavity flow at Mach 0.74 and

achieved 10 dB suppression of a single Rossiter model. Other modes in the cavity spectrum

were unaffected. Insufficient actuator bandwidth and authority limited the control performance

to a single mode. Williams and Morrow (2001) applied an adaptive filtered-X LMS algorithm to

the cavity problem and demonstrated multiple cavity tone suppression at Mach number up to

0.48. However, this was accompanied by simultaneous amplification of other cavity tones.

Numerical simulations using the least mean squares (LMS) algorithm were shown by Kestens

and Nicoud (1998) to minimize the output of a single error sensor. The reduction was associated

with a single Rossiter mode, but only within a small spatial region around the error sensor.

Kegerise et al. (2002) implemented adaptive system ID algorithms in an experimental cavity

flow at a single Mach number of 0.275. They also summarized the typical finite-impulse

response (FIR) and infinite-impulse response (IIR) based system ID algorithms. They concluded

that the FIR filters used to represent the flow-induced cavity process were unsuitable. On the

other hand, IIR models were able to model the dynamics of the cavity system. LMS adaptive

algorithm was more suitable for real-time control than the recursive-least square (RLS) adaptive

algorithm due to its reduced computational complexity. Recently, more advanced controllers,









such as direct and indirect synthesis of the neural architectures for both system ID and control

(Efe et al. 2005) and the generalized predictive control (GPC) algorithm (Kegerise et al. 2004),

have been implemented on the cavity problems.

From a physical point of view, the closed-loop controllers have no effect on the mean

velocity profile (Cattafesta et al. 1997). However, they significantly affect streamwise velocity

fluctuation profiles. This control effect eliminates the strength of the pressure fluctuations

related to flow impingement on the trailing edge of the cavity. Although closed-loop control has

provided promising results, the peaking (i.e., generation of new oscillation frequencies), peak

splitting (i.e., a controlled peak splits into two sidebands) and mode switching phenomena (i.e.,

non-linear interaction between two different Rossiter frequencies) often appear in active closed-

loop control experiments (Cattafesta et al. 1997, 1999 b; Williams et al. 2000; Rowley et al. 2002

b, 2003; Cabell et al. 2002; Kegerise et al. 2002, 2004a).

Explanations of these phenomena are provided by Rowley et al. (2002b, 2006), Banaszuk

et al. (1999), Hong and Bernstein (1998), and Kegerise et al. (2004). Rowley et al. (2002b,

2003) concluded that if the viewpoint of a linear model was correct, a closed-loop controller

could not reduce the amplitude of oscillations at all frequencies as a consequence of the Bode

integral constraint. Banaszuk et al. (1999) gave explanations of the peak-splitting phenomenon.

They claimed that the peak splitting effect was caused by a large delay and a relatively low

damping coefficient of the open-loop plant. Cabell et al. (2002) explained these phenomena by

the combination of inaccuracies in the identified plant model, high gain controllers, large time

delays and uncertainty in system dynamics. In addition, narrow-bandwidth actuators and

controllers may also lead to a peak-splitting phenomenon (Rowley et al. 2006).









Hong and Bernstein defined the closed-loop system disturbance amplification (peaking)

phenomenon as spillover. They illustrated that the spillover problem was caused by the

collocation of disturbance source and control signal or the collocation of the performance and

measurement sensors. For this reason, the reduction of broadband pressure oscillations was not

possible if the control input was collocated with the disturbance signal at the leading edge of

cavity. Therefore, Kegerise et al. (2004) suggested a zero spillover controller which utilized

actuators at both the leading and trailing edges of the cavity for closed-loop flow control.

Unresolved Technical Issues

Although the flow-induced cavity oscillation problem has been explored extensively, there

are still some unresolved issues that need to be studied further.

* A suitable theoretical model does not exist that estimates both the discrete frequencies as
well as the amplitude of the peaks.

* A feedback controller that reduces both broadband and tonal noise over a wide range of
Mach numbers has not been achieved. An adaptive zero spillover control algorithm may
reduce both the tones and broadband acoustic noise associated with cavity oscillations.

* The necessity for a high-order system model is a critical problem for controller design and
implementation, because this high-order system results in significant computational
complexity for application in digital signal processing (DSP) hardware. As such, the
convective delays between the control inputs and the pressure sensor outputs must be
specifically addressed in the control architecture.

* Closed-loop control of high subsonic and supersonic flows is an unexplored area.

Technical Objectives

According to those unresolved technical issues, the ultimate goals of this dissertation are

summarized as follows.

* A feedback control methodology will be developed for reducing flow-induced cavity
oscillation and broadband pressure fluctuations.

* Adaptive system ID and control algorithms will be combined and implemented in real-
time.









* The relevant flow physics and the design of appropriate active flow control actuators will
be examined in this research.

* The performance, adaptability, costs (computational and energy), and limitations of the
algorithms (spillover, etc.) will be investigated.

Approach and Outline

In order to achieve these objectives, some design and application approaches warrant

additional consideration. First, a potential theoretical model of cavity acoustic resonances is

derived based on the global model of Kerschen and Tumin. (2003). This model (derived in

Appendix D) provides the framework to estimate the amplitudes and frequencies of the cavity

tones. This model has a low system order and also accounts for the convective delay between

the disturbance input and the output pressure measurement. Second, during the controller

design, the controlled system is a continuous system; therefore, all the sensors measurements and

the actuators inputs are analog signals. However, for the present real-time application, the

control algorithms are implemented using a DSP. For this reason, additional hardware, such as

analog-to-digital converters (ADC), digital-to-analog converters (DAC), anti-aliasing filters, and

power amplifier, must be included in the whole control design procedure. Finally, multiple

actuators and multiple sensors are employed in this study in order to design an adaptive zero

spillover control algorithm to explore the possibility of achieving broadband acoustic noise

reduction in addition to suppression of the cavity tones themselves.

This active control method development procedure can be summarized as the following

stages according to Elliott (2001).

* Study the simplified analytical system model and understand the fundamental physical
limitations of the proposed control strategy.

* Obtain the sensor output and derive the states or coefficients from the system ID
algorithms using off-line or on-line methods.









* Calculate the optimum performance using different control strategies and find the control
law for realization.

* Simulate the different control strategies and tune the candidate controller for different
operating conditions.

* Implement the candidate controller in real-time experiments.

The thesis is organized as follows. Several SISO IIR system ID algorithms and a more

general MIMO system ID algorithm are derived and discussed in the next Chapter. Then the

MIMO adaptive GPC algorithm is described in Chapter 3. This is followed by a description of

the sample experimental setup and the discussion of preliminary experimental results. Chapter 5

describes the wind tunnel facilities and the data processing methods. Wind tunnel experimental

results for both open-loop (baseline) and closed-loop are then presented and discussed in Chapter

6. Finally, the conclusions and future work are presented in Chapter 7.










Turbulent
Boundary Layer
M>1 ES E
X S


-D
D


< L
A


Turbulent -'
Boundary Layer

M <1 V S
........................


D
D


-L >


B
Figure 1-1. Schematic illustrating flow-induced cavity resonance for an upstream turbulent
boundary layer. A) In supersonic flow.B) In subsonic flow.

y A
U.
Simulated Line Source


E F
D


4< L >


Figure 1-2. Tam and Block (1978) model of acoustic wave field inside and outside the
rectangular cavity.










Flow Control
Approaches


Figure 1-3. Classification of flow control. (after Cattafesta et al. 2003)

Uncertainties


Figure 1-4. Block diagram of system ID and on-line control.









CHAPTER 2
SYSTEM IDENTIFICATION ALGORITHMS


This chapter provides a detail discussion of the system identification algorithms.

Several typical adaptive SISO IIR structure filters are chosen as the candidate digital

filters. These algorithms are applied to an example from Johnson and Larimore (1977)

for simulation analysis. Then, four interested aspects of these filters, accuracy,

convergence, computational complexity, and robustness, are examined and summarized.

Finally, a more general MIMO system ID algorithm is derived from one of the promising

SISO system ID algorithms. The resulting model is used to combine with the MIMO

adaptive GPC model which is discussed in next chapter.

Overview

As discussed in the first chapter, IIR structure filter is an applicable mathematical

model to capture the cavity dynamics. Furthermore, this kind of structure can be a

starting point and easily combined with many controller design strategies. Therefore, in

this Chapter, several system ID algorithms based on the IIR filter structure are examined.

The ideal of the system ID is to construct a predefined IIR structure filter, which has the

similar frequency response of the actual dynamic system, using the information from the

previous and present input and output time series data of the dynamic system. In general,

the system ID algorithms fall into two big categories, the batch method and the recursive

method. The batch method directly identifies the final system parameters in one-time

calculation using a block data from the input and a block data from the output.

Nevertheless, the recursive method updates the estimated system parameters within each

sampling period using the latest input and output data in time domain. At each iteration

of calculation, the system parameters may not be the optimal values. However, these









estimated parameters will finally converge to the true values of the system internal states.

Successful identifying the system internal states depends on two major assumptions.

First, the input signal and the output signal must have a good correlation. Then, the

system ID model has the same structure of that of the estimated system model. The

recursive method is more attractive for present experiment, because this updating method

is more suitable for on-line implementation and it can also track the change of the system

dynamics. Furthermore, the computational complexity of recursive method is much

lower than the batch method.

SISO IIR Filter Algorithms

Netto and Diniz (1995) give a summary of some popular adaptive IIR filter

algorithms. In this section, the Output Error (OE), Equation Error (EE), Steiglitz and

McBride (SM), and Composite Error (CE) algorithms are selected and illustrated. The

general structure of an IIR filter is shown in Figure 2-1. The filter output may be

expressed as


S(k)= aZ(k i) bx(k- j)
=\1 J=0 (2-1)
= i(k)0(k)

where represents the 'estimation' values. a and b/ are the adjustable coefficients of

the model, while ha and hb is the estimated order of the feedback loop and forward path,

T
respectively. o(k)=[f(k-i) x(k-j)]T, 0(k)= a b] and i -,...,,;j= 0,1,..., b.

This IIR filter structure, Equation 2-1, is the same as the autoregressive and moving-

average (ARMA) model (Haykin 2002).









Based on different error, the value difference between the filter output and the

system output, definitions, quite a few IIR filter algorithms have been presented by Netto

and Diniz (1995). In their simulations, they use an "insufficient" model, which models a

second-order system using a first-order system to test each algorithm. The results from

their paper show that the Modified Output Error (MOE) algorithm may converge to a

meaningless stationary point. The same result is also shown by Johnson and Larimore

(1977). The Simple Hyper-stable Algorithm for Recursive Filters (SHARF) algorithm,

the modified SHARF algorithm, and the Bias Remedy Least-Mean-Square Equation

Error algorithm (BRLE) also show poor convergence rates. The Composite Regressor

(CR) algorithm has similar problems as the MOE algorithm, since this algorithm

combines the EE and MOE methods. Therefore, in this section, tests of these poor

performing algorithms are not discussed.

Fundamentally, there are two approaches for an adaptive IIR filter, the OE

algorithm and the EE algorithm, which have been derived by Haykin (2002) and

Larimore et al. (2001), respectively. Many other adaptive IIR filter algorithms are mainly

derived from these algorithms, or combine some good features from the OE and the EE

filters. Therefore, a summary of each of these two algorithms is provided in the

following section. Two other algorithms, the Steiglitz-McBride algorithm (SM) and the

Composite Error algorithm (CE), are also introduced, because both these algorithms also

show good performance in our Simulink simulations.

IIR OE Algorithm

The IIR OE algorithm is summarized in Table 2-1. To ensure the stability of the

algorithm, generally, the upper bound of step size / is set to 2/ where Amax is the
/ max









maximum eigenvalue of the autocorrelation matrix of the regress vector OE (k). The

step sizes of the following algorithms are also satisfying this criterion. Furthermore, in

order to guarantee the convergent approximation of a, and /f, this algorithm requires

slow adaptation rates for small values of n, and hb (Haykin 2002).

IIR EE Algorithm

The IIR EE algorithm is summarized in Table 2-2. Since the desired response is

the supervisory signal supplied by the actual output of plant during the training period,

the EE algorithm may lead to faster convergence rate of the adaptive filter (Haykin

2002).

IIR SM Algorithm

The IIR SM algorithm is summarized in Table 2-3. Since the EE algorithm and the

OE algorithm possess their own advantages as well as drawbacks (discuss later), the

motivation of the SM algorithm is to combine the desirable characteristics of the OE and

the EE methods.

IIR CE Algorithm

This algorithm tries to combine both the EE algorithm and the OE algorithm in

another way. As shown in Table 2-4, a parameter / is used to switch this algorithm

between the EE algorithm and the OE algorithm.

Recursive IIR Filters Simulation Results and Analyses

In adaptive control experiments, the accuracy, the convergent rate, the

computational complexity and the robustness are the main issues of the system ID

algorithms. Here, computer simulations are examined in order to compare these aspects

of the four system ID algorithms.









The setup for the following Simulink simulation is shown in Figure 2-2. A

Gaussian broad band white noise with zero mean and unity variance is chosen as the

reference input signal. The prototype test model is a second-order dynamical system

(Johnson and Larimore 1977) with the transfer function

0.05 -0.4z
H(z-1')= (2-2)
1-1.1314z 1 +0.25z 2

From the z-plane plot (Figure 2-3), it clearly shows that this test model is a stable

and non-minimum phase system, which has two real poles at z = 0.3011 and z = 0.8303,

and two zeros at z = 0 and z = 8.

In the following simulations, a sufficient order identification problem is firstly

examined, which means a second-order system model with the transfer function

b, (k) + b,(k):-
H(z1, k) = o (k) + is used to estimate the test model. Then, an
1 -a (k)z -a2(k)z

insufficient order identification problem is investigated. This approach uses a first-order

bo(k)
system model with the transfer function H(z k) (k) to estimate the test
1-a (k)z1

model. The mean square output error (MSOE) surface of this insufficient order

dynamical system is obtained by Shynk (1989)

2 ^ b
MSOE= cr2 -2b0H(c,)+ b (2-3)
1\a1

A 3D surface plot and a contour plot of the MSOE performance surface are shown

in Figure 2-4 and Figure 2-5, respectively. The plots show that the MSOE surface of the

test model is bimodal with a global minimum (denoted by "*")









at(b,a ) = (-0.311,0.906), which yieldsMSOE* = 0.277, and a local minimum (marked

by "+") at (b, a) = (0.114,-0.519), which corresponds toMSOE+ = 0.976.

The input x(k) and the test model output y(k) (with or without disturbance v(k))

are introduced to the adaptive IIR filter algorithms at the same time. The adaptive IIR

filter algorithms calculate the error signal and update the weights at each iteration.

Accuracy comparison for sufficient system

Table 2-5 and Figure 2-6 show the simulation results and weight tracks of the four

IIR algorithms for the sufficient case, respectively. For the sufficient case, the algorithms

minimize the mean square error between the system output and the filter output, and the

estimated weights converge to the original coefficients of the test model.

Accuracy comparison for insufficient system

The simulation results and weight tracks of the IIR algorithms for the insufficient

case are shown in Table 2-6 and Figure 2-7, respectively. The OE algorithm starts from

two different initial conditions. One point is closer to the global minimum, and the other

one is closer to the local minimum. This method adjusts its weights via stochastic

gradient estimation to the closest stationary point of the initial condition.

Similarly, two initial conditions are selected for EE algorithm. One of them is

close to the global minimum, and the other one is much closer to the local minimum.

This algorithm can avoid the local minimum and adjust its weights to let the final mean

square error value arrive at the area near the global minimum. However, for this

insufficient order situation, the final solution exhibits bias compared to the optimum

solution.









The SM algorithm combines the advantages of the OE algorithm and the EE

algorithm. This algorithm avoids the local minima and converges to the global minimum

with different initial points, which is like the EE algorithm. At the same time, the final

solution for this algorithm is very close to the optimum solution.

As addressed above, the CE algorithm is a combination of the OE algorithm and

the EE algorithm. It uses a weighting parameter / to switch and weight between the OE

algorithm and the EE algorithm. For this insufficient identification problem, this method

performs well. If the weighting parameter / is close to 0, this algorithm is more like the

OE algorithm, and the interesting feature of this algorithm shows that it converges to the

global minimum in the MSOE surface. However, when / is close to 1, the biased

characteristic of the EE algorithm is apparent in the results.

Convergence rate

For convergence rate comparison, the same step size and number of iterations are

chosen for simulations. The simulation conditions and the learning cures of the IIR

algorithms for the sufficient case are shown in Table 2-7 and Figure 2-8, respectively.

Obviously, the EE and SM algorithms converge faster than the OE and CE algorithms.

Computational complexity

In order to apply the ID algorithm on an adaptive control algorithm for real-time

implementation, the computational complexity for one iteration of the ID algorithm have

to be less than the sampling time of the DPS processor used for real-time experiment.

Four algorithms are compared for computational complexity by the turnaround time with

the increase of the number of unknown for each algorithm (Figure 2-9). The hardware

used for experiment is PowerPC 750 (480MHz) microprocessor (12.6 SPECfp95). The









experimental results are shown in Figure 2-9. The computational complexity of all of the

IIR algorithms is approximately linear. And the CE algorithm needs more computational

time for each iteration than time requirements for the other three algorithms.

Conclusions

Varies of IIR adaptive filters are examined in this Chapter, the objective of these

digital filters is to identify the system coefficients (internal states) from the input and

output signals. The OE algorithm and the EE algorithm are two basic structures of an

adaptive IIR filter. Beyond that, two other algorithms, the SM algorithm and the CE

algorithm, are also examined. Simulation results show that the mean square error value

calculated by the OE algorithm converges to the optimum solution for both the sufficient

case and the insufficient case if the "proper" initial condition is chosen. This means that

the OE algorithm may converge to local minima in the MSOE surface. Furthermore, this

algorithm does not guarantee that the poles of the ARMA model always lie inside the unit

circle in the z-plane. Thus, the OE method may become unstable (Haykin 2002) during

the experiment. Therefore, a small enough step size and stability monitoring are required

to ensure the convergence of the algorithm. However, the optimum step size is unknown,

and the stability monitoring highly increases the computational complexity. These are

the main drawbacks that should be considered in applications.

The mean square error value calculated by the EE algorithm avoids the local

minima and converges to the global minimum in the MSOE surface. The convergent rate

and the computational complexity are good for real-time implementation. Unfortunately,

the final solution is biased when the test model uses a lower-order system to model a

higher-order system (Shynk 1989, Netto and Diniz 1995).









Both the SM algorithm and the CE algorithm can find the global minimum in the

MSOE surface. However, the good performance of the SM algorithm does not occur in

general and, in fact, cannot be assured in practice (Netto and Diniz 1992). Moreover, the

CE algorithm produces good results when 0.04
algorithm is unimodal, and the bias is negligible (Netto and Diniz 1992). However, the

stability of the CE algorithm must still be monitored, and the computational complexity is

also high for this algorithm.

A summary of the four algorithms is given in Table 2-8. The robustness results of

each ID algorithms come from the experiment discussed in Chapter 4. As the results, the

EE algorithm is the best algorithm comparing to the other three ID algorithms.

Therefore, in next step, a MIMO IIR filter is going to be derived based on this algorithm.

MIMO IIR Filter Algorithm

In this section, a MIMO system ID algorithm is developed based on the SISO IIR

EE algorithm. First, a linear system model is summed with the r inputs [u]rl and the m

outputs [y] For simplification, the order p of the feedback loop is assumed the same

as the order of the forward path. At specific time index k, the system can be expressed

as

y(k) = ay(k -1) + ay(k 2) + .. + a y(k p)
(2-4)
+ 0u(k)+ P+u(k- 1)+ f2u(k- 2) + ... + f (k p)

where









u, (k) y, (k)
u ,,(k) y,(k)
u(k) = [(k)], = ( ,y(k) [y(k)]m 1

Ur(k)J rl y, (k)
ta = [al]mmI.' C2 = [ak 2,mm< ap Lap]mM

fo =['8,0 mx 'l=[AL,-, '. =[ ,]P1gI


Define the observer Markov parameters

o(k) = a -


a I )60 P ]m[m.pr(p+i)]


and the regression vector


y(k 1)

y(k p)
(p(k) =
u(k)

u(k ) [m*p+r*(p+l)]

substituting Equation 2-6 and Equation 2-7 into Equation 2-4 yields a matrix equation for

the filter outputs

[y(k)]ml = [L(k) [(pr(p))] (k)][(m pr(p ))]

Furthermore, the errors are defined as

[ )(k)]l = [j(k)]m -[y(k)]m

Finally, the observer Markov parameters 2-6 can be identified recursively by

0(k + 1) = (k)-/ p6(k)'T (k)

In order to automatically update the step size, choose


C + PL2


(2-5)







(2-6)


(2-7)









(2-8)




(2-9)



(2-10)


(2-11)









where c is a small number to avoid the singularity when ( 2 is zero.

The main steps of the MIMO identification for one iteration are summarized as

follows

Step 1: Initialize [\(k)] m[(mp+r*(p+l))] = [0].


Step 2: Construct regression vector [(p(k)][m*p+r (p1)] 1 according to Equation 2-7.

Step 3: Calculate the output error [E(k)]mx1 according to Equation 2-9.

Step 4: Calculate the step size according to Equation 2-11.

Step 5: Update the observer Markov parameters matrix [9(k)] m[m*p+r(p+l)] according to

Equation 2-10. Then, the calculation for the next iteration goes back to step 2.

The detail derivation for this MIMO ID algorithm is given in Appendix B. And the

experimental results of the algorithm, the computational complexity, and the disturbance

effects will be discussed in later Chapters. The calculation result of this MIMO ID gives

an estimated model of the system with the form of Equation 2-4. In the following

Chapter, a MIMO control algorithm is developed based on this MIMO ID model.









Table 2-1. Summary of the IIR OE algorithm.
Initialization: 0(k) = [a (0) b (0)]r = 0, where i
Computation: For k = 1,2,...

OE(k)= Z a E(k i) + bx(k j)
=) y ( =
eOE (k) = y(k) 0E (k)


a, (k)& E(k -i)+ Z^,r(k (-), for
Define: 1=
,(k) & x(k j)+ a a, (k -), for
=1
OE (k)= [a, (k) |1 (k)]T
0(k + 1) (k) + /eo (k)oo (k)
where / is the step size.
In practice:


i 1,..., ,

j =0,1,...,b


yfoE(k -i) yoE(k -i)+akyfoE(k -i- l), for i = ,...,h
Define: 1
n,
tfOE(k -j) =x(k -j)+ ak fE(k -j-l),for J = 0, .b
1=1
OEf(k) [f E(k -i) I oE(k- j)]l
(k + 1) (k) +/e, (k)O (k)


1,...,ha,j = o,1,...,IA










Table 2-2. Summary of the IIR EE algorithm.
Initialization: 0(k) [ a (0) b (0) 0
where i= 1,...,n,j = 0,1,...,Ib
Computation: For k = 1,2,...

iY(k)= C^y(k -i) + x(k j)
t=1 ]=0
e (k)= y(k)- Y(k)
0__ (k) = [y(k i) I x(k j)]T
+(k + 1) = 0(k)+ +/e_ (k)EE, (k)
where /u is the step size.










Table 2-3. Summary of the IIR SM algorithm.
Initialization: 0(k) [ a (0) b (0)] 0
where i= 1,..., 0a,j = 0,1,...,1* b
Computation: For k = 1,2,...

YiE(k))= C^Zy(k -i) + x(k j)
t=1 ]=0




es (k)= 1 e, (k)
Y=1
ha
y,(k i)= j(k i)+ takyj (k -i ),for i= ,...,
Define:
f s(k-j)= x(k-j)+ ,akiSM(k-j-1),for j=0,1,...,hb
/=1
Osu (k) ['s (k- i)I fM (k- j)]
0(k + 1) = 0(k) +-u-es (k)osM (k)










Table 2-4. Summary of the IIR CE algorithm.
Initialization: 0(k) [, (0) b(0)] 0
where i= 1,..., ,, j = 0,1,..., hb
Computation: For k = 1,2,...

Step 1:

OE (k) = Z a oE(k -i) + Z x(k j)

eoE(k) = y(k)- YoE(k)

foE (k ) = OE(k -i) + E (k l-),for i = 1,...,
Define: /
IxE(k- j)= x(k- j)+ E(k-j-1), for = 0,1,. .,b
1=1
~OEf (k)= [LfE(k-i)| OfE(k j)]'

Step 2.

y(k)= Z ,y(k- i) + x(k- j)

eEE(k)= y(k)- E(k)
EE(k) = [y(k i) I x(k j)]T

Step 3:
eCE (k) = fle (k) + (1 )eOE (k)
wCE (k) = PEE (k) + (1- P)E (k)
O(k + 1) = (k) + pec (k)CE (k)
where 0 < < 1









Table 2-5. Simulation results of IIR algorithms for sufficient case.
Initial Point Final Point
Adaptive Algorithm (0) (0) Numberof () (n)
Structure Parameters iterations () (




=0000 004 [0.05 ,-0.4007]- ,
OE = 0.005 12000 .,0.4 ]
,0 0 [-1.13,0.2484]
EE U = 0.01 8000 [0.05 0.4]
L0 0] [-1.131 0.2493]
SM u 0.005 K 011000 [0.05022,-0.4006]
F 007 [0.050,-0.4006]
u = 0.005,/ = 0.04 12000 [-1.13,0.2485]

u 0.01, 0.60 08000 [0.04997,-0.400]
L0 0] [-1.131,0.2496]












Table 2-6. Simulation results of IIR algorithms for insufficient case.


Initial Point
[O (0),"a,(0)]

[-0.5,0.1]
[-0.5,-0.2]
[-0.5,0.1]
[0.11,-0.52]
[-0.5,0.1]
[0.5,-0.2]
[0.11,-0.52]
[-0.5,0.1]
[0.5,-0.2]


Number of
iterations

5500
12000
7000
7000
3000
4500
5500
5000
19000


Global Min.
[-0.311,0.906]
Final Point
[-0.3098,0.8998]
[0.0928,-0.4896]
[0.04577,0.8755]
[0.05003,0.8719]
[-0.3132,0.9039]
[-0.3062,0.8992]
[-0.2967,-0.9031]
[-0.3037,0.9112]
[-0.315,0.9181]


Adaptive
Structure


Algorithm
Parameters


OE

EE


SM


CE


0.001
0.003
0.001
0.001
0.0005
0.0005
0.0005
0.001, 8
0.003, 8


0.04
0.04










Table 2-7. Simulation conditions of IIR algorithms for sufficient case.
Initial Point
SAlgorithm hO (0) (0) Number of
Adaptive Structure Paramet(0) (0) iterations


OE / = 0.005 o 12000
10 01
EE / = 0.005 o 12000
10 01
SM o = 0.005 o 12000
CE 0
CE u = 0.005,/8= 0.04 0 0 12000












Table 2-8. Summary of the IIR/LMS algorithms. Rank Order:
A(High or Good) -> B -> C -> D(Low or Bad)


Computational
Convergent Rate Complit
Complexity


Robustness


Accuracy


SM
CE









Output y(k)


Figure 2-1. Linear time-invariant (LTI) IIR Filter Structure.


Additive White Noise
v(k)


Test Model


Output
y(k)


Figure 2-2. Simulation structure of the adaptive IIR filter.


Input x(k)


c(k)






















E 1
-
_ 0
"C
m-


z-plane of Selected System










---------- ---- -- -----------------------------


-1 0 1 2 3 4 5 6 7 8
Real Part
Figure 2-3. z-plane of the test model.







18
16
14
12
10
8
6
4
2
66


a -0.4
-6.8


Figure 2-4. 3D plot of the MSOE performance surface of the insufficient order test
system.

















0.8



0.4

0.2

S0-

-0.2

-0.4

-0.6

-0.8 Local Minimum

-1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
b

Figure 2-5. Contour plot of the MSOE performance surface.


IIR OE method


5


1


) 2000 4000 6000 8000 10000
number of iterations
IIR SM method


12000


IIR EE method
0.5





-0.5


-1


-1.5
0- 2000 4000 6000 8000 10000 120
number of iterations
IIR CE method


00


-0.5 -0.5

-1

-1 -1.5

-5 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000
number of iterations number of iterations
Figure 2-6. Simulation results of weight track of the IIR algorithms for sufficient case.











IIR OE method


IIR SM method


0 02 04 06
b


IIR CE method


Figure 2-7. Simulation results of weight track of the IIR algorithms for insufficient case.


Learning Curve


LIuJ
04
C-
03


02


0 1


0 2000 4000 6000 8000 10000 12000
number of iterations

Figure 2-8. Learning curve of IIR algorithms for sufficient case.


IIR EE method










100
70
50


1 10 20 30 40 50 64
Number of Unknown
Figure 2-9. Computational complexity results from the experiment.









CHAPTER 3
GENERALIZED PREDICTIVE CONTROL ALGORIHTM


This Chapter describes the background of the generalized predictive control (GPC)

algorithm. Then, the GPC algorithm is developed based on the MIMO system ID model

(discussed in Chapter 2). Both batch method and recursive version are given and discussed.

Introduction

The generalized predictive control (GPC) algorithm belongs to a family of the most

popular model predictive control (MPC). The MPC algorithm is a feedback control method,

different choices of dynamic models, cost functions and constraints can generate different MPC

algorithms. It was conceived near the end of the 1970s and has been widely used in industrial

process control. The methodology of MPC is represented in Figure 3-1, where k is the time

index number, u(k) are the input sequences, and y(k) are the actual output sequences. The y(k)

and y, (k) are estimated output and reference signals, respectively.

Two comments are made here to describe all MPC algorithms. First, at each time step, a

specific cost function is constructed by a series of future control signals up to u(k+s ) and a

series of future error signals, which are the differences between the estimated output signals

y(k+j) and the reference signals y,(k+j). Second, a series of future inputs u(k+j) are

calculated by minimizing this cost function, and only the first input signal is provided to the

system. At the next sampling interval, new values of the output signals are obtained, and the

future control inputs are calculated again according to the new cost function. The same

computations are repeated.

Some important MPC algorithms, such as model algorithmic control (MAC), dynamic

matrix control (DMC) and GPC, have become popular in industry. MAC explicitly uses an









impulse response model and DMC applies the step response process model in order to predict the

future control signals (Camacho 1995). The GPC method, which is inherited from generalized

minimum variance (GMV) (Clarke 1979), was proposed and explained by Clarke (1987 a, b).

The GPC algorithm is an effective self-tuning predictive control method (Clarke 1988). It uses

controlled auto-regressive and integrated moving average (CARIMA) model to derive a control

law and can be used in real time applications. Juang et al. (1997, 2001) give the derivation of the

adaptive MIMO GPC algorithm. This algorithm is an effective control method for systems with

problems of non-minimum phase, open loop unstable plants or lightly damped systems. It is also

characterized by good control performance and high robustness. Furthermore, the GPC

algorithm can deal with the multi-dimension case and can easily be combined with adaptive

algorithms for self-tuning real-time applications. The problem of flow-induced open cavity

oscillations exhibit several theses issues, therefore, the GPC is considered as a potential

candidate controller.

Two modifications are made for this algorithm. First, a input weight matrix is integrated

into the cost function, this control matrix can put the penalty for each control input signal and

further to tune the performance for each input channels. Second, a recursive version of GPC is

developed for real-time control application.

MIMO Adaptive GPC Model

In this section, a MIMO model, which has the same form of the MIMO ID algorithm, is

considered. A linear and time invariant system with r inputs [u]rl and m outputs [y]l. at the

time index k can be expressed as

y(k)= ay(k -1) + ay(k -2).. + ay(k p)
+ u(k) 3u(k -1) + u(k 2) + u(k (3-1)
+ nu(k)+ Pzu(k- 1)+ Pu(k- 2)+... + Pu(k- p)








where


Su (k) y, (k)
u(k)= [u(k)] y(k) [y(k)

ur (k)J rl ym(k)Jmi

fla = [,8,0 l, = ,[ ]x,,, *., p [p p ,
{a=[oA]mm=[a2 [2]mm,<,ap=[ [a9]

Shifting j step ahead from the Equation 3-1, the output vector y(k + j) can be derived as

y(k+ j)= aJ) y(k 1) +... + cr y(k- p + 1)
+a, Jy(k- p) + fu(k + j)+ ,1)u(k + j -1) +.
+jfl )u(k) + AP, )u(k -1) + Jp + Pu(k p)


where


[al(j lmxm
[a2(J)m]m


[ap(j) ]mXm
[Cl!;m


1)a,2 + a0 1)


-1) Ot p -1)

a,-1 p


[fo(j' ]m r (al(J- )fi + PA(J-1)
[fi(I) ]mrM, (a1(J 1) A + l2(j 1))

[,A", (a,(J -)f,1 + fiA 0-1)
^-"'L"r "-. a1" )8


and with initial


[a(O)]mxm
[2Jm]m


(3-5)


LLap- 1()Jmm a -, LAp()m, 1 j ,
[ (O)]m am [ rp(O]mx__

The quantities 8,(k) (k = 0,1,..-) are the impulse response sequence of the system. Defining the

following the vector form


(3-2)


(3-3)


(3-4)









Su(k-p) u(k)
u(k p + 1) u(k + 1)
up(k-p)= k ,u(k)= u(k
u(k -) ,,x u(k + j) r.( )x
u(k 1 (rp) 1 k + 1/ (r*(j+))xl (3-6)
(3-6)
(y (k p)

yp(k p)= y(k -p+)

Sy(k- 1) 9(m*p)x1

the predictive index j = 0,1, 2, .q, q +1, s -1, and



usk) u(k + 1) ,y(k) y(k1) (3-7)

\u(k + s- 1) ( y(k + s 1), (.m*

Finally, the predictive model for future outputs, ys, is obtained, this future outputs consists of a

weighted summation of future inputs, us, previous inputs, up, and previous outputs, yp

y (k)= Tu (k) + Bup (k p)+ Ayp (k p) (3-8)

where

p o0 ... 0
T (1) fl ... 0(3-9)

o(s-1) () (s-2) f. 0 (m*s)(r*s)



B p() Bp (1) f il) ( 1
B .= (3-10)
p(s-1) p l(s-\1) ... 1) (m*s)x(r*p)









ap ap1 a,

= : .. : (3-11)
(s-1) (s-1) (s-1)
-a a,^ ... a,
SP P 1 1 (m*s)x(m*p)

The detail derivation of the GPC model is given in Appendix B.

MIMO Adaptive GPC Cost Function

Assume the control inputs (present input and future inputs) depend on the previous inputs

and output and can be expressed as


up(k- p)
(s*r) 1 (s*r) [p*(m+r)] y(k p)-12)
Lp P][p*(m+r)] 1

Two potential cost functions are list below. The first one consists terms of future outputs and a

trace of the feedback gain matrix

J(k)= yf,(k)Qy,(k)+ 7tr(H'H) (3-13)

and the second definition of cost function based on the total energy of future outputs as well as

the inputs

J(k) = (y (k)Qy, (k) + uf (k)Ru, (k)) (3-14)

The output weight matrix Q, input weight matrix R and the control horizon s are important

parameters for tuning the controller. The horizon s is usually chosen to be several times longer

than the rise time of the plant in order to ensure a stable feedback controller (Gibbs et al. 2004).

Also, if the predict horizon range is from zero to infinity, the resulting controller approaches the

steady-state linear quadratic regulator (Phan et al. 1998).

MIMO Adaptive GPC Law

In order to minimize the cost function, three approaches are considered as follows.









* Based on Equation 3-13, the control coefficients can be update using adaptive gradient
algorithm.

* Based on Equation 3-14, the optimum solution can be derived. However, this method
requires the calculation of a matrix inverse, so the computational complex is higher.

* Based on Equation 3-14, the control coefficients can be updated using an adaptive gradient
algorithm.

The first approach is examined by Kegerise et al. (2004). In the next section, the latter two

approaches are derived.

MIMO Adaptive GPC Optimum Solution

Based on the cost function 3-14, the goal is to find [H](s*r)x[p*(m+r)] or [u (k)](sr)a 1 to

minimize the cost function. We will show that both minimizing the cost function 3-14 respect to

control matrix [H](s*r)x[p*(m+r)] and input vector [u (k)](s r)l will provide the same result. To

simplify the expression, let's define

= Up (k p) (3-15)
P [p*(m+r)]x1 y (k p) v
L"p P l[p*(m+r)]x1

Substituting the predictive model 3-8 and control law 3-12 into the cost function 3-14 gives

J(k) = (y, (k)Qy,(k) + u (k)Ru, (k))

=I(Tu +[B A][v ])Q(Tu++[B A][vp]) (3-16)

+ ([H][1]) R([H][v)

with some algebraic manipulation, the gradient of cost function respect to the control matrix

[H](s*r)[p* (m+ r) can be obtained. The optimum solution is obtained when the gradient equal to

zero.









J(k)= T T, [V P]'T +R([H][H v )[v]

=TTQ((Tu+[B A][vp ]) [VP]T +Ru [Iv ] (3-17)

=(TQT+R)u [v] +T TQ[B A][vvP][ V
=0

thus,

u =- (TTQT+R) TTQ[B A][v] (3-18)

Alternatively, from Equation 3-16, setting the gradient of the cost function with respect to

the input vector [u (k)](s ,) to zero gives


J(k) =(Tu, +[B A][v ) QT+ R (3-19)
9u (3-19)
=0

thus,

u =-(T TQT+R) TTQ[B A][v] (3-20)

A comparison of Equation 3-20 to Equation 3-18 shows that these two approaches yield

the same result.

It is easy to apply the optimal solution of the Equation 3-20 on the cavity problem.

However, the matrix inversion calculation has high computational complexity. Only if the

model order is low enough, the optimal input can be used in real-time application.

MIMO Adaptive GPC Recursive Solution

To avoid calculating the inverse of the matrix in Equation 3-20, the stochastic gradient

descend method can be used to update the control matrix H using the following algorithm

"J(k)
H(k + 1) = H(k) -u- (3-21)
H (k)








Substituting Equation 3-12 into Equation 3-17 gives

OJ(k) (TTQT+R)[H[] vp ] [VP ]T
OH(k)
+TTQ[B A][vp][vP ] (3-22)
={(TQT+R)[H] +TQ[B A]}[v ]v [V

therefore, the recursive solution is given by

H(k 1) =H(k) (TTQT+R)H(k)+TTQ[B A]}[vp][v]T (3-23)

Since only present r controls [u(k)] i are applied to the system, only the first r rows in

Equation 3-23 are used

[h(k+1)]=[h(k)] -u(T'QT+R)[H(k)]+TQ[B A][v][v ] (3-24)
first r rows

In next chapter, this adaptive feedback controller, which is the combination of the MIMO

system ID (discussed in Chapter 2) and the GPC algorithm, is implementation on a vibration

beam test bed. The output weight matrix Q, input weight matrix R and the control horizon s

are tuning for testing their effects to the control performance.















Horizon


k-2 k-1 k k+l k+2 k+3


Figure 3-1. Model predictive control strategy.


k+j k+s









CHAPTER 4
TESTBED EXPERIMENTAL SETUP AND TECHNIQUES


In this Chapter, the MIMO system ID (discussed in Chapter 2) algorithm and the GPC

algorithm (discussed in Chapter 3) are implemented on a vibration beam test bed. Since the

objective idea of this sample experiment is similar to the flow-induced cavity oscillation, which

is the disturbance rejection problem, the results of this vibration beam experiment will give us

some insights to guide the later flow control applications of using this real-time adaptive control

mythology. First, computational complexity of ID algorithm, ID results in time domain and

frequency domain, and the disturbance effect for ID algorithm are examined. Then, the output

and input weight matrices as well as the control horizon are tuned for testing the control

performance with varies of these parameters.

Schematic of the Vibration Beam Test Bed

Figure 4-1 shows a detailed sketch of the whole vibration control testbed setup. A thin

aluminum cantilever beam with one piezoceramic (PZT-5H) plate bonded to each side is

mounted on a block base and connected to an electrical ground. The two piezoceramic plates are

used to excite the beam by applying an electrical field across their thickness. The piezoceramic

plate bonded to the right side of the beam is called the "disturbance piezoceramic" because it is

used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the left

side of the beam is called the "control piezoceramic" because it is supplied with the controller

output signal to counteract the unknown disturbance actuator. The beam system has a natural

frequency of about 97 Hz.

The goal of the disturbance rejection controller is to mitigate the tip deflection of the

aluminum beam generated by an external unknown disturbance signal. The controller tries to

generate a signal to counteract the vibration of the aluminum beam generated by the "disturbance









piezoceramic". The performance signal and the feedback signal of the controller are collocated,

which is measured at the center of the tip of the beam by a laser-optical displacement sensor

(Model Micro-Epsilon OptoNCDT 2000). This device gives an output sensitivity of 1 V/mm

with a resolution of 0.5/um and a sample rate of 10kHz. The performance signal is filtered by a

high pass filter (Model Kemo VBF 35) with f, = 1 HZ to filter out the dc offset of the

displacement sensor and then amplified by a high-voltage amplifier (Model Trek 50/750) with a

gain of 10.

The disturbance and control signals are generated by dSPACE (Model DS1005) DSP

system with 466MHz Motorola PowerPC micro-processor and amplified by two separate

channels of the power amplifier by the same gain of 50. The types and conditions of the signals

are discussed in detail in the next section. The dSPACE system has a 5-channel 16-bit ADC

(DS2001) and a 6-channel 16-bit DAC (DS2102) board. The signals are acquired using

Mlib/Mtrace programs in MATLAB through the dSPACE system. The block diagram of the

vibration beam test bed is shown in Figure 4-2.

System Identification Experimental Results

Computational Complexity

During the real-time adaptive control of flow-induced cavity oscillations, computational

complexity is an important issue. Kegerise et al. (2004) use 80 order estimated model for the

system ID and 240 prediction horizon for the recursive GPC algorithm to capture the dynamics

of the cavity system. Therefore, the computational complexity of the on-line adaptive controller

have to reach or beyond these lengths of parameters. Figure 4-3 shows the changes of turn

around time with the increasing estimated system order of the MIMO system ID algorithm. It is

clear that the computational complexity of this algorithm is approximately linear, and the time









requirement to estimate the same system order for the two inputs and two outputs (TITO) system

is approximately three times longer than the requirement of the SISO system. And both cases

have enough turn around time for subsonic cavity experiment.

System Identification

Before identifying the parameters of the system, the system order has to be estimated.

Using the ARMARKOV/LS/ERA algorithm (Akers et al. 1997; Ljing 1998), the eigenvalues of

the triangle matrix calculated from singular value decomposition (SVD) of the vibration beam

system is shown in Figure 4-4. This plot shows that the minimum reasonable estimated order of

the system is 2. For different ID algorithms, the estimated system order may be different in

practice. Therefore, the resulting identified system transfer function should be checked in order

to match the experimental data shape in both time domain and frequency domain.

A periodic swept sine signal (Figure 4-5) is chosen as the ID input signal (without external

disturbance). The sampling frequency is 1024 Hz, the sweep frequency produced by dSPACE

system is from 0 Hz to 150 Hz, and the amplitude of the sweep sine signal is 0.25 volt. The

performance of the system ID algorithm improves with increasing estimated system order. For

this case, the estimated order of system ID block is set to 10. Figure 4-5 shows that the output of

the system ID algorithm matches the system output very well in the time domain. The coherence

function (Figure 4-6) shows good correlation between the input and output signals.

