<%BANNER%>

Adaptive Control of Separated Flow

Permanent Link: http://ufdc.ufl.edu/UFE0021285/00001

Material Information

Title: Adaptive Control of Separated Flow
Physical Description: 1 online resource (162 p.)
Language: english
Creator: Tian, Ye
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: closed, disturbance, optimization, separation, synthetic, system, znmf
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 separation has severe adverse effects on performance of flow-related devices (e.g., lift loss of aircraft). Active control of separated flow has received extensive attention as it is able to mitigate or eliminate flow separation effectively. Most research has been open-loop in nature (i.e., manually adjusting control inputs to achieve best results). Closed-loop control of separated flow has many potential advantages over open-loop control, namely optimization in multi-dimensional domain with constraints, adaptability to changing flow conditions, etc. In this research, adaptive closed-loop control is used to reattach the separated flow over a NACA 0025 airfoil using multiple zero-net-mass-flux (ZNMF) actuators that cover the central 33% of the airfoil span. In particular, two distinct approaches are used. Adaptive disturbance rejection algorithms are used to apply dynamic feedback control of separated flow. The closed-loop control results show ~ 7 x improvements in the lift/drag ratio, with a corresponding increase in lift and reduced drag and concomitant reductions in the fluctuating surface pressure spectra. On the other hand, a simplex optimization approach uses the lift and drag measured by a strain-gauge balance for feedback and searches for the optimal actuation parameters in a closed-loop fashion. The constrained optimization results seeking to maximize lift-to-drag ratio are promising and reveal the importance of forcing nonlinear interactions between the shear layer and wake instabilities. To the author's knowledge, this is the first time that these types of closed-loop control schemes are implemented to control separated flow.
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.
Statement of Responsibility: by Ye Tian.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Cattafesta III, Louis N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021285:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021285/00001

Material Information

Title: Adaptive Control of Separated Flow
Physical Description: 1 online resource (162 p.)
Language: english
Creator: Tian, Ye
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: closed, disturbance, optimization, separation, synthetic, system, znmf
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 separation has severe adverse effects on performance of flow-related devices (e.g., lift loss of aircraft). Active control of separated flow has received extensive attention as it is able to mitigate or eliminate flow separation effectively. Most research has been open-loop in nature (i.e., manually adjusting control inputs to achieve best results). Closed-loop control of separated flow has many potential advantages over open-loop control, namely optimization in multi-dimensional domain with constraints, adaptability to changing flow conditions, etc. In this research, adaptive closed-loop control is used to reattach the separated flow over a NACA 0025 airfoil using multiple zero-net-mass-flux (ZNMF) actuators that cover the central 33% of the airfoil span. In particular, two distinct approaches are used. Adaptive disturbance rejection algorithms are used to apply dynamic feedback control of separated flow. The closed-loop control results show ~ 7 x improvements in the lift/drag ratio, with a corresponding increase in lift and reduced drag and concomitant reductions in the fluctuating surface pressure spectra. On the other hand, a simplex optimization approach uses the lift and drag measured by a strain-gauge balance for feedback and searches for the optimal actuation parameters in a closed-loop fashion. The constrained optimization results seeking to maximize lift-to-drag ratio are promising and reveal the importance of forcing nonlinear interactions between the shear layer and wake instabilities. To the author's knowledge, this is the first time that these types of closed-loop control schemes are implemented to control separated flow.
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.
Statement of Responsibility: by Ye Tian.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Cattafesta III, Louis N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021285:00001


This item has the following downloads:


Full Text

PAGE 1

ADAPTIVE CONTROL OF SEPARATED FLOW By YE TIAN 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 2007 1

PAGE 2

2007 Ye Tian 2

PAGE 3

ACKNOWLEDGMENTS First, I thank my advisor, Dr. Louis Catta festa, for his contin uous support throughout my research; his insightful advice a nd comments made this dissertati on possible. Second, I thank my Ph.D. committee members: Warren Dixon, Bruce Carroll, Toshikaza Nishida for their advice and recommendations. The funding for this rese arch is from the NASA Langley Research Center and the Air Force Office of Scientific Re search. Additionally, I thank the student body in the Interdisciplinary Microsystems Group in the University of Flor ida for their great help to my research. Finally and most importantly, I thank my wife, Bei Wang, for her unconditional support. 3

PAGE 4

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ............................................................................................................... 3LIST OF TABLES ...........................................................................................................................7LIST OF FIGURES .........................................................................................................................8ABSTRACT ...................................................................................................................... .............16 CHAPTER 1 INTRODUCTION ................................................................................................................ ..18Overview ...................................................................................................................... ...........18Motivation ...............................................................................................................................21Background .................................................................................................................... .........21Two-Dimensional Separation Flow Physics ...................................................................21Effects of Flow Separation ..............................................................................................22Control of Flow Separation .............................................................................................23Open-loop separation control ...................................................................................24Closed-loop separation control .................................................................................31Closed-Loop Control Algorithms ....................................................................................34Optimization algorithms ...........................................................................................34System identification and disturbance rejection algorithms ....................................36Objectives .................................................................................................................... ...........39Approach ...................................................................................................................... ...........39Outline of This Dissertation ....................................................................................................402 THEORETICAL BACKGROUND........................................................................................44Optimization Algorithms ....................................................................................................... .44Downhill Simplex Algorithm ..........................................................................................44Extremum Seeking Algorithm .........................................................................................45System Identification Algorithms ...........................................................................................48ARMARKOV/LS Algorithm ..........................................................................................49ARMARKOV/LS/ERA Algorithm .................................................................................52Recursive ARMARKOV/Toeplitz Algorithm .................................................................53Adaptive Disturbance Rejection Algorithms ..........................................................................55ARMARKOV Disturbance Rejection Algorithm ...........................................................563 SIMULATION AND VALI DATION EXPERIMENTS .......................................................63Optimization Simulations ...................................................................................................... .63Downhill Simplex Simulation Results ............................................................................63 4

PAGE 5

Extremum Seeking Si mulation Results ...........................................................................64Vibration Control Testbed Setup ............................................................................................65Results of the Vibration Control Tests ...................................................................................67Computational Tests ........................................................................................................67System Identification .......................................................................................................68Adaptive Disturbance Rejection ......................................................................................704 EXPERIMENTAL SETUP AND DATA ANALYSIS METHOD ........................................92NACA 0025 Airfoil Model .....................................................................................................92Synthetic Jet Actuators ...........................................................................................................93Experimental Methods .......................................................................................................... ..94Flow Visualization ...........................................................................................................9 4Lift/Drag Balance ............................................................................................................94Dynamic Pressure Transducers .......................................................................................96Hot Wire Anemometry ....................................................................................................97Control System Hardware and Software ................................................................................98Higher Order Statistical Analysis (HOSA) .............................................................................985 RESULTS AND DISCUSSION ...........................................................................................109Dynamic Feedback Control ..................................................................................................109Experimental Configuration ..........................................................................................109System Identification .....................................................................................................109Coherent flow structures ........................................................................................109Linear prediction ....................................................................................................110Frequency response and the performance of system ID.........................................111Acoustic contamination ..........................................................................................112Disturbance Rejection ...................................................................................................113Closed-loop control ................................................................................................113Effect of control on su rface pressure signals .........................................................114Quantitative flow visualization ..............................................................................115Control input ..........................................................................................................116Discussion ..............................................................................................................117Nonlinear Control .................................................................................................................118Experimental Configuration ..........................................................................................118Flow Instabilities ...........................................................................................................119Actuator Calibration ......................................................................................................120Frequency response ................................................................................................120Types of actuation waveforms ...............................................................................121C and electrical power calibration .......................................................................123Adaptive Control Results ..............................................................................................1256 SUMMARY AND FUTURE WORK ..................................................................................140APPENDIX 5