The zero-pole location and the transfer function between the input and sensor output are

shown in Figure 4-7 and Figure 4-8 respectively. Three system ID methods are used for

comparison. Two batch methods calculate transfer function in frequency domain using the

experimental data and the FRF method to fit the frequency domain data. One recursive method

updates the system coefficients in real-time. Notice that all of these three methods give the









similar shape in frequency domain and capture the two dominant poles of the system (Figure 4-

7). However, the FRF fit batch method gives a lower order model than the recursive method.

Disturbance Effect

External disturbance degrades the performance of the system ID algorithm. Figure 4-9

shows the vibration shim experiment system ID result with different external disturbance levels.

A larger SNR (lower external disturbance level) in the input signal generally give more accurate

identified system models. However, although the lower SNR input signal may result in a

suboptimal system model, the closed-loop control implementation based on this model still

works well. The results are shown later.

Closed-Loop Control Experiment Results

Computational Complexity

The controller design block is the most time consuming blocks in the entire adaptive

control implementation. Estimated model order and the length of the predict horizon are two

main parameters effecting the computational complexity. Figure 4-10 illustrates the

computational complexity of the main C code S-function block which maps the observer Markov

parameters (discussed in Chapter 2) to predict model coefficients (discussed in Chapter 3). The

result shows that the turnaround time increases more quickly with increasing estimated system

order than increasing of the prediction horizon.

Closed-Loop Results

The experimental parameters for the closed-loop control are list in Table 4-1, and some

result plots are presented here. Figure 4-11 shows the sensor output signal in time domain, in

which the control signal is initiated at time 0.









Power spectra of open loop (base line) vs. closed-loop sensor output and the closed-loop

sensitivity are shown in Figure 4-12 and Figure 4-13, respectively. The sensitivity function is

define as


S= (f (4-1)


Equation 4-1 provides a scalar measurement of disturbance rejection. A value less than one

(negative log magnitude) indicates disturbance attenuation, while a value greater than one

(positive log magnitude) indicates disturbance amplification. Although the resonance of the

open loop system can be mitigated by the closed-loop controller, a spillover phenomenon is also

observed in Figure 4-13. As discussed in Chapter 1, the spillover problem is generated because,

for this special case, the performance sensor output and the measurement sensor output

(feedback signal) are collocated.

Next, the effects of the adaptive GPC parameters are examined. Figure 4-14 shows the

effect of the changes of the estimated model order. Figure 4-15 shows the effect of the changes

of the predict horizon. Figure 4-16 shows the effect of the changes of the input weight. Figure

4-18 shows the effect of the different level of disturbance (SNR) during the system ID. The

results for each case are discussed below.

Estimated Order Effect

In general, increasing the estimated order of the GPC, up to a certain point, can improve

the performance of the closed-loop control (Figure 4-14). The experimental result shows that

when the estimated model order is greater than 4, the closed-loop controller can not improve the

performance any more.









Predict Horizon Effect

It is clearly see that increasing the predict horizon can improve the performance of the

closed-loop controller (Figure 4-15).

Input Weight Effect

The input weight penalizes the magnitude of the input signal. For this experiment,

+0.75 volt saturation is given to the input signal to avoid the damage of the actuator. In order to

restrict the input signal within the limits of the saturation, the input weight should be carefully

tuned to obtain a realizable GPC. Although a smaller input weight improve performance of the

closed-loop controller (Figure 4-16), it also generates a larger control signal (Figure 4-17).

Therefore, the tuning idea is to decrease the input weight as low as possible under the input

saturation constraints.

Disturbance Effect for Different SNR Levels During System ID

As mentioned above, the level of the external disturbance signal (different SNR) is an

important issue for the accuracy of the system ID (Figure 4-9). However, the adaptive closed-

loop controller gives the surprising results (Figure 4-18). Three cases are examined and

compared in this section. First, the open loop (base line) case is the power spectrum of the

output measurement of laser sensor without any control input. Second, the external disturbance

is turned off during the system ID. Finally, the external disturbance is turned on with some level

during the system ID. The result shows that the higher disturbance level (low SNR) does not

have a detrimental effect on the performance of the closed-loop system. In fact, the performance

of the closed-loop controller with low SNR is improved slightly.









Summary

Table 4-2 gives a summary of the experimental results of adaptive GPC algorithm. It can

be seen that the GPC algorithm gives the better control performance with the larger estimated

system order, the higher prediction horizon and the lower input weight.

In Chapter 6, the similar control approach combining the system ID algorithm and the GPC

algorithm will be implemented on the flow-induced cavity oscillations problem. Since the

control ideas for both the vibration beam problem and the cavity oscillations problem are

disturbance rejection, the successful implementation of the system ID algorithm and GPC

algorithm to the vibration beam test bed may give the guidance to the flow control.









Table 4-1. Parameters selection of the vibration beam experiment.
Fs Disturbance GPC


Low Pass Filter
White Noise (IIR Butterworth,
4th order)
1024Hz Var = 0.09 fc = 150Hz


Input
Weight


Prediction
Estimated H
Horizon
Order











Table 4-2. Summary of the results of the adaptive GPC algorithm.


Input Weight


Prediction
Horizon


Integrated
Reduction
(In dB)
7.0
9.2
8.4
2.8


Reduction at
Resonance
(In dB)
11.2
20.1
13.4
4.2


Estimated
Order








































Figure 4-1. Schematic diagram of the vibration beam test bed.


Figure 4-2. Block diagram of the vibration beam test bed.


Analog Part





Digital Part












x 10-a
dt d1


II


0.8



.E 0.6
-0
C
i-

< 0.4



0.2


C"


Computational Complexity of MIMO ID


0 100 200 300 400 500 600 700 800 900 1000
Estimated System Order
Figure 4-3. Computational complexity of the MIMO system ID.

Eigenvalues of the System
1.5
**












++


++



+


0 5 10 15 20 25
Number of Eigenvalues
Figure 4-4. Eigenvalues of the triangle matrix obtained by the SVD method of the vibration
beam system. (Calculated by ARMARKOV/LS/ERA algorithms with 50 Markov parameters and
estimated order of the denominator is 10).




81


/ TITO





--I ------ ------ ---- ----- T------I------- ------- ------- -
------- ------ -------------- ----- ----- -- --------- -- -








---I ---- ---- --------------- --- -- ------- - -
I - -- - T - I - - - r -


1













0.4
0-24--- -------








S 0 01 02 0.3
System









0
-05 --- --- --


1
r
(-0.5


-1
0 0.1 0.2 0.3


Figure 4-5.
(bottom).


Input


0.4 0.5 0.6 0.7 0.8 0.9

n Output and ID Output


0.4 0.5 0.6 0.7 0.8 0.9 1


Times)

Input time series (top), system output and system ID algorithm output time series


Coherence



--V V-;2_
--------------------..--.. ...----------.------ -- .. ..


.------------------------ ----------------------- -----------------------


--------------------------------------
.-----_---_----_---_------ _---_----_---_-------_--_- ---_----------_----------

________________________ __-- _---- _-- _-------- -- ---_---------- --- -----



.----------------------- ----------------------- ----------------------
--L
--
,, -,,
- - - - - - - - -


1

0.9

0.8

07
E
LU 0.6

0.5
L.L
i 0.4

-- 0.3
0

0.2

0.1

n


Frequency (Hz)

Figure 4-6. Coherence function of system input and system output.










Z-plane (FRF:top, ID: bottom)




------------------------ ---------- ---------- ------------------------



I I I [ I I I


-3 -2 -1 0 1 2 3
Real Part




----- -


I Oy


-4 -3 -2 -1 0
Real Part


1 2 3 4


Figure 4-7. Zero-pole location of FRF (top) and system ID algorithm. The estimated order of
FRF fit function is 2 and the estimated order of system ID algorithm is 10.
TF Magnitude
40
S* experiment
OD 20 -r C f --------------------f- ---------------------- -
0 FRF fit


r0 -2--- -- ------ ---------------------

-60 ---------------L-------------------
-0 -50 1

0 50 100 150


200
(; 100

(D 0


c -200

-300
0


Frequency (Hz)

S experiment
-------------- --- --------------- FRF fit
ID

----'r
S.-------- ------- _.-------------- ----

-----------------------5 1---------------------- ----------------------

50 100 1
50 100 15


Frequency (Hz)
Figure 4-8. Identified transfer function using the experiment data by frequency response
function (experiment), frequency response function fit(FRF fit) and time domain system ID
algorithm (ID). The estimated order of FRF fit function is 2 and the estimated order of system
ID algorithm is 10.


83


0.5

0

-0.5


-1ik










Learning Curve (Fs=1024Hz)


10-


ro

104

10


10 L

0
-51


5 10
Times)


Figure 4-9. Learning curve of system ID with different input SNR. The estimated system order
is 10, sampling frequency is 1024 Hz.


dt 1



0.8


x 10-3 Computational Complexity of Controller Design (SISO)


0 5 10 15 20 25 30 35 40
Prediction Horizon
A
Figure 4-10. Computational complexity of the main controller design C S-function. A) For SISO
case. B) For TITO case.


84


-- 2nd order
-*- 3rd order
a-- 5th order /
S-- 10th order
-e- 16th order






IEstim ated Order
BEstimatedt Order


-- - -^j -/ -- T- -- -- -- T-- -- --- -- -- -T- -- -- -











x 10- Computational Complexity of Controller Design (TITO)
dt 1
-- 2nd order
-- 3rd order
-- 5th order
-e- 10th order -
-e- 12th order

.E 0 -- -- ----
| _0 6 - -- -- --------- -- -----^. --- ------^
F-


Estimated Order
< 0.4 -----------L------------ ----- ------------- -------- 1 ----------_



0 .2 ----------- -- ... ...........- ----------- ---
0.2




0 5 10 15 20 25 30
Prediction Horizon
B
Figure 4-10. Continued
System Output

w/o:control with :c ntrol
0.6 ---

0.4



2!



-0.2

-0.4 -

-0.6 --

-15 -10 -5 0 5 10 15
Time(s)
Figure 4-11. System output time series data. The control signal is introduced at 0 second, the
estimated order is 10, predict horizon is 10, and the input weight is 1.


85










Power Spectrum of performance
-10 i
-I- open loop (w/o control)
20 losed-loop (with control)


-30^[ ------------------------ -------------------- --IL-----------------------

E
40 ----- ------ ------




0 ---------------------- ------------- -- ------ --------


-70

-80
0 50 100 150
Frequency (Hz)
Figure 4-12. Power spectrum of output signal with control and without control signal. The
estimated order is 10, predict horizon is 10, and the input weight is 1.
Sensitivity



0.2


0. -- --- ----
O -0.2
--
0
Cl)
0 -0.2


-0.4
(.9
0
S-0.6
ry

-0.8


-1
0 50 100 150
Frequency (Hz)
Figure 4-13. Sensitivity function of the system. The estimated order is 10, predict horizon is 10,
and the input weight is 1.










Power Spectrum of performance
-10
-- open loop
4th order
-20 -- 6thorder -------------
0 6th order
10th order
-30--- --------- ------ -


-40-----------------


S-50----------


a- -60------ -------


-70 -
-70 -------------------------------------------


-80
0 50 100 150
Frequency (Hz)
Figure 4-14. Power spectrum of output signals for different estimated order. Predict horizon is
10, and the input weight is 1.
Power Spectrum of performance
-10
-'- open loop
-20 -- predict horizon is 4..
predict horizon is 10

-30

E





0 -60------------ ----- ---


-70-----------

-80
0 50 100 150
Frequency (Hz)

Figure 4-15. Power spectrum of output signals for different predict horizon. The estimated
order is 4, and the input weight is 1.



87










Power Spectrum of performance
-10
-- open loop
-20 -inputweightis 1 -..---
input weight is 10

-30 ----------------------- .-------------------- ----------------------










-70
j -40 ---------------


-50 ----- ------ --



-8------- ----------------------------
C - - -





-80
0 50 100 150
Frequency (Hz)
Figure 4-16. Power spectrum of output signals for different input weight. The estimated order is
10, predict horizon is 10.
Control Signal (input weight 1)
0.2





-0.1
0 .0 1 ----------------------- ----------------------- ------------------------







Control Signal (input weight 10)
0.04

> 0.00 4 ---- ------ ---------- - -- -----
-o 0



-0.02 ------------ ---------------------

-0.04
0 5 10 15
Times)
Figure 4-17. Control signals for different input weight. The estimated order is 10, predict
horizon is 10.






88











Power Spectrum of performance
-10
--- open loop
-- IDw/o dist.
20--
ID with dist. (SNR=0)

30 ---


-40




a_ -60 --


-70
-50 --------------------------------



-80
0 50 100 150
Frequency (Hz)
Figure 4-18. Power spectrum of output signals for different system ID disturbance conditions.
The estimated order is 10, predict horizon is 10, and the input weight is 1.









CHAPTER 5
WIND TUNNEL EXPERIMENTAL SETUP


The experimental facilities and instruments used in this study are described in detail in this

Chapter. These devices consist of a blowdown wind tunnel with a test section and cavity model,

unsteady pressure transducers, data acquisition systems, and a DSP real-time control system.

Finally, the actuator used in this study is described.

Wind Tunnel Facility

The compressible flow control experiments are conducted in the University of Florida

Experimental Fluid Dynamics Laboratory. A schematic of the supply portion of the

compressible flow facility is shown in Figure 5-1. This facility is a pressure-driven blowdown

wind tunnel, which allows for control of the upstream stagnation pressure but without

temperature control.

The compressed air is generated by a Quincy screw compressor (250 psi maximum

pressure, Model 5C447TTDN7039BB). A desiccant dryer (ZEKS Model 730HPS90MG) is used

to remove the moisture and residual oil in the compressed air. The flow conditioning is

accomplished first by a settling chamber. The stagnation chamber consists of a 254 mm

diameter cast iron pipe supplied with the clean, dry compressed air. A computer controlled

control valve (Fischer Controls with body type ET and Acuator Type 667) is situated

approximately 6 meters upstream of the stagnation chamber with a 76.2 mm diameter pipe

connecting the two. A flexible rubber coupler is located at the entrance of to the stagnation

chamber to minimize transmitted vibrations from the supply line. The stagnation chamber is

mounted on rubber vibration isolation mounts. A honeycomb and two flow screens are located

at the exit of the settling chamber and the start of the contraction section, respectively. The

honeycomb is 76.2 mm in width (the cell is 76.2 mm long) with a cell size of 0.35 mm. Two









anti-turbulence screens spaced 25.4 mm apart are used; these screens have 62% open area and

use 0.1 mm diameter stainless steel wire. For the current experiment, the facility was fitted with

a subsonic nozzle that transitions from the 254 mm diameter circular cross-section to a

50.8 mm x 50.8 mm square cross-section linearly over a distance of 355.6 mm. The profile

designed for this contraction found in previous work provides good flow quality downstream of

the contraction (Carroll et al. 2004). The overall area contraction ratio from the settling chamber

to the test section is 19.6:1. For the present subsonic setup, the freestream Mach number can be

altered from approximately 0.1 to 0.7, and the facility run times are approximately 10 minutes at

the maximum flow rate due to the limited size of the two storage tanks, each with volume of

3800 gallons.

Test Section and Cavity Model

A schematic of the test section with an integrated cavity model is shown in Figure 5-2.

The origin of the Cartesian coordinate system is situated at the leading edge of the cavity in the

mid-plane. The test section connects the subsonic nozzle exit and the exhaust pipe with 431.8

mm long duct with a 50.8 mm x 50.8 mm square cross section.

The cavity model is contained inside this duct and is a canonical rectangular cavity with a

fixed length of L = 152.4 mm and width of W = 50.8 mm and is installed along the floor of the

test section. The depth of the cavity model, D, can be adjusted continuously from 0 to

50.8 mm. This mechanism provides a range of cavity length-to-depth ratios, L/ D, from 3 to

infinity. The cavity model spans the width of the test section W. However, a small cavity width

is not desirable, because the side wall boundary layer growth introduces three-dimensional

effects in the aft region of the cavity. As a result, the growth of the sidewall boundary layers in

the test section may result in modest flow acceleration. The boundary layers have not been









characterized in this study. Nevertheless, the cavity geometry applied in this study is consistent

with previous efforts in the literature (Kegerise et al. 2007a,b) considered to be shallow and

narrow, so two-dimensional longitudinal modes will be dominant (Heller and Bliss 1975).

Removable, optical quality plexiglas windows with 25.4 mm thickness bound either side of the

cavity model to provide a full view of the cavity and the flow above it. The floor of the cavity is

also made of 14 mm thick plexiglas for optical access.

Two different wind tunnel cavity ceiling configurations are available. The first one is an

aluminum plate with 25.4 mm thickness that can be considered a rigid-wall boundary condition.

This boundary condition helps excite the cavity vertical modes and the "cut-on" frequencies of

the cavity/duct configuration (Rowley and Williams 2006). The performance of this ceiling is

discussed in the next chapter.

In order to simulate an unbounded cavity flow encountered in practical bomb-bay

configurations, a flush-mounted acoustic treatment is constructed to replace the rigid ceiling

plate. The new cavity ceiling modifies the boundary conditions of the previous sound hard

ceiling. This acoustic treatment consists of a porous metal laminate (MKI BWM series,

Dynapore P/N 408020) backed by 50.8 mm thick bulk pink fiberglass insulation (Figure 5-2).

This acoustic treatment covers the whole cavity mouth and extends 1 inch upstream and

downstream of the leading edge and trailing edge, respectively. This kind of acoustic treatment

reduces reflections of acoustic waves. The performance of this treatment is assessed in the next

chapter.

The exhaust flow is dumped to atmosphere via a 5 angle diffuser attached to the rear of

the cavity model for pressure recovery. A custom rectangular-to-round transition piece is used to

connect the rectangular diffuser to the 6 inch diameter exhaust pipe.









Three structural supports are used to reduce tunnel vibrations (Carroll et al. 2004). Two of

these structural supports attach to both sides of the test section inlet flange, and the additional

structural support is installed to support the iron exhaust pipe (Figure 5-2).

Pressure/Temperature Measurement Systems

Stagnation pressure and temperature are monitored during each wind tunnel run and

converted to Mach number via the standard isentropic relations with an estimated uncertainty of

0.01. The reference tube of the pressure transducer is connected to static pressure port (shown

in Figure 5-2) using 0.254 mm ID vinyl tubing to measure the upstream static pressure of the

cavity. The stagnation and static pressures are measured separately with Druck Model DPI145

pressure transducers (with a quoted measurement precision of 0.05% of reading). The stagnation

temperature is measured by an OMEGA thermocouple (Model DP80 Series, with 0.10C

nominal resolution).

Two pressure transducers are located in the test section to measure the pressure

fluctuations. The first transducer is a flush-mounted unsteady Kulite dynamic pressure

transducer (Model XT-190-50A) and is an absolute transducer with a measured sensitivity

(2.64 0.06) x 10 7 V/Pa with a nominal 500 kHz natural frequency, 3.447 x 105 Pa (50 psia)

max pressure, and is 5 mm in diameter. This pressure transducer is located on the cavity floor

(y = -D) 0.6 inch upstream from the cavity real wall (x = L), and 8.89 mm (z = 8.89 mm)

away from the mid-plane. This position allows optical access from the mid-plane of cavity floor

for flow visualization and avoids the possibility of coinciding with a pressure node along the

cavity floor (Rossiter 1964). The second pressure transducer is also an Kulite absolute

transducer (with measured sensitivity (5.13 + 0.03) x10 7 V/Pa and nominal 400 kHz natural

frequency, 1.724 x105 Pa (25 psia) max pressure, 5 mm in diameter), and it is flush mounted in









the tunnel side wall 63.5 mm downstream of the cavity as shown in Figure 5-2. From a series of

vibration impact tests performed in a previous study (Carroll et al. 2004), the results indicated

that the pressure transducer outputs are not affected by the vibration of the structure. An

experiment to validate this hypothesis is discussed in the next Chapter. Due to a modification of

the experimental setup, the second pressure sensor is moved to the cavity floor (Figure 5-2) for

both open-loop control and closed-loop control.

A PC monitors the upstream Mach number, stagnation pressure, and stagnation

temperature, as well as the static pressure. This computer is also used for remote pressure valve

control (Figure 5-1) in order to control the freestream Mach number using a PID controller. In

addition, an Agilent E1433A 8-channel, 16-bit dynamic data acquisition system with built-in

anti-aliasing filters acquires the unsteady pressure signals and communicates with the wind

tunnel control computer via TCP/IP for synchronization. The code for both data acquisition and

remote pressure control output generation are programmed in LabVIEW. The pressure sensor

time-series data are also collected for both the baseline and controlled cavity flows for post-test

analysis.

Facility Data Acquisition and Control Systems

The schematic of the control hardware setup is shown in Figure 5-3. For the real-time

digital control system, the voltage signals from the dynamic pressure transducers are first pre-

amplified and low-pass filtered using Kemo Model VBF 35. This filter has a cutoff range 0.1 Hz

to 102 kHz, and three filter shapes can be used. Option 41 with nearly constant group delay

(linear phase) in the pass band and 40dB/octave roll-off rate is chosen. The cutoff frequency is 4

kHz for a sampling frequency of 10.24 kHz. The signal is then sampled with a 5-channel, 16-bit,

simultaneous sampling ADC (dSPACE Model DS2001).









The control algorithms are coded in SIMULINK and C code S-functions and are compiled

via Matlab/Real-Time Workshop (RTW). These codes are uploaded and run on a floating-point

DSP (dSPACE DS1006 card with AMD OpteronTM Processor 3.0GHz) digital control system.

The DSP was also used to collect input and output data from the DS2001 ADC boards as well as

computing the control signal once per time step. At each iteration, the computed control effort is

converted to an analog signal accomplished using a 6-channel 16-bit DAC (DS2102). This

signal is passed to a reconstruction filter (Kemo Model VBF 35 with identical settings to the

anti-alias filters) to smooth the zero-order hold signal from the DAC. The output from this filter

is then sent to a high-voltage amplifier (PCB Model 790A06) to produce the input signal for the

actuator. The computer is also able to access the data with the dSPACE system via the Matlab

mlib software provided by dSPACE Inc.

Actuator System

In order to achieve effective closed-loop flow control, high bandwidth and powerful (high

output) actuators are required. The following issues should be considered for selecting the

actuators (Schaeffler et al. 2002).

* The selected actuators must produce an output consisting of multiple frequencies at any
one instant in time.

* The bandwidth of the actuators should enable control of all significant Rossiter modes of
interest.

* The control authority must be large enough to counteract the natural disturbances present
in the shear layer.

According to Cattafesta et al. (2003), one kind of actuator called "Type A" has these

desirable properties. Such actuators include piezoelectric flaps and have successfully been used

for active control of flow-induced cavity oscillations by Cattafesta (1997) and Kegerise et al.

(2002). Their results show that the external flow has no significant influence of the actuator









dynamic response over the range of flow conditions. Their later work (Kegerise et al. 2004;

2007a,b) also shows that one bimorph piezoelectric flap actuator is capable of suppressing

multiple discrete tones of the cavity flow if the modes lie within the bandwidth of the actuator.

Therefore, the piezoelectric bimorph actuator is a potential candidate for the present cavity

oscillation problem.

Another candidate actuator is the synthetic or zero-net mass-flux jet (Williams et al. 2000;

Cabell et al. 2002; Rowley et al. 2003, 2006; Caraballo et al. 2003, 2004, 2005; Debiasi et al.

2003, 2004; Samimy et al. 2003, 2004; Yuan et al. 2005). This actuator can be used to force the

flow via zero-net-mass flux perturbations through a slot in the upstream wall of the cavity.

Although the actuator injects zero-net-mass through the slot during one cycle, a non-zero net

momentum flux is induced by vortices generated via periodic blowing and suction through the

slot.

In this research, a piezoelectric-driven synthetic jet actuator array is designed. This type of

synthetic jet based actuators normally gives a larger bandwidth than the piezoelectric flap type of

actuators. A typical commercial parallel operation bimorph piezoelectric disc (APC Inc., PZT5J,

Part Number: P412013T-JB) is used for this design. The physical and piezoelectric properties of

the actuator material are listed in Table 5-1. The composite plate is a bimorph piezoelectric

actuator, which includes two piezoelectric patches on upper and lower sides of a brass shim in

parallel operation (Figure 5-4). The final design of the actuator array consists of 5 single

actuator units. Each actuator unit contains one composite plate and two rectangle orifices shown

in Figure 5-5. The designed slot geometries for the actuator array are shown in Figure 5-6.

Another advantage of this design is that it avoids the pressure imbalance problem on the

two sides of the diaphragm during the experiment. Since the two cavities on either side of a









single actuator unit are vented to the local static pressure, the diaphragm is not statically

deflected when the tunnel static pressure deviates from atmosphere. The challenge is whether

these actuators can provide strong enough jets to alter the shear layer instabilities in a broad

Mach number range and also whether the actuators produce a coherent signal that is sufficient

for effective system identification and control.

A lumped element actuator design code (Gallas et al. 2003) was used together with an

experimental trial-and-error method to design the single actuator unit. The final designed

geometric properties and parameters of the single actuator unit are listed in Table 5-2. To

calibrate this compact actuator array, the centerline jet velocities from each slot are measured

using constant-temperature hotwire anemometry (Dantec CTA module 90C10 with straight

general purpose 1-D probe model 55p 1 and straight short 1-D probe support model 55h20). A

Parker 3-axis traverse system is used to position the probe at the center of actuator slots. The

sinusoidal excitation signal from the Agilent 33120A function generator is fed to the 790A06

PCB power amplifier with a constant gain of 50 V/V. The piezoceremic discs are driven at three

input voltage levels: 50 Vpp, 100 Vpp, and 150 Vpp, respectively, over a range of sinusoidal

frequencies from 50 Hz to 2000 Hz in steps of 50 Hz. Each bimorph disc serves as a wall

between two cavities labeled side A and side B. The notation used to identify each bimorph and

its corresponding slots is shown in Figure 5-7. The rms velocities of the slots 3A and 3B located

in the centerline of the cavity are shown in Figure 5-8 as an example. The maximum centerline

velocities measured at the three excitation voltages for each slot are listed in Table 5-3. A

summary of the measurements of the centerline velocities and currents to the actuator array for

each slot are provided in Appendix E. The piezoelectric plate is tested over a range of

frequencies and amplitudes to determine the current saturation associated with the amplifier.









Figure 5-9 shows the simulation result calculated by the LEM actuator design code and is

superposed on the experimental result, Figure 5-8. The results show that, the LEM actuator

design code provides a pretty accurate rms velocity estimation of synthetic jet over a large

frequency range between 50 Hz and 2000 Hz. Finally, the measure input current level to the

actuator array after the amplifier is measured. The results are shown in Figure 5-10 and indicate

that the input current will saturate above 136mApp, which means if the input voltage is larger

than 100 Vpp, the current to the actuator will keep a constant value. During the closed-loop

experiments, an upper limit of 150 Vpp is used since the current probe is unavailable.

Figure 5-11 shows the spectrogram of the pressure measurement on the cavity floor with

acoustic treatment. The Rossiter modes (Equation 1-2) with ca=0.25, K=0.7 are superimposed on

this figure. The experimental details are explained in the next chapter. For this dissertation, the

lower portion of the Mach number range (from 0.2 to 0.35) is our control target as an extension

to previous work by Kegerise et al. (2007a,b). The desired bandwidth of the designed actuator

should cover the dominant peaks of Rossiter mode 2, 3 and 4, which is between 500 Hz and 1500

Hz. (Rossiter mode 1 is usually weaker compared to Rossiter modes 2, 3 and 4.) Over this

frequency range, the designed actuator can generate large disturbances. In addition, the array

produces normal oscillating jets that seek to penetrate the boundary layer, resulting in streamwise

vortical structures. In essence, it acts like a virtual vortex generator. A simple schematic of the

actuator jets interacting with the flow vortical structures is shown in Figure 5-12. The approach

boundary layer contains spanwise vorticity in the x-y plane (the coordinate is shown in Figure 5-

2). By interacting with the ZNMF actuator jets, the 2D shape of the vortical structures transform

to a 3D shape with spanwise vortical structures. These streamwise vortical disturbances seek to

destroy the spanwise coherence of the shear layer, and the corresponding Rossiter modes are









disrupted (Arunajatesan et al. 2003). Alternatively, the introduced disturbances may modify the

stability characteristics of the mean flow, so that the main resonance peaks may not be amplified

(Ukeiley et al. 2003). Unfortunately, the flow interaction was not characterized in this

dissertation and will be addressed in future work.

Instead of using one specific amplitude and one frequency in open-loop control, a closed-

loop control algorithm is used in this study to exam the effects of the disturbance with multiple

amplitudes and multiple frequencies. Thus, the present actuator represents a hybrid control

approach, in which we seek to reduce both the Rossiter tones and the broadband spectral level.









Table 5-1. Physical and piezoelectric properties of APC 850 device.
Shim (Brass) Piezoceramic Bond
Elastic Modulus (Pa) 8.963 x 1010 5 x1010 3.98 x108
Poisson's Ratio 0.324 0.31 0.3446
Density (kg /m3) 8700 7400 1060
Relative Dielectric Const. 2400
d31 (m/V) -200 x1012
200 Vpp/mm
for 0.15 mm
Maximum Voltage Loading r 0.15
thickness
is 30 Vpp
Resonant Resistance (Q) 200
Electrostatic Capacitance (pF) 210, 000 30%
Operating Temp. (C) --20-70










Table 5-2. Geometric properties and parameters for the actuator.
Geometric Properties of the Diaphragm APC PZT5J, P412013T-JB
Piezo. Configuration Bimorph Disc Bender
Shim Diameter (mm) 41
Clamped Diameter (mm) 37
Shim Thickness (mm) 0.1
Piezoceramic Diameter (mm) 30
Piezoceramic Thickness (mm) 0.15
Ag Electrode Diameter (mm) 29
Total Bond Thickness (mm) 0.03 ( 0.015 on each side)
Radius a0 (mm) 1/2
Length of the Orifice L (mm) 1
Width of the Orifice wd (mm) 3
Volume in side A (mm3) 4064
Volume in side B (mm3) 2989
Damping 4 0.04









Table 5-3. Resonant frequencies with respective centerline velocities for each input voltage.
Input Voltage
50Vpp 100 Vpp 150Vpp
Slot Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s)
1A 1150 35. 1100 58 1150 62
1B 1150 44 1100 69 1150 74
2A 1150 27 1100 47 1150 52
2B 1150 37 1100 60 1150 66
3A 1150 27 1100 45 1150 49
3B 1150 38 1100 60 1150 65
4A 1150 36 1150 58 1200 62
4B 1150 44 1150 68 1200 73
5A 1150 40 1150 60 1150 65
5B 1150 49 1100 72 1150 78













Pressure Valve


Manual Valve


Honeycomb


Settling Chamber


Subsonic
Nozzle


Connect to
Test Section


Screens


Figure 5-1. Schematic of the wind tunnel facility.


Fiberglass


Exhaust
Structural
Support
L



Dynamic Pressure
Sensor P2


Static Pressure Port


Cavity Model


Dynamic Pressure
Sensor P1


Unit: Inch J


Figure 5-2. Schematic of the test section and the cavity model. Dimensions are inches.


Inlet
Structural
Supports


Perforated
Metal Plate









Reconstruction
Filter


Anti-aliasing
Filter


Fc = 4 kHz


Power
Amplifier

Gain = 50x


Actuator
Array


Cavity
Flow

Fc = 4 kHz


Figure 5-3. Schematic of the control hardware setup.

o Piezoceramic


Figure 5-4. Bimorph bender disc actuator in parallel operation. The physical and geometric
properties are shown in Table 5-1 and Table 5-2.


16-bit Cont e 16-bit c
F-- Controller k s-
ADC ADC


Fs=10.240 kHz dSPACE System


R
i S


















Side A Side B





A B














C D

Figure 5-5. Designed ZNMF actuator array. A) Operation plot. B) Assembly diagram of single
unit. C) Singe unit of the actuator. D) Actuator array.























57.15(4irnch)


I- -


iirrur nwardprcrn
u- wrsirncn-wr






."". Steel


&u r~l-ur Ii~r -.L -


Unit: mm


TITLE




SEE D'WC. N<. REV

A cover
SCALE 1'1 WEI HT SHEET1 0F1


Figure 5-6. Dimensions of the slot for designed actuator array.






















































106


A I


"'"'**'C'*'*'L
"*~'*E ~~~"""'~~'""
'*r c~.r**r.rr.. **

.*r E~. rl**l. rr.
~r~


r 4ir































Figure 5-7. ZNMF actuator array mounted in wind tunnel.












50V
PP
100 V
PP
150V
PP


S30
0- 2

I 20


500


1000
Frequency [Hz]


1500


2000


50V
PP
100 V
PP
150V
PP


40

o 30
(U
;>


0 500 1000 1500 2000
Frequency [Hz]
B
Figure 5-8. Bimorph 3 centerline rms velocities of the single unit piezoelectric based synthetic
actuator with different excitation sinusoid input signal. A) For side A. B) For side B.













Sim 50 Vpp
70 Exp. 50 Vpp
Sim 100 Vpp

60 0 Exp. 100 Vpp


E 50 -
>o

40


> 30-


20


10


0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (Hz)

Figure 5-9. The comparison plot of the experiment and simulation result of the actuator design
code for bimorph 3. The output is the centerline rms velocities of the single unit
piezoelectric based synthetic actuator with different excitation sinusoid input signal
for side B.













160- 50 Vpp
100 Vpp
140 _150 Vpp

120-

g 100

80-

60-

40

20

0
200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (Hz)

Figure 5-10. Current saturation effects of the amplifier.












5000
150
4500
140
4000 ..:...

3500 130
N ..
3000 120

(D2500 110





10 0 0 .. .







Figure 5-11. Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with
500 7080
070

0.1 0.2 0.3 0.4 0.5 0.6
Mach number
Figure 5-11. Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with
acoustic treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25,
K=0.7).











Leading Edge of the Cavity


Figure 5-12. Schematic of a single periodic cell of the actuator jets and the proposed interaction
with the incoming boundary layer.









CHAPTER 6
WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION


Experimental results for the baseline uncontrolled and controlled cavity flows are

presented in this chapter. First, the effects of structural vibrations on the unsteady pressure

transducers are illustrated. Then, a joint time-frequency analysis of the unsteady pressure

measurement for an uncontrolled cavity flow is shown. Flow-acoustic features are deduced from

the results. An improved test environment is established by replacing the original hard-wall

ceiling of the wind tunnel with an acoustic liner. This new test section minimizes the effects of

the vertical acoustic modes. Finally, the results of both open-loop and adaptive closed-loop

control experiments using the ZNMF actuator array is presented and discussed in detail. The

ability of the actuator to alter both broadband and tonal content of the unsteady pressure spectra

is demonstrated at low Mach numbers.

Background

As discussed in the first Chapter, flow-induced cavity oscillations are often analyzed via

unsteady pressure measurements in and around the cavity. However, these measurements are

often contaminated by other dynamics associated with the specific characteristics of the wind

tunnel test section. As a result, the unsteady pressure spectrum may be due to the cavity

oscillations or other phenomena.

The experimental results of Cattafesta et al. (1999), Debiasi and Samimy (2004), and

Rowley et al. (2005) show that some of the resonant frequencies measured within the cavity

track or lock on to vertical acoustic duct modes at some test conditions. This effect can be

reduced by adding acoustic treatment at the ceiling above the mouth of the cavity (Cattafesta et

al. 1998; Williams et al. 2000; Ukeiley et al. 2003; Rowley et al. 2005). This acoustic treatment

modifies the sound-hard boundary condition and thus mitigates the contribution of the cavity









vertical resonance modes to the unsteady pressure measurements. Consequently, the modified

cavity model will ideally exhibit the behavior of an unbounded cavity flow and be dominated by

Rossiter modes. Alvarez et al. (2005) developed a theoretical prediction method and showed that

the wind tunnel walls lead to a significant increase in the growth rate of a resonant mode for

frequencies near the cut-on frequency of a cross-stream mode.

In the present baseline (i.e., uncontrolled) experimental study, flow-acoustic resonances in

the test section region and in the cavity region are examined. A schematic of the simplified wind

tunnel model and the cavity region of the experimental setup are shown in Figure 6-1.

Using the same nomenclature of Alvarez et al. (2005), the domain is divided into three

regions: an upstream tunnel region (x < 0), a cavity region (0 < x
(x > L). The Rossiter modes (R,, i = 0,1,...) are the combined result of a receptivity process at

x = 0, instability growth in the unstable shear layer, sound generation due to impingement of the

shear layer at x = L, and upstream and downstream propagating acoustic waves within the cavity

region. The resulting flow oscillations are interesting targets for fluid dynamics and control

researchers to analyze and mitigate.

Additional vertical cavity acoustic modes (V,, i = 0,1,...) and cavity cut-on modes

(C,, i = 0,1,...) can also be present, as discussed above. These acoustic modes are generated by

the reflections from the ceiling and area changes of the cavity model. During the wind tunnel

experiments, the vertical modes (V) are undesirable and should be reduced in order to mimic the

unbounded bomb bay problem more accurately.

As explained in Alvarez et al. (2005), the upstream region can support duct cut-on modes

DI (i = 0,1,...) and upstream propagating duct modes (T', i = 0,1,...) due to the acoustic


114









scattering process. Similarly, the downstream region can support duct cut-on modes Dd

(i = 0,1,...) and downstream propagating tunnel modes ( Td,i = 0,1,...) due to the scattering

process. Here, we focus our attention on the propagating modes in the cavity and the

downstream tunnel regions.

Data Analysis Methods

A schematic of unsteady pressure transducer locations for this study was presented in

Chapter 5 (Figure 5-2). P] and P, measure the unsteady pressure fluctuations in the cavity and

downstream regions, respectively.

The cavity and wind tunnel acoustic modes can be obtained experimentally using two

approaches. One way is to measure the output of each unsteady dynamic pressure sensor for

different fixed freestream Mach numbers and then find the spectral peaks for each discrete Mach

number. However, with this method it is difficult to track the gradual frequency changes with

Mach number. The other choice is to record each unsteady pressure sensor output continuously

as the Mach number is increased gradually over the desired range. Then, ajoint-time frequency

analysis (JTFA) (Qian and Chen 1996) is applied to these recorded pressure time series data.

JTFA provides information on the measurement in both the time and frequency domains.

Finally, the time axis is converted to Mach number via synchronized measurements of the Mach

number versus time. Similar analysis methods can be found in Cattafesta et al. (1998), Kegerise

et al. (2004), and Rowley et al. (2005). In this study, the sampling frequency for experimental

data collection is 10.24 kHz and the frequency resolution is 5 Hz. The cut-off frequency of the

anti-aliasing filter is 4 kHz, and 500 continuous blocks of time series data are used in the

analysis. During the experiment, Mach number sweeps from 0.1 to 0.7 in about 100 seconds.









Noise Floor of Unsteady Pressure Transducers

The effective in-situ noise floor of the two unsteady pressure transducers is presented in

Figure 6-2. Each noise floor measurement is compared with the spectra obtained at different

discrete Mach numbers for the acoustically treated L/D=6 cavity. Within the tested frequency

range, the signal-to-noise ratio (SNR) is in excess of 30 dB, which demonstrates adequate

resolution of unsteady pressure transducers for the present experiments despite their large full-

scale pressure ranges.

Effects of Structural Vibrations on Unsteady Pressure Transducers

A series of initial impulse impact tests are performed before the baseline experiments. As

discussed in Chapter 5, with the wind tunnel turned off, the pressure transducer outputs are not

affected by hammer- or shaker-induced structural vibrations. A simple test is described here to

investigate the effects of structural vibrations while the wind tunnel is running. To avoid

confounding cavity oscillations, the cavity floor is mounted flush with the tunnel floor (D = 0).

A piezoceramic accelerometer (PCB Piezotronics Model 356A16) is used to measure the

structural vibrations. It is attached to the test section outer wall using wax at the location

indicated in Figure 5-2, which is close to one of the pressure transducers (P2). This

piezoceramic accelerometer is connected to a multi-channel signal conditioner (PCB

Piezotronics Model 481A01). Three channels of the piezoceramic accelerometer corresponding

to the x, y, and z directions are measured. The coordinate directions of x and y are shown in

Figure 5-2, and z is the corresponding lateral direction using the right-hand rule. The

accelerometer is calibrated with a reference shaker (PCB Piezotronics, Model: 394C06) that

provides 1 g (rms) at 1000 rad/s (159.2 Hz).


116









The JTFA results (Figure 6-3 to Figure 6-5) for all components of the accelerometer

measurements show that the power of the structural vibration spreads is broadband with a few

spectral peaks. A modest peak at 1000 Hz is present in the lateral (z) and vertical (y)

directions. In addition, some higher frequency peaks (i.e., 2450 Hz in the x or flow direction,

1880 Hz in the lateral direction and 3200 Hz in the vertical direction) can also be detected.

However, the JTFA results of P, (Figure 6-6) and P2 do not display any of these resonances.

These results confirm that the unsteady pressure transducers are not affected by structural

vibrations.

Baseline Experimental Results and Analysis

The rigid ceiling plate (no acoustic treatment) above the mouth of the cavity is considered

first. JTFA results of the unsteady pressure transducer measurement for this case are shown in

Figure 6-7. Numerous flow-acoustic resonances can be observed in the plots. For easy reference

in the subsequent discussion, these features are numbered 1 and 2. The final goal of the baseline

experiment is to simulate the unbounded weapon bay using the cavity model in the test section.

Therefore, the active flow control scheme targets the Rossiter modes (feature 1 in Figure 6-7).

The other unknown acoustic features 2 in Figure 6-7 are undesirable features that we wish to

eliminate. These acoustic modes come from the bounded wind tunnel walls, the mismatched

acoustic impedance due to area change, and the leading and trailing edges of the cavity.

In order to better mimic an unbounded cavity flow in a closed wind tunnel, the boundary

condition of the cavity ceiling must be altered to eliminate the unexpected modes within the

cavity region. A flush-mounted acoustic treatment (discussed in Chapter 5) is fabricated to

replace the previous solid tunnel ceiling. The new cavity ceiling modifies the zero normal

velocity boundary condition of the previous sound hard top plate.









The unsteady pressure transducer JTFA measurement for the trailing edge floor of the

cavity is shown in Figure 6-8. The results illustrate a very clean flow field below Mach 0.6. The

acoustic features 2-4 in Figure 6-7 are eliminated within the cavity region. Therefore, the

experimental Rossiter modes R, shown in JTFA plot (Figure 6-8) now follow the estimated

Rossiter curves. At higher upstream Mach numbers (M > 0.6), the experimental Rossiter modes

deviate slightly from the expected Rossiter curves. This is partly because the estimated curves

use the upstream static temperature to calculate the speed of sound. This estimation does not

account for the expected significant static temperature drop due to the large flow acceleration

near the aft cavity region seen by Zhuang et al. (2003). Another possible reason for these

deviations of the flow-acoustic resonance comes from the structural vibration coupling with the

Rossiter modes. At high Mach numbers above 0.6, the structural vibrations may cause a lock-on

phenomenon with the Rossiter modes. For this study, all experiments are thus performed below

M= 0.6.

In conclusion, the observed flow-acoustic behavior of the acoustically treated cavity model

behaves as expected below M = 0.6 and is therefore suitable for application of open-loop and

closed-loop flow control.

Open-Loop Experimental Results and Analysis

The open-loop and closed-loop experimental results using the designed actuator array are

shown in this section. Before the control experiments, measurements of the pressure sensor at

the surface of the trailing edge of the cavity with the without the actuator turned on are shown in

Figure 6-9. Without the upcoming flow, the noise floor shows a significant peak at 660 Hz and a

small peak at 2000 Hz. The pressure sensor can also sense the acoustic disturbances associated

with the excitation frequency and its harmonics, and the measured unsteady pressure level can









reach 115-120 dB. The extent to which the measured levels deviate from theses values with flow

on (considered below) indicates the relative impact of the actuator on the unsteady flow.