PAGE 6

HIGHER ORDER SPECTRUM EXAMPLES ............................................................................144Example 1: Quadratic System .............................................................................................144Case 1 ............................................................................................................................144Case 2 ............................................................................................................................145Example 2: Cubic System ....................................................................................................14 5Example 3: x x ..................................................................................................................146LIST OF REFERENCES .............................................................................................................155BIOGRAPHICAL SKETCH .......................................................................................................162 6

PAGE 7

LIST OF TABLES Table page 3-1 Parameters for the simulations. ..........................................................................................7 33-2 Suppression performance of the di sturbance rejecti on algorithm with 20c and varying .....................................................................................................730.1 cn 3-3 Suppression performance of the di sturbance rejecti on algorithm with and varying 2cn 0.1 c ...................................................................................................................733-4 Suppression performance of the di sturbance rejecti on algorithm with 2cn 20c and varying ....................................................................................................733-5 Suppression performance of the disturba nce rejection algorithm at different modes with 2cn 20c and ....................................................................................740.1 5-1 Case descriptions and performance for di sturbance rejection e xperiments. (AoA=12 and Re=120,000) ..............................................................................................................1275-2 Summary of parameters in di sturbance rejection algorithm. ...........................................1275-3 Constrained optimization results using the AM, BM and PM signals. (Baseline L/D=1.01 at AoA=12 and Re=120,000) .........................................................................1275-4 Constrained optimization results using the AM, BM and PM signals. (Baseline L/D=1.1 at AoA=20 and Re=120,000) ...........................................................................1285-5 Results using sinusoidal excitations. ................................................................................128 7

PAGE 8

LIST OF FIGURES Figure page 1-1 Separation of flow over an airfoil. .....................................................................................411-2 Types of velocity profiles as a func tion of pressure grad ient (White 1991). .....................411-3 Lift and drag coefficien ts of NACA 0025 airfoil at Re100,000 ..................................421-4 Classification of flow contro l. (Cattafesta et al. 2003) ......................................................421-5 Characterization of possible frequency scales in separated flow (Mittal et al. 2005). ......431-6 Comparison between the lumped elemen t model (-) and experimental frequency response () measured using phase-locked LDV for a prototypical synthetic jet (Gallas et al. 2003). ......................................................................................................... ...432-1 Flow chart of downhill simplex algorithm. ........................................................................612-2 Block diagram for the extremum seeking control..............................................................622-3 Block diagram of disturbance rejection control. ................................................................623-1 One-dimensional example of the downhill simplex algorithm. .........................................743-2 Two-dimensional example of the downhill simplex algorithm. ........................................753-3 Simulation block diagram for extremum seeking control. .................................................753-4 Single global maximum test model: 2 *ff* where *10f and .......76*5 3-5 Double hump model with one local ma ximum and one global maximum. The function is fitted by a polynomial: -128-107-76-55 -34-23-12-1.2e-1.6e+1.4e-2.3e +1.5e-4.4e+3.8e2.35.4f ..........763-6 converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while ..............77*f10 a50 w 3-7 converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while .........77*f10 w 0.001 a 3-8 converges to the local optim al input (see Figure 3-5) *14 ( ). ...780.001,50 aw 3-9 f converges to the local maximum (see Figure 3-5) 40 f (0.001,50 aw ). ........78 8

PAGE 9

3-10 Vibration Control Testbed. ............................................................................................... .793-11 Block diagram of vibration cont rol with ID, then control. .............................................793-12 Block diagram of vibration c ontrol with ID and control. ...............................................803-13 Effects of varying or n p on the computational intensity of the ARMARKOV/Toeplitz system ID. ...................................................................................803-14 Effects of varying p on the computational intensity of the ARMARKOV/Toeplitz system ID. .................................................................................................................... ......813-15 Effects of varying p on the growth rate of the computational intensity of the ARMARKOV/Toeplitz system ID with respect to n ................................................813-16 Effects of varying or cnc on the computational intensity of the ARMARKOV disturbance rejection. ........................................................................................................ .823-17 Zero-pole map of the non-parametric fit of the frequency response. ................................823-18 Measured output and fitted output by the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. ..................................................................................833-19 Weight tracks of the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. .......................................................................................................................833-20 Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 1 and SNR=20 dB. ...................843-21 Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. .................843-22 Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 20 and SNR=20 dB. .................853-23 Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 30 and SNR=20 dB. .................853-24 Comparison of MSE of the ARMARKOV/Toeplitz system ID with SNR=20 dB with different 2 pn ...........................................................................................863-25 Comparison of MSE of the ARMARKOV/Toeplitz system ID with 2 pn 20 with different SNR. ................................................................................................86 9

PAGE 10

3-26 Performance signal of the ARMARKOV disturbance rejection to band-limited white noise (0-150 Hz) with 2cn 20c and 1 ............................................................873-27 Control signal of the ARMA RKOV disturbance rejection with 2cn 20c and ...................................................................................................................................871 3-28 Power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 20c and ........................................................................................8813-29 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection with 0.1 20c and different .........................................88cn3-30 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection with 2cn 0.1 and different c ...........................................893-31 Comparison of convergence of the ARMARKOV disturba nce rejection with 2cn 20c and different ...................................................................................................893-32 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection with 2cn 20c and different ...........................................903-33 Control signal of the AR MARKOV disturbance rejection at ID and control mode with 2cn 20c and ....................................................................................900.1 3-34 Comparison of convergence of the ARMARKOV disturbance reje ction at different modes with 2cn 20c and 0.1 .........................................................................913-35 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection at different modes with 2cn 20c and ....................910.1 4-1 NACA 0025 airfoil model with actuators and pressure transd ucers installed. (Adapted from Holman et al. 2003) .................................................................................1004-2 Schematic of a synthetic jet actuator. ..............................................................................1004-3 Synthetic jet array. (Adapt ed from Holman et al. 2003).................................................1014-4 Forces on NACA0025 airfoil. ..........................................................................................1014-5 Closer view of th e strain gauges. .....................................................................................1024-6 Wheatstone bridge configur ation of the balance. ............................................................1024-7 Normal force vs. balance output. .....................................................................................103 10

PAGE 11

4-8 Axial force vs. balance output. ........................................................................................1034-9 Static pressure dist ributions on the airfoil su rface at different AOA at ..104Re150,000 4-10 Comparison of lift and drag coefficients measured by the static pressure and the balance at .................................................................................................105Re100,000 4-11 Comparison of lift and drag coefficients measured by the static pressure and the balance at .................................................................................................106Re150,000 4-12 Picture of a Kulite transducer. .........................................................................................1 074-13 Linear response (at 500Hz) of a typical Kulite transducer. .............................................1074-14 Frequency response of a typical Kulite transducer. .........................................................1084-15 Hot wire calibration curve. .............................................................................................. 1085-1 NACA 0025 airfoil model with actuators, sensors and closed-loop control system. ......1295-2 Phase averaged pulse response measur ed by six pressure sensors. The slow propagation velocity of the coherent flow structures is clearly visible. ..........................1295-3 Comparison between measured signal from the pressure sensor (#6 in Figure 5-1) and the fitted output by ARMARKOV system ID algorithm for long and short time intervals. Results show a reasonable match at low frequencies between measured and fitted outputs. For ARMARKOV ID: p=1, n=2, =10. .................................................1305-4 Mean Squared Error (Running MSE) between measured and fitted outputs. Results show that the ARMARKOV ID algorithm converges, i.e. error being minimized. .........1305-5 Comparison between frequency respons e (FR) and fitted response by ARMARKOV ID algorithm. Parameters for FR: s f =4096 Hz, NFFT=1024, 75% overlap and Hanning window. For ARMARKOV system ID: p=1, n=2, =10. ...............................1315-6 Dual signal paths from the actuator to the pressure sensor (acoustic and hydrodynamic). A digital filter is introduced to remove the acoustic component by turning off the flow to isolate the acoustic path. ..............................................................1315-7 Actual measured and predicted acoustic noise using a band-limited random signal to the actuator. ......................................................................................................................1315-8 Actual measured and predicted acoustic noise using the same filter as in Figure 5-7 but with one half of the input amplitude. .........................................................................1315-9 Power spectra of the sensor signals (w ith wind tunnel running) before and after applying acoustic filter. ....................................................................................................1 32 11