First, open-loop active control is explored. The purpose of the open-loop experiments is to

verify if the synthetic jets generated from the designed actuator array can affect and control the

cavity flow. A parametric study for the open-loop control is explored first. A sinusoidal signal

is chosen as the excitation input with the frequency swept from 500 Hz to 1500 Hz. The open-

loop experimental results are shown in Appendix F. The open-control performance is best over

the frequency range 1000 Hz to 1500 Hz, which corresponds to the resonance frequencies of the

actuator array. Since at the resonance frequency, the actuator array can generate larger velocity

jet, and the blow coefficient Be = l/(pUoAc ,) increases. As a result, the control effect

increases.

For these open-loop tests, the upstream flow Mach number is varied from 0.1 to 0.4. For

illustration purposes, results are examined here for two sinusoidal signals with 200 Vpp and

excitation frequencies at either 1.05 kHz or 1.5 kHz to drive the actuator array. The 1.05 kHz

excitation frequency is close to the resonance frequency of the actuator, while the 1.5 kHz

frequency lies between the second and third Rossiter modes. The experimental results shown in

Figure 6-10 illustrate that this actuator array can successfully reduce multiple Rossiter modes,

particularly at Mach number 0.2 and 0.3. In addition, the pressure fluctuation is mitigated at the

broadband level on the surface of the cavity floor for all the tested flow conditions. However,

new peaks are generated by the excitation frequencies and their harmonics, especially at low

Mach number 0.1. With increasing upstream Mach number, the unsteady pressure level also

increases and the effect of the control is reduced. Note the synthetic jets introduce temporal and

spatial disturbances to modify the mean flow instabilities and destroy the coherence structure in


119









spanwise, respectively. The effectiveness of the actuator scales with the momentum coefficient,

which is inversely proportional to the square of the freestream velocity. So, as the upstream

Mach number increases, the synthetic jets are eventually not strong enough to penetrate the

boundary layer and the control effect is reduced. Future work should perform detailed

measurements to validate this hypothesis.

The results of the open-loop control suggest that this kind of actuator array can generate

significant disturbances not only along the flow propagation direction but also in the spanwise

direction of the cavity. The combination of these effects disrupts the Kelvin-Helmholtz type of

convective instability waves, which are the source of the Rossiter modes. As a result, multiple

resonances are reduced via active control. The experimental results also show the limitation of

the open-loop control.

Closed-Loop Experimental Results and Analysis

The open-loop control results suggest that this compact actuator array may be effective for

adaptive closed-loop control. As discussed above, the synthetic jets add disturbances to disrupt

the spanwise coherence structure of the shear layer and result in a broadband reduction of the

oscillations. However, at the same time, the coherence between the drive signal and the unsteady

pressure transducer will be reduced. High coherence is considered essential for accurate system

identification methods. To exam the accuracy of the system ID algorithm with the change of the

estimated order, an off-line system ID analysis is first performed. The nominal flow condition is

chosen at M = 0.275 (to match that of Kegerise et al. 2007a,b) with a L/D=6 cavity, and two

system ID signals, one with white noise (broadband frequency and amplitude 0.29 Vrms ) and

the other with a chirp signal (amplitude 0.86 Vrms and fL = 25 Hz to fH = 2500 Hz in T = 0.05

sec), are used as a broadband excitation source to identify the system. The running error

variances the system ID are shown in Figure 6-11. It is clear that the larger the estimated order,


120









p, the more accurate is the system ID algorithm. However, due to the limitations of the DSP

hardware, we cannot choose very large values of the estimated order for system ID algorithm on-

line.

One potential advantage of the closed-loop adaptive control algorithm is that it does not

rely exclusively on accurate system ID. Figure 6-12 shows the result of the closed-loop real-time

adaptive system ID together with the GPC control algorithm for an upstream Mach number 0.27.

Based on the above system ID results, due to the DSP hardware limitation, the estimated GPC

order and the predictive horizon are chosen as 14 and 6, respectively. The breakdown voltage of

the actuator array restricts the excitation voltage level; therefore, the diagonal element of the

input weight penalty matrix R (Equation 3-14) is chosen as 0.1. This research represents an

extension of Kegerise et al. (2007b) where the system ID algorithm and the closed-loop

controller design algorithm are used simultaneously in a real-time application. It is important to

note that only the system ID white noise or chirp signal is used to identify the open-loop

dynamics, and the feedback signal is not used for this purpose. Clearly, the results show that the

GPC controller can generate a series of control signals to drive the actuator array resulting in

significant reductions for the second, third, and fourth Rossiter modes by 2 dB, 4 dB, and 5 dB,

respectively. In addition, the broadband background noise is also reduced by this closed-loop

controller; the OASPL reduction is 3 dB. The input signal is shown in Figure 6-13.

The sensitivity function discussed in Chapter 4 (Equation 4-1) is shown in Figure 6-14. A

negative amplitude value indicates disturbance attenuation, while a positive value indicates

disturbance amplification. The results show that all the points are negative, which indicates the

closed-loop controller reduces the pressure fluctuation power at all frequencies. The spillover

phenomenon (Rowley et al. 2006) is not observed in Figure 6-14. As discussed in Chapter 1, the









spillover problem is generated because either the disturbance source and control signal or the

performance sensor output and the measurement sensor output (feedback signal) are collocated.

The Bode's integral formula is shown in 6-1.


flog S(ico) do = r- Re(pk) (6-1)
k

where Pk are the unstable poles of the loop gain of the closed-loop system. So, for a stable

system, any negative area at the left hand side of the Equation 6-1 must be balanced by an equal

positive area at the left hand side of the Equation 6-1. However, for present closed-loop control

study, the left hand side of the Bode's integral formula is -38 rad/sec, which shows that Bode's

integral formula does not hold here. Since this formula is valid for a linear controller, the

combination of the adaptive system ID and controller is apparently nonlinear. A more detailed

study is required in the future to validate this hyothesis.

A parametric study of the GPC is then studied by varying the estimated order and the

predictive horizon. Figure 6-15 and Figure 6-16 show that the control effects improve with

increasing order and predictive horizon. This trend matches the simulation results shown in

Chapter 4.

The comparison between the open-loop and closed-loop results is shown in Figure 6-17 for

the same flow condition. Notice that the baseline measurement for a same flow condition can

vary a little from case to case. The open-loop control uses a sinusoidal input signal at 1150 Hz

forcing and 150 Vpp and the rms value of the input is 53V. The closed-loop control uses the

estimated order 14, the predictive horizon 6, and the input weight 0.1, and the input rms value is

43 V.











Upstream Region -.


Cavity Region


- Downstream Region


Figure 6-1. Schematic of simplified wind tunnel and cavity regions acoustic resonances for
subsonic flow.


0.4
0.5
0.55
- 0.58
-0.6
0.65
0.69
SNoise Floor


3000
frequency [Hz]


4000


5000


6000


Figure 6-2. Noise floor level comparison at different discrete Mach numbers with acoustic
treatment at trailing edge floor of the cavity with L/D=6.


150


140------- ---
t, I
130 '.


120 '
.I r


SNoise Floor


1000


2000


;IliLa ~d~












5000
100
4500

4000 9

3500

80
4 3000

2500
) 70
2000

1500 60

1000
50
500


0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mach number

Figure 6-3. x -acceleration unsteady power spectrum (dB ref Ig) for case with acoustic
treatment and no cavity.


124












5000
100
4500
90
4000 9

3500. 80

S3000
70
= 2500

60
2000 60

1500 50

1000-
40
500 -

0. 30
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mach number

Figure 6-4. y -acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic
treatment and no cavity.






























125









































I~ ~ ~~. I;~ ~~~;~~~~


0.1 0.2


0.3 0.4
Mach number


Figure 6-5. z -acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic
treatment and no cavity.


126


5000


4500


4000


3500


T 3000


S2500


L 2000
LI.


1500o


1000


500


0


U.b U.6 0.7


~ .s,
;. r ..


.
".;r 'S
~~;.;
.Rr~~-P~
11~1












5000


4500140
140
4000

3500 130

T 3000
S.120
= 2500

2000
:'" -:. 'i:.. 1 10
1500

1000 .- 100

500
90
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mach number

Figure 6-6. Spectrogram of pressure measurement (dB ref. 20e-6 Pa) on the trailing edge floor
of the cavity for the case with acoustic treatment and no cavity. Noise spike near 600
Hz is electronic noise.












5000 -.: :: '

4500 .. .. 2 1 150
4500-, 150

4000 .
140
3500

N 3000 130

2500-
120
2000

1500
1.110
1000-
00 100


0.1 0.2 0.3 0.4 0.5 0.6
Mach number

Figure 6-7. Spectrogram of pressure measurement (ref 20e-6 Pa) on the trailing edge cavity floor
without acoustic treatment at L/D=6. Unknown acoustics features are denoted as "2,"
while the Rossiter modes are denoted as "1."












5000
150
4500
140
4000 ..:...

3500 130
N ..
3000 120

(D2500 110





10 0 0 .. .







Figure 6-8. Spectrogram of pressure measurement (ref20e-6 Pa) on the cavity floor with
500 7080
070

0.1 0.2 0.3 0.4 0.5 0.6
Mach number
Figure 6-8. Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with
acoustic treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25,
K=0.7).


129












- TE noisefloor
TE 1050Hz-150Vpp


1000 2000 3000
Frequency [Hz]


4000 5000 6000


- TE noisefloor
TE 1500Hz and 150Vpp


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]


Figure 6-9. Noise floor of the unsteady pressure level at the surface of the trailing edge of the
cavity with and without the actuator turned on. A) The exciting sinusoidal input has
frequency 1050 Hz and amplitude 150 Vpp. B) The exciting sinusoidal input has
frequency 1500 Hz and amplitude 150 Vpp. The peaks near 600 Hz and 2000 Hz are
electronic noise.


130













115

110


105


100

95


90


85

80


75
0


500 1000 1500 2000 2500
frequency [Hz]


125

120


115


110


105


100


95


0 500 1000 1500 2000 25
frequency [Hz]


3000 3500 4000


Mach 0.2, Baseline
1.05 kHz, 200 Vpp
1.50 kHz, 200 Vpp














00 3000 3500 4000



00 3000 3500 4000


Figure 6-10. Open-loop sinusoidal control results for flow-induced cavity oscillations at trailing
edge floor of the cavity. A) At Mach number 0.1. B) At Mach number 0.2. C) At
Mach number 0.3. D) At Mach number 0.4. The cavity model with 6 inch long and
L/D=6.


- Mach 0.1, Baseline
- 1.05 kHz, 200 Vpp
1.50 kHz, 200 Vpp
















Mach 0.3, Baseline
1.05 kHz, 200 Vpp
1.50 kHz, 200 Vpp


frequency [Hz]

C


Mach 0.4, Baseline
1.05 kHz, 200 Vpp
1.50 kHz, 200 Vpp


0 500 1000 1500 2000 2500
frequency [Hz]


3000 3500 4000


Figure 6-10. Continued


"'
r

i
















p=2
........... p = 4
p=6
p=8
p=14
p =50
p= 100


Time(s)

A


x 10-7
1.4


1.2


p=2
.......... p = 4
-p=6
p=8
p=14
p =50
p= 100


1.5
Time(s)


Figure 6-11. Running error variance plot for the system identification algorithm. A) With chirp
signal as input. B) With white noise signal as input. Upstream Mach number is
0.275, L/D=6.


x 107
1.4


0.8

0)
._ 0.6


0.4


0.2












TE Baseline, M=0.27
TE Closed-Loop, p=14, s=6
130


125



120



115



110



105
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]

Figure 6-12. Closed-Loop active control result for flow-induced cavity oscillations at Mach 0.27
at the trailing edge floor of the L/D =6 cavity. The control algorithm uses an
estimated order of 14 for both the system ID and GPC algorithms, and the predictive
horizon is chosen as 6. A chirp signal is used as the system ID excitation source.


134















10/ V



0
\




S-10


-20



-30



-40
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]

Figure 6-13. Input signal of the Closed-Loop active control result for flow-induced cavity
oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity. The control
algorithm uses an estimated order of 14 for both the system ID and GPC algorithms,
and the predictive horizon is chosen as 6. A chirp signal is used as the system ID
excitation source.













Disturbance Amplification


0



-0.1
UO
0)
-j

M -0.2

U)
SD isturban
-0.3



-0.4


0 500 1000 1500 2000 2500
Frequency [Hz]


ce Attenuation


3000 3500 4000


Figure 6-14. Sensitivity function (Equation 4-1) of the closed-loop control for M=0.27 upstream
flow condition. The estimated order is 14, prediction horizon is 6, and the input
weight R is 0.1. This sensitivity is calculated based on Figure 6-12.


136












- TE Baseline, M=0.27
TE Closed-Loop, p=2, s=6
TE Closed-Loop, p=14, s=6


0 500 1000 1500 2000 2500
Frequency [Hz]


3000 3500 4000


Figure 6-15. Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream
flow condition with varying estimated order. The prediction horizon is 6, and the
input weight is 0.1. The excitation source for the system ID is a swept sine signal.


135


130


125


120


115


110


105










135
S-- TE Baseline, M=0.27
TE Closed-Loop, p=14, s=2
130 TE Closed-Loop, p=14, s=6


125


120


115


110


105
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]

Figure 6-16. Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream
flow condition with varying predictive horizon s. The estimated order of the system
is 6, and the input weight is 0.1. The excitation source for the system ID is a swept
sine signal.











135
S- TE Open-Loop
130-- TE Baseline, M=0.27
TE Closed-Loop

125


-' 120
-e

0 115 ---------


110 -



1050-------------------------T--------------------
100
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]

Figure 6-17. Unsteady pressure level comparison between the open-loop control and closed-loop
control for M=0.27 upstream flow condition. The open-loop control uses a sinusoidal
input signal at 1150 Hz forcing and 150 Vpp and the rms value is 53 V. The closed-
loop control uses an estimated order 14, predictive horizon 6, input weight 0.1, and
the input rms value is 43 V.


139









CHAPTER 7
SUMMARY AND FUTURE WORK


This chapter summarizes the previously discussed work and presents contributions from

this study. Future work is summarized that addresses detailed measurements of the actuator

system, a systematic experimental analysis of the flow using various flow diagnostics, and a

more detailed parametric study of open-loop and closed-loop control.

Summary of Contributions

The contributions of this research are summarized here. First, a global model of flow-

induced cavity oscillation is derived that provides insight into the required structure for a plant

model used for subsequent control. When simplified, this model matches the Rossiter model.

Second, a novel piezoelectric-driven synthetic jet actuator array is designed for this research.

The resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well

as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes

of interest at low subsonic Mach numbers. This actuator array produces normal zero-net mass-

flux jets that seek to penetrate the boundary layer, resulting in streamwise vortical structures.

These streamwise vortical disturbances destroy the spanwise coherence of the shear layer, and

the corresponding Rossiter modes are disrupted. Alternatively, the introduced disturbances

modify the stability characteristics of the mean flow, so that the main resonance peaks may not

be amplified.

Next, a MIMO system ID IIR-based algorithm is developed based on the structure inferred

from the global model. This system ID algorithm combined with a GPC algorithm is applied to a

validation vibration beam problem to demonstrate its capabilities. The control achieves -20 dB

reduction at the single resonance peak and -9 dB reduction of the integrated vibration levels.

Finally, this control methodology is extended and applied to subsonic cavity oscillations for on-









line adaptive identification and control. Open-loop active control uses a sinusoidal signal with

200 Vpp and an excitation frequency of either 1.05 kHz or 1.5 kHz, which are detuned from the

Rossiter frequencies, to drive the actuator array. Multiple Rossiter modes and the broadband

level at the surface of the trailing edge floor are reduced. However, when the upstream Mach

number increases (greater than Mach number 0.4), the effects of the synthetic jets from this

actuator are gradually reduced. Adaptive closed-loop control is then applied for an upstream

Mach number of 0.27; the estimated GPC order is 14 and the predictive horizon is 6. To avoid

saturation in the control signal, the input weight penalty is chosen as 0.1. The GPC controller

can generate a series of control signals to drive the actuator array resulting in dB reduction for

the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition,

the broadband background noise is also reduced by this closed-loop controller (i.e., the OASPL

reduction is 3 dB). However, unlike previously reported closed-loop cavity results, a spillover

phenomenon is not observed in the closed-loop control result. As discussed in Chapter 1, the

spillover problem is generated by a linear controller because the disturbance source and control

signal or the performance sensor output and the measurement sensor output (feedback signal) are

collocated. The nonlinear nature of the adaptive system may be responsible for this effect.

Future Work

Recommended future work consists of the following items.

* The phase-locked centerline velocity of the actuator array should be measured using, at
least, by hot-wire anemometry.

* The actuator system for the active flow control needs to be explored in detail. Since the
size of the rectangular slots are small (1 mm by 3 mm), the present size of the hotwire (-1
mm) cannot provide spatially-resolved measurements. The hot wire is also not suitable to
decipher the 3-D velocity field resulting from the interaction of the jets with the boundary
layer. Laser Doppler Velocimetry (LDV) or stereo Particle Image Velocimetry (PIV)
measurements can provide good spatial resolution of the 3-D, turbulent velocity field that
results from the interaction between the ZNMF jets and the grazing boundary layer.









* The turbulent boundary layer characteristics (e.g., incoming boundary layer thickness) at
the leading edge of the cavity and the mean flowfield of the baseline uncontrolled case
should be measured.

* The impedance of the ceiling liner should be measured using an acoustic impedance tube.

* Another potential system order estimation algorithm from information theory of empirical
Bayesian linear regression by Stoica (1989, 1997) should be applied to this problem.

* Parametric studies are recommended to analyze the performance, adaptability, cost
function, and limitations (spillover, etc.) of the adaptive MIMO control algorithms over
and above that of the present SISO experiments.

* The effectiveness of the adaptive closed-loop controller should be evaluated with changes
in the upstream Mach number.









APPENDIX A
MATRIX OPERATIONS


Vector Derivatives

In this appendix, finite length vectors derivatives are illustrated. During most optimization

method development, they are the fundamental tools to find the optimum value. Only real

numbers are considered in this appendix.

Definition of Vectors

Define the vectors u and y as following

u1 Y, (u) y, (ul, u")
u2 y,(u) (u,-,u,)
u = ,y y= = (A-l)

u n-1 Y.m(") M A ym(u,"---'n)-u x

For special case, if n 1 or m = 1, the vector u or y is reduced to scalar, respectively.

Derivative of Scalar with Respect to Vector


au = [ u I(A -2)


Note the derivative of scalar is a row vector.

Derivative of Vector with Respect to Vector



a y ly OYl y ... OY2

y= 0u = 0U2 n (A-3)
du

-Ym aym 2Ym aY,
1 2 n mxn









Second Derivative of Scalar With Respect to Vector

ay 82y


a2y 0 ay 'y2
a2 OU2


-uu 1uaouy
_^. 8u,


(Hessian Matrix)

a2y a2y
Ou1 cU2 c1 nSu"
a2y a2y
cOu2 Ou2 u28un



aUJaU2 On"O _


Example 1

Given:


Y3 1 L 2 1 2
u= u2, y= 1
22 2x1 32 +3u2
3 3A


Find:
l u

Solution:


From the Equation A-3, the derivative of matrix is computed as
Qu


1i 1 1 1, ay,
y _yu yu,1 u2 93

anu u k u 3 3 (A-6)
1 -2 3 2I3
2u, -1 0
0. 3 2u3 23

Table of Several Useful Vector Derivative Formulas

Table A-i lists the most common vector derivative results, these results are very useful for

MIMO controller design. If [A, ,] is symmetric, the last formulas in Table A-i can be


expressed as


(A-4)


(A-5)











'= 2([A7) [A x]


Proof of the Formulas

Proof (a)


Ynxl

y1
Y2


- n nxl


[A,,,,][u,,,1

a11 a12 a1 n U1
a21 a22 a2n a 2

a a a *
_ n1 n2 nn _nx n nxl
a11+a1 12u2 + .. +ain
a21,u +a2, u +--- +a,2Un


an^lH +anu 2 U 2 + + annUn n


(A-8)


According to Equation A-3


a11
a21




[A nl
az1 <
aK J


(A-9)


. aln

n -


nn nxn


yn,, =[u,, n]T [Anxn]

all

S[ i ia2
=[ 1u 2 U "]L x

an

=[auI +a7212 +--- +a -lnln


a12 aIn
a22 a2n


an2 ". nn nxn
... alnu" +a2nu2 +' +"+annUnnu


According to Equation A-3


Proof (b)


(A-10)


(A-7)









FaI a21 anl1

au i i i (A-11)
Aln a2n nn n,,n
= [A nx ]

Proof (c)

Yix =- [ ].Ti [". x]


=[ 2 (A-12)
[1 .2 ""n n 1

LUn -Jnxl
= [2 +U22 +... + Un21]

According to Equation A-2

=[2u, 2u2 2u
pu "
=2[u1 u2 ]l, (A-13)
=2[u n1 ]

Proof (d)

Y>xi = [uxll]T [AI,][u x]

1 1 1 lxn "
12 1






= [al 21 + a.U + + t.n).n ... 2ln.l +ni7 +... + tl 2n2 n

LiE i 1
= [(aul, +a2lu2 + +aunlUn)u 1 + ( + (aUl +a 2nU2 +- 1+ann,,un) n ]

According to Equation A-2









y (~1 +au +a 21u2 +a + lun)+(a1ual +a12u2 + + +alnun) "
u (a, u, +a2 +- +a, )+ (a,,,u +a + + --+a,,]n) i,

= [aI,+u +C L21U +a+C un anuI + +72A + +a+annUn, n +
[a11, +a21U2 +. +au, ... a,,,u +anu + +annUn ] n
[ +ai2n2+'"+a" ... an+an2u2+...+annun]
a11 a12 aln
a a 2a2 a2
=[u U2 .. 21 u22 (A-15)

ani an2 "-" ann
al 1 a21 n
+[u1 22 u]n a12 a 22 a2
+ u 2 ... *** n

an a2n ann nxn

=[un ]T [Ann + [u nx ]T [An ]T

Example 2

Given:

uz 1 -1 0
u= u2 ,A= 1 0 1
3 3I0 0 1 (A-16)

Y 1x [31s [A3x3[1[3 1]

Find:
Bu

Solution:

Y = [U 31 ]T [A 33 ][u3x1]
1 -1 0 u I
= u2, 2 13 1 0 1 u2

(A-17)

u12

2 3x 2
= 112 + U 23 + U 2










According to Equation A-2, the derivative of is computed as
8u

--= [2u1 u3 u2 +2u3 ]1
au

Now, calculate the gradient of using the Table A-1
Qu

=[uI ]T [A 3]+[uI]T [A 3]
au
[1 1 -1" 02 1 1 0

=[u1 u u1 u2 ] 3]0 1 +[ 2 [ 1u u3 3]-1 0 0
-0 0 1- 0 1 1-3x3


=[2u, u3 u +2u3 ]3

The result of Equation A-19 is the same as that in Equation A-18.

The Chain Rule of the Vector Functions

Define the vectors u, y, z and w as following

u1 y,(u)
u, y2(u)
u= ,y=

-n l -y (")l
zI (y) (z)
z,(y) w2(z)
Z= ,W= ,(z
Lzr(y).rl w ,(z


From Equation A-3


(A-18)


(A-19)


(A-20)



















ouZ


Each element of Equation A-21 may be expanded using chain rule as

Oz1 OzC 1 Oz1 0y2 Oz 0Ym
Oz, Oz, y-, + z, -y, + Oz, C

Si &Y, Y j

k=l C Uk


i--, 2,...,r
where 1, 2,,n
j=Substitute Equati,2,on A-n

Substitute Equation A-22 into Equation A-21


1 Oz
k=ml k au1


k=k



kz1 2Yk U
m OZ,
I--

az az
Ol OY2
ay, ay2
Oz, Oz,
ay, ay2


k1
k=l @k


az,
k=lk


m O!r
k-I-
k=l k

&l
az2

aYm


"L rxm Lu ]mxn


k=l @k 1n


k=I iyk 1n
5$z 2 k

y kn


k=I Ok 1n
m aZ ar


ay1
4 1 21

Oy1 11/2
chi^ 8 i


rxm [L 1 2


(A-21)


(A-22)


LIz_~
]rn^


aYm
au"


(A-23)


Jmxn









Similarly, for more vectors, the chain rule just builds the new derivatives to the right.


Iu = -z ] ~ m u(A-24)
Qu ,,, z y Qu a

The Derivative of Scalar Functions Respect to a Matrix

Define a matrix

1 42 ... 4n
h, h,, h,
H= 22 (A-25)

hl hm2, hmn- n

and a scalar function

J = f(H). (A-26)

The gradient of J with respect to H represented by

8J aJ 8J


aJ 8J 8J
8J
h= h h ah22 (A-27)
OH

8J aJ 8J
ah,, ah, aknh

Example 3

Find the gradient matrix, if J is the trace of a square matrix H.

J tr(H)
= f h (A-28)


According to Equation A-27 the gradient of J with respect to H is











SJ F 0 1 ... 0

O 8H ~ = i (A-29)
0 0 1 -- 1
Kxn

= [I]ln n

Example 4

Find the gradient matrix, if J is the trace of a square matrix H'H, where H is defined by

Equation A-25 and need not be square. The scalar function J can be expressed as

J = tr(HH)

t h1 ,, 21 h h,11_ h12 ... l ,n
h r 12 h22 hm 2 21 h 22 ... h2n
= tr i i : i .

h,, h2,, h ,,,,, n h,, h 2,, h -,,,,,n n
= (h2 +h22, +...+h2,l)+(h21 +h22, +...+h2m2)+... (A-30)
+ (h +2h 2 +--- h +h 2)
n .
=zh2
J 1 1

and the gradient of J is

24, 22 ... 2h
8J 2kh, 2h2 .. 2h
-H .,, : i mI (A-31)
22h, 2hm2 2 h ,n
2[H] n


Example 5

Define the vectors A, B, and H as following










a, b,

A= a ,B= b2

a b
nImxl nxl

H.1 _, ,1 f I

hl h,, ... h/)
Al k2 mn.my.,

Find the derivatives of a scalar function J respect to matrix H.

Express the scalar function as

J = AHB
hI1 h1, ... h,n b,
h,, h22,, hn b2
=[al a2^ a- 4H= K Z

hl hm2 ... hmn -.. b
bi

= (alhl +a2h2 +...+ahl) ... (alhn +a2h2n +...+ahal)] b2

bn
m m
=Zakhklbl +...+ Zakhhb
k=l k=1
n m
=ZZakhk,b,
-l= k=l


So, the derivative of J respect to the matrix H is


a,b, a,




aamb, am
a,
Sa2 [b




= [A]ml B


a bn
ab


a2bn,,
a1mbn mn


b
b,


,b2


T lxn


(A-32)





















(A-33)


(A-34)


1 b -.. b, ]x









Note that since J is a scalar function, J = J' = BTHTA, and the following equation is also

hold.

=a O (BTHT A)J
LH lOH [ (A-35)
=[A],[BT








Table A-1. Vector derivative formulas.


[Anxn]
[A ,jn+u]T
2[unxI
[nx ]T [A nxn I + [u nx ]T [A nxn ]T


nxI, = [A nx, [u nxl ]
yIxn = [unx,]T [Ann]
IxI, = [uxi]T [unxI]
ixI, = [I"nx,] [A nxn ] [unxI









APPENDIX B
CAVITY OSCILLATION MODELS


Rossiter Model

The derivation of the Rossiter Model depends on the following assumptions (Rossiter

1964)

* Frequencies of the acoustic radiation are the same as the vortex shedding frequency.

* There are mV complete wave length of the vertex motion at the time t = to + t'. (e.g.
m =1,2,...)

The schematic of the Rossiter model is illustrated in Figure B-l, and the symbols in the

plot are listed as follows

L,D Cavity length and depth

U, Free stream velocity

a Mean sound speed inside the cavity

m, Mode number (integer number 1,2...)

a Phase lag factor between impact of the large scale structure on the trailing edge

and the generation of the acoustic wave.

K Proportion of the convective vortices speed to the free stream speed

A, Spacing of the vortices

At specific initial time t = to, an acoustic wave forms at the trailing edge with the distance

aAi (Figure B-1). Appropriately choose the time t', such that, the propagating wave front just

reaches the cavity leading edge. By the assumption, the mode number m, is an integer number.

Then,

L = at' (B-l)









L = mnA, aAl cKUj '


And the frequencies of the oscillations are related the phase speed and the wavelength of

the vortical disturbance.


f U (B-3)


Substituting the Equation B-l in to Equation B-2,


L =m, a, KU L
a (B-4)
1+KUjL =(m,-a)A,
a

Then, combining the Equation B-3 and Equation B-4 resulting


1+aUj L=(m -a) KU.
a f
fL (m, -a) (B-5)

a

The Rossiter model is defined by


St f
U.



IK a
(m a
St = ( m- :=1,2,...
+M


Linear Models of Cavity Flow Oscillations

The block diagram of the linear model of the flow-induced oscillations is show in Figure

B-2 (Rowley et al. 2002). And the closed-loop cavity transfer function can be expressed as


(B-2)










G(s)S(s)A(s)
P(s) = (B-7)
1- G(s)S(s)A(s)R(s)

Furthermore, the shear layer model is considered as a second-order system with a time


delay


G(s)= Go0(s)e


w2+2w+
s2 + 2wo + o2


L
KU~


where


wo Natural frequency of second order system


Damping ratio

r, Time delay inside the shear layer

The acoustics model A(s) can be represented as a reflection model (Figure B-3)

where

Ta Time delay inside the cavity

r Reflection coefficient

The closed-loop transfer function of the reflection model can be written as

es,
A(s)= -2
1-- re 2S

and

L
a =-
a


To recover the Rossiter formula, additional assumptions are required


(B-8)


(B-9)


(B-10)


(B-1)









* Impingement model S(s) and receptivity model R(s) are unit gains.

* No reflections in acoustic model in B-10, r = 0.

* The shear layer model is only a constant phase delay

Go(s) =e -2 (B-12)

Depending on these assumption, combine the Equation B-7, Equation B-10 and Equation

B-12,

G(s)A(s)
1- G(s)A(s) (B-13)
(B-13)
e12-~a -s(r+r,)
1-e e12 2Tes(z+z" )


In order to find the resonant of the system, substitute the poles locations s = iw into the

characteristic function of the Equation B-13, and then combine the Equation B-9 and Equation

B-11 resulting

1- e -2Tae -w(e+z,) = 0
e-12n, 2 e-12~a -w(r+r,)
L L
2.(m, a)= w(+ -) (B-14)

wL m, -a
2.crUo U+ 1
+-
a K

Define


St f
U.
(m a (B-15)
m m 1,B2,..



The linear model results Equation B-15 matches the Rossiter formula Equation B-6.









Global Model for the Cavity Oscillations in Supersonic Flow

The global model for cavity oscillations in supersonic flow is shown in Figure B-4. To be

consistent with the notation of Kerschen et al. (2003), the global model consists of a downstream

traveling shear layer instability wave of amplitude S, upstream propagating cavity acoustic

modes U and downstream propagating cavity acoustic modes D, and 'fast' modes Ef and

'slow' modes E near-field acoustic waves in the supersonic stream. The local amplitudes of all

quantities at the leading edge are denoted by the decoration while quantities at the trailing

edge do not have the decoration.

The scattering processing at the leading edge is modeled by

S CSU

b CDU^
SC U (B-16)

_E _CEU

Csu, CDU, CE and CEsU are the four scattering coefficients for the upstream end of the cavity.

The first subscript denotes the output, while the second subscript denotes the input. Similarly,

the scattering process at the trailing edge is modeled by

S

U=[cus CUD C CUE E (B-17)



Cus, CUD, CU and CuE are the four scattering coefficients for the downstream end of the

cavity. The downstream propagating components from the leading edge to the trailing edge are

given by









S = SelaL,D= D eedL
3 3 (B-18)
Ef =L 2Efe zL1L,Es =L 2Ee l2

where a and Td are the complex wavenumbers of the shear layer instability wave and the

downstream propagating acoustic cavity mode, respectively. M, and M2 are the wavenumbers

of the 'fast' and 'slow' downstream propagating near-field acoustic waves, respectively. Finally,

the upstream propagating acoustic cavity mode is

U= UezlL (B-19)

where r, is the complex wavenumber of this mode.

The global model can be expressed in a block diagram (Figure B-5).

Substituting Equation B-19 into Equation B-17 results

le- L = U = CusS + C UDD + CUE +C, E

U
S (B-20)
[e'"L -CUs -CUD -CUE -CU D =0
Ef



Also, substituting Equation B-18 into Equation B-16, the following formula can be obtained

Se -'L C
De -dL
= 3 L C= U (B-21)
E, L 2Efe CEfU
S-3 C
_E_ L 2 Ee2L -C EHu

These four equations can be written in matrix form











-CsU
-CDU

-C


-CE,U
EsU


e laL


0 0


0 e ldL


0 0 L 2e lML 0


0 0 0


L 2e 2L


(B-22)


Now, combine Equation B-20 and Equation B-22 yields


Define


e-,L _CU _CUD

-Cs e 0
-C. 0 e"rL


CUEf

0
0


3
0 0 L 2 eL1L


-CEU









A=









X=


-CU

0
0

0


3
0 0 0 L 2 e 2L





eL _Us -CUD _CUE

-CSU e -a 0 0
-CDU 0 e -dL 0
3
-C,,, 0 0 L 2 e L


-CEsU

U
S
D
Ef
E
A.


U
S
D
Ef

E


-c^
C
0
0

0


3
0 0 0 L 2e M2L


(B-23)


(B-24)


Therefore, Equation B-23 can be written as


AX = 0


(B-25)









Notice that the quantities of X are the incident waves on the two ends of the cavity. The global

mode has to satisfy Equation B-25 which corresponds to the condition det(A) = 0. Calculating

the determinant and simplifying,

3
L 2 (CE fUCUEfe -MlL CEU CUEs- -2L (B-26)
(B-26)
+L (e + CsuCuseU U + CDUCUDe L) 0

Assume a simple case where only Cus 0, Cs, # 0, and all other scattering coefficients are

zeros. Therefore, Equation B-26 can be simplified to

-e -' L + CSUC Se aL 0

Csu Cse (a+-)L = 1 (B-27)

(Csue'L (CuseL) I

L'
Enforce the phase criterion and notice that the length is normalized by L = and
U

Equation B-27 results

CsuCse(a+,)L= 1
(B-28)
Arg(Csu )+ Arg(Cus)+ Re [(a + ,,)]L = 27rm

or

L' 2z7m m-Arg(Cs) -Arg(Cus)
-L-
U Re[a + (B-29)
(B-29)
oL' m -(Arg(Cs)+ Arg(Cus)) /2~
2;r-U Re[a +r,]

Consider the normalized wave number


Re[a']=- /
L JKU/U^, K
(B-30)
Re [r,- 0 = M
c / U









and define


St- oL'
2;r U (B-31)
a = (Arg(Cs )+ Arg(Cs ))/ 2r

Therefore, Equation B-29 can be written as

oL'= L = 2rm Arg(Csu )- Arg(Cus)
-L-
U Re [a + r,
L' m -a (B-32)
St= ,m = 1, 2,...
2.r -U 1
+M


This matches the Rossiter formula Equation B-6.

Global Model for the Cavity Oscillations in Subsonic Flow

The global model for cavity oscillations in subsonic flow is shown in Figure B-6. To be

consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream

traveling shear layer instability wave of amplitude S, upstream modes U and downstream

modes D propagating acoustic modes in the cavity, and upstream modes E^ and downstream

modes Ed propagating near-fields acoustic waves in the subsonic flow. The local amplitudes of


all quantities at the leading edge are denoted by the decoration ^), while quantities at the


trailing edge do not have the decoration.

The scattering processing at the leading edge is modeled by

SSU SE,"
D = CDU CDE,, (B-33)

d EdU Ed E,









Csu CDu, CEdu, CSE, C DE and CEdE are scattering coefficients for the upstream end of the

cavity. Again, the first subscript denotes the output, while the second subscript denotes the

input. Similarly, the scattering process at the trailing edge is modeled by


FU C CUD CUE d
=CS CD C D (B-34)


Cus, CUD CUE > CEus CEuD and CEEd are scattering coefficients for the downstream end of the

cavity. The downstream propagating components from the leading edge to the trailing edge are

given by

3
S = Sea, D = De L, ,Ed = L 2Ed edL (B-35)

where a and Td are the complex wavenumbers of the shear layer instability wave and the

downstream propagating acoustic cavity mode, respectively. Md is the complex wavenumber of

the downstream propagating near-field acoustic wave. Finally, the upstream propagating

acoustic cavity modes are

3
U= Ue^,E =L 2EUeL (B-36)

where zr and M are the complex wavenumbers of the upstream traveling cavity mode and the

upstream traveling acoustic near-field mode, respectively.

The global model can be represented in a block diagram (Figure B-7). Again, substituting

Equation B-35 into Equation B-34and Equation B-36 into Equation B-33, and combine the

results, a matrix equation AX = 0 can be obtained, where











-1 0 useaL CUD eL L 2CU eLdL
3 U
0 -1 CESe C edL L 2C edL E
EE,D E,,Ed

A= eL L2C eL 1 0 0 X= S (B-37)
3 1D
CDU eL L 2CDE, e ML 0 -1 0
3
C edUL L2C elL 0 0 -1
EdEdE,

The global mode has to satisfy AX = 0 which corresponds to the condition det(A) = 0.

Calculating the determinant and simplifying,

3
CSU se (a+US )L CDUCUDe(d -)L 'L CEd CUEd (d e )L
3 3
L 2CSE' C ESe +M)L L CDE, CE, e'(d+M,)L +L 3CE CE,-Ed' e(Md+M,)L
+LC C C C C C C
(a+d+M+,)L SU ES DE, UD DU ED SEUS
+Le '
CSUCUSCDE, E,,D CDUCUDCSE ES (B-38)

(C C` C +C C Cs CUS
L 3 e(a+Md ,+M )L CSU ES EdE, UEd EdU EEd SEHUS
-C SUCUS CEd -CE CEd CSEd CSE,S

+Lc c c c +3C( c c+ c
L3 e'(dd+M,+M )L CDUCED EdE UEd EdU E,E DE CUD
-CDU UD EdE E,,Ed EdU UEd DE E,,D

Assume a simple case where only Cus # 0, Csu # 0, and all other scattering coefficients are


zero. Therefore, Equation B-38 can be simplified to

CsuCuse (a+z,)L =
(B-39)
(Cssue ) (Cuse L)=

This matches the supersonic case results Equation B-27










'a -~ t-a'


a, + KU t ---

) ( ) (


t= t+t'


Figure B-l. Schematic of Rossiter model.


Shear Layer


Actuator
Input


(Trailing Edge)
Impingment


Figure B-2. Block diagram of the linear model of the flow-induced cavity oscillations.


Acoustics
Feedback


Sensor


~I
3 3



















Figure B-3. Block diagram of the reflection model.


Turbulent
Boundary Layer
M > 1 9Ts


-D
D

I(


<- L


Figure B-4. Global model for the cavity oscillations in supersonic flow.


CUS D CUEf CUEs


Figure B-5. Block diagram of the global model for a cavity oscillation in supersonic flow.


y


X S


I )D U










Turbulent
Boundary Layer

..M <.1
,X S


-..< L


Figure B-6. Global model for a cavity oscillation in subsonic flow.


Figure B-7. Block diagram of the global model for a cavity oscillation in subsonic flow.


D-
D

I(








APPENDIX C
DERIVATION OF SYSTEM ID AND GPC ALGORITHMS

MIMO System Identification

Assume a linear and time invariant system, with the r inputs [u]r1l and the m outputs

[y]m,, at the time k, the system can be expressed as

y(k) = ay(k 1)+ay(k 2)+-. +ay(k p)
(C-1)
+f,u(k)+ APu(k -1) + ,u(k 2) + ...+ ,u(k- p)

where

u, (k) y, (k)
S ,((k) )=[y(k), y2(k)
u(k) = [u(k)] = k) ,y(k) = [y(k)]= Yk)

S(,k). y,(k) (C-2)
{a [a I...mIa2 =[a'2I...mI<... lap =[La P mXm
Ao, [fAo,,,,A= [AL,8,,= ,p =[p'~m

Define

B(k)=[a, --- a, P, -- f] (C-3)
(k) 1 o P lmx[m*p+r*(p+l)] (C-3)

and

y(k 1)

y(k- p)
(P(k) = (C-4)
u(k)

u(k p) [m*p+r*(p+l)]xl

and these yield the filter outputs

[j)(k)]m = [L (k)]m[(m, p+,(p+l))] [o(k)][ p+r(,,,))] (C-5)








Therefore, the error between the two outputs is defined as

[s(k)]l = [j(k)].1 -[Y(k)]m, (C-6)
(C-6)
= (k)q(k) y(k)

and the scalar error cost function is defined by

J(k)= e (k)e(k) (C-7)
2

To identify the observer Markov parameters C-3, the following equations based on the gradient

descend method is developed

OJ(k)
(k + 1) = (k) -/ ^ (C-8)
a0(k)

where / is the step size. Substituting Equation C-6 into Equation C-7

J(k) = (k)e(k)
2
= l[(k)-p(k) y(k)] [j(k)p(k) y(k)]
(C-9)


= [O p'O y yOp + yT y]


and the gradient of error cost function is

aJ(k) 1 T T T-
80(k) 21 Y (

1= O[p9P + (pO' 2yCpT

2= LO[rp + pp'T 2ypT] (C-10)

=(0op-,)opT

[F (k)][sub g T (k) E [m*p+* (p+ l)

Finally, substituting Equation C-10 into Equation C-8 and yielding









+(k +1) = 0(k) (k)q' (k) (C-11)

In order to automatically update the step size, choose

P = (C-12)
cr+ k2

where c is a small number to avoid the infinity number when qp 2 is zero.

Here the main steps of the MIMO identification are given as follows

Step 1: Initialize [\(k) lm[mp+rpl))] =[01.


Step 2: Construct regression vector [q,(k) ][mp+r*p+l)]1 according to Equation C-4.

Step 3: Calculate the output error [e(k)]m1 according to Equation C-6.

Step 4: Calculate the step size according to Equation C-12

Step 5: Update the observer Markov parameters matrix [O(k)] m[m pr(p)] according to Equation

C-11. And then go back to step 2 for next iteration.

Generalized Predictive Control Model

In this section, a MIMO model, which is the same as the model of the MIMO ID C-l, is

considered. Assume a linear and time invariant system, with the r inputs [u]r and m outputs

[y]mm', at the time index k, the system can be expressed as

y(k)= ay(k 1)+ay(k 2)+--..+ay(k- p)
(C-13)
+flu(k)+ Pu(k- 1)+ fPu(k- 2)+.- + flpu(k- p)

where








u, (k) y, (k)
u (k) y (k)
u(k) = [(k)]= ( y(k) [y(k) Yk)

ur(k)Jrl y,(k)j (C-14)
{tI = [al]ma2 = [a2]m... ap lap ]mXm
fa = [1, ],, l a = [a,, ],, ..., p [= p, L,

Shifting Equation C-13 one time step ahead and can be expressed as

y(k +1) = acy(k) + azy(k -1) + + apy(k p +1)
(C-15)
+P,u(k + 1) + P,u(k) + P,u(k 1) + + fpu(k p + 1)

Substituting Equation C-13 into Equation C-15

y(k +1) = alaty(k -1) + aazy(k 2) +.. + alapy(k p)
+a2y(k -1) + ay(k 2) +. + ay(k p +1)
+apfu(k) + a1Pu(k -1) +.. + atpu(k p)
+P,u(k + 1) + P,u(k) + P,u(k 1) + + Pu(k p+ 1)
= (a + a) y(k- )+(aa, + a) y(k- 2)+-..
+(alap + ac) y(k- p +1) + (acla) y(k p)+ flu(k +1)
+(aP0 + P1)u(k)+(c~a, + P2)u(k 1) +-... (C-16)
+(a,3p + 3,)u(k- p+ 1)+(aflp)u(k- p)
= [a x y(k 1)+ a2 ] y(k-2)+

[a(k p +1)+ [a]x y(k p)
+ [o u(k + 1)+ lo') ]m, u(k)+ [PA(')I u(k 1) +..
+ ,< u-')]mr,(k p + 1)+ [p(')]mx, uk p)

where










[a(2] :

[ Jm] m
L< m


a1a, + a
aa2 +a3


a ^ap + a
= a,a,


[ o() mxr
[A(')Jmxr


(all l fl2)
:(W+A)


(C-17)


[1)] = a +fm


The output vector y(k + 1) is the linear combination of the past outputs, the past inputs and

future inputs. By induction, the output vector y(k + j) can be derived as

y(k + j) = [a(J ] 4 y(k -1)+ [a2) ] x y(k 2)+ .. + [a,-0 ] x y(k p +1)
+ [ap)m] y(k -p)+[Pfo ] u(k + j)+[f,: ]m u(k +j .-1)+...
[ ](C-18)
+[lomw] u(k)+[P)] m u(k-1)+ [Pl)]mr (k-2)+

p+ [ f fJxr -fx u(k p)


where


[ apl()] m
[ p() ]m m


-1) +a~'2J1)



1) a, +a a 1)
-1)api + ap0-1)

a,0 (


[,6I]-mxr


(a 1( l)flo + fij1))


(C-19)


[/ 1(j ) 1 x = (a1(Jl1)'# 1 + 18( 01)
58p (j)Lmxr a"(-.-^-"


[1aO ]mxr
[Pl mxr


[f- 1(0')]m

[f(O()] rmxrl


The quantities [fpA(k) (k = 0,1, ) are the impulse response sequences of the system.


with initial


[ai(0) ]m
[a2 ]m

a, 1i0l ]m

[a (0) ]m


a=
m



m
x. = apl
x. = ap


(C-20)








Define the following the vector form


u,(k -p)= [u (k p)], ) [U(k+)],:
[u-)(k P1)] 1 [u(k + j)]ri

[y(k- p)] x





Substituting Equation C-21 into Equation C-18 and express it in matrix form as

[y(k+ j)]mxl [" [m xr)m (r*(j 1)) l(k)](r*(j+l))x1

S( l r 2 mr r mx(r*p) I (k p) 1()x (C-22)
+ p([a ] ... [a2] 1 [a,]mxm ) qy -(k-p)](m*p)x1

Now, let the predictive index = 0,c1,2, n, q +1, -, s 1, and define d bye

/I [u (k)L rx l
us,(k) [u(k +1)Ix],x

\[u(k+ s -1 ) ] rx1 (r*s)x1 (C-23)
[y(k)]mxl

Ys(k)= [y(k + 1)]mxl

A[y(k + pm m can, b(m)xl

A predictive model can be expressed by
















([a mxr [O]mxr



([fo(1) mxr [0 lmxr


[af (1)]m


[/0(s-2) ]mxr


[, -2) ].
L mxr


[Y(k)]m 1
[y(k + 1)]mx
x,
L[y(k+s -1)]mx J(m*s)x1
=I + II+III

[(k)],"/
L'" [O]mxr )mx(r .s) [O]x l
[O rxl ( .(r .)xl


[O]m -0lr m(r*s)


[u(k)Jrxl
[u(k + 1)]rx1

[O]rx1 (r*s)xl


S [u(k)]r 1
[ mxr )mx(r (k +1)]rxI
[u(k+s- 1)]Irx s


[0]mr
[O]mxr
... [0]
-- 0imx

-' w e_


[u(k)], .
[u(k +)]r 1


(m*s)(rs) (k+1)


or for simplification


I [T](ms)x(r*s) [u (k)](rs)xl


where


(C-24)


(['0(s-1) ]m


[ ]mxr [Olmxr
[a(1) [ ]mxr
[P: ,:


(C-25)


(C-26)


[ y. (k) ](m.s)











[Tf1 (m-*sx(r*-s


[Lf lmxr [OLmxr



[l(s-1) [m o(s-2) mxr


... [0]L
... [L0]
SJOmxr
0 mxr J


The matrix [T] is called Toeplitz matrix. And the second part of Equation C-24 is


[u(k-P)]L




[u(k -p+)],
[u(k +l)]

S[u(k -1)],)L (r)1


(5s-1) Ir I'flp-1(S-1) Ir

[P mxr mxr ,_

fLP mxr pl-1 mxr
fp (1) mxr 'fp-1 (1) mxr
5p(s-1) mr [p-1(s-1) mr
W ^~s!]~ [apl(ljm


[u(k-P))L
[u(k P +1)]r

[u(k 1)]r1 ) (r*p) (m*)

[u(k p)]
[u(k -P+)]L (C-28)

[u(k -1)] /Jr (rp)xl


or for simplification


II = [B](ms)x(r*p) [u(k p)]()xl 1


(C-27)


[PLlsOmr ) mx(r/n p)




[(1(1) ]m


1(s-1) mxr (m*sx(r*


where


(C-29)


(m*s)x(r*s)


/Il


r


(1) M-1(1) [( m mx [ I mxr mx(rlp)











[B](m*s)x(r*p)


[ ]mxr
c1(1) ]mx

[f1 (-1) ]m
L Im,


(C-30)


LfP Imxr Lp-1 mxr
_lp) m, J fp-1 (1) Im,


[fp(s-1) mr p-1 (s-1) mr,


and the third part of Equation C-24Is


r [y(k p)],x

mam [y(k p +1)]r
[y(k 1)]rx1 )(m*p)xl
[y(k- p)]i x
) m () [y-(k p+1)]

I[y(k-1)]rl (m*p)x1


(l i 1) mm [ap-(s-1) Im



[iaP mxm lm P-1 \mxm
lap(1) mxm [p-1(1) Imm

[ap (1) ,m [ap-(s-1) Im


[a mxm
[a(1)mxm


[y(k-p)],r1
[y(k-p+1)]l



S[y(k p)],r
[y(k +1)]r-

[y(k 1)]rxi J(m.p)x1


(m*s)x(m*p)


or for simplification


III [A](ms)(m*p) [yp(k p)](m*p)l


where


m*s)xl


(C-31)


(C-32)


(m*s)x(r*p)


(ap,, p almxm




([a<'L],,, [cu,-."']


S(s-1) mxmmx(m*p)











[A] (m*s)x(n*p)


[P mxm [a-1 /mm
[a) [ p-1)]( Im


_ (s-1) mxm L p-1s -1)mxm


... [a ,]xi


. [ (1) ] m
L (s-1) mxmI


(C-33)


Combine Equation C-26, Equation C-29 and Equation C-32 in to Equation C-24,
[ys (k)](m.s) = [T],(ms)(r]s) [u (k)](rs) 1 +[B](m*s)(r*p) [IP(k- p)](-.)x1

+[A](m*sx(m*p) [y(k p)](m*p) 1


(C-34)


l(m*s)x(m*p)









APPENDIX D
A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES


In this section, a potential theoretical model of cavity acoustic resonance is derived based

on the model ofKerschen et al. (2003). The model combines scattering analyses for the two

ends of the cavity and the propagation analyses of the cavity shear layer, internal region of the

cavity, and acoustic near-field. Kerschen et al. solve a matrix eigenvalue problem to identify the

frequencies of the cavity oscillation. A different approach for characterizing the same model is

illustrated in this section. A signal flow graph is first constructed from a block diagram of the

physical model, and then Mason's rule (Nise 2004) is applied to obtain the transfer function from

the disturbance input to the selected system output. This method gives a prediction for the

resonant frequencies of the flow-induced cavity oscillations. In addition, this method also

provides a linear estimate for the system transfer function.

Mason's Rule

Mason's rule reduces a signal flow graph to a transfer function between of any two nodes

in the network. A signal flow graph connects "nodes," used to represent variables, by line

segments, called "branches." First, some definitions are given as follows

* Input node: a node that has only outgoing branches.

* Output node: a node that has only incoming branches.

* Path: a string of connected branches and nodes. It contains the same branch and node only
once.

* Forward path: a path that traverses from the input node to the output node of the signal
flow graph in the direction of signal flow. It touches the same node only once.

* Forward path gain: the product of gains found by traversing the path from the input node to
the output node of the signal flow graph in the direction of signal flow.

* Loop: a path that starts at a node and ends at the same node without passing through any
other node more than once and follows the direction of the signal flow.









* Loop gain: the product of branch gains in a loop.

* Touching: two loops, a path and a loop, or two paths that have at least one common node.

* Nontouching loop: Loops that do not have any nodes in common.

* Nontouching loop gain: The product of loop gains from nontouching loops.

The closed loop transfer function, T(s), of a linear dynamic system represented by a signal

flow graph is (Nise 2004)

N
PkAk
T(s)= k- (D-l)


where

N: number of forward paths

pk: the kth forward path gain

A : 1 E(loop gains)+ E(nontouching loop gains taken two at a time) -
(nontouching loop gains taken three at a time)+ E(nontouching loop
gains taken four at a time) -...

Ak : A (loop gain terms in A that touch the kth forward path). In other
words, Ak is found by eliminating from A those loop gains that touch
the kth forward path.

Global Model for a Cavity Oscillation in Supersonic Flow

The global model for cavity oscillations in supersonic flow is shown in Figure D-1. To be

consistent with the notation of Kerschen et al. (2003), the global model consists of a downstream

traveling shear layer instability wave of amplitude S, upstream modes U and downstream

modes D propagating acoustic modes in the cavity, and 'fast' modes Ef and 'slow' modes E

near-field acoustic waves in the supersonic stream. The local amplitudes of all quantities at the









leading edge are denoted by the decoration (), while quantities at the trailing edge do not have

the decoration. The scattering processing at the leading edge is modeled by

S CSU
S CDU
C U (D-2)

_E_ CEU

Csu, CDU CEU and CE,. are the four scattering coefficients for the upstream end of the cavity.

The first subscript denotes the output, while the second subscript denotes the input. Similarly,

the scattering process at the trailing edge is modeled by

S

U =[Cs CD Cr CuE D (D-3)

E

CUs, CUD, CU and CUE are the four scattering coefficients for the downstream end of the

cavity. The downstream propagating components from the leading edge to the trailing edge are

given by

S = eL, D=bedL
3 3 (D-4)
Ef =L 2Efe zLL,E =L2Ee llL

where a and Td are the complex wavenumbers of the shear layer instability wave and the

downstream propagating acoustic cavity mode, respectively. M, and M2 are the wavenumbers

of the 'fast' and 'slow' downstream propagating near-field acoustic waves, respectively. Finally,

the upstream propagating acoustic cavity mode is

U = UeL (D-5)









where r, is the complex wavenumber of this mode.


The global model can be represented in a block diagram (Figure D-2) or a signal flow

graph (Figure D-3). From the signal flow graph (Figure D-3), the transfer function between the

disturbance input N and the upstream propagating cavity acoustic mode U' can be found. First,

identify the components of Equation D-l. The results are listed in Table D-l. The characteristic

function of the system can be identified from Equation D-l

4
A= k
k=l1
S -3 (D-6)
=1 Csuuse L "+CDUCUDe(+ + L 2 (CEUCUE e(-M)L +E CEuC, e -M2


The numerator of the transfer function can be derived from D-1. The Ak terms are formed

by eliminating from A those loop gains that touch the kth forward path.

4 3
SPk Ak=CsuCuse + DUCUDe +L EUUE -ML +CEUUEe -2 (D-7)
k=l

Finally, the transfer function between the disturbance input N and the upstream traveling wave

U' in the cavity is



4
PkAk
TU'N(S) k=l

CC3L C L (CEfL C eI (D-8)

CsuCuse' +CDuCuDe +L 2 CEfUUE f eL u uCEU C UE e2
1- CsuCuse( +CDuCuDe (+)L + L 2 E U E, Hf- L E+ u CuEe e('-M2


The characteristic function A in Equation D-6 is the same as the eigenvalue relation

derived by Kerschen et al. (2003). And the transfer function in Equation D-8 gives more









information concerning the physical model. For example, this model can predict the resonant

frequencies as well as the nodall" regions or zeros of the flow field.

Global Model for a Cavity Oscillation in Subsonic Flow

The global model for cavity oscillations in subsonic flow is shown in Figure D-4. To be

consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream

traveling shear layer instability wave of amplitude S, upstream modes U and downstream

modes D propagating acoustic modes in the cavity, and upstream modes E and downstream

modes Ed propagating near-fields acoustic waves in the subsonic flow. The local amplitudes of


all quantities at the leading edge are denoted by the decoration ^), while quantities at the

trailing edge do not have the decoration.

The scattering processing at the leading edge is modeled by

s CSU CSEu-
D = CDU CDE (D-9)
E.
Ed EdU -EdEj

Csu Dv,DU > CE, Cs E, CDE and CEdE, are scattering coefficients for the upstream end of the

cavity. Again, the first subscript denotes the output, while the second subscript denotes the

input. Similarly, the scattering process at the trailing edge is modeled by



L L ECS D CUE, C D (D-10)


Cus, CUD, CE CES CEuD and CEE are scattering coefficients for the downstream end of the

cavity. The downstream propagating components from the leading edge to the trailing edge are

given by









3
S = SeL, D = )De" Ed = L EddL (D-11)

where a and Td are the complex wavenumbers of the shear layer instability wave and the

downstream propagating acoustic cavity mode, respectively. Md is the complex wavenumber of

the downstream propagating near-field acoustic wave. Finally, the upstream propagating

acoustic cavity modes are

3
U = UeL =L E eeA iL (D-12)

where ,, and M are the complex wavenumbers of the upstream traveling cavity mode and the

upstream traveling acoustic near-field mode, respectively.

The global model can be represented in a block diagram (Figure D-5) or a signal flow

graph (Figure D-6). From the signal flow graph (Figure D-6), the transfer functions between the

disturbance input N and the upstream propagating cavity acoustic modes, U' and E ', can be

found. The transfer function between the disturbance input N and the upstream propagating

cavity acoustic mode U' is calculated by first identifying the components of Equation D-1. The

results are listed in Table D-2.

The characteristic function of the system can be identified from Equation D-1










12 6
A= 1--/k 2k
k=l k=l

C,,C,,e"' ) +C DUC (+UD )L
3 3
+L CE EU UE(Md + L 2SUC CEHSCDE CUDI1
3
+L-3CSUCES CEdE CUUEd 2 + L 2CDU CED Cs CUSI1
+-L3CDU CEC CE C UEd 3CL U CEd C C US2
3
+L 3CU CEE CDE CUD 3 +L 2CS C se+M)L
3
+L 2CD C D(e +M,)L L -3CE CE e(Md +Mu)L
L 2CCDEu CED L CE E Ed

L 2 SU US D^u EuD^ SU US Edu EuEd 2
3
+ +L 2CDUCUDCSE CESI1 L+, 3CDU CUD CEdE CEE 13

+L3CEU CEUC SCE + L3CEI U CUE DE CEDI3



CsuCuse + CDU CeiUD +)L
3 3
= 1- +L2CE uUEde(Md +L L 2CSEu CES e(+M)L
3
+L 2CD C (e (d+Mu)L + L3C CE e'(Md+,)L
DEu EuD EdEu EuEd

CSUCES DECUD DU ED SE, US



(C C C C 3C C C (D-13)
DSU ED EdE, UEd EdU EEd SE, US
-C C C C -CC E,
SU US EdE, ^ E,'Ed EdU C CCES)
(CDUCECE CUE +CUC C C (D-13)
L3I 3LC +jED ECC EE E^ EDE, UD
-CDU UD EdE, EEd EdU UEd DE ED 1


where


I, = e'(a+Td,+r,)L ,2 = e(a+Md+Mu+z )L ,3 e'(z"+Md+M,"" +)L (D-14)

The numerator of the transfer function can be derived from D-1









9
SPkAk
k=l

= CsuCuse ) 1-L CDE, EDe EdEL ECEd C de

3
"(CDUCUD dL L 2CSE C e(a+M,)L -L C CEEd

3 3 3


+L E C UE Su ES (a+M)L 3DE E +M)L

S SUCES DE, UD + ES EdE" C UEd d

3
lDUEuDSE rUSez(-+Mu+a)L 3 DU u EdEud +Md)L

"+( 3CE_ CEuEd SEuUS (Md mu + (L-3EdU EuEd CDu D +Mu dL
3


Li3 (ed )KC+C C C C C
SCsuCuseL +CDUCUe ZdL + L 2C UEd uE dL +


-CSUCUSCDE, ED -DUCUD SE, E2S
-c c c c +-c cc c -


+L-3e (a+Md+Mu)L SUESEdEUEd CEdUCEEdSECUS (D-15)
-CCC C C -C C CS CC
c 3 e(d+Md+Mc)L c c c
L-3 ,(-+Md+M,)L DU ED EdECUEd EdU EuEd DE, UD
CDU UD EdE, EEd EdU JUEd DE, ED

Finally, the transfer function between the disturbance input N and the upstream traveling

wave U' in the cavity is










T7' (s)
9
YPkAk
k=l
A
3
Cs uCse'L +C C DeLL + L 2CEu C eMdL +

+L-2 e(a+d+M)L SUCESCDEu UD DU E,uD SE US
-CsuCu DE, ED DU UD SEuEuS



-CsuCUSCE-Eu-C d Ed d E EUC-UEd SEES
(C C C C+C C C C
L-3 (+Md+M)L CDU E-D EdEu- UEd EdU -E,Ed DE, CUD
-CdDU CEUDE- CEEEEd CEdU CUEd CDE CEuD
3
L C su C use a + C D UM UD ( L +C D )L C L _)2 EdU U L CEd C E e (
+L 2CSe (a+Mu )L L 2 C L +M)L LE 3 C M(MdL


_-L e(a+zd+M,+,)L CSUCESCDECUD+ CDUC E.DC SE CUS
e-'_CsuCusC DE CHED DU UD CSE CES (D-16)

--3 (a+Md +M,+,)L SU ES EdE UEd E+C EdE dSE US


-3 ,(-d+Md+Mu,-)L CDU EuD EdEUE d EdUd EuEd DEu UD
-CDU UD EdE, E,,Ed EdU UECdDEu EH D

The characteristic function in Equation D-13 is the same as the eigenvalue relation derived

by Alvarez et al. (2003), but the transfer function in Equation D-16 gives more information

concerning the physical model. For example, this model can predict the resonant frequencies as

well as the nodall" regions or zeros of the flow field.

Then, the transfer function between the disturbance input N and the upstream propagating

acoustic mode E', is calculated. Similarly, first, identify the components of Equation D-1. The

results are listed in Table D-3. Because the loop gains and the nontouching loop gains are the

same as before (Table D-2), only the forward path gains are listed.










The characteristic function of the system is the same as Equation D-13. And the numerator

of the transfer function can be derived from D-1

3 3
Z pkAk SUCEse L + DU CEDe +L 2 E-CEE edL (D-17)
k=l

Finally, the transfer function between the disturbance input N and the upstream traveling

wave E', in the cavity is


TEN(S)
3
PkAk
k=l
A
3
CsuCESeL +CDU CE1D eZ L +L 2CEd CE edL

+L CSECESe (+L DE CEDe L C3 E CEE, 2E( M (Md+L)L
1 CSUCuse" +CDU UD e+L L 2EdU UE d

i +L 2C (a+ML L 2- 2 r '(qrd +Mu)L +L-3 jr r (Md+M )L
'- SEuEuS DEu EuD EdEu CuCC

_L e(a+d+,+,)L CSUCESCDE CUD+ DU ED SE, US
-CSUCU CD -CD DUC C (D-18)

--3 e(a+MdM )L SU ES EE UEd EdU EUEd SE US
-CSUCUSCEdE, ECEd -EdUUEd SECE5S

_L-3 ,(-,+M,+M,+ )L DUED EEUEE d EdU EuEd DECUD
-DU UD EdE EC Ed-C CUEd DE ED
^ ^DU^UD^ ,C,* E^ CEdU UE^D IE CED )










Table D-1. Components of the Mason's formula for supersonic case.
Index Forward Paths Forward Path Gains
1 0 1 2 6 10 11 p, = CsuCse
2 0 -1 3 7 10 11 2 = CDUCUDeLL


0 1 44 810 11
0->1 ->54 ->8 ->10-> 11
0 1- 5 9 10 11

Loops
1 2 3- 6 3- 10 1

1-3 7 -10-1

1 4 3-8 10 1

1 5 -9 10 1


3= L 2CEU -CU e IML
3
4 = L 2 C C e- 2L
Loop Gains
1I = Csu~se a)L
/12 = CDU UDe( d)L
3
,/1 3L 2C H CuL e Ml
13=L ^EfU UEf
3
14= L2 Cu e (,-M2)L


Index
1











Table D-2. Components of the Mason's formula for subsonic case.
Index Forward Paths Forward Path Gains
1 0 1 3 6 9 11 p = CsuCuseaL
2 0 -1 4 7 ~9 -11 P2 =CDUCUDedL
3
3 0- 1 5 8 9 11 3= L 2 CE Ee dL
3
4 0 1 3 36 10 2 4 7 9 11 p4 =L 2CSUCECDE'UD +,+)L

5 0-1->3- >6-10 ->2->5->8->9- >11 ps =L 3CSUESC CUE, E a+ d)L
3
6 01- >4->710-2->2-> 6911 p6 = LCDCE DC Cue+M"+)L

7 0->1->4->7->10->2->5->8->9->11 p7=L CDUC C -E uE(M, Md)L
7 0 1 447 10 2 5 8 911 p = 3DUCDciE E (q,+A"+Ac)L

8 0o1>5->8-10->2->3->6->911 p8 =-3 CEU E CE CSE_ USC(Mdu++a)L

9 0->1->5 ->8->10->2->4->7->9->11 p9 = L3CCE ECDE CuCDeM(d+,+d)L
Index Loops Loop Gains
1 1 3 6 9 1 1/ = CCsuuse (+)L

2 14->479->7 1 12 CCDUCDe(",)L
3
3 1 5 8 59-1 /13 L2 EC UE (Md+ ,)L
3
4 13 6 10 2 4 7 9 1 14 = U~CEUCS DE, C UDe'a+ ,+-d+,)L
5 1 3 6 10 ~4 2 5 8 9 1 15 3 (+M+d+ )L

6 1->4->7->10->2-3->->9-1 /1 =L- C, C CU e' ,)L
66 L" 2CDU CE,,DC SE CUS

7 1 4->7 10 2 5 87 = 310 9C C5 8DCEIE9 C UE( _,+A'+M,+Md+ r,)L

8 1 5 8 10 2 3691 1-8 = L-3 CEU CEc _SCECSE CCS(+,+a+- )L

9 15->85- 10 2 4->4791 /19 = L 3CUC C C,, CuDe (Md"M"+,++,)L
3
10 2-> 3-> 6 -> 10 ->2 /11 L- 2 Cs,,(+M)L
10 SE E, S^-^ +
3
11 24->4710-2 111 L-2 (DE ECEz( )L
S11 CD-, CDE-1, De
12 2 5 -8-102 12 3= C_3 EC CE, et(d +eM+,, )L
Index Nontouching Loops Nontouching Loop Gains
3
1 Loop 1 and Loop 11 /21 L 2CSU USCDE CE"D (+" ++M)L

2 Loop 1 and Loop 12 122 = 3C SU CUSCEE, dEEd e( +l,
3
3 Loop 2 and Loop 10 /23 L 2CDUCUDSE C C, ~(++c+M)L










Loop 2 and Loop 12

Loop 3 and Loop 10

Loop 3 and Loop 11


124 -3C DU UD C E C E (d e'd d ,)L
/2 = Lr3C C CU C Sel(Md+r,+a+M )L
125 EdU UEd SE ES
126 L3C EdUE DE EuDe(Md ur Td M












Components of the Mason's formula for subsonic case.
Forward Paths Forward Path gains
0 1 -3 6 10 12 p, = CsCCEseaL
0 -1 4 7 10 12 P2 = CDUCDe dL


0->1 ->5 ->8 ->10 ->12


p3 = L 2CEU CEe_ edL


Table D-3.
Index
1
2









Turbulent
Boundary Layer
M>1 19T


x ES


L


Figure D-1. Global model for a cavity oscillation in supersonic flow.


Figure D-2. Block diagram of the global model for a cavity oscillation in supersonic flow.


D
D

Ir


~t~ E,.


tALE






























Figure D-3. Signal flow graph of the global model for a cavity oscillation in supersonic flow.


Turbulent Y
Boundary Layer





S




L


Figure D-4. Global model for a cavity oscillation in subsonic flow.



























Figure D-5. Block diagram of the global model for a cavity oscillation in subsonic flow.


S ,L S


Figure D-6. Signal flow graph of the global model for a cavity oscillation in subsonic flow.









APPENDIX E
CENTER VELOCITY OF ACTUATOR ARRAY

In this appendix, the center velocity of each slot (notation see Chapter 5) and

corresponding current measurement of the actuator array are shown.









slot la


50V
PP
100 V
PP
150V
PP


40

o 30

20

10


500


1000
Frequency [Hz]


A

slot la


0.06
0.0 o 50 V
PP
0.05 + 100 Vpp
+ 150 Vp
0.04

0.03

0.02


0.01


0- I
0 500 1000
Frequency [Hz]
B
Figure E-1. Hot-wire measurement for actuator array slot la.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


1500


2000










slot lb


50V
PP
100 V
PP
150V
PP


.X 40
20



20


1000
Frequency [Hz]


0.06 o 50V
pp

0.05 +100 Vpp
150V +
PP
0.04
+

0.03 +


0.02 +


0.01 ++ o000

0

0 500


slot lb






+ +



o O
00
OO o0





1000
Frequency [Hz]
B


Figure E-2. Hot-wire measurement for actuator array slot lb.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


500


1500


2000


a


a,
(U










slot 2a


1000
Frequency [Hz]
A

slot 2a


0 1 I
0 500 1000
Frequency [Hz]
B
Figure E-3. Hot-wire measurement for actuator array slot 2a.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


S40


30
0
> 20


50V
PP
100 V
PP
150V
PP


500


1500


2000


50V
PP
100 V
pp
PP
150V
PP


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01









slot 2b


500


1000
Frequency [Hz]
A

slot 2b


o 50V
pp
100 V
PP
+ 150V
PP


0 1 I
0 500 1000
Frequency [Hz]
B
Figure E-4. Hot-wire measurement for actuator array slot 2b.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


200


50

S40

o 30


50V
PP
100 V
PP
150V
PP


1500


2000


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01


:++++++










slot 3a


1000
Frequency [Hz]
A

slot 3a


0 1 I
0 500 1000
Frequency [Hz]
B
Figure E-5. Hot-wire measurement for actuator array slot 3a.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


40


30


20
,- 20


50V
PP
100 V
PP
150V
PP


500


1500


2000


50V
PP
100V
PP
150V
PP


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01










slot 3b


1000
Frequency [Hz]
A

slot 3b


0 1 I
0 500 1000
Frequency [Hz]
B
Figure E-6. Hot-wire measurement for actuator array slot 3b.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


202


50

S40

o 30


50V
PP
100 V
PP
150V
PP


500


1500


2000


50V
PP
100V
PP
150V
pp


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01










slot 4a


60

50

40

o 30
>


0 1 I
0 500 1000
Frequency [Hz]
B
Figure E-7. Hot-wire measurement for actuator array slot 4a.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


203


++ 000
o
0






1000
Frequency [Hz]
A

slot 4a


50V
PP
100 V
PP
150V
PP


500


1500


2000


50V
PP
100 V
PP
150V
pp


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01










slot 4b


50V
PP
100 V
PP
150V
PP


. 40
0


1000
Frequency [Hz]
A


slot 4b


0 1 I
0 500 1000
Frequency [Hz]
B
Figure E-8. Hot-wire measurement for actuator array slot
Current measurement of the actuator array.


1500


2000


14b. A) Center RMS velocity. B)


204


500


1500


2000


50V
PP
100V
PP
150V
PP


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01













60

50

40

o 30

20-

10


500


1000
Frequency [Hz]
A

slot 5a


50V
PP
100V
PP
150V
PP


0- I
0 500 1000
Frequency [Hz]
B
Figure E-9. Hot-wire measurement for actuator array slot 5a.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


205


slot 5a


50V
PP
100 V
PP
150V
PP


1500


2000


0.06

0.05


S0.04
<

0.03

S0.02


0.01




























500


1000
Frequency [Hz]
A


o 50V
pp
+100 V

150V
PP


0 500 1000
Frequency [Hz]
B
Figure E-10. Hot-wire measurement for actuator array slot 5b.
Current measurement of the actuator array.


1500


2000


A) Center RMS velocity. B)


206


slot 5b


60



. 40
0


50V
PP
100 V
PP
150V
PP


1500


2000


slot 5b

+t**t


0.06

0.05


S0.04
<

0.03

0.02
U 0.02


0.01









APPENDIX F
PARAMETRIC STUDY FOR OPEN-LOOP CONTROL

In this appendix, a parametric study results for open-loop control are shown. To illustrate

the open-loop control, a fixed flow condition (M=0.31) is chosen for all experimental cases. The

frequencies of the excitation input signals to the actuator array are varied from 500 Hz to 1500

Hz, and for each frequency, two excitation voltage levels, 100 Vpp and 150 Vpp, are chosen.


207













TE Baseline
TE Open Loop




- -- -


-- -- -'----Ir--------


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-1. Open-Loop control result for M=0.31 and excitation sinusoidal
500 Hz and 100 Vpp voltage.


140

130 r


120


110-


100


input with frequency


TE Baseline
TE Open Loop


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]


Figure F-2. Open-Loop control result for M=0.31 and excitation sinusoidal
500 Hz and 150 Vpp voltage.


input with frequency


208


130
I I


S100


II




..











TE Baseline
TE Open Loop


100
S100


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-3. Open-Loop control result for M=0.31 and excitation sinusoidal
600 Hz and 100 Vpp voltage.


140

130 -

120

110

100

90-


input with frequency


TE Baseline
TE Open Loop


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]


Figure F-4. Open-Loop control result for M=0.31 and excitation sinusoidal
600 Hz and 150 Vpp voltage.


input with frequency


209


:"i












TE Baseline
TE Open Loop


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-5. Open-Loop control result for M=0.31 and excitation sinusoidal
700 Hz and 100 Vpp voltage.


140

130 -

120

110-

100


input with frequency


TE Baseline
TE Open Loop


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]


Figure F-6. Open-Loop control result for M=0.31 and excitation sinusoidal
700 Hz and 150 Vpp voltage.


input with frequency


210


130

120
';


100

90

80

70











TE Baseline
TE Open Loop


-- -- -- --
'- -^A~rWil--- ---------- --------r--------
- --
I I^

:: : : : : : : : : : : : I : : :: :


100

90

80

70


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-7. Open-Loop control result for M=0.31 and excitation sinusoidal
800 Hz and 100 Vpp voltage.


140

130 -

120

110-

100


input with frequency


TE Baseline
TE Open Loop


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]


Figure F-8. Open-Loop control result for M=0.31 and excitation sinusoidal
800 Hz and 150 Vpp voltage.


input with frequency


1











TE Baseline
TE Open Loop


100


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-9. Open-Loop control result for M=0.31 and excitation sinusoidal
900 Hz and 100 Vpp voltage.


input with frequency


130

120


TE Baseline
TE Open Loop


K


110

100


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-10. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 900 Hz and 150 Vpp voltage.


212


"(:












TE Baseline
TE Open Loop


110
-o
100

90

80

70


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-11. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1000 Hz and 100 Vpp voltage.


140

130 ,


TE Baseline
TE Open Loop


120


100

90

80


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-12. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1000 Hz and 150 Vpp voltage.


213


''
:'~













TE Baseline
TE Open Loop


- - - -


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-13. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1100 Hz and 100 Vpp voltage.


TE Baseline
TE Open Loop


100

90

80

70


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-14. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1100 Hz and 150 Vpp voltage.


214


I JUV

120

110---------


100O













STE Baseline
TE Open Loop

--------------

---------------- -- ------

I I I


130

120

110

-0
100


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-15. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1200 Hz and 100 Vpp voltage.


130

120--

110

100---------

90

80-

70-


TE Baseline
TE Open Loop


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-16. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1200 Hz and 150 Vpp voltage.


215













- TE Baseline
- TE Open Loop





7 -- - -


S100


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-17. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1300 Hz and 100 Vpp voltage.


140

130 -

120 ''


TE Baseline
TE Open Loop


100


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-18. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1300 Hz and 150 Vpp voltage.


216


' I v~va











TE Baseline
TE Open Loop


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-19. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1400 Hz and 100 Vpp voltage.


140

130 ,


TE Baseline
TE Open Loop


120

110

100


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-20. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1400 Hz and 150 Vpp voltage.


217


100


--
, I_ _





I4 I


1~












TE Baseline
TE Open Loop


100


1000 2000 3000
Frequency [Hz]


4000 5000 6000


Figure F-21. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1500 Hz and 100 Vpp voltage.


TE Baseline
TE Open Loop


100

90


0 1000 2000 3000 4000 5000 6000
Frequency [Hz]
Figure F-22. Open-Loop control result for M=0.31 and excitation sinusoidal input with
frequency 1500 Hz and 150 Vpp voltage.


218









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Zhuang, N., Alvi, F. S., Alkislar, M. B., and Shih, C. 2003 Aeroacoustic Properties of Supersonic
Cavity Flows and Their Control. AIAA 2003-3101, 9th AIAA/CEAS Aeroacoustics
Conference.


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BIOGRAPHICAL SKETCH

Qi Song was born in Beijing, P. R. China. He entered the Beijing University of

Aeronautics and Astronautics, Beijing China, in 1992 and received his BE. degree in the

Department of Mechanical Engineering in August 1996. After several years working at Air

China, the flag carried airline of China, he entered the University of Florida, Gainesville, Florida

in August 2001 as a research assistant. In May 2004, he earned a Master of Science degree in

electrical engineering. He received his Ph.D. degree with an aerospace engineering major in

May 2008 at the University of Florida. His research focuses on the active closed-loop control of

the cavity oscillations.


227





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CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS By QI SONG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1

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2008 Qi Song 2

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To my wife, Jingyan Wang; and my lovely son, Lawrence W. Song 3

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ACKNOWLEDGMENTS This study was performed while I was a member of the Interdiscip linary Microsystems Group (IMG) in the Department of Mechanical a nd Aerospace at the University of Florida in Gainesville, Florida, USA. First, I sincerely acknowledge my advisor, Dr. Lou Cattafesta, for providing me with this opportuni ty and giving me so much precious advice during my course time at UF. His guidance and encouragement al ways gave me sufficient confidence to conquer any difficulty. I thank all of my colleagues in the IMG group for their invaluable assistance. Finally, I appreciate my friends and my dear family for their tremendous consideration and unselfish support during my journey. 4

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TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 LIST OF ABBREVIATIONS........................................................................................................14 ABSTRACT...................................................................................................................................16 CHAPTER 1 INTRODUCTON.................................................................................................................. .18 Literature Review.............................................................................................................. .....20 Physical Models...............................................................................................................2 1 Physics-Based Models.....................................................................................................21 Numerical Si mulations....................................................................................................27 POD-Type Models...........................................................................................................28 On-Line System ID and Active Cl osed-Loop Control Methodologies...........................28 Unresolved Technical Issues...........................................................................................33 Technical Objectives........................................................................................................... ...33 Approach and Outline.............................................................................................................34 2 SYSTEM IDENTIFICATION ALGORITHMS....................................................................38 Overview....................................................................................................................... ..........38 SISO IIR Filter Algorithms....................................................................................................39 IIR OE Algorithm............................................................................................................40 IIR EE Algorithm............................................................................................................41 IIR SM Algorithm...........................................................................................................41 IIR CE Algorithm............................................................................................................41 Recursive IIR Filters Simulation Results and Analyses..................................................41 Accuracy comparison for sufficient system.............................................................43 Accuracy comparison for insufficient system..........................................................43 Convergence rate......................................................................................................44 Computational complexity.......................................................................................44 Conclusions.....................................................................................................................45 MIMO IIR Filter Algorithm...................................................................................................46 3 GENERALIZED PREDICTI VE CONTROL ALGORIHTM...............................................62 Introduction................................................................................................................... ..........62 MIMO Adaptive GPC Model.................................................................................................63 5

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MIMO Adaptive GPC Cost Function.....................................................................................66 MIMO Adaptive GPC Law....................................................................................................66 MIMO Adaptive GPC Optimum Solution......................................................................67 MIMO Adaptive GPC Recursive Solution......................................................................68 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES..............................................71 Schematic of the Vibration Beam Test Bed...........................................................................71 System Identification Experimental Results...........................................................................72 Computational Complexity.............................................................................................72 System Identification.......................................................................................................73 Disturbance Effect...........................................................................................................74 Closed-Loop Control Experiment Results..............................................................................74 Computational Complexity.............................................................................................74 Closed-Loop Results.......................................................................................................74 Estimated Order Effect....................................................................................................75 Predict Horizon Effect.....................................................................................................76 Input Weight Effect.........................................................................................................76 Disturbance Effect for Different SNR Levels During System ID...................................76 Summary.................................................................................................................................77 5 WIND TUNNEL EXPERIMENTAL SETUP........................................................................90 Wind Tunnel Facility........................................................................................................... ...90 Test Section and Cavity Model...............................................................................................91 Pressure/Temperature Measurement Systems........................................................................93 Facility Data Acquisition and Control Systems......................................................................94 Actuator System................................................................................................................ ......95 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION..............................113 Background...........................................................................................................................113 Data Analysis Methods.........................................................................................................115 Noise Floor of Unsteady Pressure Transducers....................................................................116 Effects of Structural Vibrations on Unsteady Pressure Transducers....................................116 Baseline Experimental Results and Analysis.......................................................................117 Open-Loop Experimental Results and Analysis...................................................................118 Closed-Loop Experimental Results and Analysis................................................................120 7 SUMMARY AND FUTURE WORK..................................................................................140 Summary of Contributions...................................................................................................140 Future Work..........................................................................................................................141 APPENDIX A MATRIX OPRATIONS.......................................................................................................143 6

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Vector Derivatives................................................................................................................143 Definition of Vectors.....................................................................................................143 Derivative of Scalar w ith Respect to Vector.................................................................143 Derivative of Vector with Respect to Vector................................................................143 Second Derivative of Scalar With Resp ect to Vector (Hessian Matrix).......................144 Table of Several Useful Vector Derivative Formulas...................................................144 Proof of the Formulas....................................................................................................145 Proof (a)..................................................................................................................145 Proof (b).................................................................................................................145 Proof (c)..................................................................................................................146 Proof (d).................................................................................................................146 The Chain Rule of the Vector Functions..............................................................................148 The Derivative of Scalar Func tions Respect to a Matrix......................................................150 B CAVITY OSCILLATION MODELS..................................................................................155 Rossiter Model................................................................................................................. .....155 Linear Models of Cavity Flow Oscillations..........................................................................156 Global Model for the Cavity Oscillations in Supersonic Flow.............................................159 Global Model for the Cavity Oscillations in Subsonic Flow................................................163 C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS............................................169 MIMO System Identification................................................................................................169 Generalized Predictive Control Model.................................................................................171 D A POTENTIAL THEOR ETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES....................................................................................................................179 Masons Rule........................................................................................................................179 Global Model for a Cavity Oscillation in Supersonic Flow.................................................180 Global Model for a Cavity Oscillation in Subsonic Flow....................................................183 E CENTER VELOCITY OF ACTUATOR ARRAY..............................................................196 F PARAMETRIC STUDY FO R OPEN-LOOP CONTROL..................................................207 LIST OF REFERENCES.............................................................................................................219 BIOGRAPHICAL SKETCH.......................................................................................................227 7

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LIST OF TABLES Table page 2-1 Summary of the IIR OE algorithm.........................................................................................49 2-2 Summary of the IIR EE algorithm..........................................................................................50 2-3 Summary of the IIR SM algorithm.........................................................................................51 2-4 Summary of the IIR CE algorithm.........................................................................................52 2-5 Simulation results of IIR algorithms for sufficient case.........................................................53 2-6 Simulation results of IIR algorithms for insufficient case......................................................54 2-7 Simulation conditions of IIR algorithms for sufficient case...................................................55 2-8 Summary of the IIR/LMS algorithms.....................................................................................56 4-1 Parameters selection of the vibration beam experiment.........................................................78 4-2 Summary of the results of the adaptive GPC algorithm.........................................................79 5-1 Physical and piezoelectric properties of APC 850 device....................................................100 5-2 Geometric properties and parameters for the actuator..........................................................101 5-3 Resonant frequencies with respective cen terline velocities for each input voltage.............102 A-1 Vector derivative formulas................................................................................................ ..154 D-1 Components of the Mason s formula for supersonic case...................................................189 D-2 Components of the Masons formula for subsonic case......................................................190 D-3 Components of the Masons formula for subsonic case......................................................192 8

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LIST OF FIGURES Figure page 1-1 Schematic illustrating flow-induced cav ity resonance for an upstream turbulent boundary layer...................................................................................................................36 1-2 Tam and Block (1978) model of acoustic wave field inside and outside the rectangular cavity......................................................................................................................... .........36 1-3 Classification of flow control............................................................................................ .....37 1-4 Block diagram of syst em ID and on-line control...................................................................37 2-1 Linear time-invariant (LTI) IIR Filte r Structure.....................................................................57 2-2 Simulation structure of the adaptive IIR filter........................................................................57 2-3 z-plane of the test model................................................................................................. ........58 2-4 3D plot of the MSOE performance surf ace of the insufficient order test system..................58 2-5 Contour plot of the MSOE performance surface....................................................................59 2-6 Simulation results of weight track of the IIR algorithms for sufficient case..........................59 2-7 Simulation results of weight track of the IIR algorithms for insufficient case.......................60 2-8 Learning curve of IIR algorithms for sufficient case.............................................................60 2-9 Computational complexity results from the experiment........................................................61 3-1 Model predictive control strategy......................................................................................... ..70 4-1 Schematic diagram of the vibration beam test bed.................................................................80 5-1 Schematic of the wind tunnel facility...................................................................................10 3 5-2 Schematic of the test section and the cavity model..............................................................103 5-3 Schematic of the control hardware setup..............................................................................104 5-4 Bimorph bender disc actua tor in parallel operation..............................................................104 5-5 Designed ZNMF actuator array............................................................................................10 5 5-6 Dimensions of the slot for designed actuator array..............................................................106 5-7 ZNMF actuator array mounted in wind tunnel.....................................................................107 9