PAGE 12

5-10 Performance surface pressure (S1) and c ontrol input signals (in Volt) before and after the ID and control is initiated for case #2. Control is established within 1 second or <100 convective time scales. ...........................................................................1325-11 Power spectra of the pressure transdu cer output for the baseline and the closed-loop control cases measured by S6 (performance). .................................................................1325-12 Contours of streamwise velocity uU for (a) baseline and (b) closed-loop control case #2 at AoA = 12 and Rec=120,000. .........................................................................1335-13 Contours of vorticity for baseline and closed-loop control case #2 at AoA = 12 and Rec=120,000. ....................................................................................................................1335-14 Voltage and electrical po wer spectra of the actuator A1 input signal for the closedloop control case. ............................................................................................................ .1335-15 Performance comparison at different AoA. .....................................................................1345-16 Power spectra of the pressure transdu cer output for the baseline and the closed-loop control cases measured by S6 (performance) at AoA = 20 and Rec=120,000. ..............1345-17 NACA 0025 airfoil model with actuators, sensors and closed-loop control system. ......1345-18 Flow structures in separated flow. ...................................................................................1355-19 Wake (1 chord aft of TE) and shear la yer (near separation) power spectral density functions at peak rms location. ........................................................................................1355-20 Auto-bicoherence of the same velocity si gnal analyzed in Figure 5-19 using the same parameter settings. The auto-bicoheren ce is zero except where nonlinear phase quadratic phase coupling occurs due to in teractions between the shear layer and wake instabilities. ........................................................................................................... ..1365-21 Frequency response of ZNMF actuator A1. ....................................................................1365-22 Various waveforms of unit amplitude 1 A that can be used to excite multiple instabilities or modes in a separated flow. .......................................................................1375-23 Velocity response (a) and its power spectra l density (b) subject to an AM excitation for the ZNMF actuator A1. A 50 Vpp 50mf Hz and 1180cf Hz. Measurements were made outside the region of reverse flow. ........................................1375-24 C (a) and electrical power (b) profile subject to AM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. ...............................1375-25 C (a) and electrical power (b) profile subject to BM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. ...............................138 12

PAGE 13

5-26 C (a) and electrical power (b) profile subject to PM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. ...............................1385-27 Constrained search using AM: Cases (a ) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. ...................................................................................1395-28 Constrained search using BM: Cases (a ) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. ...................................................................................1395-29 Constrained search using PM: Cases (a) a nd (b). AOA = 20 deg. and Re = 120,000. ...139A-1 Test model #1 for higher order spectrum analysis. ..........................................................147A-2 Power spectrum of y for test model #1. ...........................................................................147A-3 Auto-bicoherence of y for test model #1. ........................................................................148A-4 Cross-bicoherence between x and y for test model #1. ...................................................148A-5 Test model #2 for higher order spectrum analysis. ..........................................................149A-6 Power spectrum of y for test model #2. ...........................................................................149A-7 Auto-bicoherence of y for test model #2. ........................................................................150A-8 Cross-bicoherence between x and y for test model #2. ...................................................150A-9 Test model #3 for higher order spectrum analysis. ..........................................................151A-10 Power spectrum of y for test model #3. ...........................................................................151A-11 Auto-bicoherence of y for test model #3. ........................................................................152A-12 Cross-bicoherence between x and y for test model #3. ...................................................152A-13 Test model #4 for higher order spectrum analysis. ..........................................................153A-14 Power spectrum of y for test model #4. ...........................................................................153A-15 Auto-bicoherence of y for test model #4. ........................................................................154A-16 Cross-bicoherence between x and y for test model #4. ...................................................154 13

PAGE 14

NOMENCLATURE c Airfoil chord length D C Drag coefficient ( Dqc) LC Lift coefficient ( Lqc) pC Static pressure coefficient ( p p q ) C Steady momentum coefficient ( Jqc) C Oscillatory momentum coefficient ( Jqc) D Drag (pdC dxc) Pressure recovery coefficient c f Filter cutoff frequency e f Excitation frequency m f Modulation frequency s ep f Shedding frequency of separated flow ( TEUX) wake f Wake shedding frequency ( wakeUW) F Reduced excitation frequency ( TE f XU ) h Slot width J Steady jet momentum ( ) 2 jUh J Oscillatory jet momentum ( ) 2 juhL Lift p Static local pressure p Free stream pressure q Free stream dynamic pressure ( 22U) ju Oscillatory jet velocity jU Mean jet velocity U Free stream velocity SEP X Distance from separation points to trailing edge TE X Distance from excitation slot to trailing edge Air density Boundary layer momentum thickness 2 s Variance of signal 2n Variance of noise 14

PAGE 15

ABBREVIATIONS AM Amplitude Modulation AOA Angle Of Attack BM Burst Modulation DAQ Data AcQuisition DSP Digital Signal Processing ID IDentification LDV Laser Doppler Velocimetry MSE Mean Square Error PIV Particle Image Velocimetry PM Pulse Modulation PSD Power Spectral Density SJA Synthetic Jet Actuator SNR (dB) Signal to Noise Ratio ( 22 1010log() s n ) 15

PAGE 16

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 B ADAPTIVE CONTROL OF SEPARATED FLOW By Ye Tian August 2007 Chair: Louis Cattafesta Major: Aerospace Engineering Flow separation has severe adverse effects on performance of flow-related devices (e.g., lift loss of aircrafts). Active control of separa ted flow has received extensive attention as it is able to mitigate or eliminate flow separation eff ectively. Most research has been open-loop in nature (i.e., manually adjusting control inputs to achieve best results). Closed-loop control of separated flow has many potential advantages over open-loop c ontrol, namely optimization in multi-dimensional domain with constraints, adaptabi lity to changing flow conditions, etc. In this research, adaptive closed-loop control is used to reattach the separa ted flow over a NACA 0025 airfoil using multiple zero-net-mass-flux (ZNMF) actuators that cover the central 33% of the airfoil span. In particular, tw o distinct approaches are used. Adaptive disturbance rejection algorithms are used to apply dynamic feedback control of separated flow. The closed-loop control results show ~ 7 x improvements in th e lift/drag ratio, with a corresponding increase in lift and reduced drag and concomitant reductions in the fluctuating surface pressure spectra. On the other hand, a simplex optimization approach uses the lift and drag measured by a straingauge balance for feedback and searches for th e optimal actuation parameters in a closed-loop fashion. The constrained optimization result s seeking to maximize lift-to-drag ratio are promising and reveal the importance of forcing nonlinear interactions between the shear layer 16

PAGE 17

and wake instabilities. To the authors knowledg e, this is the first time that these types of closed-loop control schemes are implemen ted to control separated flow. 17