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5-8 Bimorph 3 centerline rms velocities of the single unit piezoele ctric based synthetic actuator with different excita tion sinusoid input signal...................................................108 5-9 The comparison plot of the experiment a nd simulation result of the actuator design code for bimorph 3.................................................................................................................. .109 5-10 Current saturation effects of the amplifier..........................................................................110 5-11 Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7).........111 5-12 Schematic of a single periodic cell of th e actuator jets and th e proposed interaction with the incoming boundary layer...................................................................................112 6-1 Schematic of simplified wind tunnel a nd cavity regions acoustic resonances for subsonic flow...................................................................................................................123 6-2 Noise floor level comparison at differe nt discrete Mach nu mbers with acoustic treatment at trailing edge floor of the cavity with L/D=6................................................123 6-3 x -acceleration unsteady power spectrum (dB ref. 1g) for case with acoustic treatment and no cavity....................................................................................................................124 6-4 y -acceleration unsteady power spectrum (dB re f. 1g) for case with acoustic treatment and no cavity....................................................................................................................125 6-5 z -acceleration unsteady power spectrum (dB re f. 1g) for case with acoustic treatment and no cavity....................................................................................................................126 6-6 Spectrogram of pressure measurement (dB ref. 20e-6 Pa) on the trailing edge floor of the cavity for the case with ac oustic treatment and no cavity..........................................127 6-7 Spectrogram of pressure measurement (re f 20e-6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D=6...............................................................................128 6-8 Spectrogram of pressure measurement (re f 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7).........129 6-9 Noise floor of the unsteady pressure level at the surface of the trailin g edge of the cavity with and without the actuator turned on..........................................................................130 6-10 Open-loop sinusoidal control results for flow-induced cavity oscillations at trailing edge floor of the cavity....................................................................................................131 6-11 Running error variance plot for the system identification algorithm.................................133 6-12 Closed-Loop active control result for flow-induced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity.....................................................................134 10

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6-13 Input signal of the Closed-Loop activ e control result for flow-induced cavity oscillations at Mach 0.27 at the trai ling edge floor of the L/D =6 cavity........................135 6-14 Sensitivity function (Equation 4-1) of the closed-loop control for M=0.27 upstream flow condition..................................................................................................................136 6-15 Unsteady pressure level of the clos ed-loop control for M=0.27, L/D=6 upstream flow condition with varying estimated order...........................................................................137 6-16 Unsteady pressure level of the clos ed-loop control for M=0.27, L/D=6 upstream flow condition with varying predictive horizon s....................................................................138 6-17 Unsteady pressure level comparison be tween the open-loop c ontrol and closed-loop control for M=0.27 upstream flow condition...................................................................139 B-1 Schematic of Rossiter model............................................................................................... 166 B-2 Block diagram of the linear model of the flow-induced cavity oscillations........................166 B-3 Block diagram of the reflection model................................................................................167 B-4 Global model for the cavity os cillations in supersonic flow................................................167 B-5 Block diagram of the global model for a cavity oscillation in supersonic flow..................167 B-6 Global model for a cavity os cillation in subsonic flow.......................................................168 B-7 Block diagram of the global model for a cavity oscillation in subsonic flow.....................168 D-1 Global model for a cavity os cillation in supersonic flow....................................................193 D-2 Block diagram of the global model for a cavity oscillation in supersonic flow..................193 D-3 Signal flow graph of th e global model for a cavity osci llation in supersonic flow.............194 D-4 Global model for a cavity os cillation in subsonic flow.......................................................194 D-5 Block diagram of the global model for a cavity oscillation in subsonic flow.....................195 D-6 Signal flow graph of the global model for a cavity oscillation in subsonic flow................195 E-1 Hot-wire measurement for actuator array slot 1a.................................................................197 E-2 Hot-wire measurement for actuator array slot 1b................................................................198 E-3 Hot-wire measurement for actuator array slot 2a.................................................................199 E-4 Hot-wire measurement for actuator array slot 2b................................................................200 11

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E-5 Hot-wire measurement for actuator array slot 3a.................................................................201 E-6 Hot-wire measurement for actuator array slot 3b................................................................202 E-7 Hot-wire measurement for actuator array slot 4a.................................................................203 E-8 Hot-wire measurement for actuator array slot 14b..............................................................204 E-9 Hot-wire measurement for actuator array slot 5a.................................................................205 E-10 Hot-wire measurement for actuator array slot 5b..............................................................206 F-1 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 500 Hz and 100 Vpp voltage...........................................................................................208 F-2 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 500 Hz and 150 Vpp voltage...........................................................................................208 F-3 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 600 Hz and 100 Vpp voltage...........................................................................................209 F-4 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 600 Hz and 150 Vpp voltage...........................................................................................209 F-5 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 700 Hz and 100 Vpp voltage...........................................................................................210 F-6 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 700 Hz and 150 Vpp voltage...........................................................................................210 F-7 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 800 Hz and 100 Vpp voltage...........................................................................................211 F-8 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 800 Hz and 150 Vpp voltage...........................................................................................211 F-9 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 900 Hz and 100 Vpp voltage...........................................................................................212 F-10 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 900 Hz and 150 Vpp voltage...........................................................................................212 F-11 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1000 Hz and 100 Vpp voltage.........................................................................................213 F-12 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1000 Hz and 150 Vpp voltage.........................................................................................213 12

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F-13 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1100 Hz and 100 Vpp voltage.........................................................................................214 F-14 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1100 Hz and 150 Vpp voltage.........................................................................................214 F-15 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1200 Hz and 100 Vpp voltage.........................................................................................215 F-16 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1200 Hz and 150 Vpp voltage.........................................................................................215 F-17 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1300 Hz and 100 Vpp voltage.........................................................................................216 F-18 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1300 Hz and 150 Vpp voltage.........................................................................................216 F-19 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1400 Hz and 100 Vpp voltage.........................................................................................217 F-20 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1400 Hz and 150 Vpp voltage.........................................................................................217 F-21 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1500 Hz and 100 Vpp voltage.........................................................................................218 F-22 Open-Loop control result for M=0.31 and ex citation sinusoidal input with frequency 1500 Hz and 150 Vpp voltage.........................................................................................218 13

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LIST OF ABBREVIATIONS D Cavity depth L Cavity length M Freestream flow Mach number U Freestream flow velocity a Mean sound speed inside the cavity vm Mode number (integer number 1,2) 0w Natural frequency of second order system r Reflection coefficient Damping ratio Phase lag factor Proportion of the vortices speed to the freestream speed Ratio of the specific heats v Spacing of the vortices a Time delay inside the cavity s Time delay inside the shear layer ADC Analog to digital converter ARMA Autoregressive and moving-average CARIMA Auto-regressive and integrated moving average CE Composite error DAC Digital to analog converter DNS Direct Numerical Simulations DSP Digital signal processing EE Equation error 14

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FFT Fast Fourier transform FIR Finite impulse response FRF Frequency response function GPC Generalized predictive control ID Identification IIR Infinite impulse response JTFA Joint-time frequency analysis LES Large Eddy Simulations LMS Least mean square LQG Linear quadratic Gaussian LTI Linear time-invariant MIMO Multiple-input multiple-output MPC Model predictive control MSOE Mean square output error OE Output error PDF Probability density function POD Proper orthogonal decomposition RANS Reynolds Averaged Navier-stokes RLS Recursive least square SISO Single-input single-output SM Steiglitz and McBride SNR Signal to noise ratio SPL Sound pressure level STR Self-tuning regulator TITO Two-input Two-output 15

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Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS By Qi Song May 2008 Chair: Louis Cattafesta Major: Aerospace Engineering Flow-induced cavity oscillations are a coupl ed flow-acoustic problem in which the inherent closed-loop system dynamics can lead to large unsteady pressure levels in and around the cavity, resulting in both broadband noise and di screte tones. This problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, and automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been working on this problem for more than fifty year s. This is because not only is the physical nature of this problem rich and complex, but also it has become a standard test bed for controller deign and implementation in flow control. The ultimate goal of this research is to minimize the cavity acoustic tones and the broadband noise level over a range of freestr eam Mach numbers. Although open-loop and closed-loop control methodologies have been explored extensively in recent years, there are still some issues that need to be studied further. For example, a low-order theoretical model suitable for controller design does not exist. Most recen t flow-induced cavity models are based either on Rossiters semi-expirical formula or a proper orthogonal decomposition (POD) based models. These models cannot be implemented in adaptive controller design directl y. In addition, closedloop control of high subsonic and supers onic flows remains an unexplored area. 16

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17 In order to achieve these objectives, an analytical system model is first developed in this research. This analytical model is a transfer function based model and it can be used as a potential model for controller design. Then, a MIMO system identificat ion algorithm is derived and combined with the generalized prediction control (G PC) algorithm. The resultant integration of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics and reduce the flow-induced oscillations. A novel piezoelectric-driven synt hetic jet actuator arra y is designed for this research. The resulting actuator produces high ve locities (above 70 m/s) at the cen ter of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. A validation vibration beam problem is used to demonstrate the combin ation of system ID and GPC algorithms. The result shows a ~20 dB re duction at the single resonance peak and a ~9 dB reduction of the integr ated vibration levels. Both open-loop control and cl osed-loop control ar e applied to the flow-induced cavity oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced for both cases. The GPC controlle r can generate a series of control signals to drive the act uator array resulting in dB re duction for the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB respectively. In a ddition, the broadband background noise is also reduced by this closed-loop cont roller (i.e., the OASPL reduction is 3 dB). The relevant flow physics and active flow control actuators are exam ined and explained in this research. The limitations of the present setup are discussed.

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CHAPTER 1 INTRODUCTON Equation Section 1 Flow-induced cavity oscillations have been st udied for more than fifty years, and the problem has attracted researchers in both fluid dynamics and controls. Firs t, this problem exists in many practical environments, such as land ing gear bays and weapon delivery systems on aircraft, sunroofs and windows buffeting in auto mobiles, and junctions between structural and aerodynamic components in both (Kook et al. 1997). The flow-acoustic coupling inherent in cavity resonance can lead to high unsteady pressu re levels (both broadba nd noise and discrete tones), and can cause fatigue failure of the cavity and its contents. For example, the measured sound pressure levels in and around a weapons bay can exceed 170 dB ref 20 For this reason, researchers are usually inte rested in suppression of flow-induced open cavity oscillations. Furthermore, this problem has become a sta ndard test problem for designing, testing, and analyzing real-time feedback control systems. Although the standard rectangular cavity geometry is relatively simple, the physical nature of this problem is both rich and complex. Several good review articles on the flow-induced cavity oscillati on problem are available in the literature (Rockwell, 1978, Komerath 1987, Colonius 2001, Cattafesta 2003). Pa Figure 1-1 is a simplified schematic for two t ypical flow situations, corresponding to external (a) supersonic and (b) subsonic fl ow over a rectangular cavity with length, depth, and width, W. The cavity oscillation process can be summarized as follows. A (usually) turbulent boundary layer with thickness, L D and momentum thickness, separates at the upstream edge of the cavity. Both a turbulent boundary layer and laminar boundary layer generate the discrete tones cau sed by the external flow. Howe ver, a laminar boundary layer has been shown to produce louder tones, presumably b ecause a turbulent flow generally results in a 18

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thicker shear layer with broadband disturban ces, which leads to overall lower levels of oscillations (Tam and Bl ock 1978; Colonius 2001). Following the description of Kerschen and Tumin (2003) and Alvarez et al.(2004), when the turbulent boundary layer separates at the upstream edge of the cavity, the resulting high speed or fast acoustic wave, and the low speed or slow acoustic wave, fE s E, propagate downstream in the supersonic flow case. In the subsonic flow case, only the so-called disturbance wave, propagates downstream. In both cas es, the shear layer instability, S, develops based upon its initial conditions coupled by the upstream traveling acoustic feedback wave, U, inside the cavity (and outside the cavity for subsonic flow). Kelvin-Helmholtz-type (Tam and Block 1978) convective instability wave s develop and amplify in the shear layer as they propagate downstream and finally saturate due to nonlinearity. In part icular, the instability waves grow and form large-scale vortical structur es that convect downstream at a fraction of the freestream velocity. These structures then impi nge near the trailing edge of open cavities ( ). The reattachment region acts as the primary acoustic source, which has been modeled as a monopole (Tam and Block 1978) or a dipole (Bilanin and Covert 1973) source. dEuE/1 LD 0 As a result, an upstream traveling acoustic wave, U, is generated inside the cavity. In subsonic flow, an additional acoustic wave, propagates upstream outsi de the cavity. In this description, the acoustic fee dback is modeled via acoustic waves that travel in the uE x direction. Finally, the loop is closed by a receptivity proces s, in which the upstream traveling waves are converted to downstream traveling instability an d acoustic waves. The initial amplitude and phase of these waves are set by the incident acoustic disturbances through this receptivity process. Physically, some of the acoustic dist urbance energy is converted to the instability waves at the upstream separation edge. Since the wavelength and the veloc ity of the instability 19

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waves and the acoustic disturba nces differ, only those waves that are in-phase ensure reinforcement of disturbances at that frequency. Therefore, this process is normally considered an introduction of a disturbance into the system, ultimately resulting in large-amplitude discrete tones inside and around the cavity. The measured broadband noise component is mainly due to the turbulent shear layer. The relevant dimensionless parameters are: LDLW, L and shape factor *H with the freestream flow parameters, Reynolds number Re and Mach number M all of which lead to tones (with Strouhal number StfLU ) characterized by their strength as unsteady pressure normalized by the freestream dynamic pressure, rmspq In this study, three dimensional effects are not considered, since the cavity tones are generated by the interaction of the freestream flow and the lo ngitudinal modes (coupled with vertical depth modes). Width mode s are not relevant in this feedback loop if the width is small enough to prevent higher-order spanwise modes but large enough so that the mean flow over the cavity length is approximately two dimensional. No te that the width of the cavity does affect the amplitude of the cavity oscillations (Rossiter 1966) but is of secondary importance (Cain 1999). Therefore, a two-dimensional model is reasonable from a physical perspective even though the unsteady turbulent motion is inevitably thr ee-dimensional (Bilanin and Covert 1973). Literature Review In this section, some publis hed results related to the physics of flow-induced cavity oscillations are discussed. Since the ultimate go al of this research is to minimize the cavity acoustic tones and perhaps the broadband noise level, potential control methodologies and algorithms are also reviewed. A recent review pa per by Cattafesta et al. (2003) gives a summary of the various passive and ope n-loop cavity suppression studies. 20

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Physical Models In order to suppress the disc rete tones and the broadband aco ustic level of flow-induced open cavity resonance, an understanding of the phys ics is essential. From a control engineers point of view, a simplified and low-order model is desirable in order to predict the resonant frequencies and amplitudes over a broad range of the governing dimensionless parameters. Physics-Based Models Rossiter (1964) performed an extensive expe rimental study on the measurement of the unsteady pressure in and around a rectangular open cavity ( 21.5 f tft ) in a subsonic and transonic freestream air flow (0.). He observed broadband noise and a series of unsteady acoustic tones generated in th e cavity. For the deeper cavities ( ), there was usually a single dominant tone, and the domin ant frequency was observed to jump between different cavity tones. For the shallower cavities ( ), two or more peaks were often observed and were approximately equal in magnit ude. He proposed that the flow entering the cavity caused the external stream to accelerate, and then the flow decelerated near the reattachment region. As a result, pressure wa s lower near the separa tion region (leading edge) and higher near the reattachment region (trailing edge). As a re sult, he suggested that large eddies developed within the cavity due to this pre ssure gradient. He also used shadowgraphs to illustrate that the shear layer separates from th e cavity leading edge, and the instability waves develop into discrete vortices that are shed at regular time internals from the front lip of the cavity (at Mach number 0.6 and with two-dimensional cavity 51.2M /LD 4/ LD 4/4LD and a laminar boundary layer). He postulated that there were some connections between the vortex shedding and the acoustic feedback, and this phenomenon produced a se ries of periodic pre ssure fluctuations. 21

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When the frequency of one of these components is close to the natural fr equency of the cavity, resonance occurs. In his study, Rossiter gave a semi-empiric al formula for predicting the resonant frequencies of these peaks at a specific Mach nu mber. The derivation of the Rossiter equation is given in Appendix B, and the resulting form ula for the dimensionless Strouhal number is 1v mm fL St U M (1-1) where m f is the resonance frequency for integer mode is the length of the cavity, U is the freestream velocity, vmL is the phase lag factor (i n fractions of a wavelength), is the ratio of the vortex propagation speed to the freestream velocity, and M is the freestream M number. Empirical constant values of ach 0.57 and 0.25 are shown to best fit the measured frequencies of resonances over a wide range of the Mach numbers for his experiment. These experimental constants account for the phase sh ift associated with the coupling between the shear layer and acoustic waves at the two ends of the cavity, and this phase shift is approximately independent of frequency. The phase speed U of the vortices is a weak function of M L and D (Colonius 2001). Different integer values give different frequencies, commonly referred to as shear layer or Rossiter mode s. In conclusion, Rossiters formula is based on an integer number of vm2 phase shifts, 2 k around a resonant feedback loop consisting of a downstream unstable shear layer di sturbance and an upstream feedb ack acoustic wave inside the cavity. This phase shift is a necessary condition fo r self-sustaining oscilla tions (Cattafesta et al. 1999a). However, Rossiters expression does not account for the depth or width of the cavity and only successfully predicts the longitudinal ca vity resonant frequencies at moderate-to-high Mach numbers. It also does not predic t the amplitude of the oscillations. 22

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Heller and Bliss (1975) corrected the Rossiter equation for the higher sound speed in the cavity, in which the static temperature in the cavity was assumed to be the stagnation temperature of the upstream. Th e modified Rossiter formula is 21 1 1 2v mm fL St U M M (1-2) where is defined as the ratio of specific heat s. They gave a discussion on the physical mechanisms of the oscillation process based on wa ter table visualization experiments. They suggested that the unsteady motion of the shear layer leads to a periodic mass addition and removal at the cavity trailing edge, leading to subsequent modeling efforts that employ an acoustic monopole source. In addition, the wave motion of the shear layer and the wave structure within the cavity were strongly coupled. Bilanin and Covert (1973) modeled the cavit y problem by splitting th e domain into two parts outside and inside the cavity. These two flow fields were separated by a thin mixing layer, which was approximated by a vortex sheet, and th e flow was assumed to be inviscid. The dominant pressure oscillations at the trailing edge were modele d by a single periodic acoustic monopole. They also assumed that the pressure field from the trailing ed ge source had no effect on the vortex sheet itself. Hence, the main distur bance was introduced at the leading edge of the shear layer. Kegerise et al. (2004) illustrated the agreement be tween the disturbance sensitivity function defined in control systems and the perf ormance measurement of output disturbances. Their analysis confirmed the notion that the disturbances were mainly introduced into the cavity at the cavity leading edge. 23

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Tam and Block (1978) carried out extensive expe rimental investigations at low subsonic Mach numbers ( ) and postulated that vortex sheddi ng was probably not the main factor for cavity resonance over the en tire Mach number range. They made two key assumptions, namely that the rectangular cavity flow was twodimensional, and the mean flow velocity inside the cavity was zero. These two assumptions were based on experimental evidence of little correlation between the mean flow and the acoustic feedback inside the cavity. Tam and Block proposed a process of flow-induced cavity oscillations as follows. The shear layer oscillated up and down at the trailing edge of the cavity. The upward movement was uncorrelated with the generation of the acoustic waves, because if the shear layer covered the trailing edge, then the external flow passes over the trailing edge without impingeme nt. They argued that only the downward motion of the shear la yer into the cavity caused signi ficant generation of pressure waves and subsequent radiation of acoustic waves in all directions (0.4MFigure 1-2 ). For example, some of the waves radiating in to the external flow (e.g., wave A) were argued to have minor effects on th e oscillations inside the cavity. However, the effect of the waves propagating inside of the cavity was deemed more significant. The resulting acoustic waves included the upstream propagating waves (e .g., wave C) and the reflected waves from the floor (e.g., wave F) and the upstream wall (e.g., wave E). Subsequent reflections of the acoustic waves by the walls, the cavity, or the shear layer we re deemed negligible. They concluded that the directly radiated wave and the first reflecte d waves by the floor and upstream end wall of the cavity provided the energy to excite the instability waves of the shear layer. These disturbances within the shear layer were then amplified as the instability waves propa gate downstream. When the disturbances amplitudes became large, nonlinear effects were important and ultimately established the amplitude of the discrete tones. A mathematical model of the cavity oscillation 24

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and acoustic field were developed. In order to calculate the phas es and waves generated at the trailing edge, a periodic line sour ce was simulated at the trailing edge of the cavity. In addition, the reflections of the acoustic waves by the cavity walls were m odeled by periodic line image sources about the cavity walls. Their model account s for the finite shear layer thickness effects and produces a more accurate estimation of the re sonance frequencies than Rossiters model. However, their resulting model is complicated and difficult to employ for control law design. Rowley et al. (2002 b, 2003, 2006) provided an alternative viewpoint for understanding flow-induced cavity oscillations. They showed that self-sustained oscillations existed only under certain conditions. The resonant frequencies were due to the instabiliti es in the shear layer interacting with the flow and acoustic fields. The amplitude of the oscillations was determined by nonlinear saturation. However, at other c onditions, the cavity oscillations could be represented as a lightly damped but stable linear system. The oscillat ions were caused by the amplification of external disturbances via the cl osed-loop dynamics of the cavity. The amplitude of each mode was determined by the amplitude of the external forcing disturbances and some frequency-dependent gain of the system. They modeled the dynamics of the shear layer as a second-order system and the acoustic propagation process via a one-dimensional, standing-wave model. The impingement and receptivity procedures were simply modeled as a constant unity gain. Finally, the Rossiter formula was derive d under some specific conditions. The derivation of this model is provided in Appendix B. They also used Gaussian white noise as input and examined the probability density function (PDF) and the phase portrait of the output pressure signal at different Mach numbers. Their results showed that under some conditions, the self-sustained regime of Rossiter modes was valid. However, at other conditions, called the forced regime, open cavity oscillations may 25

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be represented as lightly damped stable linear sy stems. External random forces drove the finiteamplitude cavity oscillations, which implies they will disappear if the external forces were removed. This physical linear model was also proposed as a potential model for controller design. Kerschen and Tumin. (2003) and Alvarez et al. (2004) provided a pr omising global model to describe the flow-induced cavity oscillation problem for two different flow patterns ( Figure 11). Their model combined scattering analyses for the two ends of the cavity and a propagation analysis of the cavity shear layer, internal re gion of the cavity, and acoustic near-field. They solved a matrix eigenvalue problem to identify th e resonant frequencies of the cavity oscillation. From their resulting characteristic functions, four and twelve closed loops could be identified for the supersonic flow and subsonic flow cases, re spectively. One more feedback loop makes the subsonic flow much more complex than the supers onic flow. For example, some of these closed loops were major loops, such as closed loop and (, SU UDFigure 1-1 ), while the other closed loops were considered minor loops. The combin ed effects of these loops caused the cavity resonances in the cavity flow. Besides these closed loops, the fo rward propagation paths, such as and also have critical effects on the amplit ude of the oscillations. This global model provides more insights for controller desi gn. A detailed derivation of this model is provided in Appendix B. dSDEEsfEClearly, the physics-based models described a bove provide physical insight concerning flow-induced cavity oscillations. However, the original Rossiter mode l and the global model derived by Kerschen and Tumin. (2003) can only estimate the resonance frequencies of the cavity flow. The linear model derived by Rowley et al. (2002b, 2003, 2006) is transfer function based model but is not sufficiently accurate to de sign a control system. A transfer function based 26

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model, which is an extension of Kerschen et al. s model, of cavity acoustic resonances is derived and given in Appendix D. For this approach, a signal flow graph is firs t constructed from the block diagram of the Kerschen et als physical model, and then Masons rule (Nise 2004) is applied to obtain the transfer f unction from the disturbance input to the selected system output. This method can give predictions for both the re sonant frequencies of the flow-induced cavity oscillations and the amplitude of the cavity tones. In addition, this method also provides a linear estimate for the system transfer function from th e disturbance input at th e leading edge and the pressure sensor output within the cavity walls. Th erefore, this model is a potential global model for controller design in this research. Numerical Simulations Some computational fluid dynamic (CFD) method s, such as Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) and Reynolds Averaged Navier-stokes (RANS), provide useful information for understand ing the issues of phys ical modeling of cavity oscillations. A review paper by Colonius (2001) gives a summary of issues related to each of these topics. More recent research on these topics can be found by Rizzetta et al. (2002, 2003) and Gloerfelt (2004). The Detached Eddy Simulation (DES) method, whic h is a involves a hybri d turbulence modeling methodology, has also been used to calculate the flow and acoustic fields of the cavity (Allen and Mendonca 2004; Hamed et al. 2003, 2004). Another hybrid RANS-LES turbulence modeling approach is presented by Arunajate san and Sinha (2001, 2003). They model the upstream boundary layer flow field and the sh ear layer region via RANS and LES models, respectively. All of thes e computational methods provide, at a minimum, good flow visualization and physical insight, and, at a maxi mum, quantitative information on the details of flow dynamics. 27

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POD-Type Models The previous analytical physical models are not accurate enough to design a control system. Furthermore, CFD methods are far too co mputationally intensive at the present time to provide a reasonable framework to design and test potential controllers. This translates into the need for new methods to develop more accurate reduced-order models. Therefore, simulation and experimental data based models were propos ed and later used for the controller design. Rowley et al. (2001) introduced a nonlinear dynamical model for flow-induced rectangular cavity oscillations, which was based on the method of vector-valued proper orthogonal decomposition (POD) and Galerkin projection. The POD method obtains low-dimensional descriptions of a high-order system (Chatterj ee 2000). For the cavity flow problem, data resulting from the temporal-spatial evolution of the numerical simu lations or experiments is used to construct a low-order subspace system that capt ures the main features (coherent structures) of the cavity flow. A more detailed explanati on of POD methods for cavity flow are given by Rowley et al. (2000, 2001, 2002c, 2003a). Some of the control methodologies discussed in next section can be constructed based on the result ant model obtained by POD (Caraballo et al. 2003, 2004, 2005; Samimy et al. 2003, 2004; Yuan et al. 2005). Instead, we turn our attention to an alternat ive experimental-based modeling approach that employs system identification techniques. He re, the nonlinear infini te-dimensional governing equations are modeled by a reduced set of differential (in continuous) or difference (in discrete time) equations. This method is the focus of this study and is discussed in the following section. On-Line System ID and Active Closed-Loop Control Methodologies Previous studies aimed at suppression of the flow-induced cavity tones have employed mainly passive or open-loop active flow control methodologies. The standard classification of the flow control techniques is shown in Figure 1-3 The review paper by Cattafesta et al. (2003) 28

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provides a detailed overview of various passive and open-loop control methodologies. However, passive and open-loop approaches are only effective for a limite d range of flow conditions. Active feedback flow control has recently been applie d to the flow-induced cavity oscillation problems. The closed-loop control approaches have advantages of reduced energy consumption (Cattafesta et al. 1997), no a dditional drag penalty, and robustn ess to parameter changes and modeling uncertainties. In genera l, closed-loop flow control measures and feeds back pressure fluctuations at the surf ace of the cavity walls (or floor) to an actuator at the cavity leading edge to suppress the cavity oscillati ons in a closed-loop fashion. In general, past active control strategies have taken one of two approaches for the purpose of reducing cavity resonance. First, they can thicken the boundary layer in order to reduce the growth of the instabilities in the shear layer. A lternatively, they can be us ed to break the internal feedback loop of the cavity dynamics. Most cl osed-loop schemes exploit the latter approach. Early closed-loop control applications used manual tuning of the gain and delay of simple feedback loops to suppress resonance (Gharib et al. 1987; Williams et al. 2000a,b). Mongeau et al. (1998) and Kook et al. (2002) used an active spoiler driven at the leading edge and a loopshaping algorithm to obtain significant attenuation with small actuation effort. Debiasi et al. (2003, 2004) and Samimy et al. (200 3) proposed a simple logic-ba sed controller for closed-loop cavity flow control. Low-order model-based co ntrollers with different bandwidths, gains and time delays have also been designed and implemented (Rowley et al. 2002, 2003, Williams et al. 2002, Micheau et al. 2004, Debiasi et al. 2004). Linear optimal c ontrollers (Cattafesta et al. 1997, Cabell et al. 2002, Debiasi et al. 2004, Samimy et al. 2004, Caraballo et al. 2005) have been successfully designed for operation at a sing le flow condition. These models are all based on reduced-order system models, and most of these controller design methods are based on 29

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model forms of the frequency response function, rational discrete/continu ous transfer function, or state-space form. However, the coefficients of these model forms are assumed to be constant, and this assumption requires that the system is time invariant or at least a quasi-static system with a fixed Mach number. Although the physical models of flow-induced cavity oscillations have been explored extensively, they are not convenien t for control realization. This is because these models are highly dependent on the accuracy of the estimated internal states of the cavity system. In addition, cavity flow is known to be quite sensitive to slight changes in flow parameters. So a small change in Mach number can deteriorat e the performance of a single-point designed controller (Rowley and Williams 2003). Therefore, adaptive control is certainly a reasonable approach to consider for reducing oscillations in the flow past a cavity. Adaptive control methodology combines a general control strategy and system identification (ID) algorithms. This method is thus potentially able to adapt to the changes of the cavity dimension and flow conditions. It updates the cont roller parameters for optimum performance automatically. The structure of this method is illustrated in Figure 1-4 Two distinct loops can be observed in the controller. The outer loop is a standard feedback control system comprised of the process block and the controlle r block. The controller operates at a sample rate that is suitable for the discrete process under control. The inner loop consists of a parameter estimator block and a controller design block. An ID algorithm and a specified cost function are then used to design a controller that will minimize the output. The steps for real-time flow control include: (i) Use a broadband system ID input from the actua tor(s) and the measured pressure fluctuation output(s) on the walls of the cavity to estimate the sy stem (plant and disturbance) parameters. (ii) 30

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Design a controller based on the estimated system parameters. (iii) Control the whole system to minimize the effects of the disturbance, measur ed noise, and the uncertainties in the plant. Based on this adaptive control methodol ogy, some adaptive algorithms adjust the controller design parameters to track dynamic changes in the system. However, only a few researchers have demonstrated the on-line adap tive closed-loop control of flow-induced cavity oscillations. Cattafesta et al. (1999 a, b) applied an adaptive di sturbance rejection algorithm, which was based upon the ARMARKOV/Toeplitz models (Akers and Bernstein 1997; Venugopal and Bernstein 2000, 2001), to identify and control a cavity flow at Mach 0.74 and achieved 10 dB suppression of a single Rossiter model. Other modes in the cavity spectrum were unaffected. Insufficient act uator bandwidth and authority limited the control performance to a single mode. Williams and Morrow (2001) applied an adaptive filtered-X LMS algorithm to the cavity problem and demonstrated multiple ca vity tone suppression at Mach number up to 0.48. However, this was accompanied by simultaneous amplification of other cavity tones. Numerical simulations using the least mean squares (LMS) algorithm were shown by Kestens and Nicoud (1998) to minimize the output of a singl e error sensor. The reduction was associated with a single Rossiter mode, but only within a small spatial region around the error sensor. Kegerise et al. (2002) implemen ted adaptive system ID algorithms in an experimental cavity flow at a single Mach number of 0.275. They also summarized the typical finite-impulse response (FIR) and infinite-impulse response (IIR) based system ID algorithms. They concluded that the FIR filters used to represent the flow -induced cavity process were unsuitable. On the other hand, IIR models were able to model the dynamics of the cavity system. LMS adaptive algorithm was more suitable for real-time control than the recursive-least square (RLS) adaptive algorithm due to its reduced computational comple xity. Recently, more advanced controllers, 31

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such as direct and indirect synthesis of the ne ural architectures for bot h system ID and control (Efe et al. 2005) and the generalized predictiv e control (GPC) algorithm (Kegerise et al. 2004), have been implemented on the cavity problems. From a physical point of view, the closed-loop controllers have no effect on the mean velocity profile (Cattafesta et al. 1997). Howeve r, they significantly affect streamwise velocity fluctuation profiles. This control effect eliminates the strength of the pressure fluctuations related to flow impingement on the trailing edge of the cavity. Although closed-loop control has provided promising results, the peaking (i.e., gene ration of new oscillation frequencies), peak splitting (i.e., a controlled peak splits into two sidebands) and mode switching phenomena (i.e., non-linear interaction between two different Rossiter frequencies) often appear in active closedloop control experiments (Cattafesta et al. 1997, 1999 b; Williams et al. 2000; Rowley et al. 2002 b, 2003; Cabell et al. 2002; Kegerise et al. 2002, 2004a). Explanations of these phenomena are provi ded by Rowley et al. (2002b, 2006), Banaszuk et al. (1999), Hong and Bernstei n (1998), and Kegerise et al (2004). Rowley et al. (2002b, 2003) concluded that if the viewpoint of a lin ear model was correct, a closed-loop controller could not reduce the amplitude of oscillations at all frequencies as a consequence of the Bode integral constraint. Banaszuk et al. (1999) gave explanations of the peak-splitting phenomenon. They claimed that the peak splitting effect was caused by a large delay and a relatively low damping coefficient of the open-loop plant. Cabell et al. (2002) expl ained these phenomena by the combination of inaccuracies in the identified plant model, hi gh gain controllers, large time delays and uncertainty in system dynamics. In addition, narrow-bandwidth actuators and controllers may also lead to a peak-splitting pheno menon (Rowley et al. 2006). 32

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Hong and Bernstein defined the closed-loop sy stem disturbance amplification (peaking) phenomenon as spillover. They illustrated that the spillover problem was caused by the collocation of disturbance source and control signal or the co llocation of the performance and measurement sensors. For this reason, the redu ction of broadband pressure oscillations was not possible if the control input was collocated with the disturbance signal at the leading edge of cavity. Therefore, Kegerise et al. (2004) s uggested a zero spillover controller which utilized actuators at both the leading a nd trailing edges of the cavity fo r closed-loop flow control. Unresolved Technical Issues Although the flow-induced cavity oscillation prob lem has been explored extensively, there are still some unresolved issues th at need to be studied further. A suitable theoretical model does not exist that estimates both the disc rete frequencies as well as the amplitude of the peaks. A feedback controller that re duces both broadband and tonal noise over a wide range of Mach numbers has not been achieved. An ad aptive zero spillover control algorithm may reduce both the tones and broa dband acoustic noise associated with cavity oscillations. The necessity for a high-order system model is a critical problem for controller design and implementation, because this high-order syst em results in significant computational complexity for application in digital signal processing (DSP) hardware. As such, the convective delays between the control inputs and the pressure sensor outputs must be specifically addressed in the control architecture. Closed-loop control of high subsonic and s upersonic flows is an unexplored area. Technical Objectives According to those unresolved t echnical issues, the ultimate goa ls of this dissertation are summarized as follows. A feedback control methodology will be developed for reducing flow-induced cavity oscillation and broadband pressure fluctuations. Adaptive system ID and control algorithms will be combined and implemented in realtime. 33

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The relevant flow physics and the design of appropriate active flow control actuators will be examined in this research. The performance, adaptabilit y, costs (computational and ener gy), and limitations of the algorithms (spillover, etc.) will be investigated. Approach and Outline In order to achieve these objectives, some design and application approaches warrant additional consideration. First, a potential theoretical model of cavity acoustic resonances is derived based on the global model of Kerschen and Tumin. (2003). This model (derived in Appendix D) provides the framework to estimate the amplitudes and frequencies of the cavity tones. This model has a low system order and also accounts for the c onvective delay between the disturbance input and the output pressure measurement. Second, during the controller design, the controlled system is a continuous syst em; therefore, all the sensors measurements and the actuators inputs are analog signals. However, for the present real-time application, the control algorithms are implemented using a DSP. For this reason, additional hardware, such as analog-to-digital converters (ADC), digital-to-ana log converters (DAC), anti-aliasing filters, and power amplifier, must be included in the whol e control design procedure. Finally, multiple actuators and multiple sensors are employed in this study in order to design an adaptive zero spillover control algorithm to explore the possibility of achieving broadband acoustic noise reduction in addition to suppression of the cavity tones themselves. This active control method development proc edure can be summarized as the following stages according to Elliott (2001). Study the simplified analytical system model and understand the fundamental physical limitations of the proposed control strategy. Obtain the sensor output and derive the states or coefficients from the system ID algorithms using off-line or on-line methods. 34

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Calculate the optimum performance using differe nt control strategies and find the control law for realization. Simulate the different control strategies a nd tune the candidate c ontroller for different operating conditions. Implement the candidate controlle r in real-time experiments. The thesis is organized as follows. Severa l SISO IIR system ID algorithms and a more general MIMO system ID algorithm are derived and discussed in the next Chapter. Then the MIMO adaptive GPC algorithm is described in Chapter 3. This is followed by a description of the sample experimental setup a nd the discussion of preliminary e xperimental results. Chapter 5 describes the wind tunnel facilities and the da ta processing methods. Wind tunnel experimental results for both open-loop (baseline) and closed -loop are then presented and discussed in Chapter 6. Finally, the conclusions and future work are presented in Chapter 7. 35

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L Turbulent Boundary Layer D 1 M s E f E S D U x y A 1 L Turbulent Boundary LayerD M dEuESDU x y B Figure 1-1. Schematic illustra ting flow-induced cavity resona nce for an upstream turbulent boundary layer. A) In supersonic flow.B) In subsonic flow. L D U Simulated Line Source A C E F x y Figure 1-2. Tam and Block (1978) model of acoustic wave fi eld inside and outside the rectangular cavity. 36

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37 Flow Control Approaches Passive Control Active Control Open-Loop Closed-Loop Quasi-Static Dynamic Figure 1-3. Classification of flow control. (after Cattafesta et al. 2003) input u(k) output y(k) disturbance reference Controller Plant Controller Design Parameter Estimator process part ID part control part ID input u(k) System ParametersControl Parameters Measured Noise Uncertainties Figure 1-4. Block diagram of sy stem ID and on-line control.

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CHAPTER 2 SYSTEM IDENTIFICATION ALGORITHMS Equation Section 2 This chapter provides a detail discussion of the system identification algorithms. Several typical adaptive SISO IIR structure fi lters are chosen as the candidate digital filters. These algorithms are applied to an example from Johnson and Larimore (1977) for simulation analysis. Then, four interested aspects of these filters, accuracy, convergence, computational complexity, and robustness, are examined and summarized. Finally, a more general MIMO sy stem ID algorithm is derive d from one of the promising SISO system ID algorithms. The resulting model is used to combine with the MIMO adaptive GPC model which is discussed in next chapter. Overview As discussed in the first chapter, IIR struct ure filter is an applicable mathematical model to capture the cavity dynamics. Furt hermore, this kind of structure can be a starting point and easily combined with many controller design strategi es. Therefore, in this Chapter, several system ID algorithms ba sed on the IIR filter structure are examined. The ideal of the system ID is to construct a predefined IIR structure filter, which has the similar frequency response of the actual dynamic system, using the information from the previous and present input and output time series data of the dynamic system. In general, the system ID algorithms fall into two big cat egories, the batch me thod and the recursive method. The batch method directly identifies the final system parameters in one-time calculation using a block data from the i nput and a block data from the output. Nevertheless, the recursive method updates the estimated system para meters within each sampling period using the latest input and output data in time domain. At each iteration of calculation, the system parameters may not be the optimal values. However, these 38

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estimated parameters will finally converge to th e true values of the system internal states. Successful identifying the system internal states depends on two major assumptions. First, the input signal and the output signa l must have a good correlation. Then, the system ID model has the same structure of that of the estimated system model. The recursive method is more attractive for pres ent experiment, because this updating method is more suitable for on-line implementation and it can also track the change of the system dynamics. Furthermore, the computational complexity of recursive method is much lower than the batch method. SISO IIR Filter Algorithms Netto and Diniz (1995) give a summary of some popular ad aptive IIR filter algorithms. In this section, the Output E rror (OE), Equation Erro r (EE), Steiglitz and McBride (SM), and Composite Error (CE) algo rithms are selected and illustrated. The general structure of an IIR filter is shown in Figure 2-1 The filter output may be expressed as 10 ()()() ()()abnn ij ij Tykaykibxkj kk (2-1) where represents the estimation values. and are the adjustable coefficients of the model, while and is the estimated order of the feedback loop and forward path, respectively. ia jb an bn ky-i)|x(k-j) () (k ()T ij|b ka and This IIR filter structure, Equation abi=1,...,n;j=0,1,...,n 2-1 is the same as the au toregressive and movingaverage (ARMA) model (Haykin 2002). 39

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Based on different error, the value diffe rence between the filter output and the system output, definitions, quite a few IIR filter algorithms have been presented by Netto and Diniz (1995). In their simulations, they us e an insufficient model, which models a second-order system using a first-order system to test each algorithm. The results from their paper show that the Modified Out put Error (MOE) algorithm may converge to a meaningless stationary point. The same result is also shown by Johnson and Larimore (1977). The Simple Hyper-stable Algorithm for Recursive Filters (SHARF) algorithm, the modified SHARF algorithm, and the Bias Remedy Least-Mean-Square Equation Error algorithm (BRLE) also show poor convergence rates. The Composite Regressor (CR) algorithm has similar problems as the MOE algorithm, since this algorithm combines the EE and MOE methods. Therefore, in this section, tests of these poor performing algorithms are not discussed. Fundamentally, there are two approaches for an adaptive IIR filter, the OE algorithm and the EE algorithm, which ha ve been derived by Haykin (2002) and Larimore et al. (2001), respectiv ely. Many other adaptive IIR filter algorithms are mainly derived from these algorithms, or combine some good features from the OE and the EE filters. Therefore, a summary of each of these two algorithms is provided in the following section. Two other algorithms, th e Steiglitz-McBride algorithm (SM) and the Composite Error algorithm (CE), are also in troduced, because both these algorithms also show good performance in our Simulink simulations. IIR OE Algorithm The IIR OE algorithm is summarized in Table 2-1 To ensure the stability of the algorithm, generally, the upper bound of step size is set to max2 where max is the 40

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maximum eigenvalue of the autocorrelat ion matrix of the regress vector The step sizes of the following algorithms are also satisfying this criterion. Furthermore, in order to guarantee the co nvergent approximation of ()OEkj andi this algorithm requires slow adaptation rates for small values of and (Haykin 2002). an bnIIR EE Algorithm The IIR EE algorithm is summarized in Table 2-2 Since the desired response is the supervisory signal supplied by the actual out put of plant during the training period, the EE algorithm may lead to faster converg ence rate of the adaptive filter (Haykin 2002). IIR SM Algorithm The IIR SM algorithm is summarized in Table 2-3 Since the EE algorithm and the OE algorithm possess their own advantages as well as drawbacks (discuss later), the motivation of the SM algorithm is to combine the desirable characteristics of the OE and the EE methods. IIR CE Algorithm This algorithm tries to combine both th e EE algorithm and the OE algorithm in another way. As shown in Table 2-4 a parameter is used to switch this algorithm between the EE algorithm and the OE algorithm. Recursive IIR Filters Simulation Results and Analyses In adaptive control experiments, the accuracy, the convergent rate, the computational complexity and the robustness are the main issues of the system ID algorithms. Here, computer simulations are ex amined in order to compare these aspects of the four system ID algorithms. 41

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The setup for the following Simulink simulation is shown in Figure 2-2 A Gaussian broad band white noise with zero mean and unity variance is chosen as the reference input signal. The prototype test model is a s econd-order dynamical system (Johnson and Larimore 1977) with the transfer function 1 1 10.050.4 () 11.13140.25 z Hz zz 2 (2-2) From the z-plane plot ( Figure 2-3 ), it clearly shows that this test model is a stable and non-minimum phase system, which has two real poles at 0.3011 z and and two zeros at and 0.8303 z 0 z 8 z In the following simulations, a sufficient order identification problem is firstly examined, which means a second-order syst em model with the transfer function 1 1 01 1 12 ()() (,) 1()() bkbkz Hzk akzakz 2 is used to estimate the test model. Then, an insufficient order identification problem is investigated. This approach uses a first-order system model with the transfer function 1 0 1 1 () (,) 1() bk Hzk akz to estimate the test model. The mean square output error (MSOE) surface of this insufficient order dynamical system is obtained by Shynk (1989) 2 0 01 2 1 2() 1yb MSOEbHa a (2-3) A 3D surface plot and a contour plot of the MSOE performance surface are shown in Figure 2-4 and Figure 2-5 respectively. The plots show that the MSOE surface of the test model is bimodal with a global minimum (denoted by *) 42

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at which yields and a local minimum (marked by +) at( which corresponds to **(,)(0.311,0.906) ba ,)(0.114, ba*0.277 MSOE 0.519) 0.976 MSOE The input () x k and the test model output () y k (with or without disturbance ) are introduced to the ad aptive IIR filter algori thms at the same time. The adaptive IIR filter algorithms calculate th e error signal and update the weights at each iteration. () vkAccuracy comparison for sufficient system Table 2-5 and Figure 2-6 show the simulation results a nd weight tracks of the four IIR algorithms for the sufficient case, respectiv ely. For the sufficient case, the algorithms minimize the mean square error between the system output and the filter output, and the estimated weights converge to the origin al coefficients of the test model. Accuracy comparison for insufficient system The simulation results and weight tracks of the IIR algorithms for the insufficient case are shown in Table 2-6 and Figure 2-7 respectively. The OE algorithm starts from two different initial conditions. One point is closer to the global minimum, and the other one is closer to the local minimum. This method adjusts its weights via stochastic gradient estimation to the closest stat ionary point of the initial condition. Similarly, two initial conditions are sele cted for EE algorithm. One of them is close to the global minimum, and the other on e is much closer to the local minimum. This algorithm can avoid the local minimum an d adjust its weights to let the final mean square error value arrive at the area near the global minimum. However, for this insufficient order situation, the final soluti on exhibits bias compared to the optimum solution. 43

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The SM algorithm combines the advantages of the OE algorithm and the EE algorithm. This algorithm avoids the local minima and converges to the global minimum with different initial points, which is like th e EE algorithm. At the same time, the final solution for this algorithm is very close to the optimum solution. As addressed above, the CE algorithm is a combination of the OE algorithm and the EE algorithm. It uses a weighting parameter to switch and weight between the OE algorithm and the EE algorithm. For this in sufficient identification problem, this method performs well. If the weighting parameter is close to 0, this algorithm is more like the OE algorithm, and the interesting feature of th is algorithm shows that it converges to the global minimum in the MSOE surface. However, when is close to 1, the biased characteristic of the EE algorithm is apparent in the results. Convergence rate For convergence rate comparis on, the same step size and number of iterations are chosen for simulations. The simulation c onditions and the learning cures of the IIR algorithms for the sufficient case are shown in Table 2-7 and Figure 2-8 respectively. Obviously, the EE and SM algorithms converge faster than the OE and CE algorithms. Computational complexity In order to apply the ID al gorithm on an adaptive control algorithm for real-time implementation, the computational complexity for one iteration of the ID algorithm have to be less than the sampling time of the DPS processor used for real-time experiment. Four algorithms are compared for computa tional complexity by th e turnaround time with the increase of the number of unknown for each algorithm ( Figure 2-9 ). The hardware used for experiment is PowerPC 750 (480M Hz) microprocessor (12.6 SPECfp95). The 44

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experimental results are shown in Figure 2-9 The computational complexity of all of the IIR algorithms is approximately linear. A nd the CE algorithm needs more computational time for each iteration than time requirements for the other three algorithms. Conclusions Varies of IIR adaptive filters are examined in this Chapter, the objective of these digital filters is to identify the system coe fficients (internal stat es) from the input and output signals. The OE algorithm and the EE algorithm are two basic structures of an adaptive IIR filter. Beyond that, two other algorithms, the SM algorithm and the CE algorithm, are also examined. Simulation resu lts show that the mean square error value calculated by the OE algorithm converges to the optimum solution for both the sufficient case and the insufficient case if the proper initial condition is chosen. This means that the OE algorithm may converge to local minima in the MSOE surface. Furthermore, this algorithm does not guarantee that the poles of the ARMA model always lie inside the unit circle in the z-plane. T hus, the OE method may become unstable (Haykin 2002) during the experiment. Therefore, a small enough st ep size and stability monitoring are required to ensure the convergence of the algorithm. However, the optimum step size is unknown, and the stability monitoring highly increases the computational complexity. These are the main drawbacks that should be considered in applications. The mean square error value calculat ed by the EE algorithm avoids the local minima and converges to the global minimum in the MSOE surface. The convergent rate and the computational complexity are good for real-time implementation. Unfortunately, the final solution is biased when the test model uses a lower-order system to model a higher-order system (Shynk 1989, Netto and Diniz 1995). 45

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Both the SM algorithm and the CE algorith m can find the global minimum in the MSOE surface. However, the good performan ce of the SM algorithm does not occur in general and, in fact, cannot be assured in practice (Netto and Diniz 1992). Moreover, the CE algorithm produces good results when 0.041 Within these limits, the algorithm is unimodal, and the bias is neglig ible (Netto and Diniz 1992). However, the stability of the CE algorithm mu st still be monitored, and the computational complexity is also high for this algorithm. A summary of the four algorithms is given in Table 2-8 The robustness results of each ID algorithms come from the experiment discussed in Chapter 4. As the results, the EE algorithm is the best algorithm compari ng to the other three ID algorithms. Therefore, in next step, a MIMO IIR filter is going to be derived based on this algorithm. MIMO IIR Filter Algorithm In this section, a MIMO system ID al gorithm is developed based on the SISO IIR EE algorithm. First, a linear system model is summed with the r inputs 1 ru and the m outputs 1 my. For simplification, the order p of the feedback loop is assumed the same as the order of the forward path. At specific time index the system can be expressed as k 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuu p (2-4) where 46

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1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (2-5) Define the observer Markov parameters 10 **(1)()pp mmprpk (2-6) and the regression vector **(1)(1) () () () ()mprpk kp k k kp1 y y u u (2-7) substituting Equation 2-6 and Equation 2-7 into Equation 2-4 yields a matrix equation for the filter outputs 1 (**(1)) ()() ()m mmprpkkk y( ( 1 ) ) 1 m p r p (2-8) Furthermore, the errors are defined as 11 ()()()mmkkk1 m yy) k (2-9) Finally, the observer Markov parameters 2-6 can be identified recursively by (2-10) (1)()()(TkkkIn order to automatically upda te the step size, choose 21 (2-11) 47

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where is a small number to avoid the singularity when 2 is zero. The main steps of the MIMO identification for one iteration are summarized as follows Step 1: Initialize (**(1)) ()mmprpk 0. Step 2: Construct regression vector **(1)1()mprpk according to Equation 2-7 Step 3: Calculate the output error 1()mk according to Equation 2-9 Step 4: Calculate the step size according to Equation 2-11. Step 5: Update the observe r Markov parameters matrix *(1)()mmprpk according to Equation 2-10. Then, the calculation for the ne xt iteration goes back to step 2. The detail derivation for this MIMO ID al gorithm is given in Appendix B. And the experimental results of the algorithm, the co mputational complexity, and the disturbance effects will be discussed in later Chapters. The calculation result of this MIMO ID gives an estimated model of the system with the form of Equation 2-4 In the following Chapter, a MIMO control algorithm is de veloped based on this MIMO ID model. 48

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Table 2-1. Summary of the IIR OE algorithm. Initialization: where ()[(0)|(0)]T ijkab 0 1,...,,0,1,...,abinj n Computation: For 1,2,... k 10 ()()()abnn OE iOE j ij y kaykibxkj a b) k ()()()OE OEekykyk Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an iO Eki l n jk j lkykiaklforin kxkjaklforjn ()[()|()]T OE ijkkk (1)()()(OEOEkkek where is the step size. In practice: Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an ff OE OE kOE a l n ff OE kOE b l y kiykiaykilforin x kjxkjaxkjlforjn ()[()|()] f ff OE OE OEkykixkj T) k (1)()()(f OEOEkkek 49

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Table 2-2. Summary of the IIR EE algorithm. Initialization: T ij(k)=a(0)|b(0)= 0 where 1,...,,0,1,...,abinj n Computation: For 1,2,... k 10 ()()()abnn EE i j ij y kaykibxkj )k ()()()EE EEekykyk ()[()|()]T EEkykixkj (1)()()(EEkkekEE where is the step size. 50

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Table 2-3. Summary of the IIR SM algorithm. Initialization: T ij(k)=a(0)|b(0)= 0 where 1,...,,0,1,...,abinj n Computation: For 1,2,... k 10 ()()()abnn EE i j ij y kaykibxkj ()()()EE EEekykyk 1 11 () () aSM EE n i i iekek az Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an ff SM kSM a l n ff SM kSM b l y kiykiaykilforin x kjxkjaxkjlforjn ()[()|()] f fT SM SM SMkykixkj (1)()()(SMkkekSM )k 51

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Table 2-4. Summary of the IIR CE algorithm. Initialization: T ij(k)=a(0)|b(0)= 0where 1,...,,0,1,...,abinj n Computation: For 1,2,... k Step 1: 10 ()()()abnn OE iOE j ij y kaykibxkj ()()()OE OEekykyk Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an ff OE OE kOE a l n ff OE kOE b l y kiykiaykilforin x kjxkjaxkjlforjn ()[()|()] f ff OE OE OEkykixkj T Step 2: 10 ()()()abnn EE i j ij y kaykibxkj ()()()EE EEekykyk ()[()|()]T EEkykixkj Step 3: ()()(1)()CE EE OEekekek ()()(1)() kkCE EE OE fk)k (1)()()(CEkkekCE where 0 1 52

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Table 2-5. Simulation results of IIR algorithms for sufficient case. Adaptive Structure Algorithm Parameters Initial Point 01 12 (0)(0) (0)(0) bb aa Number of iterations Final Point 01 12 ()() ()() bnbn anan OE 0.005 00 00 12000 [0.05,.4007] [-1.13,0.2484] EE 0.01 00 00 8000 [0.05 .4] [-1.131 0.2493] SM 0.005 00 00 11000 [0.05022,-0.4006] [-1.13,0.249] 0.005,0.04 00 00 12000 [0.050,-0.4006] [-1.13,0.2485] CE 0.01,0.60 00 00 8000 [0.04997,-0.400] [-1.131,0.2496] 53

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Table 2-6. Simulation results of IIR algorithms for insufficient case. Adaptive Structure Algorithm Parameters Initial Point 01 (0),(0) ba Number of iterations Global Min. 0.311,0.906 Final Point A 0.001 [-0.5,0.1] 5500 [-0.3098,0.8998] OE B 0.003 [-0.5,-0.2] 12000 [0.0928,-0.4896] A 0.001 [-0.5,0.1] 7000 [0.04577,0.8755] EE B 0.001 [0.11,-0.52] 7000 [0.05003,0.8719] A 0.0005 [-0.5,0.1] 3000 [-0.3132,0.9039] B 0.0005 [0.5,-0.2] 4500 [-0.3062,0.8992] SM C 0.0005 [0.11,-0.52] 5500 [-0.2967,-0.9031] A 0.001,0.04 [-0.5,0.1] 5000 [-0.3037,0.9112] CE B 0.003,0.04 [0.5,-0.2] 19000 [-0.315,0.9181] 54

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Table 2-7. Simulation conditions of IIR algorithms for sufficient case. Adaptive Structure Algorithm Parameters Initial Point 01 12 (0)(0) (0)(0) bb aa Number of iterations OE 0.005 00 00 12000 EE 0.005 00 00 12000 SM 0.005 00 00 12000 CE 0.005,0.04 00 00 12000 55

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Table 2-8. Summary of the IIR /LMS algorithms. Rank Order: (High or Good) (Low or Bad) AB C D Accuracy Convergent Rate Computational Complexity Robustness OE C D B C EE B A A A SM A B B C CE A C D C 56

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1Z 1Z 1Z Output y(k)Input x(k) c(k) ana2 a 1ana1 a bnb 1bnb 2 b 1 b 0 b Figure 2-1. Linear time-invari ant (LTI) IIR Filter Structure. Input x(k) Test Model Additive White Noise v(k) Test Model Output y(k) Adaptive IIR Filter Optimization MethodError e(k)LTI Discrete FilterFilter Output y(k) Figure 2-2. Simulation structure of the adaptive IIR filter. 57

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Figure 2-3. z-plane of the test model. Figure 2-4. 3D plot of the MSOE performance surface of the insufficient order test system. 58

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Figure 2-5. Contour plot of the MSOE performance surface. Figure 2-6. Simulation results of weight track of the IIR algorithms for sufficient case. 59

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Figure 2-7. Simulation results of weight track of the IIR algorithms for insufficient case. Figure 2-8. Learning curve of IIR algorithms for sufficient case. 60

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61 Figure 2-9. Computational comple xity results from the experiment.

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CHAPTER 3 GENERALIZED PREDICTIVE CONTROL ALGORIHTM Equation Section 3 This Chapter describes the background of the generalized predictive control (GPC) algorithm. Then, the GPC algorithm is deve loped based on the MIMO system ID model (discussed in Chapter 2). Both batch method and recursive version are given and discussed. Introduction The generalized predictive cont rol (GPC) algorithm belongs to a family of the most popular model predictive control (MPC). The MPC algorithm is a feedback control method, different choices of dynamic models, cost functi ons and constraints can generate different MPC algorithms. It was conceived near the end of th e 1970s and has been widely used in industrial process control. The methodology of MPC is represented in Figure 3-1 where is the time index number, are the input sequences, and are the actual output sequences. The and are estimated output and reference signals, respectively. k()uk()yk ()yk()rykTwo comments are made here to describe all MPC algorithms. First, at each time step, a specific cost function is cons tructed by a series of futu re control signals up to and a series of future error signals, which are the differences between the estimated output signals and the reference signals uks ) ) (ykj (rykj Second, a series of future inputs are calculated by minimizing this cost function, and only the first input signal is provided to the system. At the next sampling interval, new valu es of the output signals are obtained, and the future control inputs are calculated again accord ing to the new cost function. The same computations are repeated. ()ukjSome important MPC algorithms, such as model algorithmic control (MAC), dynamic matrix control (DMC) and GPC, have become p opular in industry. MAC explicitly uses an 62

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impulse response model and DMC applies the step response process model in order to predict the future control signals (Camacho 1995). The GPC method, which is inherited from generalized minimum variance (GMV) (Clarke 1979), was propos ed and explained by Clarke (1987 a, b). The GPC algorithm is an effective self-tuning pred ictive control method (Clarke 1988). It uses controlled auto-regressive and integrated moving average (CARIMA) model to derive a control law and can be used in real time applications. Juang et al. (1997, 2001) gi ve the derivation of the adaptive MIMO GPC algorithm. Th is algorithm is an effective control method for systems with problems of non-minimum phase, open loop unstable plants or lightly damped systems. It is also characterized by good control performance a nd high robustness. Furthermore, the GPC algorithm can deal with the multi-dimension case and can easily be combined with adaptive algorithms for self-tuning real-time applicati ons. The problem of flow-induced open cavity oscillations exhibit several theses issues, ther efore, the GPC is considered as a potential candidate controller. Two modifications are made for this algorithm. First, a input weight matrix is integrated into the cost function, this control matrix can put the penalty for each control input signal and further to tune the performance for each input channels. Second, a recursive version of GPC is developed for real-time control application. MIMO Adaptive GPC Model In this section, a MIMO model, which has th e same form of the MIMO ID algorithm, is considered. A linear and time invariant system with inputs r 1ru and outputs m 1my at the time index k can be expressed as 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuup (3-1) 63

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where 1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (3-2) Shifting j step ahead from the Equation 3-1 the output vector () kj y can be derived as (3-3) () () 11 () (1) 00 () () () 01()(1)(1) ()()(1) ()(1)()jj p j p jj j pkjk kp kpkjkj kkkp yyy yuu uuu where () (1)(1) () (1)(1) 0101 11 1 2 () (1)(1) () (1)(1) 21 2 311 1 2 () (1) (1)() 11 11 () (1) 1,jjj jjj mr mm jjjjj mm mr jjjj pp p p mm jj pp mm + + ++ + (1) (1) 11 () (1) 1 jj pp mr jj pp mr + j 0 1 1 p p (3-4) and with initial (0) (0) 110 (0) (0) 221 (0) (0) 11 1 (0) (0),mm mr mm mr pp p mm mr ppp mm mr (3-5) The quantities () 0k ( ) are the impulse response sequence of the system. Defining the following the vector form 0,1, k 64

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1 1 1() () (1)(1) () ,() (1) () () (1) () (1)pj rp rj p mpukp uk ukp uk kp k uk ukj ykp ykp kp yk ( 1 ) 1 uu y s (3-6) the predictive index and 0,1,2,,1,,1 jqq 11() () (1) (1) () () (1)(1)ss rs msuk yk uk yk k, k uks yks uy (3-7) Finally, the predictive model for future outputs, s y is obtained, this future outputs consists of a weighted summation of future inputs, s u, previous inputs, and previous outputs, pupy (3-8) ()()()()ssppkkkpkpyTuBuAywhere 0 (1) 00 (1)(2) 00000 0ss msrs T (3-9) 11 (1) (1) (1) 11 (1)(1) (1) 11 pp pp sss pp msrp B (3-10) 65

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11 (1) (1) (1) 11 (1)(1) (1) 11 pp pp sss pp msmp A (3-11) The detail derivation of the GPC model is given in Appendix B. MIMO Adaptive GPC Cost Function Assume the control inputs (present input and future inputs) depend on the previous inputs and output and can be expressed as (*)1(*)[*()] [*()]1() () ()p s sr srpmr p pmrkp k kp u uH y (3-12) Two potential cost functions are li st below. The first one consis ts terms of future outputs and a trace of the feedb ack gain matrix ()()()T ssJkkktrT y Q y HH (3-13) and the second definition of cost function based on the total energy of future outputs as well as the inputs 1 ()()()()() 2TT ssssJkkkkk yQyuRu (3-14) The output weight matrix Q, input weight matrix and the control horizon are important parameters for tuning the controller. The horizon is usually chosen to be several times longer than the rise time of the plant in order to ensure a stable fee dback controller (Gibbs et al. 2004). Also, if the predict horizon range is from zero to infinity, the resu lting controller approaches the steady-state linear quadratic regulator (Phan et al. 1998). Rs sMIMO Adaptive GPC Law In order to minimize the cost function, th ree approaches are considered as follows. 66

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Based on Equation 3-13 the control coefficients can be update using adaptive gradient algorithm. Based on Equation 3-14 the optimum solution can be derived. However, this method requires the calculation of a ma trix inverse, so the comput ational complex is higher. Based on Equation 3-14 the control coefficients can be updated using an adaptive gradient algorithm. The first approach is examined by Kegerise et al. (2004). In the next section, the latter two approaches are derived. MIMO Adaptive GPC Optimum Solution Based on the cost function 3-14, the goal is to find (*)[*()] s rpmrH or (*)1()s srku to minimize the cost function. We will show that both minimizing the cost function 3-14 respect to control matrix (*)[*()] s rpmrH and input vector (*()sk)1sr u will provide the same result. To simplify the expression, lets define [*()]1 [*()]1() ()p p pmr p pmrkp kp u v y (3-15) Substituting the predictive model 3-8 and control law 3-12 into the cost function 3-14 gives 1 ()()()()() 2 1 2 1 2TT ssss T sps ppJkkkkk p TyQyuRu TuBAvQTuBAv HvRHv (3-16) with some algebraic manipulation, the gradient of cost function respect to the control matrix (*)[*()] s rpmrH can be obtained. The optimum solution is obtained when the gradient equal to zero. 67

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() 0TT sp pp TT sp ps p TT sp ppJk T T TTTQyvRHvv H TQTuBAvvRuv TQT+RuvTQBAvv (3-17) thus, 1 s p opt TTuTQT+RTQBA v (3-18) Alternatively, from Equation 3-16 setting the gradient of the cost function with respect to the input vector (*)1()s srku, to zero gives ()T sp sJk TTuBAvQTuR u =0s (3-19) thus, 1 s p opt TTuTQT+RTQBA v (3-20) A comparison of Equation 3-20 to Equation 3-18 shows that these two approaches yield the same result. It is easy to apply the optim al solution of the Equation 3-20 on the cavity problem. However, the matrix inversion calculation has high computational complexity. Only if the model order is low enough, the optimal inpu t can be used in real-time application. MIMO Adaptive GPC Recursive Solution To avoid calculating the invers e of the matrix in Equation 3-20, the stochastic gradient descend method can be used to update the control matrix using the following algorithm H () Jk (k+1)(k) (k) HH H (3-21) 68

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Substituting Equation 3-12 into Equation 3-17 gives ()T pp T pp T ppJk (k) T T TTTQT+RHvv H TQBAvv TQT+RHTQBAvv (3-22) therefore, the recursiv e solution is given by T pp(k+1)(k) (k) TT H HTQT+RHTQBAvv (3-23) Since only present controls r 1()ruk are applied to the system, only the first rows in Equation r 3-23 are used first r rows T pph(k+1)h(k) (k) TTTQT+RHTQBAvv (3-24) In next chapter, this adaptive feedback controller, which is the combination of the MIMO system ID (discussed in Chapter 2) and the GPC algorithm, is implementation on a vibration beam test bed. The output weight matrix Q, input weight matrix R and the control horizon are tuning for testing their effects to the control performance. s 69

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70 k1 k 3 k 2 k ks 1 k 2 k () yk () yk ()ryk () uk Prediction Horizon PastFuturekj Figure 3-1. Model pred ictive control strategy.

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CHAPTER 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES Equation Section 4 In this Chapter, the MIMO system ID (dis cussed in Chapter 2) algorithm and the GPC algorithm (discussed in Chapter 3) are implemente d on a vibration beam test bed. Since the objective idea of this sample experiment is similar to the flow-induced cavity oscillation, which is the disturbance rejection problem, the results of this vibration beam experiment will give us some insights to guide the later flow control applications of using this real-time adaptive control mythology. First, computational complexity of ID algorithm, ID results in time domain and frequency domain, and the distur bance effect for ID algorithm are examined. Then, the output and input weight matrices as well as the c ontrol horizon are tuned for testing the control performance with varies of these parameters. Schematic of the Vibr ation Beam Test Bed Figure 4-1 shows a detailed sketch of the whole vi bration control testbed setup. A thin aluminum cantilever beam with one piezoce ramic (PZT-5H) plate bonded to each side is mounted on a block base and connected to an elec trical ground. The two piezoceramic plates are used to excite the beam by applying an electrical field across their thickness. The piezoceramic plate bonded to the right side of the beam is called the disturb ance piezoceramic because it is used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the left side of the beam is called the control piezocer amic because it is supplied with the controller output signal to counteract the unknown disturba nce actuator. The beam system has a natural frequency of about 97 Hz. The goal of the disturbance rejection controller is to mitigate the tip deflection of the aluminum beam generated by an external unknown disturbance signal. Th e controller tries to generate a signal to counteract the vibration of the aluminum beam generated by the disturbance 71

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piezoceramic. The performance signal and the fee dback signal of the controller are collocated, which is measured at the center of the tip of the beam by a laser-optical displacement sensor (Model Micro-Epsilon OptoNCDT 2000). This device gives an output sensitivity of 1 / with a resolution of Vmm0.5 m and a sample rate of 10. The performance signal is filtered by a high pass filter (Model Kemo VBF 35) with kHz1cz f H to filter out the offset of the displacement sensor and then amp lified by a high-voltage amplifie r (Model Trek 50/750) with a gain of 10. dcThe disturbance and control signals ar e generated by dSPACE (Model DS1005) DSP system with 466MHz Motorola PowerPC micr o-processor and amplif ied by two separate channels of the power amplifier by the same gain of 50. The types and conditions of the signals are discussed in detail in the next section. The dSPACE sy stem has a 5-channel 16-bit ADC (DS2001) and a 6-channel 16-bit DAC (DS2102) board. The signals are acquired using Mlib/Mtrace programs in MATLAB through the d SPACE system. The block diagram of the vibration beam test bed is shown in Figure 4-2 System Identification Experimental Results Computational Complexity During the real-time adaptive control of flow -induced cavity oscillations, computational complexity is an important issue. Kegerise et al. (2004) use 80 order estimated model for the system ID and 240 prediction horizon for the re cursive GPC algorithm to capture the dynamics of the cavity system. Therefore, the computati onal complexity of the online adaptive controller have to reach or beyond these lengths of parameters. Figure 4-3 shows the changes of turn around time with the increasing estimated system order of the MIMO system ID algorithm. It is clear that the computational complexity of this algorithm is approximately linear, and the time 72

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requirement to estimate the same system order for the two inputs and two outputs (TITO) system is approximately three times longer than the requ irement of the SISO system. And both cases have enough turn around time fo r subsonic cavity experiment. System Identification Before identifying the parameters of the system, the system order has to be estimated. Using the ARMARKOV/LS/ERA algorithm (Akers et al. 1997; Ljing 1998) the eigenvalues of the triangle matrix calculated from singular va lue decomposition (SVD) of the vibration beam system is shown in Figure 4-4 This plot shows that the minimum reasonable estimated order of the system is 2. For different ID algorithms, the estimated system order may be different in practice. Therefore, the resulti ng identified system transfer func tion should be checked in order to match the experimental data shape in both time domain and frequency domain. A periodic swept sine signal ( Figure 4-5 ) is chosen as the ID input signal (without external disturbance). The sampling frequency is 1024, the sweep frequency produced by dSPACE system is from to 150, and the amplitude of the sweep sine signal is The performance of the system ID algorithm improves with increasing estimated system order. For this case, the estimated order of system ID block is set to 10. Hz 0 Hz Hz 0.25 voltFigure 4-5 shows that the output of the system ID algorithm matches the system output very well in the time domain. The coherence function ( Figure 4-6 ) shows good correlation between th e input and output signals. The zero-pole location and the transfer func tion between the input and sensor output are shown in Figure 4-7 and Figure 4-8 respectively. Three system ID methods are used for comparison. Two batch methods calculate tran sfer function in frequency domain using the experimental data and the FRF method to fit th e frequency domain data One recursive method updates the system coefficients in real-time. Notice that all of these three methods give the 73

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similar shape in frequency domain and capture the two dominant poles of the system ( Figure 47). However, the FRF fit batch method gives a lower order model than the recursive method. Disturbance Effect External disturbance degrades the perfo rmance of the system ID algorithm. Figure 4-9 shows the vibration shim experiment system ID re sult with different extern al disturbance levels. A larger SNR (lower external disturbance level) in the input signal generally give more accurate identified system models. However, although the lower SNR input signal may result in a suboptimal system model, the closed-loop cont rol implementation base d on this model still works well. The results are shown later. Closed-Loop Control Experiment Results Computational Complexity The controller design block is the most tim e consuming blocks in the entire adaptive control implementation. Estimated model order and the length of the predict horizon are two main parameters effecting the computational complexity. Figure 4-10 illustrates the computational complexity of the main C code S-function block which maps the observer Markov parameters (discussed in Chapter 2) to predict mo del coefficients (discussed in Chapter 3). The result shows that the turnaround time increases more quickly with increasing estimated system order than increasing of the prediction horizon. Closed-Loop Results The experimental parameters for th e closed-loop control are list in Table 4-1 and some result plots are presented here. Figure 4-11 shows the sensor output signal in time domain, in which the control signal is initiated at time 0. 74

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Power spectra of open loop (base line) vs. cl osed-loop sensor output and the closed-loop sensitivity are shown in Figure 4-12 and Figure 4-13 respectively. The sensitivity function is define as 2 2() ()cl op y f s y f (4-1) Equation 4-1 provides a scalar measurement of distur bance rejection. A value less than one (negative log magnitude) indicates disturbance attenuation, while a value greater than one (positive log magnitude) indicates disturbance amplification. Although the res onance of the open loop system can be mitigated by the closed-loop controller, a spillover phenomenon is also observed in Figure 4-13 As discussed in Chapter 1, the spillover problem is generated because, for this special case, the performance sens or output and the measurement sensor output (feedback signal) are collocated. Next, the effects of the adaptive GPC parameters are examined. Figure 4-14 shows the effect of the changes of the estimated model order. Figure 4-15 shows the effect of the changes of the predict horizon. Figure 4-16 shows the effect of the ch anges of the input weight. Figure 4-18 shows the effect of the different level of disturbance (SNR) during the system ID. The results for each case are discussed below. Estimated Order Effect In general, increasing the estimated order of the GPC, up to a certain point, can improve the performance of the closed-loop control ( Figure 4-14 ). The experimental result shows that when the estimated model order is greater than 4, the closed-loop controll er can not improve the performance any more. 75

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Predict Horizon Effect It is clearly see that increasing the predic t horizon can improve th e performance of the closed-loop controller ( Figure 4-15 ). Input Weight Effect The input weight penalizes the magnitude of the input signal. For this experiment, saturation is given to the input signal to avoid the damage of the actuator. In order to restrict the input signal within the limits of th e saturation, the input we ight should be carefully tuned to obtain a realizable GPC. Although a sm aller input weight improve performance of the closed-loop controller (0.75 volt Figure 4-16 ), it also generates a larger control signal ( Figure 4-17 ). Therefore, the tuning idea is to decrease the input weight as low as possible under the input saturation constraints. Disturbance Effect for Different SNR Levels During System ID As mentioned above, the level of the external disturbance sign al (different SNR) is an important issue for the accuracy of the system ID ( Figure 4-9 ). However, the adaptive closedloop controller gives the surprising results ( Figure 4-18 ). Three cases are examined and compared in this section. First, the open l oop (base line) case is th e power spectrum of the output measurement of laser sensor without any control input. Second, th e external disturbance is turned off during the system ID. Finally, the external disturbance is turned on with some level during the system ID. The result shows that th e higher disturbance leve l (low SNR) does not have a detrimental effect on the performance of th e closed-loop system. In fact, the performance of the closed-loop cont roller with low SNR is improved slightly. 76

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Summary Table 4-2 gives a summary of the experimental results of adaptive GPC algorithm. It can be seen that the GPC algorithm gives the better control performance with the larger estimated system order, the higher prediction horizon and the lower input weight. In Chapter 6, the similar control approach combining the system ID algorithm and the GPC algorithm will be implemented on the flow-induced cavity oscillations problem. Since the control ideas for both the vibra tion beam problem and the cavity oscillations problem are disturbance rejection, the successful implem entation of the system ID algorithm and GPC algorithm to the vibration beam test bed ma y give the guidance to the flow control. 77

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Table 4-1. Parameters selection of the vibration beam experiment. Fs Disturbance GPC White Noise Low Pass Filter (IIR Butterworth, 4th order) Input Weight Estimated Order Prediction Horizon 1024 Hz 0.09 Var 150c f Hz 1 10 10 78

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Table 4-2. Summary of the results of the adaptive GPC algorithm. Estimated Order Input Weight Prediction Horizon Integrated Reduction (In dB) Reduction at Resonance (In dB) 4 1 4 7.0 11.2 4 1 10 9.2 20.1 10 1 10 8.4 13.4 10 10 10 2.8 4.2 79

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PC MLIB/MTRACE DS1005 Processor Board DS2102 D/A Board OptoNCDT 2000 Laser-Optical Displacement Sensor Kemo VBF35 Reconstruction Low Pass Filer Trek 50/750 Power Amplifier Aluminum Shim Disturbance Piezoceramic Control Piezoceramic Base Kemo VBF35 Anti-aliasing Low Pass Filer DS2001 A/D Board Disturbance Signal ID Input or Control Input Trek 50/750 Power Amplifier Figure 4-1. Schematic diagram of the vibration beam test bed. 16-bit ADC LPF LPF AMP AMP System Sensor 16-bit ADC System ID Controller Design Controller W(k) ID Input U(k) Y(k) Y(t) U(t) W(t) Actuator Analog Part Digital Part AMP BPF Figure 4-2. Block diagram of the vibration beam test bed. 80

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Figure 4-3. Computational complexity of the MIMO system ID. Figure 4-4. Eigenvalues of the triangle matr ix obtained by the SVD method of the vibration beam system. (Calculated by ARMARKOV/LS/ERA algorithms with 50 Markov parameters and estimated order of the denominator is 10). 81

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Figure 4-5. Input time series (top), system output and system ID algorithm output time series (bottom). Figure 4-6. Coherence function of system input and system output. 82

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Figure 4-7. Zero-pole location of FRF (top) and system ID algorithm. The estimated order of FRF fit function is 2 and the estimated order of system ID algorithm is 10. Figure 4-8. Identified transfer function usi ng the experiment data by frequency response function (experiment), frequency response functi on fit(FRF fit) and tim e domain system ID algorithm (ID). The estimated orde r of FRF fit function is 2 and the estimated order of system ID algorithm is 10. 83

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Figure 4-9. Learning curve of syst em ID with different input SNR. The estimated system order is 10, sampling frequency is 1024 Hz. A Figure 4-10. Computational complexity of the ma in controller design C Sfunction. A) For SISO case. B) For TITO case. 84

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B Figure 4-10 Continued Figure 4-11. System output time series data. Th e control signal is intr oduced at 0 second, the estimated order is 10, predict horizon is 10, and the input weight is 1. 85

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Figure 4-12. Power spectrum of output signal wi th control and without control signal. The estimated order is 10, predict horizon is 10, and the input weight is 1. 0 50 100 150 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Sensitivity Frequency (Hz)RMS Gain (in Log Scale) Figure 4-13. Sensitivity function of the system. Th e estimated order is 10, predict horizon is 10, and the input weight is 1. 86

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Figure 4-14. Power spectrum of output signals for different estimated order. Predict horizon is 10, and the input weight is 1. Figure 4-15. Power spectrum of output signals for different predict horizon. The estimated order is 4, and the input weight is 1. 87

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Figure 4-16. Power spectrum of output signals for different input weight. The estimated order is 10, predict horizon is 10. Figure 4-17. Control signals fo r different input wei ght. The estimated order is 10, predict horizon is 10. 88

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89 Figure 4-18. Power spectrum of output signals fo r different system ID disturbance conditions. The estimated order is 10, predict horiz on is 10, and the input weight is 1.

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CHAPTER 5 WIND TUNNEL EXPERIMENTAL SETUP Equation Section 5 The experimental facilities and instruments used in this study are described in detail in this Chapter. These devices consist of a blowdown wind tunnel with a te st section and cavity model, unsteady pressure transducers, data acquisition systems, and a DSP real-time control system. Finally, the actuator used in this study is described. Wind Tunnel Facility The compressible flow control experiments ar e conducted in the University of Florida Experimental Fluid Dynamics Laboratory. A schematic of the supply portion of the compressible flow facility is shown in Figure 5-1 This facility is a pressure-driven blowdown wind tunnel, which allows for control of th e upstream stagnation pressure but without temperature control. The compressed air is generated by a Qu incy screw compressor (250 psi maximum pressure, Model 5C447TTDN7039BB). A desiccan t dryer (ZEKS Model 730HPS90MG) is used to remove the moisture and residual oil in the compressed air. The flow conditioning is accomplished first by a settling chamber. Th e stagnation chamber consists of a 254 mm diameter cast iron pipe supplied with the clean, dry compressed air. A computer controlled control valve (Fischer Contro ls with body type ET and Acuator Type 667) is situated approximately 6 meters upstream of the stagnation chamber with a 76.2 mm diameter pipe connecting the two. A flexible rubber coupler is located at the entrance of to the stagnation chamber to minimize transmitted vibrations from the supply line. The stagnation chamber is mounted on rubber vibration isol ation mounts. A honeycomb and two flow screens are located at the exit of the settling chambe r and the start of the contracti on section, respectively. The honeycomb is 76.2 mm in width (t he cell is 76.2 mm long) with a cell size of 0.35 mm. Two 90

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anti-turbulence screens spaced 25.4 mm apart are used; these screens have open area and use 0.1 mm diameter stainless steel wire. For the current experiment, the facility was fitted with a subsonic nozzle that transitions from the 254 mm diameter circular cross-section to a square cross-section linearly over a distance of 355.6 mm. The profile designed for this contraction found in previous work provides good flow quality downstream of the contraction (Carroll et al. 2004). The overall area contraction ratio fr om the settling chamber to the test section is 19.6:1. For the present subsonic setup, the freestream Mach number can be altered from approximately 0.1 to 0.7, and the fa cility run times are approximately 10 minutes at the maximum flow rate due to the limited size of the two storage tanks, each with volume of 3800 gallons. 62%50.8 mm 50.8 mm Test Section and Cavity Model A schematic of the test section with an integrated cavity model is shown in Figure 5-2 The origin of the Cartesian coordinate system is s ituated at the leading edge of the cavity in the mid-plane. The test section connects the subs onic nozzle exit and the exhaust pipe with 431.8 mm long duct with a 50.8 m square cross section. m 50.8 mm The cavity model is contained inside this duct and is a canonical rectangular cavity with a fixed length of and width of 152.4 mm L 50.8 mm W and is installed along the floor of the test section. The dept h of the cavity model, can be adjusted continuously from 0 to This mechanism provides a range of cavity length-to-depth ratios, D50.8 mm / L D, from 3 to infinity. The cavity model spans the width of the test section W. However, a small cavity width is not desirable, because the side wall bounda ry layer growth introduces three-dimensional effects in the aft region of the cavity. As a result, the grow th of the sidewall boundary layers in the test section may result in modest flow acceleration. The boundary layers have not been 91

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characterized in this study. Ne vertheless, the cavity geometry a pplied in this study is consistent with previous efforts in the li terature (Kegerise et al. 2007a,b) considered to be shallow and narrow, so two-dimensional longitudinal mode s will be dominant (Heller and Bliss 1975). Removable, optical quality plexiglas windows w ith 25.4 mm thickness bound either side of the cavity model to provide a full view of the cavity and the flow above it. The floor of the cavity is also made of 14 mm thick plexiglas for optical access. Two different wind tunnel cavity ceiling configurations are available. The first one is an aluminum plate with 25.4 mm thickness that can be considered a rigid-wall boundary condition. This boundary condition helps excite the cavity vertical modes a nd the cut-on frequencies of the cavity/duct configuration (Rowley and Williams 2006). The performance of this ceiling is discussed in the next chapter. In order to simulate an unbounded cavity fl ow encountered in practical bomb-bay configurations, a flush-mounted acoustic treatment is construc ted to replace the rigid ceiling plate. The new cavity ceiling modifies the boundary conditions of th e previous sound hard ceiling. This acoustic treatment consists of a porous metal laminate (MKI BWM series, Dynapore P/N 408020) backed by 50.8 mm th ick bulk pink fiberglass insulation ( Figure 5-2 ). This acoustic treatment covers the whole cav ity mouth and extends 1 inch upstream and downstream of the leading edge a nd trailing edge, respec tively. This kind of acoustic treatment reduces reflections of acoustic waves. The perfor mance of this treatment is assessed in the next chapter. The exhaust flow is dumped to atmosphere via a 5 angle diffuser attached to the rear of the cavity model for pressure recovery. A custom rectangular-to-round trans ition piece is used to connect the rectangular diffuser to the 6 inch diameter exhaust pipe. 92

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Three structural supports are used to reduce tunnel vi brations (Carroll et al. 2004). Two of these structural supports attach to both sides of the test section inlet flange, and the additional structural support is installed to support the iron exhaust pipe ( Figure 5-2 ). Pressure/Temperature Measurement Systems Stagnation pressure and temperature are monitored during each wind tunnel run and converted to Mach number via the standard isentropic re lations with an esti mated uncertainty of The reference tube of the pressure transducer is connected to static pressure port (shown in 0.01 Figure 5-2 ) using 0.254 mm ID vinyl tubing to measure the upstream static pressure of the cavity. The stagnation and static pressures are measured separa tely with Druck Model DPI145 pressure transducers (with a quoted measurement precision of 0.05% of reading). The stagnation temperature is measured by an OMEGA thermocouple (Model DP80 Series, with nominal resolution). 0.1 CTwo pressure transducers are located in th e test section to measure the pressure fluctuations. The first transducer is a flush-mounted unsteady Kulite dynamic pressure transducer (Model XT-190-50A) and is an absolute transducer with a measured sensitivity V/Pa with a nominal 500 kHz natural frequency, Pa (50 psia) max pressure, and is 5 mm in diameter. This pressure transducer is located on the cavity floor (72.640.061053.44710 y D inch) 0.6 upstream from the cavity real wall ( x L ), and 8.89 mm ( ) away from the mid-plane. This position allows optical access from the mid-plane of cavity floor for flow visualization and avoids the possibility of coinciding with a pressure node along the cavity floor (Rossiter 19 64). The second pressure transdu cer is also an Kulite absolute transducer (with measured sensitivity 8.89 mm z 5.130.03107 V/Pa and nominal 400 kHz natural frequency, Pa (25 psia) max pressure, 5 mm in di ameter), and it is flush mounted in 51.72410 93

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the tunnel side wall 63.5 mm downst ream of the cavity as shown in Figure 5-2 From a series of vibration impact tests performed in a previous st udy (Carroll et al. 2004), the results indicated that the pressure transducer outputs are not a ffected by the vibration of the structure. An experiment to validate this hypothesis is discussed in the next Chapter. Due to a modification of the experimental setup, the second pressure sensor is moved to the cavity floor ( Figure 5-2 ) for both open-loop control and closed-loop control. A PC monitors the upstream Mach numb er, stagnation pressure, and stagnation temperature, as well as the static pressure. This computer is also used for remote pressure valve control ( Figure 5-1 ) in order to control the freestream M ach number using a PID controller. In addition, an Agilent E1433A 8-channel, 16-bit dynamic data acquisition system with built-in anti-aliasing filters acquires the unsteady pre ssure signals and communicates with the wind tunnel control computer via TCP/IP for synchroni zation. The code for both data acquisition and remote pressure control output generation are pr ogrammed in LabVIEW. The pressure sensor time-series data are also collected for both the ba seline and controlled cavity flows for post-test analysis. Facility Data Acquisition and Control Systems The schematic of the control hardware setup is shown in Figure 5-3 For the real-time digital control system, the voltage signals from the dynamic pressure transducers are first preamplified and low-pass filtered using Kemo Model VBF 35. This filter has a cutoff range 0.1 Hz to 102 kHz, and three filter shap es can be used. Option 41 with nearly constant group delay (linear phase) in the pass band a nd 40dB/octave roll-off rate is c hosen. The cutoff frequency is 4 kHz for a sampling frequency of 10.24 kHz. The si gnal is then sampled with a 5-channel, 16-bit, simultaneous sampling ADC (dSPACE Model DS2001). 94

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The control algorithms are coded in SIMULINK and C code S-functions and are compiled via Matlab/Real-Time Workshop (RTW). These codes are uploaded and run on a floating-point DSP (dSPACE DS1006 card with AMD Opteron Pr ocessor 3.0GHz) digital control system. The DSP was also used to collect input and out put data from the DS2001 ADC boards as well as computing the control signal once per time step. At each iteration, the computed control effort is converted to an analog signal accomplished us ing a 6-channel 16-bit DAC (DS2102). This signal is passed to a reconstruction filter (Kem o Model VBF 35 with identical settings to the anti-alias filters) to smooth the zero-order hold si gnal from the DAC. The output from this filter is then sent to a high-voltage amplifier (PCB Model 790A06) to produce the input signal for the actuator. The computer is also able to access th e data with the dSPACE system via the Matlab mlib software provided by dSPACE Inc. Actuator System In order to achieve effective closed-loop flow control, high bandwidth and powerful (high output) actuators are required. The following issues should be considered for selecting the actuators (Schaeffler et al. 2002). The selected actuators must produce an output consisting of multiple frequencies at any one instant in time. The bandwidth of the actuators should enable control of all signifi cant Rossiter modes of interest. The control authority must be large enough to counteract the natural disturbances present in the shear layer. According to Cattafesta et al (2003), one kind of actuator called Type A has these desirable properties. Such actuators include piez oelectric flaps and have successfully been used for active control of flow-induced cavity oscillations by Cattafe sta (1997) and Kegerise et al. (2002). Their results show that the external flow has no significant in fluence of the actuator 95