PAGE 18

CHAPTER 1 INTRODUCTION The primary goal of this research is to implement a closed-loop control system to control separated flow and to evaluate the performance of the controller. A control system that includes an array of actuators, sensors (pressure sensors or lift/drag balance) and a digital controller is proposed to control flow separati on in a closed-loop fashion. This first chapter introduces the flow physics and active co ntrol approaches of flow separation. It is organi zed as follows. First, a brief overview of separation control is provided to orient the reader, followed by the motivation. Th en a technical background section is presented to review previous work reported in the lite rature. Finally, the objectives and technical approaches of this research are presented. Overview Flow separation is identified as one of the most important flow phenomena due to its severe adverse effects on flow-related devices. Following the introduction of the concept of the boundary layer by Prandtl (1904), flow separation has received considerable attention in the fluid dynamics community. Flow separation is the breakaway or detachment of fluid from a solid surface (Greenblatt and Wygnanski 2000). Flow separation incurs a large amount of energy/lift loss and limits the performance of many flow-related devices (e.g., airplanes, diffuse rs, etc.). Researchers have been trying to eliminate or at least mitigate flow separation for over a century because of its large potential payoff in many applications. As shown in Figure 1-4 control of separated flow is di vided into two main categories: active control and passive control. Active contro l provides external energy into the flow while passive control does not. Some passive separatio n control methods, such as geometrical shaping 18

PAGE 19

and turbulators (i.e., turbulence generators), ar e commonly used because of their simplicity and feasibility. On the other hand, tremendous progre ss has been made in act ive separation control over the past twenty years. Traditional ac tive separation control methods, such as steady blowing and suction, were initially used to control flow separation (Gad-el-Hak 2000). These methods were able to control of separation to some extent. However, they were far from optimal because the overall energy required input re quired to gain a meaningful lift increase or drag reduction was comparable to the energy saved via control of separation (Greenblatt and Wygnanski 2000). Schubauer and Skramstad (1948) first introduced a breakthrough in activ e flow control: periodic excitation. This technique requires much less energy than traditional steady active methods and accelerates and regulates the genera tion of large coherent structures that are primarily responsible for the transport of momentum across the flow (Greenblatt and Wygnanski 2000). The increased large coherent structures make the flow mo re resistant to separation. Periodic excitation has subsequent ly been shown to be superior to steady boundary layer control methods by many researchers (Seifert 1996; Greenblatt and Wygnanski 2000; Nishri and Wygnanski 1998). Because of thes e reasons, periodic exc itation is now widely used to control flow separation. Optimal excitation locations, waveforms shapes, and frequencies of periodic perturbations have been systematically studied by numerous research ers (Seifert and Pack 2003A, Amitay et al. 2001). Yet none of these st udies has used feedback control to optimize the excitation waveform. One of the most important aspects of separa tion control is the actuation mechanism that introduces periodic perturbations in to the flow structure. Intern al acoustic excitation (Hsiao et al. 1990; Huang et al. 1987), speakers (Naray anan and Banaszuk 2003), oscillatory blowing 19

PAGE 20

valves (Allen et al. 2000), and MEMS-based ac tuators (Rathnasingham a nd Breuer 2003), etc. have been investigated. Among th ese, synthetic jet or zero-net mass flux (ZNMF) actuators have been the focus of significant research for the past decade due to their utility in flow control applications (Glezer and Amita y 2002). ZNMF actuators utili ze the working fluid and do not require an external fluid source, which ma kes them very attractive from a systems implementation perspective. Significant progress has been made in the modeling and design of such devices (Gallas et al. 2003, 2005). More details of the synthetic jet ac tuators used in this research are described in Chapter 4. The drivi ng frequency, location, and momentum coefficient of the actuation are the primary parameters that characterize their perf ormance (Amitay et al. 2001). Although separation control has received extensiv e attention, to date most studies have focused on open-loop separation control. In the au thors opinion, this open-loop approach is due to a fluid mechanics bias to avoid using a more complex closed-loop control approach. Closedloop separation control has the potential to save more energy than open-loop methods (Cattafesta et al. 1997) and make separation control systems adaptable to different flow conditions. Few experimental studies have focused on closed-loop separation control. For example, Allan et al. (2000) attempted to tune a PID controller for cl osed-loop separation control and showed that the integral gain was the most effective as a result of the large time constant of their low bandwidth actuator system. However, the realized model and controller were simple. Their results merely scratched the surface of what can possibly be accomp lished. Therefore, it is believed that control of flow separation using an ar ray of high bandwidth actuators and surface sensors (pressure or shear stress) is an excellent candidate for clos ed-loop separation control. Hence, implementation 20

PAGE 21

of feedback controllers including more advanc ed modeling and control algorithms to flow separation control is proposed and is the focus of this research. Motivation Numerous applications of separa tion control, each with significant potential payoffs, have been identified (Greenblatt and Wygnanski 2000). Many separation control strategies have been applied on civil and military aircrafts and unde rwater vehicles. However, most of the applications are open-loop in nature because of their simplicity. Although some closed-loop separation control resear ch has been done (Allan et al. 2000; Banaszuk et al. 2003, etc), they are not sufficiently developed to be implemented on real vehicles. The goal of this research is to design and implement various closed-loop control sy stems for control of se parated flows and to seek physical insights behind the control sche mes. The main advantages of closed-loop separation control potentially include better perf ormance, energy efficiency and adaptability to changing of flow conditions. Background Two-Dimensional Separation Flow Physics Under the circumstances of an adverse pressure gradient ( 0 dpdx ), fluid particles are retarded by both the increasing pressure as well as wall skin friction. If the adverse pressure gradient is of sufficient strength, fluid particles near the wall are likely to separate from the wall and move upstream. This is due to the fact th at these particles have finite kinetic energy and cannot penetrate far into the adve rse pressure gradient region. The flow separates from the boundary layer and forms large scale vortical structures in the separated region ( Figure 1-1 ). Assuming two-dimensional, incompressible, steady flow with negligible gravity, the streamwise ( x ) component of the momentum equation at the wall reduces to 21

PAGE 22

22 221 udpu u u x dxxy (1.1) or 22 21 udpu u ydxx2u x (1.2) where is the kinematic viscosity, y is the wall normal coordinate, x is the streamwise coordinate with a corresponding velocity and streamwise pressure gradient u dpdx. From eqn. (1.2) we can see that only an a dverse pressure gradient ( 0 dpdx ) can cause a point of inflection in the velocity profile and the curvature changing sign to make the profile Sshape. In this case, separation will occur when the adverse pressure gradient is strong enough to make the right hand side of eqn. (1.2) positive (shown in Figure 1-1 ). Effects of Flow Separation In the separation region, the normal velocity co mponent significantly in creases as well as the thickness of boundary layer. Therefore, th e boundary layer approxim ations are no longer valid and the problem can no longer be solved using boundary layer theory. Flow separation significantly changes the pre ssure distribution around the surface. Such deviations are usually detrimental. As an example, Figure 1-3 shows the and LC D C100 of a NACA0025 airfoil versus angle of attack m easured by a lift/drag balance at When the angle of attack increases from zero degree, both and Re,000LC D C increase as expected. However, drops dramatically due to flow separation at about 13 degrees of angle of attack. At the same time, LC D C continues to increase be yond the inception of stall. Both of these effects generally have a negative impact on the airplane performance. However, some applications utilize flow separation. For ex ample, the use of spoilers on ai rplanes during landing reduces the lift and increases drag to allow the brakes to work more efficiently. 22

PAGE 23

More commonly, we want to mitigate or eliminat e flow separation. Typical applications of flow separation control include: separation control of various airfoils to increase for larger payload (Greenblatt and Wygnanski 2000; Seifert and Pack 2002; etc); to reduce engine power and noise at takeoff (Gad-el-Ha k 2000); to increase efficiency of diffusers (i.e. pressure recovery) (Banaszuk et al. 2003); etc. maxLCControl of Flow Separation Because of the effects mentioned above and th e large potential payoff, researchers have been preoccupied with delaying flow separation or eliminating it entirely. As suggested by Cattafesta et al. (2003), the cl assification of flow control is chosen as shown in Figure 1-4 to be consistent with terminology used in active noise and vibration control. Active control is subdivided into open-loop versus closed-loop co ntrol. Closed-loop control can be further classified into quasi-static versus dynamic, the distinction between the tw o being whether or not the feedback control is performed on a time scale with the dynamical scales of the flow. Since fluid flows are inherently nonlin ear (Wu et al. 1998), the standa rd frequency preservation of a linear system does not hold. Consequently, non linear feedback contro l on a very slow time compared to the characteristic times scales of th e flow is, in fact, possi ble and attractive. In essence, this so-called quasistatic control becomes a nonlinea r optimization problem. This research will investigate both classe s of closed-loop control shown in Figure 1-4 Other fluid dynamic issues have been studied extensively, such as the effects of Reynolds number, frequency, actuator and sensor locations momentum coefficient, surface curvature, and compressibility, etc.. Although the topic of this research is closed-loop separation control, the results and conclusions from the open-loop contro l studies should serve as a sound physical basis for effective control and are reviewed below. 23