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dynamic response over the range of flow condition s. Their later work (Kegerise et al. 2004; 2007a,b) also shows that one bimorph piezoelectri c flap actuator is capable of suppressing multiple discrete tones of the cavity flow if the modes lie within the bandwidth of the actuator. Therefore, the piezoelectric bimorph actuator is a potential candidate for the present cavity oscillation problem. Another candidate actuator is the synthetic or zero-net mass-flux jet (Williams et al. 2000; Cabell et al. 2002; Rowley et al. 2003, 2006; Caraballo et al 2003, 2004, 2005; Debiasi et al. 2003, 2004; Samimy et al. 2003, 2004; Yuan et al. 2005) This actuator can be used to force the flow via zero-net-mass flux pertur bations through a slot in the upstream wall of the cavity. Although the actuator injects zero-net-mass through the slot during one cycle, a non-zero net momentum flux is induced by vor tices generated via periodic bl owing and suction through the slot. In this research, a piezoelectric-driven synthetic jet actuator array is designed. This type of synthetic jet based actuators normally gives a larger bandwidth than the piezoelectric flap type of actuators. A typical commercial parallel operation bimorph piezoel ectric disc (APC Inc., PZT5J, Part Number: P412013T-JB) is used for this design. The physical and piezoelectric properties of the actuator material are listed in Table 5-1 The composite plate is a bimorph piezoelectric actuator, which includes two piezoelectric patche s on upper and lower sides of a brass shim in parallel operation ( Figure 5-4 ). The final design of the actuator array consists of 5 single actuator units. Each actuator unit contains one co mposite plate and two re ctangle orifices shown in Figure 5-5 The designed slot geometries fo r the actuator array are shown in Figure 5-6 Another advantage of this design is that it avoids the pressure imbalance problem on the two sides of the diaphragm during the experiment Since the two cavities on either side of a 96

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single actuator unit are vented to the local sta tic pressure, the diaphragm is not statically deflected when the tunnel static pressure deviates from atmosphere. The challenge is whether these actuators can provide strong enough jets to alter the shear layer in stabilities in a broad Mach number range and also whether the actuator s produce a coherent signal that is sufficient for effective system identification and control. A lumped element actuator design code (Galla s et al. 2003) was used together with an experimental trial-and-error method to design the single actuator unit. The final designed geometric properties and parameters of the single actuator unit are listed in Table 5-2 To calibrate this compact actuator array, the centerl ine jet velocities from each slot are measured using constant-temperature hotwire anemomet ry (Dantec CTA module 90C10 with straight general purpose 1-D probe model 55p11 and stra ight short 1-D probe support model 55h20). A Parker 3-axis traverse system is used to positi on the probe at the center of actuator slots. The sinusoidal excitation signal from the Agilent 33120A function generato r is fed to the 790A06 PCB power amplifier with a consta nt gain of 50 V/V. The piezoceremic discs are driven at three input voltage levels: 50 Vpp, 100 Vpp, and 150 V pp, respectively, over a range of sinusoidal frequencies from 50 Hz to 2000 Hz in steps of 50 Hz. Each bimorph disc serves as a wall between two cavities labeled side A and side B. The notation used to identify each bimorph and its corresponding slots is shown in Figure 5-7 The rms velocities of the slots 3A and 3B located in the centerline of the cavity are shown in Figure 5-8 as an example. The maximum centerline velocities measured at the three excitation voltages for each slot are listed in Table 5-3 A summary of the measurements of the centerline velocities and cu rrents to the actuator array for each slot are provided in Appendix E. The pi ezoelectric plate is tested over a range of frequencies and amplitudes to determine the curre nt saturation associated with the amplifier. 97

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Figure 5-9 shows the simulation result calculated by the LEM actuator design code and is superposed on the experimental result, Figure 5-8 The results show that, the LEM actuator design code provides a pretty accurate rms veloci ty estimation of synthetic jet over a large frequency range between 50 Hz and 2000 Hz. Finally, the measure input current level to the actuator array after the amplifier is measured. The results are shown in Figure 5-10 and indicate that the input current will satu rate above 136mApp, which means if the input voltage is larger than 100 Vpp, the current to the actuator will ke ep a constant value. During the closed-loop experiments, an upper limit of 150 Vpp is used since the current pr obe is unavailable. Figure 5-11 shows the spectrogram of the pressure measurement on the cavity floor with acoustic treatment. The Rossiter modes (Equation 1-2 ) with =0.25, =0.7 are superimposed on this figure. The experimental deta ils are explained in the next chap ter. For this dissertation, the lower portion of the Mach number range (from 0.2 to 0.35) is our control target as an extension to previous work by Kegerise et al. (2007a,b). The desired ba ndwidth of the designed actuator should cover the dominant peaks of Rossiter mode 2, 3 and 4, which is between 500 Hz and 1500 Hz. (Rossiter mode 1 is usually weaker compar ed to Rossiter modes 2, 3 and 4.) Over this frequency range, the designed actua tor can generate larg e disturbances. In addition, the array produces normal oscillating jets that seek to pene trate the boundary layer, resulting in streamwise vortical structures. In essence, it acts like a vi rtual vortex generator. A simple schematic of the actuator jets interacting with the fl ow vortical structures is shown in Figure 5-12 The approach boundary layer contains spanwise vorticity in the x-y plane (the coordinate is shown in Figure 52 ). By interacting with the ZNMF actuator jets, the 2D shape of the vortical structures transform to a 3D shape with spanwise vortical structures. These streamwise vortical disturbances seek to destroy the spanwise coherence of the shear layer, and the corresponding Rossiter modes are 98

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disrupted (Arunajatesan et al. 2003 ). Alternatively, th e introduced disturbances may modify the stability characteristics of the mean flow, so th at the main resonance peaks may not be amplified (Ukeiley et al. 2003). Unfortunate ly, the flow interaction was not characterized in this dissertation and will be addressed in future work. Instead of using one specific amplitude and one frequency in open-l oop control, a closedloop control algorithm is used in this study to ex am the effects of the di sturbance with multiple amplitudes and multiple frequencies. Thus, the present actuator represents a hybrid control approach, in which we seek to reduce both the Rossiter tones and the broadband spectral level. 99

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Table 5-1. Physical and piezoelectric proper ties of APC 850 device. Shim (Brass) Piezoceramic Bond Elastic Modulus (Pa) 108.96310 10510 83.9810 Poissons Ratio 0.324 0.31 0.3446 Density ( ) 3/ kgm 8700 7400 1060 Relative Dielectric Const. 2400 31d (m/V) 1220010 Maximum Voltage Loading 200 Vpp/mm for 0.15 mm thickness is 30 Vpp Resonant Resistance () 200 Electrostatic Capacitance (pF) 210,00030% Operating Temp. (C) -20~70 100

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Table 5-2. Geometric properties and parameters for the actuator. Geometric Properties of the Diaphragm APC PZT5J, P412013T-JB Piezo. Configuration Bimorph Disc Bender Shim Diameter (mm) 41 Clamped Diameter (mm) 37 Shim Thickness (mm) 0.1 Piezoceramic Diameter (mm) 30 Piezoceramic Thickness (mm) 0.15 Ag Electrode Diameter (mm) 29 Total Bond Thickness (mm) 0.03 ( 0.015 on each side) Radius (mm) 0a 1/2 Length of the Orifice (mm) L 1 Width of the Orifice (mm) dw 3 Volume in side A ( ) 3mm 4064 Volume in side B ( ) 3mm 2989 Damping 0.04 101

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Table 5-3. Resonant frequencies with respec tive centerline velocities for each input voltage. Input Voltage 50 Vpp 100 Vpp 150 Vpp Slot Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) 1A 1150 35. 1100 58 1150 62 1B 1150 44 1100 69 1150 74 2A 1150 27 1100 47 1150 52 2B 1150 37 1100 60 1150 66 3A 1150 27 1100 45 1150 49 3B 1150 38 1100 60 1150 65 4A 1150 36 1150 58 1200 62 4B 1150 44 1150 68 1200 73 5A 1150 40 1150 60 1150 65 5B 1150 49 1100 72 1150 78 102

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Pressure Valve Settling Chamber Honeycomb Screens Manual Valve Subsonic Nozzle Connect to Test Section Figure 5-1. Schematic of the wind tunnel facility. Connect to Nozzle Cavity Model Perforated Metal Plate Fiberglass Dynamic Pressure Sensor P1 Static Pressure Port Dynamic Pressure Sensor P2 x y Exhaust Structural Support Inlet Structural Supports Unit: Inch Figure 5-2. Schematic of the test section and the cavity mode l. Dimensions are inches. 103

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16-bit ADC Actuator Array Power Amplifier Fc Cavity FlowReconstruction Filter FcAnti-aliasing Filter 16-bit ADC Controller dSPACE System Fs=10.240 kHz Fc = 4 kHz Fc = 4 kHz Gain = 50x Figure 5-3. Schematic of th e control hardware setup. pt s t Shim Piezoceramic acVp R s R ++++++ -----++++++ -----Figure 5-4. Bimorph bender disc actuator in paralle l operation. The physical and geometric properties are shown in Table 5-1 and Table 5-2 104

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Side A Side B A B C D Figure 5-5. Designed ZNMF actua tor array. A) Operation plot. B) Assembly diagram of single unit. C) Singe unit of the actuator. D) Actuator array. 105

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Figure 5-6. Dimensions of the sl ot for designed actuator array. 106

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Cavity Floor Flow Direction Figure 5-7. ZNMF actuator array mounted in wind tunnel. 107

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0 500 1000 1500 2000 0 10 20 30 40 50 Frequency [Hz]Velocity [m/s] 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s] 50 Vpp 100 Vpp 150 Vpp B Figure 5-8. Bimorph 3 centerlin e rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid inpu t signal. A) For side A. B) For side B. 108

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0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 10 20 30 40 50 60 70 80 Frequency (Hz)Jet Velocity |Vout| (m/s) Sim 50 Vpp Exp. 50 Vpp Sim 100 Vpp Exp. 100 Vpp Figure 5-9. The comparison plot of the experime nt and simulation result of the actuator design code for bimorph 3. The output is the cen terline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal for side B. 109

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200 400 600 800 1000 1200 1400 1600 1800 2000 0 20 40 60 80 100 120 140 160 Frequency (Hz)Current (mApp) 50 Vpp 100 Vpp 150 Vpp Figure 5-10. Current saturati on effects of the amplifier. 110

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 70 80 90 100 110 120 130 140 150 Ri Figure 5-11. Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7). 111

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112 x Y x z Y Leading Edge of the Cavity Figure 5-12. Schematic of a single periodic cell of the actuator jets and the proposed interaction with the incoming boundary layer.

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CHAPTER 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION Equation Section 6 Experimental results for the baseline unc ontrolled and controlled cavity flows are presented in this chapter. Firs t, the effects of structural vibrations on the unsteady pressure transducers are illustra ted. Then, a joint time-frequency analysis of the unsteady pressure measurement for an uncontrolled cavity flow is sh own. Flow-acoustic features are deduced from the results. An improved test environment is established by re placing the original hard-wall ceiling of the wind tunnel with an acoustic liner. This new test section minimizes the effects of the vertical acoustic modes. Finally, the results of both open-loop and adaptive closed-loop control experiments using the ZN MF actuator array is presented and discussed in detail. The ability of the actuator to alter both broadband and tonal content of the uns teady pressure spectra is demonstrated at low Mach numbers. Background As discussed in the first Chapter, flow-induced cavity oscillations are often analyzed via unsteady pressure measurements in and around th e cavity. However, these measurements are often contaminated by other dynamics associated with the specific characteristics of the wind tunnel test section. As a re sult, the unsteady pressure spec trum may be due to the cavity oscillations or other phenomena. The experimental results of Cattafesta et al. (1999), De biasi and Samimy (2004), and Rowley et al. (2005) show that some of the re sonant frequencies measured within the cavity track or lock on to vertical acoustic duct modes at some test conditions. This effect can be reduced by adding acoustic treatment at the ceiling above the mouth of the cavity (Cattafesta et al. 1998; Williams et al. 2000; Ukeiley et al. 2003; Rowley et al. 2005). This acoustic treatment modifies the sound-hard boundary condition and thus mitigates the contribution of the cavity 113

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vertical resonance modes to th e unsteady pressure measurements Consequently, the modified cavity model will ideally exhibi t the behavior of an unbounded cav ity flow and be dominated by Rossiter modes. Alvarez et al. (2005) developed a theoretical prediction method and showed that the wind tunnel walls lead to a significant increase in the growth rate of a resonant mode for frequencies near the cut-on fre quency of a cross-stream mode. In the present baseline (i.e., uncontrolled) expe rimental study, flow-acoustic resonances in the test section region and in the cavity region are examined. A schematic of the simplified wind tunnel model and the cavity region of th e experimental setup are shown in Figure 6-1 Using the same nomenclature of Alvarez et al. (2005), the domain is divided into three regions: an upstream tunnel region (0 x ), a cavity region (0 x L ), and a downstream region ( x L ). The Rossiter modes ( ...iRi 0,1, ) are the combined result of a receptivity process at instability growth in the unstable shear layer, sound gene ration due to impingement of the shear layer at 0 x x L and upstream and downstream propaga ting acoustic waves within the cavity region. The resulting flow osc illations are in teresting targets for fluid dynamics and control researchers to analyze and mitigate. Additional vertical cavity acoustic modes ( 0,1,...iVi ) and cavity cut-on modes ( ) can also be present, as discussed a bove. These acoustic modes are generated by the reflections from the ceiling and area changes of the cavity model. During the wind tunnel experiments, the vertical modes ( ) are undesirable and should be reduced in order to mimic the unbounded bomb bay problem more accurately. 0,1,...iCiiV As explained in Alvarez et al. (2005), the upstream region can support duct cut-on modes ( ) and upstream propagating duct modes ( ) due to the acoustic iu uD0,1,...i,0,1,...u iTi 114

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scattering process. Similarly, the down stream region can support duct cut-on modes ( ) and downstream propagating tunnel modes ( ) due to the scattering process. Here, we focus our attention on the propagating modes in the cavity and the downstream tunnel regions. id uD0,1,... i,0,1,...d iTiHzData Analysis Methods A schematic of unsteady pressure transducer locations for this study was presented in Chapter 5 (Figure 5-2). and measure the unsteady pressure fl uctuations in the cavity and downstream regions, respectively. 1P2P The cavity and wind tunnel acoustic modes can be obtained experimentally using two approaches. One way is to measure the output of each unsteady dynamic pressure sensor for different fixed freestream Mach numbers and then find the spectral peaks for each discrete Mach number. However, with this method it is difficu lt to track the gradual frequency changes with Mach number. The other choice is to record each unsteady pressure se nsor output continuously as the Mach number is increased gradually over th e desired range. Then, a joint-time frequency analysis (JTFA) (Qian and Chen 1996) is applied to these recorded pressure time series data. JTFA provides information on the measurement in both the time and frequency domains. Finally, the time axis is converted to Mach num ber via synchronized measurements of the Mach number versus time. Similar analysis methods can be found in Cattafesta et al. (1998), Kegerise et al. (2004), and Rowley et al. (2005). In this study, the samp ling frequency for experimental data collection is 10.24 and the frequency resolution is 5 The cut-off frequency of the anti-aliasing filter is 4 and 500 continuous blocks of time series data are used in the analysis. During the experiment, Mach number sweeps from 0.1 to 0.7 in about 100 seconds. kHzkHz 115

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Noise Floor of Unsteady Pressure Transducers The effective in-situ noise floor of the two uns teady pressure transduc ers is presented in Figure 6-2 Each noise floor measurement is compar ed with the spectra obtained at different discrete Mach numbers for the acoustically treated L/D=6 cavity. Within the tested frequency range, the signal-to-noise ratio ( ) is in excess of 30 which demonstrates adequate resolution of unsteady pressure tr ansducers for the present experiments despite their large fullscale pressure ranges. SNR dBEffects of Structural Vibrations on Unsteady Pressure Transducers A series of initial impulse impact tests are pe rformed before the baseline experiments. As discussed in Chapter 5, with the wind tunnel turn ed off, the pressure transducer outputs are not affected by hammeror shaker-induced structural vibr ations. A simple test is described here to investigate the effects of structural vibrations while the wind tunnel is running. To avoid confounding cavity oscillations, the cavity floor is mounted flush with the tunnel floor (0 D ). A piezoceramic accelerometer (P CB Piezotronics Model 356A16) is used to measure the structural vibrations. It is attached to the test section outer wall us ing wax at the location indicated in Figure 5-2 which is close to one of the pressure transducers ( ). This piezoceramic accelerometer is connected to a multi-channel signal conditioner (PCB Piezotronics Model 481A01). Three channels of the piezoceramic accelerometer corresponding to the 2P x and directions are measured. Th e coordinate directions of y z x and are shown in yFigure 5-2 and is the corresponding late ral direction using the right-hand rule. The accelerometer is calibrated with a reference shaker (PCB Piezotronics, Model: 394C06) that provides 1 (rms) at 1000 rad/s (159.2 ). zg Hz 116

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The JTFA results ( Figure 6-3 to Figure 6-5 ) for all components of the accelerometer measurements show that the power of the structural vibration spreads is broadband with a few spectral peaks. A modest peak at 1000 is present in the lateral ( ) and vertical ( Hz z y ) directions. In addi tion, some higher frequency peaks (i.e., 2450 in the Hz x or flow direction, 1880 in the lateral direction and 3200 in the vertical direction) can also be detected. However, the JTFA results of (Hz Hz1P Figure 6-6 ) and do not display any of these resonances. These results confirm that the unsteady pressure transducers are not affected by structural vibrations. 2PBaseline Experimental Results and Analysis The rigid ceiling plate (no acoustic treatment) ab ove the mouth of the cavity is considered first. JTFA results of the unsteady pressure tr ansducer measurement for this case are shown in Figure 6-7 Numerous flow-acoustic res onances can be observed in the plots. For easy reference in the subsequent discussion, thes e features are numbered 1 and 2. The final goal of the baseline experiment is to simulate the unbounded weapon bay using the cavity model in the test section. Therefore, the active flow control scheme targets the Rossiter modes (feature 1 in Figure 6-7 ). The other unknown acoustic features 2 in Figure 6-7 are undesirable features that we wish to eliminate. These acoustic modes come from the bounded wind tunnel walls, the mismatched acoustic impedance due to area change, and th e leading and trailing edges of the cavity. In order to better mimic an unbounded cavity flow in a closed wind tunnel, the boundary condition of the cavity ceiling must be altered to eliminate the unexpected modes within the cavity region. A flush-mounted ac oustic treatment (discussed in Chapter 5) is fabricated to replace the previous solid tunnel ceiling. Th e new cavity ceiling modifies the zero normal velocity boundary condition of the previous sound hard top plate. 117

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The unsteady pressure transducer JTFA meas urement for the trailing edge floor of the cavity is shown in Figure 6-8 The results illustrate a very clea n flow field below Mach 0.6. The acoustic features 2-4 in Figure 6-7 are eliminated within the ca vity region. Therefore, the experimental Rossiter modes i R shown in JTFA plot ( Figure 6-8 ) now follow the estimated Rossiter curves. At higher upstream Mach numbers ( ), the experimental Rossiter modes deviate slightly from the expected Rossiter curves This is partly because the estimated curves use the upstream static temperatur e to calculate the speed of sound. This estimation does not account for the expected significant static temp erature drop due to the large flow acceleration near the aft cavity region seen by Zhuang et al. (2003). Another pos sible reason for these deviations of the flow-acoustic resonance comes from the structural vibr ation coupling with the Rossiter modes. At high Mach numbers above 0.6, the structural vibratio ns may cause a lock-on phenomenon with the Rossiter modes. For this study, all experiments are thus performed below M = 0.6. 0.6 M In conclusion, the observed flow-acoustic behavi or of the acoustically treated cavity model behaves as expected below M = 0.6 and is ther efore suitable for application of open-loop and closed-loop flow control. Open-Loop Experimental Results and Analysis The open-loop and closed-loop experimental resu lts using the designed actuator array are shown in this section. Before the control experi ments, measurements of the pressure sensor at the surface of the trailing edge of the cavity with the without the actuator turned on are shown in Figure 6-9 Without the upcoming flow, the noise floor shows a significant peak at 660 Hz and a small peak at 2000 Hz. The pressure sensor can also sense the acoustic di sturbances associated with the excitation frequency a nd its harmonics, and the measured unsteady pressure level can 118

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reach 115-120 dB. The extent to which the measured levels deviate from theses values with flow on (considered below) indicates the relative impact of the actuator on the unsteady flow. First, open-loop active control is explored. The purpose of the open-loop experiments is to verify if the synthetic jets generated from the designed actuator array can affect and control the cavity flow. A parametric study for the open-loop c ontrol is explored first. A sinusoidal signal is chosen as the excitation i nput with the frequency swept from 500 Hz to 1500 Hz. The openloop experimental results are shown in Appendix F. The open-control performance is best over the frequency range 1000 Hz to 1500 Hz, which co rresponds to the resonan ce frequencies of the actuator array. Since at the res onance frequency, the actuator arra y can generate larger velocity jet, and the blow coefficient /()c cavityBmUA increases. As a result, the control effect increases. For these open-loop tests, the upstream flow Mach number is varied from 0.1 to 0.4. For illustration purposes, results are examined here for two sinusoidal signals with 200 Vpp and excitation frequencies at either 1.05 kHz or 1.5 kHz to drive the actuato r array. The 1.05 kHz excitation frequency is close to the resonance frequency of the actuato r, while the 1.5 kHz frequency lies between the second and third Ross iter modes. The experime ntal results shown in Figure 6-10 illustrate that this actuator array can successfully reduce multiple Rossiter modes, particularly at Mach number 0.2 and 0.3. In addition, the pressure fluctuation is mitigated at the broadband level on the surface of the cavity floor for all the tested flow conditions. However, new peaks are generated by the excitation freque ncies and their harmonics, especially at low Mach number 0.1. With increasing upstream Mach number, the unsteady pressure level also increases and the effect of the control is reduced Note the synthetic jets introduce temporal and spatial disturbances to modify the mean flow in stabilities and destroy the coherence structure in 119

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spanwise, respectively. The eff ectiveness of the actuato r scales with the mo mentum coefficient, which is inversely proportional to the square of the freestream velocity. So, as the upstream Mach number increases, the synthetic jets are eventually not strong e nough to penetrate the boundary layer and the control e ffect is reduced. Future work should perform detailed measurements to validate this hypothesis. The results of the open-loop cont rol suggest that this kind of actuator array can generate significant disturbances not only along the flow propagation di rection but also in the spanwise direction of the cavity. The combination of thes e effects disrupts the Kelvin-Helmholtz type of convective instability waves, which are the source of the Rossiter modes. As a result, multiple resonances are reduced via active control. The experimental results also show the limitation of the open-loop control. Closed-Loop Experimental Results and Analysis The open-loop control results sugg est that this compact actuator array may be effective for adaptive closed-loop control. As discussed above, the synthetic jets add di sturbances to disrupt the spanwise coherence structure of the shear layer and result in a broadband reduction of the oscillations. However, at the same time, the coherence between the driv e signal and the unsteady pressure transducer will be redu ced. High coherence is considered essential for accurate system identification methods. To exam the accuracy of th e system ID algorithm w ith the change of the estimated order, an off-line system ID analysis is first performed. The nominal flow condition is chosen at M = 0.275 (to match that of Kegerise et al. 2007a,b) with a L/D=6 cavity, and two system ID signals, one with white noise (bro adband frequency and amplitude 0.29 Vrms ) and the other with a chirp signal (amplitude 0.86 Vrms and fL = 25 Hz to fH = 2500 Hz in T = 0.05 sec), are used as a broadband excitation sour ce to identify the system. The running error variances the system ID are shown in Figure 6-11 It is clear that the larger the estimated order, 120

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p, the more accurate is the system ID algorithm However, due to the limitations of the DSP hardware, we cannot choose very large values of the estimated order for system ID algorithm online. One potential advantage of the closed-loop adap tive control algorithm is that it does not rely exclusively on accurate system ID. Figure 6-12 shows the result of the closed-loop real-time adaptive system ID together with the GPC contro l algorithm for an upstream Mach number 0.27. Based on the above system ID results, due to th e DSP hardware limitation, the estimated GPC order and the predictive horizon are chosen as 14 and 6, respectively. The breakdown voltage of the actuator array restricts the excitation voltage level; therefore, the diagonal element of the input weight penalty matrix R (Equation 3-14 ) is chosen as 0.1. This research represents an extension of Kegerise et al (2007b) where the system ID algorithm and the closed-loop controller design algori thm are used simultaneously in a real-tim e application. It is important to note that only the system ID white noise or chirp signal is used to identify the open-loop dynamics, and the feedback signal is not used for th is purpose. Clearly, the results show that the GPC controller can generate a seri es of control signals to drive the actuator array resulting in significant reductions for the second, third, and fourth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, th e broadband background noise is also reduced by this closed-loop controller; the OASPL reduction is 3 dB. The input signal is shown in Figure 6-13 The sensitivity function discussed in Chapter 4 (Equation 4-1) is shown in Figure 6-14 A negative amplitude value indicates disturbance attenuation, while a positive value indicates disturbance amplification. The results show that all the points are negative, which indicates the closed-loop controller reduces the pressure fluctu ation power at all frequencies. The spillover phenomenon (Rowley et al. 2006) is not observed in Figure 6-14 As discussed in Chapter 1, the 121

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spillover problem is generated because either th e disturbance source and control signal or the performance sensor output and the measurement sens or output (feedback signa l) are collocated. The Bodes integral formula is shown in 6-1 0log()Re()k kSidp (6-1) where k p are the unstable poles of the loop gain of the closed-loop system. So, for a stable system, any negative area at the left hand side of the Equation 6-1 must be balanced by an equal positive area at the left hand side of the Equation 6-1 However, for present closed-loop control study, the left hand side of the B odes integral formula is -38 ra d/sec, which shows that Bodes integral formula does not hold here. Since this formula is valid for a linear controller, the combination of the adaptive system ID and cont roller is apparently nonlinear. A more detailed study is required in the future to validate this hyothesis. A parametric study of the GPC is then studi ed by varying the es timated order and the predictive horizon. Figure 6-15 and Figure 6-16 show that the control effects improve with increasing order and predictive horizon. This trend matches the simulation results shown in Chapter 4. The comparison between the open-loop a nd closed-loop results is shown in Figure 6-17 for the same flow condition. Notice that the base line measurement for a same flow condition can vary a little from case to case. The open-loop control uses a sinusoida l input signal at 1150 Hz forcing and 150 Vpp and the rms value of the in put is 53V. The closed -loop control uses the estimated order 14, the predictive horizon 6, and the input weight 0.1, and the input rms value is 43 V. 122

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iu uDid uDi R iViCd iTu iT x y Cavity Region Downstream Region Upstream Region H L D Figure 6-1. Schematic of simplified wind tunne l and cavity regions acoustic resonances for subsonic flow. 0 1000 2000 3000 4000 5000 6000 70 80 90 100 110 120 130 140 150 frequency [Hz]Unsteady Pressure Level [dB] with Pref=20e-6 Pa 0.4 0.5 0.55 0.58 0.6 0.65 0.69 Noise Floor Noise Floor Figure 6-2. Noise floor level comparison at different discrete Mach numbers with acoustic treatment at trailing edge floor of the cavity with L/D=6. 123

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 50 60 70 80 90 100 Figure 6-3. x -acceleration unsteady power spectrum (d B ref. 1g) for case with acoustic treatment and no cavity. 124

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 30 40 50 60 70 80 90 100 Figure 6-4. y -acceleration unsteady power spectrum (dB ref. 1g) for case with acoustic treatment and no cavity. 125

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 40 50 60 70 80 90 100 Figure 6-5. -acceleration unsteady power spectrum (dB ref. 1g) for case with acoustic treatment and no cavity. z 126

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 90 100 110 120 130 140 Figure 6-6. Spectrogram of pre ssure measurement (dB ref. 20e-6 Pa) on the trailing edge floor of the cavity for the case w ith acoustic treatment and no cavity. Noise spike near 600 Hz is electronic noise. 127

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 100 110 120 130 140 150 2 2 1 Figure 6-7. Spectrogram of pre ssure measurement (ref 20e-6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D=6. Unknow n acoustics features are denoted as while the Rossiter modes are denoted as 128

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Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 70 80 90 100 110 120 130 140 150 Ri Figure 6-8. Spectrogram of pr essure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7). 129

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0 1000 2000 3000 4000 5000 6000 80 85 90 95 100 105 110 115 120 Frequency [Hz]UPL [dB] TE noisefloor TE 1050Hz-150Vpp A 0 1000 2000 3000 4000 5000 6000 80 85 90 95 100 105 110 115 120 Frequency [Hz]UPL [dB] TE noisefloor TE 1500Hz and 150Vpp B Figure 6-9. Noise floor of the unsteady pressure level at the surface of the trailing edge of the cavity with and without the actuator turned on. A) The exciting sinusoidal input has frequency 1050 Hz and amplitude 150 Vpp. B) The exciting sinusoidal input has frequency 1500 Hz and amplitude 150 Vpp. The peaks near 600 Hz and 2000 Hz are electronic noise. 130

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0 500 1000 1500 2000 2500 3000 3500 4000 75 80 85 90 95 100 105 110 115 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e-6 Pa Mach 0.1, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp A 0 500 1000 1500 2000 2500 3000 3500 4000 85 90 95 100 105 110 115 120 125 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e-6 Pa Mach 0.2, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp B Figure 6-10. Open-loop sinusoidal control results for flow-induced cavity oscillati ons at trailing edge floor of the cavity. A) At Mach number 0.1. B) At Mach number 0.2. C) At Mach number 0.3. D) At Mach number 0.4. The cavity model with 6 inch long and L/D=6. 131

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0 500 1000 1500 2000 2500 3000 3500 4000 95 100 105 110 115 120 125 130 135 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e-6 Pa Mach 0.3, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp C 0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e-6 P a Mach 0.4, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp D Figure 6-10 Continued 132

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0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10-7 Time(s)Runing Error Var p = 2 p = 4 p = 6 p = 8 p = 14 p = 50 p = 100 A 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10-7 Time(s)Runing Error Var p = 2 p = 4 p = 6 p = 8 p = 14 p = 50 p = 100 B Figure 6-11. Running error variance plot for the system identification algorithm. A) With chirp signal as input. B) With white noise si gnal as input. Upstream Mach number is 0.275, L/D=6. 133

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0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Baseline, M=0.27 TE Closed-Loop, p=14, s=6 Figure 6-12. Closed-Loop active control result for flow-induced cav ity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity. The control algorithm uses an estimated order of 14 for both the system ID and GPC algorithms, and the predictive horizon is chosen as 6. A chirp signal is used as the system ID excitation source. 134

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0 500 1000 1500 2000 2500 3000 3500 4000 -40 -30 -20 -10 0 10 20 Frequency [Hz]Input Power [dB] Figure 6-13. Input signal of the Closed-Loop active control result fo r flow-induced cavity oscillations at Mach 0.27 at the trailing edge floor of th e L/D =6 cavity. The control algorithm uses an estimated order of 14 fo r both the system ID and GPC algorithms, and the predictive horizon is chosen as 6. A chirp signal is used as the system ID excitation source. 135

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0 500 1000 1500 2000 2500 3000 3500 4000 -0.4 -0.3 -0.2 -0.1 0 0.1 Frequency [Hz]Sensitivity in Log Scale Disturbance Amplification Disturbance Attenuation Figure 6-14. Sensitivity function (Equation 4-1) of the closed-loop control for M=0.27 upstream flow condition. The estimated order is 14, prediction horizon is 6, and the input weight R is 0.1. This sensitivity is calculated based on Figure 6-12 136

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0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Baseline, M=0.27 TE Closed-Loop, p=2, s=6 TE Closed-Loop, p=14, s=6 Figure 6-15. Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream flow condition with varying estimated orde r. The prediction horizon is 6, and the input weight is 0.1. The excitation source for the system ID is a swept sine signal. 137

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0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Baseline, M=0.27 TE Closed-Loop, p=14, s=2 TE Closed-Loop, p=14, s=6 Figure 6-16. Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream flow condition with varying predictive horiz on s. The estimated order of the system is 6, and the input weight is 0.1. The ex citation source for the sy stem ID is a swept sine signal. 138

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139 0 500 1000 1500 2000 2500 3000 3500 4000 100 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Open-Loop TE Baseline, M=0.27 TE Closed-Loop Figure 6-17. Unsteady pressure level comparis on between the open-loop control and closed-loop control for M=0.27 upstream flow condition. The open-loop control uses a sinusoidal input signal at 1150 Hz forci ng and 150 Vpp and the rms value is 53 V. The closedloop control uses an estimated order 14, pr edictive horizon 6, input weight 0.1, and the input rms value is 43 V.

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CHAPTER 7 SUMMARY AND FUTURE WORK Equation Section 7 This chapter summarizes the previously disc ussed work and presents contributions from this study. Future work is summarized that a ddresses detailed measurements of the actuator system, a systematic experimental analysis of the flow using various flow diagnostics, and a more detailed parametric study of open-loop and closed-loop control. Summary of Contributions The contributions of this research are summarized here. First, a global model of flowinduced cavity oscillation is derive d that provides insight into the required structure for a plant model used for subsequent control. When simp lified, this model matches the Rossiter model. Second, a novel piezoelectric-driven synthetic jet actuator array is designed for this research. The resulting actuator produces hi gh velocities (above 70 m/s) at th e center of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) whic h is sufficient to cont rol the Rossiter modes of interest at low subsonic Mach numbers. Th is actuator array produces normal zero-net massflux jets that seek to penetrate the boundary laye r, resulting in streamwi se vortical structures. These streamwise vortical disturbances destroy th e spanwise coherence of the shear layer, and the corresponding Rossiter modes are disrupted. Alternatively, the introduced disturbances modify the stability characteristics of the mean flow, so that the main resonance peaks may not be amplified. Next, a MIMO system ID IIR-b ased algorithm is developed ba sed on the structure inferred from the global model. This system ID algorithm combined with a GPC algorithm is applied to a validation vibration beam problem to demonstrat e its capabilities. The control achieves ~20 dB reduction at the single resonance peak and ~9 dB reduction of the integrat ed vibration levels. Finally, this control methodology is extended and applied to subsonic cavit y oscillations for on140

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line adaptive identification and control. Open-l oop active control uses a sinusoidal signal with 200 Vpp and an excitation frequency of either 1. 05 kHz or 1.5 kHz, which are detuned from the Rossiter frequencies, to drive the actuator a rray. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced. However, when the upstream Mach number increases (greater than Mach number 0.4), the effects of the synthetic jets from this actuator are gradually reduced. Adaptive closed-loop control is then applied for an upstream Mach number of 0.27; the estimated GPC order is 14 and the predictive horizon is 6. To avoid saturation in the control signal, the input weight penalty is ch osen as 0.1. The GPC controller can generate a series of control signals to driv e the actuator array resulting in dB reduction for the second, third, and fouth Rossiter modes by 2 dB 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closed -loop controller (i.e., the OASPL reduction is 3 dB). However, unlike previously reported closed-loop cavity results, a spillover phenomenon is not observed in the closed-loop cont rol result. As discus sed in Chapter 1, the spillover problem is generated by a linear controller because the disturbance source and control signal or the performance sensor output and the measurement sensor output (feedback signal) are collocated. The nonlinear nature of the adaptive system may be responsible for this effect. Future Work Recommended future work consists of the following items. The phase-locked centerline velocity of the act uator array should be measured using, at least, by hot-wire anemometry. The actuator system for the active flow control needs to be explored in detail. Since the size of the rectangular slots are small (1 mm by 3 mm), the present size of the hotwire (~1 mm) cannot provide spatially-resolved measurements. The hot wire is also not suitable to decipher the 3-D velocity field resulting from the interaction of the jets with the boundary layer. Laser Doppler Velocimetry (LDV) or stereo Particle Image Velocimetry (PIV) measurements can provide good spat ial resolution of the 3-D, tu rbulent velocity field that results from the interaction between the ZNMF jets and the grazing boundary layer. 141

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142 The turbulent boundary layer characteristics (e.g., incoming boundary layer thickness) at the leading edge of the cavity and the mean flowfield of th e baseline uncontrolled case should be measured. The impedance of the ceiling liner should be measured using an acoustic impedance tube. Another potential system order estimation algor ithm from information theory of empirical Bayesian linear regression by Stoica (1989, 1997) should be applied to this problem. Parametric studies are recommended to anal yze the performance, adaptability, cost function, and limitations (spillover, etc.) of the adaptive MIMO control algorithms over and above that of the present SISO experiments. The effectiveness of the adaptive closed-loop co ntroller should be evaluated with changes in the upstream Mach number.