PAGE 24

Open-loop separation control Periodic excitation has been shown to be much more effective than steady forcing because it enhances the momentum transport across the flow domain at a substantial reduction in energy expenditure. It accelerates and regulates the generation of large coherent structures that are primarily responsible for the momentum trans port across the flow (Greenblatt and Wygnanski 2000). The enhanced momentum transport forces the separated flow to reattach to the surface and form a thick turbulent boundary layer in a tim e-averaged sense. The reattachment of the boundary layer regains the pressure suction zone on the upper surface of the airfoil and thus enhances the lift performa nce. Furthermore, the superpositi on of weak suction on the periodic excitation enhances the receptivity of the separa ted shear layer to the fundamental excitation frequency and thus the effec tiveness of periodic excitation (Seifert and Pack 2002). Given the improved performance of periodic excitation to contro l flow separation, researchers have sought to optimi ze separation control via time-cons uming parametric variations. Significant parameters or conditions that affect the performance of separation control have been identified. Although they are di scussed separately below, one should keep in mind that these factors are all coupled with each other. Actuation frequency First, consider the characteristic flow structur es associated with se parated flow. Based on previous studies, Mittal et al (2005) summarize the three situati ons with regards to separated flow, as shown in Figure 1-5 In post-stall flow (case C in Figure 1-5 ), leading-edge shear layer rollup and vortex shedding in the wake are two char acteristic features (Wu et al. 1998). Huerre and Monkewitz (1990) suggest that this type of shear flow (with a pocket of absolu te instability of sufficient size) may display in trinsic dynamics of the same natu re as in a closed-flow system, in which disturbances can grow upstream (i.e. globa l instability). Therefore, it is reasonable to 24

PAGE 25

postulate that separated flow over an airfo il acts as a nonlinear multi-frequency closed-flow system. In such a system, the shear layer instability (with characteristic frequency SL f ) and the global wake instability (with vortex shedding frequency wake f ) may interact with each other in a nonlinear fashion. In case B, a closed separa tion bubble is present at some distance downstream of the leading edge. In this case there are poten tially three characteristic flow frequencies in the separated flow: SL f wake f and s ep f where the new scale, s ep f corresponds to the characteristic frequency of the separation bubble. The scales of the three frequencies are ~SLSLfU ~SLsep f UL and ~wake wakefUW, where SL is the shear layer thickness, s epL is the length of the separation bubble and is the width of wake. Prasad and W illiamson (1996) also show that wakeWReB SL wake f Af where and Since there are different relevant leng th scales that are included in the three characteristic frequencies, one should expect a significant variation in the observed frequency scales and the correspon ding optimal frequency. 0.0235 A 0.67 B The present study is focused on how flow sy stems respond to modulated (e.g. AM, BM, PM) unsteady excitations by ZNMF de vices targeting the inherent flow instabilities that lead to the presence of these characteristic flow frequenc ies. The goal is to search for optimal forcing schemes that most effectively mitigate flow separation via nonlinear interaction of the instabilities. Much research has been conducted to determ ine what excitation frequencies are most effective for separation control. However, ex cept for the general agreement that periodic excitation is far more effective than steady blowi ng, the range of optimal actuation frequencies is a current subject of intense deba te. A dimensionless actuation fr equency is typically defined for this purpose. However, three slightly differen t definitions have been given for a so-called 25

PAGE 26

dimensionless frequency : 1) F eTEFfXU where e f is the excitation frequency, TE X is the distance from the excitation slot to the trailing edge and U is the free stream velocity; 2) esepFfLU where s epL is the distance from separation to reattachment; and 3) eFfcU where c is the chord length. These three ar e nearly identical for post-stall flow (where the separation bubble leng th is approximately the airfoil chord), but they scale very differently if a closed separation bubble of finite extent is present. One should notice that none of these definitions is related to the shear layer frequency (SL f ). Most researchers implicitly ignore this important frequency wh en studying separation control. Herein, some results regarding actuation fre quency in previous studies are summarized. Among studies that define eTEFfXU Wygnanski and his colleag ues conclude that the optimal excitation frequency is of order unity (Seifert et al. 1996, Nishri and Wygnanski 1998, Greenblatt and Wygnanski 2000) and have found that so-called high frequency fording is ineffective for their airfoil (NACA 0015) and fl ow conditions. Conversely, using the same definition of (1)FO (10)FOF Amitay et al. (2001) found that when the excitation frequency the lift-to-pressure drag ratio was larger than that when the excitation frequency Honohan et al (2000) also suggest ed that higher reduced frequencies ( ) can be effective. They argued that it is because the high frequency excitation produces a virtual aerodynamic surface modificati on that thins the turbulent boundary layer and results in a local favorable pressure gradient. (10)FO 4F10 FBesides this argument, there may be two other possible reasons accounting for this interesting discrepancy. First, the length-scale TE X may not be appropriate for their airfoil 26

PAGE 27

because of the formation of a closed separation bubble. Instead, if s epL were used, this discrepancy might not exist. Second, as mentioned earlier, the shear layer frequency SL f may also be important (Mittal et al 2005). Here, SL f U where is the boundary layer momentum thickness and not TE X or s epL. The different frequency scales are indicative of different flow instabilities that may exist in the flow and, if present, may compete with each other (Wu et al 1998). When periodic excitation is introduced, one or more of these instabilities may be energized. The controlled flow may then be regulated, and thus lift performance may be enhanced. This may explain the observed variations of the optimal excitation frequency. Along these lines, an innovative forcing appro ach that uses multiple harmonically related frequencies is presented by Narayanan and Bana szuk (2003). They demonstrated improvements of this new approach versus single frequency si nusoidal forcing in control of separation in a diffuser, although its effectiveness requires further investigation. To extend this idea further, one can use excitations with multiple frequency components corresponding to the characteristic frequencies mentioned above. This idea will be investigated in this research. Excitation amplitude Another key control parameter in a ZMNF device is jet velocity J V (some characteristic velocity measure, e.g. the peak or an average ve locity). In the literature, the jet frequency is usually non-dimensionalized as /UsepFfL where s epL is, for example, the length of separation region and U is the free stream velocity. The jet velocity is usually nondimensionalized by U. Various researchers have show n that control authority varies monotonically with for a sinusoidal excitation up to so me maximum value (Seifert et al. 1993, 1996, 1999; Glezer and Amitay 2002; Mittal and Rampunggoon 2002). In practice, /JVU 27

PAGE 28

especially in high speed flows, c ontrol authority is often lacking. From an efficiency standpoint, it is desirable to control a flow with minimal actuator input. Modulation signals Piezoelectric actuators have fast dynamic res ponse and low power consumption. However, the use of piezoelectric actuators has been lim ited because of the diminution in their response outside a narrow frequency band around their resonance frequency and the need for testing over a wide frequency range due to the issues discussed in the last section. Wiltse and Glezer (1993) introduced a clever amplitude modulation method to flow control problems to overcome this problem. The piezoelect ric actuator is resonantly driven with a carrier waveform, which is amplitude modulated with a time-harmonic wave train: () et ()[1sin()]sin()mmrcet tAt (1.3) where is the amplitude of the carrier signal, rA is the degree of modulation (01 ), c is the carrier frequency (or the resona nt frequency of the actuator) in /radsm is the modulation frequency (which is also the desired excitation fr equency or receptive frequency of the flow) in and /rad sm is the phase of the modula ting signal. By using trigono metric identities, one can show that contains frequency components at etc and cm However, when the excitation amplitude is high enough, et is demodulated by the nonlinear fluid dynamical system that is associated with the formation and coalescence of nominally spanwise vortices. This nonlinearity results in the presence of c and cm and also m in the flow. In practice, c is set at the resonance frequency of th e piezoelectric actuator (which is usually m ) and m is set at the desired low frequency corre sponding to the desired excitation frequency e f 28

PAGE 29

Along these lines, other modulation signals such as burst modulation and pulse modulation can also be used. This modula tion technique allows the actuato r operating at its resonant frequency to generate a significant flow distur bance while effectively manipulating flows at characteristic frequencies of the flow. It provid es a much more flexible approach than matching the resonant frequency of the actuator with the receptive frequencies of the flow. However, some features of the technique should al so be kept in mind. First, the actuator is driven continuously near its resonant frequency, so the probabili ty of mechanical failure is greater than when it is driven off resonance. Second, as mentioned above, demodulation of the waveform is due to nonlinearities of the flow and actuator. As a result, feedback controllers designed based on a linear assumption may not work as desired. This aspect will be studies in this research. Actuation location It is argued by many researchers that the optimal actuation location is at the vicinity of the point of separation (Amitay et al. 2001, Seifert et al. 1996, Seifert and Pack 2003). This is physically plausible since the disturbances intr oduced at this location can most effectively transport momentum between the free shear layer and the separated region. However, this has not been systematically studied because of some practical limitations, namely the difficulty of installing multiple actuators inside an airfoil. Amitay et al. (2001) used an unconventional airfoil that had an aft portion of a symmetric airfoil attached to a circular cylinder forebody with a synthetic jet slot that could be adjusted by rotating the cylinder. They state that the closer the control is located to the observed separation point, the less power is required to reattach the flow. They also made an interesting point that if ei ther the separation locat ion is unknown or practical 29

PAGE 30

limitations preclude control near the separa tion location, the momentum coefficient C may be manipulated to achieve optimal performance. Besides the effects of actuation location di scussed above, the inte raction of adjacent synthetic jet actuators has been investigated by Holman et al. (2003). They found that relative phasing between adjacent actuators does not appear to affect the effec tiveness of separation control significantly for their airf oil (NACA 0025) and flow conditions ( and ). 510eR12 AOAIn summary, based on the previous studies it is suggested that sli ghtly upstream of the separation location is the best place to introduce actuation. Furthermore, a combination of upstream leading edge and downstream trailing edge actuations may also be a good candidate and remains to be investigated (Mittal et al 2005). Wu et al (1998) discuss this idea in the context of the Kutta-Joukowski lift formula (LU L ), which assumes the flow is incompressible and steady. In the formula, is the lift, U is free stream velocity and is the circulation (a counterclockwise circulation is assumed positive). Although the separation is an unsteady process, this formula still holds in a time-averaged sense for the entire flow. Based on these arguments, if the combination of leading e dge and trailing edge actuation can be designed to alter the circulation of the airf oil, it should be able to control flow separation in some manner. Effects of Reynolds number and compressibility It is shown that control of flow separation is insensitive to the Reynolds number at high chord Reynolds numbers of 11~30 million (Sei fert and Pack 2003 A, B, Greenblatt and Wygnanski 2000). The Reynolds number has a very weak effect on pressure distributions around the surface, regardless of the Mach number. 30

PAGE 31

On the other hand, strong Reynolds number e ffects are identified in the airfoil baseline performance at moderately compressible flow conditions (Seifert and Pack 2001). Reynolds number effects weaken as the Mach number in creases and a stronger sh ock wave develops. Compressibility tends to elongate the separation bubble and reduce the capability of periodic excitation to shorten the separa tion bubble with similar excitation frequencies and momentum (Seifert and Pack 2001). It is also suggested by Seifert and Pack ( 2001) that in the presence of shock waves the excitation location should be sligh tly upstream of the shock wave. If the excitation is introduced well upstream of the shock wave, it has a detrimental effect on lift, drag a nd wake steadiness. Closed-loop separation control Closed-loop experimental separa tion control has not yet received significant attention. This section first reviews some development of the micro-electro-mechanical systems (MEMS) based actuators because of th eir potential importance to high bandwidth closed-loop control systems. Then the limited previous work on closed-loop separation control is presented. For closed-loop flow control systems, the desire d actuators should be fast, power efficient, and reliable. In previous separation control st udies, acoustic excitation (Hsiao et al. 1990 and Huang et al. 1987) seems facility dependent be cause the acoustic drivers stimulate the wind tunnel resonant modes to excite the separated flow ; oscillatory blowing valv es (Allen et al. 2000) appear to have slow dynamic response; active flexible wall transducers (Sinha 2001) have complicated structures despite its high actuation ef ficiency and ability to actuate and sense with the same hardware. These drawbacks have limited the use of these actuators. On the other hand, synthetic jet (ZNMF) act uators have been th e focus of significant research activities for the past decade due to thei r utility in flow control applications (Glezer and Amitay 2002). They utilize the working fluid and do not need external fluid injection. They can 31

PAGE 32

force the momentum transfer across the flow w ithout net mass flux (thus the name synthetic). The design of synthetic jets is also flexible and the working frequency range can be tuned according to different flow control applications. In addition, the recent paper by Gallas et al. (2003) presents a lumped element model of a piez oelectric-driven syntheti c jet actuator. They provide a novel method to design and model synthetic jets, which makes them very suitable for closed-loop separation control. In lumped element modeling ( LEM), the individual components of a synthetic jet are modeled as elements of an equivalent electrical circuit using conjugate power variables (i.e., power = generalized flow x generalized effort). The frequency response function of the circuit is derive d to obtain an expression for outACQV, the volume flow rate per applied voltage. The compar ison between the LEM and experimental frequency response is shown in Figure 1-6 For a variety of reasons, closed-loop contro l in a real-time experiment has been traditionally difficult to achieve. In reduced-s cale laboratory experiments, the characteristic frequencies of separated turbulent flows are proportionally higher th an those on full-scale models, which requires high frequency sensing and actuating capabilities. Furthermore, realtime experiments require the digital control system to sample at a minimum of twice of the highest frequency of interest. Th e availability of hardware (inclu ding actuators, sensors and realtime control systems) therefore imposes significan t limitations on the complexity of the closedloop control system. Lower order system models are typically required to reduce the complexity of the system. Many model-based approaches are being develo ped and have shown promising results. Proper Orthogonal Decomposition (POD) based low order models have been studied extensively (Holmes et al. 1998; Tadmor et al. 2007) owi ng to their relatively high resolution and low 32