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APPENDIX A MATRIX OPRATIONS Equation Section 1 Vector Derivatives In this appendix, finite length vectors derivatives are illustra ted. During most optimization method development, they are the fundamental t ools to find the optimum value. Only real numbers are considered in this appendix. Definition of Vectors Define the vectors and u y as following (A-1) 111 1 222 1 1 11()(,,) ()(,,) ()(,,)n n nmmn nmuyyuu uyyuu uyyuu u u uy u1mFor special case, if or 1 n1 m the vector or is reduced to scalar, respectively. uyDerivative of Scalar with Respect to Vector 12 1n nyyyy uuu u (A-2) Note the derivative of scalar is a row vector. Derivative of Vector with Respect to Vector 111 1 12 222 2 12 12 n n m mmm n mnyyy y uuu yyy y uuu y yyy uuu u y u u u (A-3) 143

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Second Derivative of Scalar With Re spect to Vector (Hessian Matrix) 222 2 1121 1 222 2 2 2122 2 222 12 n T n n nnnn nnyyy y uuuuu u y yyy u uuuuuu y yyy u uuuuuu 2yy uuuu (A-4) Example 1 Given: 1 2 1 12 2 2 2 32 21 3 31, 3 u y uu u y uu u uy (A-5) Find: y u Solution: From the Equation A-3, the derivative of matrix y u is computed as 111 1 123 2222 123 23 1 3 23210 032 yyy y uuu yyyy uuu u u y u u u (A-6) Table of Several Useful Vector Derivative Formulas Table A-1 lists the most common vector derivative results, these results are very useful for MIMO controller design. If nnA is symmetric, the last formulas in Table A-1 can be expressed as 144

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12T nnAn y u u (A-7) Proof of the Formulas Proof (a) 11 11 11 211 22 12 222 12 11 111122 1 211222 2 1122 1nnnn n n nnnn nn nn nn nn nnnnn nA yaaau yaaau yaaau auauau auauau auauau yun n (A-8) According to Equation A-3 1112 1 2122 2 12 n n nnnn nn nnaaa aaa y aaa A u (A-9) Proof (b) 11 1112 1 2122 2 12 1 12 11121211122 1 T nnnn n n n n nnnn nn nnnn nnn nA aaa aaa uuu aaa auauauauauau yu (A-10) According to Equation A-3 145

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1121 1 1222 2 12 n n nnnn nn T nnaaa aaa y aaa A u (A-11) Proof (c) 1111 1 2 12 1 1 222 12 11 T nn n n n n ny u u uuu u uuu uu (A-12) According to Equation A-2 12 1 12 1 1222 2 2n n n n T ny uuu uuu u u (A-13) Proof (d) 111 1 1112 1 1 2122 2 2 12 1 12 1 1 2 11121211122 1 1 111212 11 1 T nnnn n n n n nnnnn nnn nnnn nnn n n n nn nyA aaau aaau uuu aaau u u auauauauauau u auauauuau uu122 11 nn n nauauu n (A-14) According to Equation A-2 146

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111212 1 111122 1 1122 1122 1 11121211122 1 111122 1 1122 1 1112 12 1 nn nn nnnnnnnnnn n nnnn nnn n nnnn nnn n n nauauauauauau y auauauauauau auauauauauau auauauauauau aaa uuu u 1 2122 2 12 1121 1 1222 2 12 1 12 11 n n nnnn nn n n n n nnnn nn TT T nnnnnnaaa aaa aaa aaa uuu aaa AA uu (A-15) Example 2 Given: 1 2 3 31 11313331110 ,101 001Tu uA u yA u uu (A-16) Find: y u Solution: 11313331 1 123 2 13 3 3331 1 121232 13 3 31 22 1233110 101 001TyA u uuu u u u uuuuuu u uuuu uu (A-17) 147

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According to Equation A-2, the derivative of y u is computed as 1323 1322 y uuuu u (A-18) Now, calculate the gradient of y u using the Table A-1 31333133 123 123 13 13 33 33 1212312133 13 13 1323 13110 110 101 100 001 011 22TT Ty AA uuu uuu uuuuuuuuuu uuuu uu u (A-19) The result of Equation A-19 is the same as that in Equation A-18. The Chain Rule of the Vector Functions Define the vectors u y z and as following w 11 22 1 1 1 2 2 11() () () () () () () () ()nm n s r r1m s uy uy uy w z w z w z u u uy u z y z y zw z y (A-20) From Equation A-3 148

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111 12 222 12 12 n n rrr n rnzzz uuu zzz uuu zzz uuu z u (A-21) Each element of Equation A-21 may be expanded using chain rule as 12 12 1 iii i jjjm m ik k kjzzzzy yy uyuyuyu zy yu m j (A-22) where 1,2,, 1,2,ir j n Substitute Equation A-22 into Equation A-21 111 111 12 222 111 12 111 12 mmm kkk kkk kkk n mmm kkk kkk kkk n rn mmm kkk rrr kkk kkk n rnyyy zzz yuyuyu yyy zzz yuyuyu yyy zzz yuyuyu z u 111 111 12 12 222 222 12 12 12 12 n m n m mmm rrr m n rm mn mn rmyyy zzz uuu yyy yyy zzz uuu yyy yyy zzz yyyuuu zy yu (A-23) 149

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Similarly, for more vectors, the chain rule just builds the new derivatives to the right. s ns r rm m n wwzy uzyu (A-24) The Derivative of Scalar Functions Respect to a Matrix Define a matrix 1112 1 2122 2 12 n n mmmn mnhhh hhh hhh H, (A-25) and a scalar function ()Jf H (A-26) The gradient of J with respect to H represented by 1112 1 2122 2 12 n n mmmn mnJJJ hhh JJJ J hhh JJJ hhh H. (A-27) Example 3 Find the gradient matrix, if J is the trace of a square matrix H 1n ii iJ=tr h H (A-28) According to Equation A-27 the gradient of J with respect to H is 150

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100 010 001nn nn nnJ I H (A-29) Example 4 Find the gradient matrix, if J is the trace of a square matrix T H H, where H is defined by Equation A-25 and need not be square. The scalar function J can be expressed as 1121 11112 1 1222 2 2122 2 12 12 222222 1121 11222 2 222 12 2 11 mn mn nnmnmmmn nm mn nn m nnmn nm ij jiJtr hhhhhh hhhhhh tr hhhhhh hhhhhh hhh hm THH (A-30) and the gradient of J is 1112 1 2122 2 12222 222 222 2n n nn mmmn mn mnhhh hhh J hhh H H (A-31) Example 5 Define the vectors A and B H as following 151

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11 22 11 1112 1 2122 2 12,mn mn n n mmmn mnab ab ab hhh hhh hhh AB H (A-32) Find the derivatives of a scalar function J respect to matrix H Express the scalar function as 1112 1 1 2122 2 2 12 1 12 1 1 2 111221 1 1122 1 1 11 11 1 T n n m m mmmnn mnn mm nnmmn n n n mm kk kknn kk m kkll kJAHB hhhb hhhb aaa hhhb b b ahahahahahah b ahbahb ahb H1 n l (A-33) So, the derivative of J respect to the matrix H is 1112 1 2122 2 12 1 2 12 1 1 1 1 n n mn mmmn mn n n m m m nababab ababab J ababab a a bbb a H T=AB. (A-34) 152

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Note that since J is a scalar function, and the following equation is also hold. TJJ TTBHA 1 1 mn mn m nJ HHTT TBHA =AB (A-35) 153

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154 Table A-1. Vector derivative formulas. y y u 11 11 1111 111 1() () () ()nnnn T nnnn T nn T nnnnaA bA cy dyA yu yu uu uu 1 112nn T nn T n TT nnnnnnA A AA u uuT

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APPENDIX B CAVITY OSCILLATION MODELS Equation Section 2 Rossiter Model The derivation of the Rossiter Model depends on the following assumptions (Rossiter 1964) Frequencies of the acoustic radiation are th e same as the vortex shedding frequency. There are vm complete wave length of th e vertex motion at the time 0'ttt (e.g. 1,2,... ) vm The schematic of the Rossiter model is illustrated in Figure B-1 and the symbols in the plot are listed as follows ,LD Cavity length and depth U Free stream velocity a Mean sound speed inside the cavity vm Mode number (integer number 1,2) Phase lag factor between impact of the large scale structure on the trailing edge and the generation of the acoustic wave. Proportion of the convective vortices speed to the free stream speed v Spacing of the vortices At specific initial time an acoustic wave forms at the trailing edge with the distance 0tt v ( Figure B-1 ). Appropriately choose the time such that, the propaga ting wave front just reaches the cavity leading edge. By the assumption, the mode number is an integer number. Then, tvm 'Lat (B-1) 155

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'vvvLmUt (B-2) And the frequencies of the oscillations are re lated the phase speed and the wavelength of the vortical disturbance. vU f (B-3) Substituting the Equation B-1 in to Equation B-2, 1vvv vvL LmU a ULm a (B-4) Then, combining the Equation B-3 and Equation B-4 resulting 1 1v vU ULm af m fL U U a (B-5) The Rossiter model is defined by 1 ,1,2,... 1v v vfL St U m St U a m St m M (B-6) Linear Models of Cavity Flow Oscillations The block diagram of the linear model of th e flow-induced oscillations is show in Figure B-2 (Rowley et al. 2002). And the closed-loop cavity tr ansfer function can be expressed as 156

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()()() () 1()()()()GsSsAs Ps GsSsAsRs (B-7) Furthermore, the shear layer model is consid ered as a second-order system with a time delay 2 0 0 2 00()() 22 s s s sw GsGse e sww (B-8) and sL U (B-9) where 0w Natural frequency of second order system Damping ratio s Time delay inside the shear layer The acoustics model can be represented as a reflection model ( ()AsFigure B-3 ) where a Time delay inside the cavity r Reflection coefficient The closed-loop transfer f unction of the reflection model can be written as 2() 1a as se As re (B-10) and aL a (B-11) To recover the Rossiter formula, additional assumptions are required 157

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Impingement model ()Ssand receptivity model () R s are unit gains. No reflections in acoustic model in B-10, 0 r The shear layer model is only a constant phase delay 2 0()iGse (B-12) Depending on these assumption, combine the Equation B-7, Equation B-10 and Equation B-12, () 2 () 2()() () 1()() 1as ass i s iGsAs Ps GsAs ee ee (B-13) In order to find the resonant of the system, substitute the poles locations into the characteristic function of the Equation siw B-13, and then combine the Equation B-9 and Equation B-11 resulting () 2 2( 210 2()() 1 2as vaiw i im iw i v vee eee LL mw aU m wL U U a )s (B-14) Define ,1,2,... 1v vfL St U m m M (B-15) The linear model results Equation B-15 matches the Rossiter formula Equation B-6. 158

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Global Model for the Cavity Os cillations in Supersonic Flow The global model for cavity oscillations in supersonic flow is shown in Figure B-4 To be consistent with the notation of Kerschen et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude upstream propagating cavity acoustic modes U and downstream propagating cavity acoustic modes and fast modes SD f E and slow modes s E near-field acoustic waves in the supers onic stream. The local amplitudes of all quantities at the leading edge are denoted by the decoration while quantities at the trailing edge do not have the decoration. The scattering processing at th e leading edge is modeled by f sSU DU EU f EU sC S C D U C E C E (B-16) SUC, D UC, and are the four scattering coefficients for the upstream end of the cavity. The first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at th e trailing edge is modeled by fEUCsEUC fsUSUDUEUE f s S D UCCCC E E (B-17) USC, UDC f UEC, and s UEC are the four scattering coefficien ts for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by 159

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1233 22 ,diL iL iML iML ffssSSeDDe ELEeELEe (B-18) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. 1 M and 2 M are the wavenumbers of the fast and slow downstream propagating near-field acoustic waves, respectively. Finally, the upstream propagating acoustic cavity mode is uiLUUe (B-19) where u is the complex wavenumber of this mode. The global model can be expressed in a block diagram ( Figure B-5 ). Substituting Equation B-19 into Equation B-17 results (B-20) 0u fs u fsiL USUDUEfUEs iL USUDUEUE f sUeUCSCDCECE U S eCCCC D E E Also, substituting Equation B-18 into Equation B-16, the following formula can be obtained 1 23 2 3 2 d f siL SU iL DU iML EU f f EU iML s sSe C S De C D U C LEe E C E LEe (B-21) These four equations can be written in matrix form 160

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1 23 2 3 2000 000 0 000 000d f siL SU iL DU iML EU f iML s EUCe U Ce S D CL e E E CL e (B-22) Now, combine Equation B-20 and Equation B-22 yields 1 23 2 3 2 000 000 0 000 000u fs d f siL USUDUE UE iL SU iL DU iML EU f s iML EUeCCCC U Ce S Ce D CL e E E CL e (B-23) Define 1 23 2 3 2000 00 000 000 u fs d f siL USUDUE UE iL SU iL DU iML EU iML EU f seCCCC Ce Ce A CL e C U S X D E E 0 L e (B-24) Therefore, Equation B-23 can be written as 0 AX (B-25) 161

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Notice that the quantities of X are the incident waves on the two ends of the cavity. The global mode has to satisfy Equation B-25 which corresponds to the condition det( Calculating the determinant and simplifying, )0 A 123 2 30ff ss udiML iML EUUE EUUE iL iL iL SUUS DUUDLCCeCCe LeCCeCCe (B-26) Assume a simple case where only 0,0USSUCC and all other scattering coefficients are zeros. Therefore, Equation B-26 can be simplified to ()0 1 1u u uiL iL SUUS iL SUUS iL iL SU USeCCe CCe CeCe (B-27) Enforce the phase criterion and notic e that the length is normalized by L L U and Equation B-27 results ()1 Re()2uiL SUUS SU US uCCe ArgCArgC Lm (B-28) or 2 Re /2 2R eSU US u SU US umArgCArgC L L U mArgCArgC L U (B-29) Consider the normalized wave number /1 Re / / Re /uUU M cU (B-30) 162

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and define 2 /2SU USL St U ArgCArgC (B-31) Therefore, Equation B-29 can be written as 2 Re ,1,2,... 1 2SU US umArgCArgC L L U Lm St m U M (B-32) This matches the Rossiter formula Equation B-6. Global Model for the Cavity Os cillations in Subsonic Flow The global model for cavity oscillations in subsonic flow is shown in Figure B-6 To be consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude upstream modes U and downstream modes propagating acoustic modes in the cavity, and upstream modes and downstream modes propagating near-fields acoustic waves in the subsonic flow. The local amplitudes of all quantities at the leading e dge are denoted by the decoration SDdEuE while quantities at the trailing edge do not have the decoration. The scattering processing at th e leading edge is modeled by u u dd uSUSE DUDE u EUEE dS CC U DCC E CC E (B-33) 163

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SUC D UC ,dEUCuSECu D EC and are scattering coefficients for the upstream end of the cavity. Again, the first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering proce ss at the trailing edge is modeled by duEEC d uuudUSUDUE u ESEDEE dS CCC U D E CCC E (B-34) USC , and are scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by UDCdUECuESCuEDCudEEC 3 2 ,,diL iML iL ddSSeDDeELEe d (B-35) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. d M is the complex wavenumber of the downstream propagating near-field acoustic wave. Finally, the upstream propagating acoustic cavity modes are 3 2 ,uiL iML uuUUeELEeu (B-36) where u and u M are the complex wavenumbers of the upstream traveling cavity mode and the upstream traveling acoustic ne ar-field mode, respectively. The global model can be represented in a block diagram ( Figure B-7 ). Again, substituting Equation B-35 into Equation B-34and Equation B-36 into Equation B-33, and combine the results, a matrix equation can be obtained, where 0 AX 164

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3 2 3 2 3 2 3 2 3 210 01 100 010 001dd d dd uuu d uu u uu u uu dd uiL iML iL US UD UE iL iML iL ES ED EE u iL iML SU SE iL iML DU DE d iL iML EU EECeCeLCe U CeCeLCe E S AX CeLCe D CeLCe E CeLCe (B-37) The global mode has to satisfy 0 AX which corresponds to the condition det()0 A Calculating the determinant and simplifying, 3 2 33 3 22 3 2ud ud u dd ud u uu uu duud uu uu duu uu uuiLiL iML SUUS DUUD EUUE iML iML iMML SEES DEED EEEE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEESCCeCCeLCCe LCCeLCCeLCCe CCCCCCCC Le CCCCCCCC 3 31ududdudu duu duuddduu ududdudu dduu duuddduuSUESEEUEEUEESEUS iMML SUUSEEEEEUUESEES DUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEEDCCCCCCCC Le CCCCCCCC CCCCCCCC Le CCCCCCCC d u (B-38) Assume a simple case where only 0,0USSUCC and all other scattering coefficients are zero. Therefore, Equation B-38 can be simplified to ()1 1u uiL SUUS iL iL SU USCCe CeCe (B-39) This matches the supers onic case results Equation B-27 165

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L D v v U a0tt L D U a0' ttt 'vUt vvm Figure B-1. Schematic of Rossiter model. G(s) S(s) A(s)(Leading Edge) Receptivity Shear Layer (Trailing Edge) Impingment Acoustics Feedback Sensor Output Actuator Input R(s) Figure B-2. Block diagram of the linear mode l of the flow-induced cavity oscillations. 166

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a s e a s re Figure B-3. Block diagram of the reflection model. L Turbulent Boundary Layer D 1 M s EfESD U x y Figure B-4. Global model for the cavity oscillations in supersonic flow. f SUSUDUEUECCCC f SSU DU E U E UC C C C 23 2 iMLLe 13 2 iMLLe iLe N f s S D E E f s S D E EU uiLe diLe U Figure B-5. Block diagram of the global model for a cavity oscillation in supersonic flow. 167

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168 1 L Turbulent Boundary LayerD M dEuESDU x y Figure B-6. Global model for a cavit y oscillation in subsonic flow. N U uiLe 3 2uiML L e u u dd uSUSE DUDE E UEECC CC CC d uuudUSUDUE E SEDEECCC CCC 3 2diML L e diLe iLe uU E uEdS D E dS D E Figure B-7. Block diagram of the global mode l for a cavity oscillation in subsonic flow.

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APPENDIX C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS Equation Section 3 MIMO System Identification Assume a linear and time invariant system, with the inputs r 1ru and the m outputs 1my, at the time the system can be expressed as k 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuup (C-1) where 1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (C-2) Define 10 **(1)()pp mmprpk (C-3) and **(1)(1) () () () ()mprpk kp k k kp1 y y u u (C-4) and these yield the filter outputs 1 (**(1)) ()() ()m mmprpkkk y( ( 1 ) ) 1m p r p (C-5) 169

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Therefore, the error between the two outputs is defined as 11 ()()() ()()()mmkkk kkk1m yy y (C-6) and the scalar error cost function is defined by 1 ()()() 2TJkkk (C-7) To identify the observer Markov parameters C-3, the following equations based on the gradient descend method is developed () (1)() () Jk kk k (C-8) where is the step size. Substituting Equation C-6 into Equation C-7 1 ()()() 2 1 ()()()()()() 2 1 2 1 2T T T TTJkkk kkkkkk T TTyy yy y-yyy (C-9) and the gradient of er ror cost function is 1 1**(1)()1 2 () 1 2 1 2 ()()m mprpJk k kk T-y-y -2y -2y y (C-10) Finally, substituting Equation C-10 into Equation C-8 and yielding 170

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(C-11) (1)()()(Tkkk) k In order to automatically upda te the step size, choose 21 (C-12) where is a small number to avoid the infinity number when 2 is zero. Here the main steps of the MIMO identification are given as follows Step 1: Initialize (**(1)) ()mmprpk 0. Step 2: Construct regression vector **(1)1()mprpk according to Equation C-4. Step 3: Calculate the output error 1()mk according to Equation C-6. Step 4: Calculate the step size according to Equation C-12 Step 5: Update the observe r Markov parameters matrix *(1)()mmprpk according to Equation C-11. And then go back to st ep 2 for next iteration. Generalized Predictive Control Model In this section, a MIMO model, which is the same as the model of the MIMO ID C-1, is considered. Assume a linear a nd time invariant system, with the inputs r 1 ru and outputs m 1 m y at the time index the system can be expressed as k 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuu p (C-13) where 171

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1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (C-14) Shifting Equation C-13 one time step ahead and can be expressed as 12 012(1)()(1)(1) (1)()(1)(1p pkkkkp kkkkp yyyy uuuu ) (C-15) Substituting Equation C-13 into Equation C-15 12 12 1 23 1011 1 012 122 123 11 1(1)(1)(2)() (1)(2)(1) ()(1)() (1)()(1)(1) (1)(2) (1)(p p p p pp pkkkkp kkkp kkkp kkkkp kk kp k yyyy yyy uuu uuuu +y+y +y y 0 101 112 11 1 (1) (2) 12 (1) (1) 1 (1) (1) 001)( () (1) (1)() (1)(2) (1)() (1)()(1)pp p mm mm pp mm mm mr mr mrpk kk kp kp kk kp kp kkk 1) u +u+u +u u yy yy uuu (1) (1) 1(1)()pp mr mrkp kp uu (C-16) where 172

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(1) (1) 1 1120101 (2) (1) 21 2 311 1 (1) (1) 11 111 1 (1) (1) 11,mm mr mm mr pp p pp mm mr ppp mm mr ++ ++ ++ 2 p p (C-17) The output vector is the linear combination of the past outputs, the past inputs and future inputs. By induction, the output vector (1) k y() kj y can be derived as (C-18) () () () 121 () (1) 00 () () () 012 ()()(1)(2) ( ()()(1) ()(1)(2) ()jjj p mm mm mm j p mr mm mr jjj mr mr mr j p mrkj k k kp kpkjkj kkk kp yyyy yuu uuu u 1) where () (1)(1) () (1)(1) 0101 11 1 2 () (1)(1) () (1)(1) 21 2 311 1 2 () (1) (1)() 11 11 () (1) 1,jjj jjj mr mm jjjjj mm mr jjjj pp p p mm jj pp mm + + ++ + (1) (1) 11 () (1) 1 jj pp mr jj pp mr + j 0 1 1 p p (C-19) with initial (0) (0) 110 (0) (0) 221 (0) (0) 11 1 (0) (0),mm mr mm mr pp p mm mr ppp mm mr (C-20) The quantities () 0 k mr ( ) are the impulse response sequences of the system. 0,1,k 173

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Define the following the vector form 11 11 1 1 1 1 1 1 1() () (1) (1) () ,() (1) () () (1) () (1)rr rr pj r rp rj m m p m mpukp uk ukp uk kp k uk ukj ykp ykp kp yk 1 ( 1 ) 1 r uu y (C-21) Substituting Equation C-21 into Equation C-18 and express it in matrix form as (C-22) () (1) 0001 1 (1)1 (1) () () () 21 1 () () () 21 1() () () ()j j mm r rj mr mr mrj jjj pp rp mr mr mr mrp jjj pp mp mm mm mm mmpkj k kp kp yu u y Now, let the predictive index 0,1,2,,1,,1 j qqs and define 1 1 1 1 1 1 1 1() (1) () (1) () (1) () (1)r r s r rs m m s m msuk uk k uks yk yk k yks u y (C-23) A predictive model can be expressed by 174

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1 1 1 1 1() (1) () (1)m m s ms m msyk yk k yks I IIIII y (C-24) 1 1 0 1 1 1 (1) 1 00 1 1 1 (1) (2) 1 000 1() 0 00 0 () (1) 0 0 () (1) (1)r r mrmr mr mrs r rs r r mr mr mr mrs r rs r ss r mr mr mr mrs rk k k I k k ks u u u u u u 1 1 0 1 (1) 00 1 (1) (2) 1 00000 () 0 (1) (1)rs ms mr mr mr r mr mr mr r ss r rs mr mr mr msrsk k ks u u u 1 (C-25) or for simplification 1()s msrs rsIk Tu (C-26) where 175

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0 (1) 00 (1) (2) 00000 0mr mr mr mr mr mr msrs ss mr mr mr msrs T (C-27) The matrix T is called Toeplitz matrix. A nd the second part of Equation C-24 is 1 1 11 1 1 1 (1) (1) (1) 1 11 1 1 (1) (1) 1() (1) (1) () (1) (1)r r pp mr mr mr mrp r rp r r pp mr mr mr mrp r rp ss pp mrkp kp k kp kp II k u u u u u u 1 (1) 1 1 1 1 1 11 (1) (1) (1) 11 (() (1) (1)r s r mr mr mrp r rp ms pp mr mr mr pp mr mr mr s pkp kp k u u u 1 1 1) (1) (1) 1 1 11() (1) (1)r r ss r rp p mr mr mr msrpkp kp k u u u(C-28) or for simplification 1()p msrp rpII kp Bu (C-29) where 176

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11 (1) (1) (1) 11 (1) (1) (1) 11 pp mr mr mr pp mr mr mr msrp sss pp mr mr mr msrp B (C-30) and the third part of Equation C-24Is 1 1 11 1 1 1 (1) (1) (1) 1 11 1 1 (1) (1 1() (1) (1) () (1) (1)r r pp mm mm mm mmp r mp r r pp mm mm mm mmp r mp ss pp mmkp kp k kp kp III k y y y y y y 1 )( 1 ) 1 1 1 1 1 11 (1) (1) (1) 11 (() (1) (1)r s r mm mm mmp r mp ms pp mm mm mm pp mm mm mm s pkp kp k y y y 1 1 1) (1) (1) 1 1 11() (1) (1)r r ss r mp p mm mm mm msmpkp kp k y y y(C-31) or for simplification 1()p msmp mpIII kp Ay (C-32) where 177

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178 11 (1) (1) (1) 11 (1) (1) (1) 11 pp mm mm mm pp mm mm mm msmp sss pp mm mm mm msmp A (C-33) Combine Equation C-26, Equation C-29 and Equation C-32 in to Equation C-24, 11 1 1() () () ()ssp ms msrs rs msrp rp p msmp mpkk kp yTuBu Ayk p (C-34)

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APPENDIX D A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES Equation Section 4 In this section, a potential theoretical mode l of cavity acoustic resonance is derived based on the model of Kerschen et al. (2003). The model combines scattering analyses for the two ends of the cavity and the propa gation analyses of the cavity shear layer, internal region of the cavity, and acoustic near-field. Kerschen et al. so lve a matrix eigenvalue problem to identify the frequencies of the cavity oscillat ion. A different approach for characterizing the same model is illustrated in this section. A signal flow graph is first constructed from a block diagram of the physical model, and then Masons rule (Nise 2004) is applied to obtain the transfer function from the disturbance input to the se lected system output. This me thod gives a prediction for the resonant frequencies of the flow-induced cavity os cillations. In addition, this method also provides a linear estimate for th e system transfer function. Masons Rule Mason's rule reduces a signal flow graph to a transfer f unction between of any two nodes in the network. A signal flow graph connects nodes, used to represent variables, by line segments, called branches. First, so me definitions are given as follows Input node: a node that has only outgoing branches. Output node: a node that has only incoming branches. Path: a string of connected branches and nodes. It contains the same branch and node only once. Forward path: a path that traverses from th e input node to the output node of the signal flow graph in the direction of signal flow. It touches the same node only once. Forward path gain: the product of gains found by traversing the path from the input node to the output node of the signal flow graph in the direction of signal flow. Loop: a path that starts at a node and ends at the same node without passing through any other node more than once and follow s the direction of the signal flow. 179

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Loop gain: the product of branch gains in a loop. Touching: two loops, a path and a loop, or tw o paths that have at least one common node. Nontouching loop: Loops that do not have any nodes in common. Nontouching loop gain: The product of loop gains from nontouching loops. The closed loop transfer function, of a linear dynamic system represented by a signal flow graph is (Nise 2004) () Ts 1()N kk kp Ts (D-1) where N: number of forward paths k p : the forward path gain thk :1(loop gains)+(nontouching loop ga ins taken two at a time) (nontouching loop gains taken three at a time)+(nontouching loop gains taken four at a time)... th:(loop gain terms in that touch the forward path). In other words, is found by eliminating from those loop gains that touch the forward path.th k kk k Global Model for a Cavity Oscillation in Supersonic Flow The global model for cavity oscillations in supersonic flow is shown in Figure D-1 To be consistent with the notation of Kerschen et al. (2003), the global model c onsists of a downstream traveling shear layer instability wave of amplitude upstream modes U and downstream modes propagating acoustic modes in the cavity, and fast modes SD f E and slow modes s E near-field acoustic waves in the supersonic stream. The local amplitudes of all quantities at the 180

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leading edge are denoted by the decoration f s while quantities at the tr ailing edge do not have the decoration. The scattering processi ng at the leading edge is modeled by f sSU DU EU EUC S C D U C E C E (D-2) SUC D UC and are the four scattering coefficients for the upstream end of the cavity. The first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at th e trailing edge is modeled by fEUCsEUC fsUS UEUE UD f s S D UCCC E E C (D-3) USC UDC f UEC, and s UEC are the four scattering coefficien ts for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by 1233 22 iMD Ee ,diL iL L iML ffssSSeDe ELELEe (D-4) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. 1 M and 2 M are the wavenumbers of the fast and slow downstr eam propagating near-field acoustic waves, respectively. Finally, the upstream propagating acoustic cavity mode is uiLUUe (D-5) 181

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where u is the complex wavenumber of this mode. The global model can be represented in a block diagram (Figure D-2 ) or a signal flow graph ( Figure D-3 ). From the signal flow graph ( Figure D-3 ), the transfer function between the disturbance input and the upstream propagating cavity acoustic mode can be found. First, identify the components of Equation N'UD-1 The results are listed in Table D-1 The characteristic function of the system can be identified from Equation D-1 124 1 1 3 21 1uu du ff ssk k iLiL iML iML SUUS DUUD EUUE EUUEl CCeCCeLCCeCCe u (D-6) The numerator of the transfer function can be derived from D-1 The terms are formed by eliminating from those loop gains that touch the forward path. kthk 13 4 2 1d ff ssiL iML iML iL kkSUUS DUUD EUUE EUUE kpCCeCCeLCCeCCe 2 (D-7) Finally, the transfer function between the disturbance input and the upstream traveling wave in the cavity is N U 12 124 1 3 2 3 2() 1d ff ss uu duu ff sskk k UN iL iML iML iL SUUS DUUD EUUE EUUE iLiL iML iML SUUS DUUD EUUE EUUEp Ts CCeCCeLCCeCCe CCeCCeLCCeCCe (D-8) The characteristic function in Equation D-6 is the same as the eigenvalue relation derived by Kerschen et al (2003). And the transfer function in Equation D-8 gives more 182

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information concerning the physical model. For example, this model can predict the resonant frequencies as well as the nodal re gions or zeros of the flow field. Global Model for a Cavity Os cillation in Subsonic Flow The global model for cavity oscillations in subsonic flow is shown in Figure D-4 To be consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude upstream modes U and downstream modes propagating acoustic modes in the cavity, and upstream modes and downstream modes propagating near-fields acoustic waves in the subsonic flow. The local amplitudes of all quantities at the leading e dge are denoted by the decoration SDdEuE while quantities at the trailing edge do not have the decoration. The scattering processing at th e leading edge is modeled by u u dd uSUSE DUDE u EUEE dS CC U DCC E CC E (D-9) SUC D UC ,dEUCuSECu D EC and are scattering coefficients for the upstream end of the cavity. Again, the first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering proce ss at the trailing edge is modeled by duEEC d uuudUSUDUE u ESEDEE dS CCC U D E CCC E (D-10) USC , and are scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by UDCdUECuESCuEDCudEEC 183

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3 2 ,,diL iML iL ddSSeDDeELEe d (D-11) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. d M is the complex wavenumber of the downstream propagating near-field acoustic wave. Finally, the upstream propagating acoustic cavity modes are 3 2 ,uiL iML uuUUeELEeu (D-12) where u and u M are the complex wavenumbers of the upstream traveling cavity mode and the upstream traveling acoustic nea r-field mode, respectively. The global model can be represented in a block diagram (Figure D-5 ) or a signal flow graph ( Figure D-6 ). From the signal flow graph ( Figure D-6 ), the transfer functions between the disturbance input and the upstream propagating cavity acoustic modes, and can be found. The transfer function be tween the disturbance input and the upstream propagating cavity acoustic mode is calculated by first identi fying the components of Equation N U'uEN'UD-1 The results are listed in Table D-2 The characteristic function of the sy stem can be identified from Equation D-1 184

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126 12 11 33 22 1 3 3 2 21 33 32 31 1ud u du dd uu udud uu udud dudu dudukk kk iLiL SUUS DUUD iML EUUE SUESDEUD SUESEEUE DUEDSEUS DUEDEEUE EUEESEUS EUEEDEll CCeCCe LCCeLCCCCI LCCCCILCCCCI LCCCCILCCCCI LCCCC 3 2 3 3 3 2 3 3 2 12 3 3 2 13 3u uu du du uu duud uu duud uu duud dduuiML UD SEES iML iMML DEED EEEE SUUSDEED SUUSEEEE DUUDSEES DUUDEEEE EUUESEEILCCe LCCeLCCe LCCCCILCCCCI LCCCCILCCCCI LCCCC 3 23 33 22 3 3 2 3 2 11dduu ud u du u dd uu du du uu duud uu uSE U U ED EE D iLiL SUUS DUUD iML iML EUUE SEES iML iMML DEED EEEE SUESDEUDDUEDILCCCCI CCeCCe LCCeLCCe LCCeLCCe CCCCCCC LI 3 2 3 3u uu uu ududdudu duuddduu ududdudu duuddduuSEUS SUUSDEEDDUUDSEES SUESEEUEEUEESEUS SUUSEEEEEUUESEES DUEDEEUEEUEEDEUD DUUDEEEEEUUEDEEDC CCCCCCCC CCCCCCCC LI CCCCCCCC CCCCCCCC LI CCCCCCCC (D-13) where 123,,duu duu dduuiMLiMMLiMMLIeIeIe (D-14) The numerator of the transfer function can be derived from D-1 185

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9 1 3 3 2 3 3 2 333 2221 1 1du du uu duud ud u d uu duud ud d dd uu uukk k iML iMML iL SUUS DEED EEEE iML iMML iL DUUD SEES EEEE iML iML iML EUUE SEES DEEDp CCeLCCeLCCe CCeLCCeLCCe LCCeLCCeLCCe u 3 3 2 3 3 2 33ud ud uu udud du dud uu udud du dud dudu duduiML iMML SUESDEUD SUESEEUE iML iMML DUEDSEUS DUEDEEUE iMML iMML EUEESEUS EUEEDEUD SULCCCCeLCCCCe LCCCCeLCCCCe LCCCCe LCCCCe CC 3 2 3 2 3dd dd uu uu du uu uu ududdudu du duuddduuiL iML iL US DUUD EUUE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEEEEEUUESEESeCCeLCCe CCCCCCCC Le CCCCCCCC CCCCCCCC Le CCCCCCCC 3ududdudu ddu duuddduuDUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEEDCCCCCCCC Le CCCCCCCC (D-15) Finally, the transfer function between the disturbance input and the upstream traveling wave in the cavity is N U 186

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' 9 1 3 2 3 2 3()dd dd uu uu du uu uu ududdudu du dUN kk k iL iML iL SUUS DUUD EUUE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEETs p CCeCCeLCCe CCCCCCCC Le CCCCCCCC CCCCCCCC Le CCC 3 3 2 33 221uuddduu ududdudu ddu duuddduu ud ud u dd u uuEEEUUESEES DUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEED iLiL iML SUUS DUUD EUUE iML SEES DECCCCC CCCCCCCC Le CCCCCCCC CCeCCeLCCe LCCeLC 3 3 2 3du du uu duud uu uu duu uu uu ududdudu duu duudiML iMML ED EEEE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEEEEECeLCCe CCCCCCCC Le CCCCCCCC CCCCCCCC Le CCCCC 3dduu ududdudu dduu duuddduuUUESEES DUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEEDCCC CCCCCCCC Le CCCCCCCC (D-16) The characteristic function in Equation D-13 is the same as the eigenvalue relation derived by Alvarez et al. (2003), but the transfer function in Equation D-16 gives more information concerning the physical model. For example, this model can predict the resonant frequencies as well as the nodal regions or zeros of the flow field. Then, the transfer function be tween the disturbance input and the upstream propagating acoustic mode is calculated. Similarly, first, identify the components of Equation N'uE D-1. The results are listed in Table D-3 Because the loop gains and the nontouching loop gains are the same as before ( Table D-2 ), only the forward path gains are listed. 187

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The characteristic function of the system is the same as Equation D-13. And the numerator of the transfer function can be derived from D-1 3 3 2 1d uud uiL iML iL kkSUESDUED EUEE kd d p CCeCCeLCCe (D-17) Finally, the transfer function between the disturbance input and the upstream traveling wave in the cavity is N'uE 3 1 3 2 3 2 33 3 22 3 2() 1u dd uud u d ud ud u dd ud u uu uu duud dEN kk k iL iML iL SUESDUED EUEE iLiL iML SUUS DUUD EUUE iML iML iMML SEES DEED EEEE iMTs p CCeCCeLCCe CCeCCeLCCe LCCeLCCeLCCe Le 3 3uu uu uu uu uu ududdudu duu duuddduu udud dduuSUESDEUDDUEDSEUS L SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEEEEEUUESEES DUEDEEUE iMMLCCCCCCCC CCCCCCCC CCCCCCCC Le CCCCCCCC CCCCC Le d u dudu duuddduuEUEEDEUD DUUDEEEEEUUEDEEDCCC CCCCCCCC (D-18) 188

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Table D-1. Components of the Maso ns formula for supersonic case. Index Forward Paths Forward Path Gains 1 01261011 1iL SUUS p CCe 2 01371011 2diL DUUD p CCe 3 01481011 13 2 3ffiML EUUEpLCCe 4 01591011 23 2 4ssiML EUUEpLCCe Index Loops Loop Gains 1 12610 1 1 1uiL SUUSlCCe 2 13710 1 1 2udiL DUUDlCCe 3 14810 1 13 1 2 3u ffiML EUUElLCCe 4 159101 23 1 2 4u ssiML EUUElLCCe 189

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Table D-2. Components of the Ma sons formula for subsonic case. Index Forward Paths Forward Path Gains 1 013691 1 1iL SUUS p CCe 2 014791 1 2diL DUUD p CCe 3 015891 1 3 2 3d ddiML EUUE p LCCe 4 01361024791 1 3 2 4ud uuiML SUESDEUDpLCCCCe 5 01361025891 1 3 5ud ududiMML SUESEEUEpLCCCCe 6 01471023691 1 3 2 6du uuiML DUEDSEUSpLCCCCe 7 014710258911 3 7dud ududiMM DUEDEEUEpLCCCCe L1 8 01581023691 3 8du duduiMML EUEESEUSpLCCCCe 9 015810247911 3 9dud duduiMML EUEEDEUDpLCCCCe Index Loops Loop Gains 1 1369 1 1 1uiL SUUSlCCe 2 1479 1 1 2duiL DUUDlCCe 3 1589 1 3 1 2 3du ddiML EUUElLCCe 4 136102479 1 3 1 2 4udu uuiML SUESDEUDlLCCCCe 5 1361025891 13 5udu ududiMML SUESEEUElLCCCCe 6 147102369 1 3 1 2 6duu uuiML DUEDSEUSlLCCCCe 7 147102589 1 13 7dudu ududiMML DUEDEEUElLCCCCe 8 1581023691 13 8duu duduiMML EUEESEUSlLCCCCe 9 158102479 1 13 9dudu duduiMML EUEEDEUDlLCCCCe 10 23610 2 3 1 2 10u uuiML SEESlLCCe 11 247102 3 1 2 11du uuiM DEEDlLCCe L 12 258102 13 12du duudiMML EEEElLCCe Index Nontouching Loops Nontouching Loop Gains 1 Loop 1 and Loop 11 3 2 2 1udu uuiM SUUSDEEDlLCCCCe L 2 Loop 1 and Loop 12 23 2udu duudiMM SUUSEEEElLCCCCe L 3 Loop 2 and Loop 10 3 2 2 3duu uuiM DUUDSEESlLCCCCe L 190

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4 Loop 2 and Loop 12 23 4dudu duudiMM DUUDEEEElLCCCCe L 5 Loop 3 and Loop 10 23 5duu dduuiMML EUUESEESlLCCCCe 6 Loop 3 and Loop 11 23 6dudu dduuiMML EUUEDEEDlLCCCCe 191

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Table D-3. Components of the Ma sons formula for subsonic case. Index Forward Paths Forward Path gains 1 01361012 1uiL SUES p CCe 2 01471012 2d uiL DUED p CCe 3 01581012 3 2 3d dudiML EUEE p LCCe 192

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L Turbulent Boundary Layer D 1 M s EfESD U x y Figure D-1. Global model for a cavity oscillation in supersonic flow. f SUSUDUEUECCCC f SSU DU E U E UC C C C 23 2 iMLLe 13 2 iMLLe iLe N f s S D E E f s S D E EU uiLe diLe U Figure D-2. Block diagram of the global model for a cavity oscillation in supersonic flow. 193

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23 2 iMLLe 13 2 iMLLe iLe SSuiLe diLe NU U 0 1 2 3 4 5 6 7 8 9 10 USCSUCf E UCDUCS E UCUDC f UECSUEC 1 D f E s E D f E s E 1 11' U Figure D-3. Signal flow graph of the global model for a cavity oscillation in supersonic flow. 1 L Turbulent Boundary LayerD M dEuESDU x y Figure D-4. Global model for a cavit y oscillation in subsonic flow. 194

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195 N U uiLe 3 2uiML L e u u dd uSUSE DUDE E UEECC CC CC d uuudUSUDUE E SEDEECCC CCC 3 2diML L e diLe iLe uU E uEdS D E dS D E Figure D-5. Block diagram of the global mode l for a cavity oscillation in subsonic flow. 3 2uiMLLeuiLe USCSUCd E UCDUCuSECUDCu E SC U 0 1 1 2uE 6 7 8 3 4 5 3 2diMLLe iLe SSdiLe D dE D dE 9U 10uE uDECdu E ECu E DCud E ECdUEC N 11 'uE U1 1 12 Figure D-6. Signal flow graph of the global model for a cavity oscillation in subsonic flow.

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APPENDIX E CENTER VELOCITY OF ACTUATOR ARRAY In this appendix, the center velocity of each slot (notation see Chapter 5) and corresponding current measurement of the actuator array are shown. 196

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0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz] Velocity [m/s]slot 1a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 1a 50 Vpp 100 Vpp 150 Vpp B Figure E-1. Hot-wire measurement for actuator a rray slot 1a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 197

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0 500 1000 1500 2000 0 20 40 60 80 Frequency [Hz]Velocity [m/s]slot 1b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 1b 50 Vpp 100 Vpp 150 Vpp B Figure E-2. Hot-wire measurement for actuator a rray slot 1b. A) Center RMS velocity. B) Current measurement of the actuator array. 198

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0 500 1000 1500 2000 0 10 20 30 40 50 60 Frequency [Hz]Velocity [m/s]slot 2a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 2a 50 Vpp 100 Vpp 150 Vpp B Figure E-3. Hot-wire measurement for actuator a rray slot 2a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 199

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0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 2b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 2b 50 Vpp 100 Vpp 150 Vpp B Figure E-4. Hot-wire measurement for actuator a rray slot 2b. A) Center RMS velocity. B) Current measurement of the actuator array. 200

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0 500 1000 1500 2000 0 10 20 30 40 50 Frequency [Hz]Velocity [m/s]slot 3a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 3a 50 Vpp 100 Vpp 150 Vpp B Figure E-5. Hot-wire measurement for actuator a rray slot 3a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 201

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0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 3b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 3b 50 Vpp 100 Vpp 150 Vpp B Figure E-6. Hot-wire measurement for actuator a rray slot 3b. A) Center RMS velocity. B) Current measurement of the actuator array. 202

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0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 4a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 4a 50 Vpp 100 Vpp 150 Vpp B Figure E-7. Hot-wire measurement for actuator a rray slot 4a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 203

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0 500 1000 1500 2000 0 20 40 60 80 Frequency [Hz]Velocity [m/s]slot 4b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 4b 50 Vpp 100 Vpp 150 Vpp B Figure E-8. Hot-wire measurement for actuator a rray slot 14b. A) Cent er RMS velocity. B) Current measurement of the actuator array. 204

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0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 5a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 5a 50 Vpp 100 Vpp 150 Vpp B Figure E-9. Hot-wire measurement for actuator a rray slot 5a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 205

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206 0 500 1000 1500 2000 0 20 40 60 80 Frequency [Hz]Velocity [m/s]slot 5b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 5b 50 Vpp 100 Vpp 150 Vpp B Figure E-10. Hot-wire measuremen t for actuator array slot 5b. A) Center RMS velocity. B) Current measurement of the actuator array.

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APPENDIX F PARAMETRIC STUDY FOR OPEN-LOOP CONTROL In this appendix, a parametric study results for open-loop control are shown. To illustrate the open-loop control, a fixed fl ow condition (M=0.31) is chosen fo r all experimental cases. The frequencies of the excitation i nput signals to the actuator array are varied from 500 Hz to 1500 Hz, and for each frequency, two excitation volta ge levels, 100 Vpp and 150 Vpp, are chosen. 207

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-1. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 500 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-2. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 500 Hz and 150 Vpp voltage. 208

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-3. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 600 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-4. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 600 Hz and 150 Vpp voltage. 209

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-5. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 700 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-6. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 700 Hz and 150 Vpp voltage. 210

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-7. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 800 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-8. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 800 Hz and 150 Vpp voltage. 211

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-9. Open-Loop control resu lt for M=0.31 and excitation si nusoidal input with frequency 900 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-10. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 900 Hz and 150 Vpp voltage. 212

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-11. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1000 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-12. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1000 Hz and 150 Vpp voltage. 213

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-13. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1100 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-14. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1100 Hz and 150 Vpp voltage. 214

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-15. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1200 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-16. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1200 Hz and 150 Vpp voltage. 215

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-17. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1300 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-18. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1300 Hz and 150 Vpp voltage. 216

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0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-19. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1400 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-20. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1400 Hz and 150 Vpp voltage. 217

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218 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-21. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1500 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F-22. Open-Loop control result for M= 0.31 and excitation sinusoidal input with frequency 1500 Hz and 150 Vpp voltage.

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LIST OF REFERENCES Akers, J. C., and Bernstein, D. S. 1997(a) ARMARKOV Least-Squares Identification. Proceedings of the American Control Conference, Albuquerque, New Mexico. Akers, J. C., and Bernstein, D. S. 1997(b) Time-Domain Identification Using ARMARKOV/Toeplitz Models. Proceedings of the American Control Conference, Albuquerque, New Mexicl, B. Allen, R. and Mendonca, F. 2004 DES Validations of Cavity Acoustics over the Subsonic to Supersonic Range. AIAA-2004-2862. Alvarez, J. O., and Kerschen, E. J. 2005 Infl uence of Wind Tunnel Walls on Cavity Acoustic Resonances. 11th AIAA/CEAS Aeroacoustics Conference, Monterey, California. Alvarez, J. O., Kerschen, E. J., and Tumin, A. 2004 A Theoretical Model for Cavity Acoustic Resonances in Subsonic Flow. AIAA-2004-2845. Arunajatesan, S. and Sinha, N. 2001 Unified Unst eady Rans-Les Simulations of Cavity Flow Fields. AIAA-2001-0516. Arunajatesan, S., and Sinha, N. 2003 Hybrid R ANS-LES Modeling for Cavity Aeroacoustics Predictions. Aeroacoustics, Vol. 2, No. 1, pp. 65-93. Arunajatesan, S., Shipman, J. D., and Sinha, N. 2003 Mechanisms in High Frequency Control of Cavity Flows. AIAA 2003-0005. Astrom, K. J., and Wittenmark, B. 1994 Adaptive Control. ISBN: 0201558661, 2nd Edition. Banaszuk, A., Jacobson, C. A., Khibnik, A. I., and Mehta, P. G. 1999 Linear and Nonlinear Analysis of Controlled Combustion Pr ocesses. Part I: Linear Analysis. Proceedings of the 1999 IEEE International conferen ce on Control Applications. Bilanin, A.J., and Covert, E.E. 1973 Estimation of Possible Excitation Frequencies for Shallow Rectangular Cavities. AIAA Journal, Vol. 11, No. 3, pp.347-351. Cabell, R. H., Kegerise, M. A., Cox, D. E., and Gibbs, G. P. 2002 Experimental Feedback Control of Flow Induced Cavity Tones. AIAA 2002-2497. Cain, A. B., Epstein, R. E., and Kerschen, E. J. 1999 Quick Turnaround Prediction of Weapons Bay Cavity Acoustics Resonance. 5th AIAA/CEAS Aeroacoustics Conference 10-12 May, AIAA 99-1899. Camacho, E.F., and Bordons, C. 1995 Model Pred ictive Control in the Process Industry. Springer-Verlag Berlin Heidelberg New York. Caraballo, E., Malone, J., Samimy, M., and DeB onis, J. 2004 A Study of Subsonic Cavity Flows Low Dimensional Modeling. AIAA 2004-2124. 219

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BIOGRAPHICAL SKETCH Qi Song was born in Beijing, P. R. China. He entered the Beijing University of Aeronautics and Astronautics, Beijing China, in 1992 and received his BE. degree in the Department of Mechanical Engineering in A ugust 1996. After several years working at Air China, the flag carried airline of China, he entere d the University of Florida, Gainesville, Florida in August 2001 as a research assistant. In May 2004, he earned a Master of Science degree in electrical engineering. He r eceived his Ph.D. degree with an aerospace engineering major in May 2008 at the University of Florida. His res earch focuses on the active closed-loop control of the cavity oscillations. 227