PAGE 33

computational intensity. Other reduced-basis mode ls have also been studied (Coller et al. 2000; Wang et al. 2003). These models require that multiple measurements are simultaneously available in the flow field. However, this is impractical in feedback separation control and surface measurements are required in most applic ations. Ausseur et al. (2007) implemented a POD/mLSM proportional feedback c ontrol using the velocity field and surface pressure data to delay flow separation. Some non-model based control approaches ha ve gained favor because they bypass the complication of modeling separated flow while focusing on the primary control objectives. For example, Banaszuk et al. (2003) and Becker et al. (2006) used an extremum-seeking closed-loop control algorithm to optimize the pressure recovery and lift, resp ectively. The present author in Tian et al. (2006) used a multi-dimensional optim ization algorithm to optimize lift-to-drag ratio over an airfoil. These approaches are capable of training the excitation signals to be most effective in terms of the objective functions (i.e., pressure recovery lift-to-drag ratio, etc.). The main drawback of the above a pproaches is that they operate on a time scale that is much larger than that of the flow dynamics. In other words, they work on time-averaged objective functions by explicitly taking advantage of the nonlinear nature of the fl uid dynamics. This approach has the drawback of having to deal or cope with the nonlinear dynamics with no guarantee of success. This kind of approach is an exampl e of the quasi-static control scheme shown in Figure 1-4 On the other hand, the dynamic feedback control is used to model and control separated flow structures based on surface pressure data alone. The well-developed adaptive system identification (ID) algorithms in the controls co mmunity are utilized to model the flow system dynamics between the actuators and unsteady surface pressure sensors. The system ID 33

PAGE 34

algorithms generate known actuation signals and relate these signals with the surface pressure response measured by sensors. Linear dynamical equations are then used to model the relationship in a gradient descent sense (Hayki n 2002). The system therein includes the dynamics of the actuators, the flow structures excited by the actuation, and the dyna mics of the sensors. The system information is then used to pred ict the subsequent evolution of the pressure fluctuations. Control is applied using a span wise zero-net-mass-flux (ZNMF) actuator slot by attempting to reduce the power of the surface pressure fluctuations in a closed-loop fashion, thus suppressing the unsteady flow fluc tuations based on predicted flow characteristics. A similar idea has been applied to control of flow-induced cavity oscillations (Catta festa et al. 1999) and turbulent boundary layer control (Rathnasingham and Breuer 2003). This kind of approach can be categorized as a dynamic control scheme shown in Figure 1-4 Closed-Loop Control Algorithms According to the classification in Figure 1-4 the control algorithms can be divided into two categories: quasi-static and dynamic. Optimiza tion algorithms are used in this research as quasi-static algorithms. They are used to optimize target functions (such as lift, pressure recovery, etc.) in a recursive but static or time -averaged fashion. On the other hand, recursive system identification and disturba nce rejection algorithms are wide ly used in active noise control area as dynamic algorithms. No one has attempted to apply these algorithms to the closed-loop separation control problem. This section gives a br ief review of the two types of the algorithms. Details will be given in chapter 2. Optimization algorithms Optimization algorithms are widely used by decision-makers (e.g. economists, governments). They often need to choose an action to optimize target or cost functions, such as income, profit, etc. In a typically optimi zation problem, one is given a single function f that 34

PAGE 35

depends on one or more independent variables. The goal is to find the va lue of those variables where f is a maximum or a minimum value. In this research, various optimization algorithms are used to maximize/minimize different cost functi ons, such as lift, drag and pressure recovery. When using the optimization algorithms, some constraints are typi cally included in the algorithms. For example, one often seeks to limit the energy expenditure while optimizing the cost function. One should also notice that, unlike the applications used by the decision-makers, the cost functions used in this research are meas ured by sensors instead of analytical functions. Some established minimization and maximiza tion algorithms are summarized by Press et al. (1992). Most optimization algorithms can be easily implemented in a multi-dimensional space. The downhill simplex algorithm and the Powells algorithm do not require derivative calculations. Between these two algorithms, the downhill simplex algorithm is more concise and self-contained. Both of them require storage of order where is the number of dimensions or independent variables. Two other algorith ms, the conjugate gradient and quasi-Newton methods, do require the calculation of derivatives. The conjugate gradient method requires only order storage, while the quasi-Newton method requi res storage of order On the other hand, none of the algorithms mentioned above are guaranteed to find a global extremum. They can lead to local extrema. Finding a global extr emum is actually a very difficult problem. Two standard methods are typically used to improve the probability of finding a global extremum: 1) search for local extrema from various initial cond itions and pick the most extreme of these; 2) perturb a local extremum to see if the algorithm goes back to the same value or finds a better result. 2NN2NNThere are several global search algorithms that are currently ac tive in research(e.g. Genetic Algorithms (GA) (Holland 1975), Particle Sw arm Optimization (PSO) (Kennedy 1997) and 35

PAGE 36

Simulated Annealing Method (Haftka and Grdal 1992)). The genetic algorithms and the particle swarm optimization ar e both derived from biology. They are population-based algorithms, namely they generate a populati on of points at each iteration and the population approaches an optimal solution. The GA and PS O take advantage of the large search population to increase probability of approaching a global op timum. The simulated annealing method is an analogy with thermodynamics, especially with the way metals cool and anneal, in which process nature finds the minimum energy state. The essen ce of the algorithm is to allow increase of cost function with some probability to improve the changes to find a global minimum. Another optimization algorithm that has been ap plied to flow control problems is called the extremum-seeking algorithm. As a self-optimizing control algorithm, the extremum-seeking control was first introduced in the 1950s. After Krstic and Wa ng (1999) provided the stability studies, there has been a resurgen ce of interest of this control algorithm. Banaszuk et al. (2003) attempted to use this algorithm in the diffuser separation control problem. They were successful in maximizing the pressure recovery in the diffuse r. They also used this algorithm to control combustion instability (Banaszuk et al. 2000). System identification and di sturbance rejection algorithms System identification and disturbance rej ection technologies are well developed and various algorithms are available in the active noise control area. Cattafesta et al. (1999) have applied these algorithms to other flow control problems, such as cavity resonance control. No one has attempted to apply this kind of approach to the separation control problem. In this research, this approach is inve stigated. Some system identific ation and disturbance rejection algorithms are reviewed in this section. System identification algorithms 36

PAGE 37

In general, system identification (ID) uses m easured signals (i.e., inputs and outputs of the system) to identify (or estimate) the unknown syst em dynamics. It provides necessary system information for control algorithms. System iden tification algorithms can be divided into two categories: offline (or batch) and online (or recurs ive). Offline algorithms first acquire data and then try to estimate a low-order dynamical syst em model using these data offline. Online algorithms identify systems recursively while acquiring data in real-time. Online system identification is also known as adaptive filtering. Least square (LS) identification algorithm is a generally used offline algorithm. Akers and Bernstein (1997 A) applied this approach to the ARMARKOV/LS identifi cation algorithm with an ARMARKOV representation (see Chapter 2 for a detailed description of the algorithm). The ARMARKOV/LS identification algo rithm uses vectors comprised of input-output data with a least-squares criterion to estimate a weight matrix containing a specified number of Markov (i.e., pulse response) parameters of the system. Th en the eigensystem rea lization algorithm (ERA) (Juang 1994) is used to construct a minimal state sp ace realization of the syst em. This is referred to as the ARMARKOV/LS/ERA identification algorithm. The ARMARKOV/LS/ERA identification algorith m has two clear advantages compared to the ARMA/LS identification algorit hm (Akers and Bernstein 1997 A). First, eigenvalues of the ARMARKOV representation are le ss sensitive to noise compared with eigenvalues of the ARMA representation. Second, the singular value decomposition of a block Hankel matrix constructed from the estimated Markov parameters provides an efficient model order indicator (Juang 1994, pp. 139). As far as online algorithms are concerned, the least-mean-square (LMS) algorithm is the most commonly used algorithm. A more co mputationally intensive algorithm called the 37

PAGE 38

recursive-least-square (RLS) algorithm has faster convergence and smaller steady-state error than the LMS algorithm (Haykin 2002) but is mo re computationally intensive. Two different types of structure that can be applied to each of the algorithms are the finite-impulse-response (FIR) and the infinite-impulse-response (IIR) filter s. The FIR filter is widely used due to its simple architecture and inherent stability as an all-zero model. However, its simple structure introduces difficulties for a system with low dampi ng. The IIR filter can solve this problem with significantly lower-order and, therefore, lead to reduced computational expense. Unfortunately, the disadvantages of an IIR filter include more complicated adaptive algorithms compared with an FIR filter and the possible stability problems introduced by the pole(s) in the model (Haykin 2002; Shynk 1989; Netto and Diniz 1995). Applying the LMS algorithm to the ARMARKOV representation, Akers and Bernstein (1997 B) introduced the recursive ARMARKOV/Toepl itz algorithm that is based upon recursive identification of the Markov parameters of a system. It estimates the Markov parameters recursively using time-domain, i nput-output data and then constr ucts the estimated model with the Markov parameters. Disturbance rejec tion algorithms As mentioned earlier, one of the possible control schemes for closed-loop separation control is to reduce velocity and pressure fluctu ations in the separated region. This control scheme is generally called disturbance rejection. Disturbance rejection controll ers have been widely used in active noise control applications (Kuo and Morgan 1996). Recently, researchers have started to apply adaptive controllers to flow control problems. For exam ple, Cattafesta et al. (1999) used an adaptive system to suppress the disturbance induced by the flow over a weapons-bay cavity. The advantages of using adaptive controllers are that they can adapt themselves according to different 38

PAGE 39

flow conditions and that they can potentially reduce the energy co st associated with the flow control problems. Cattafesta et al. (1997) showed that the control of cavit y flow with closedloop control requires one order-of-magnitude less power than that with open loop control. Commonly used disturbance rejection algor ithms include Filtered-X LMS (FXLMS), Filtered-U LMS (FULMS), Filtered-X RLS (FXR LS) and Filtered-U RLS (FXRLS) algorithms (Kuo and Morgan 1996). Besides these, the AR MARKOV adaptive control algorithm was first introduced by Venugopal and Berstein (1997) and fu rther developed by Sane et al. (2001). The underling model structure of the ARMARKOV adaptive control algor ithm is the ARMARKOV representation, which is an exte nsion of the ARMA representati on with explicit impulse response (Markov) parameters. The ARMARKOV adaptive c ontrol algorithm doesnt require a model of the control-to-reference transfer function nor does it require a model of the transfer function from plant disturbances to sensors (S ane et al. 2001). The only transfer function needed is the control to performance transfer function, which can be identified simultaneously using the recursive ARMARKOV/Toeplitz system identi fication algorithm described in the previous section. Objectives To explore suitable linear and nonlinear contro l objectives and strate gies for closed-loop control of separated flows. To implement optimization algorithms and sy stem identification/disturbance rejection algorithms for closed-loop c ontrol of separated flow on a wind tunnel airfoil model (NACA 0025). To analyze performance, adaptability, cost s, and limitations of closed-loop separation control algorithms. To investigate the relevant flow physics of successful feedback control strategies. Approach The proposed closed-loop separa tion control includes two key parts: modeling and control strategies. As far as modeling is concerned, two types of appr oaches can be implemented to 39

PAGE 40

model the flow characteristics: 1) a reduced-o rder flow model base d on the Navier-Stokes equations, 2) system identification techniques. The first approach is widely used in computational flow control simulations. This re search will concentrate on experimental studies by using system identification tec hniques that have not yet been a pplied to the separation control problem. The dynamical systems model will include the dynamics of actuators, sensors, and the flow system. Then the disturbanc e rejection algorithm is used to suppress flow fluctuations (e.g., measured by unsteady pressure transducers). One the other hand, for the non-model based optimization algorithms, no system identification is needed. The possible cost functions for the algorithm are summarized as follows. Since the suction pressure region of the upper surface of the airfoil is primarily responsible for lift generation and drag reduction, the static pressure recovery coefficient (pdCdxc) over the upper surface of the airfoil is a reasonable candidate as a cost function to maximize for feedback separation control. Other candidates for cost func tions are lift and drag or combinations of these (e.g., li ft/drag ratio). The benefit of us ing lift/drag is that L/D is a global or integrated quantity and is less sensitiv e to sensor location. The objectives for the controller are clear, i.e. to minimize drag and to maximize lif t or the ratio of lift/drag. The experimental setup uses a lift/dra g balance for this purpose. Outline of This Dissertation A theoretical background on system identificat ion, control, and op timization algorithms will be discussed in Chapter 2. Simulation results and validation experiments of the algorithms will then be presented in Chapter 3. Chapter 4 describes the experimental setup and techniques for this research. Chapter 5 presents experiment al results and discussion Summary and future work will be presented in the last chapter. 40

PAGE 41

Figure 1-1. Separation of flow over an airfoil. Figure 1-2. Types of velocity profiles as a function of pressure gradient (White 1991). 41

PAGE 42

Figure 1-3. Lift and dr ag coefficients of NACA 0025 airfoil at R e100,000 Figure 1-4. Classificatio n of flow control. (Cattafesta et al. 2003) 42

PAGE 43

43 Figure 1-5. Characteriza tion of possible frequency scales in separated flow (M ittal et al. 2005). 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 30 35 Frequency (Hz) Magnitude of maximum velocity (m/s) Figure 1-6. Comparison between the lumped element model (-) and experimental frequency response () measured using phase-locked LDV for a prototypical synthetic jet (Gallas et al. 2003).

PAGE 44

CHAPTER 2 THEORETICAL BACKGROUND This chapter presents detailed descriptions a nd derivations of the al gorithms that are used in this research. The algorithms include optim ization, system identification, and disturbance rejection algorithms. Optimization Algorithms Some established minimization and maximiza tion algorithms are summarized by Press et al. (1992). The downhill simplex algorithm and th e Powells algorithm do not require derivative calculations, which makes them good candidates for this research since de rivative calculations are problematic for (usually no isy) experimental data. Be tween these two algorithms, the downhill simplex algorithm is more concise a nd self-contained. The so-called extremumseeking algorithm has been applied to a flow c ontrol problem by Banaszuk et al. (2003). Thus, this algorithm is also summarized here. Downhill Simplex Algorithm The downhill simplex algorithm is implemente d to minimize an objective function (e.g., drag-to-lift ratio). The benefits of the algorithm are its simplicity, applicability to multidimensional optimization and robust performa nce. The algorithm searches downhill in a straightforward fashion that makes no prior assumptions about the function. The downhill simplex algorithm requires only function evaluations not derivatives. Since it does not make any assumptions about the function, it can be very slow sometimes. However, it can be very robust in the sense that it guarantees to find a mi nimum (at least a local minimum) (Press et al. 1992). A simplex is the geometrical object consisting, in dimensions, of points (or vertices) whereas the points span a -dimensional vector space (Press et al. 1992). For N1N1 N N 44

PAGE 45

example, in two dimensions, a simplex is a tria ngle. In three dimensions, it is a tetrahedron, although not necessarily a regular tetrahedron. The downhill simplex algorithm makes use of the geometrical concept of a simplex and works its way in the local downhill direction until it encounters a (at least, local) minimum. The key steps of the downhill simplex algorithm are summarized as follows: 1) Evaluate the cost function at chosen initi al points. Note th at there should be 1 N initial points, defining an initial simplex. For two or higher dimensions, the initial points should not be li nearly dependent. 2) Take a series of steps to move in the downh ill direction. As an example, the steps for three-dimensional search are illustrated in Figure 2-1 In the figure, Reflection means that the algorithm reflects the highest (i .e. worst) point about the center of the three lower (i.e. better) points with some coef ficients to the other side of the plane and then evaluates the cost function at the refl ected point. Expansion means to expand further along the reflection direction when the Reflection point does improve (i.e., lower the cost function). Contraction mean s to move the highest (i.e. worst) point towards the plane formed by the three lower (i.e. better) points, t hus contracting the original simplex. To summarize, all the necessary steps taken here are to move the worst point reference to the plane formed by the other better points to search for a better point. 3) Stop when some termination criteria are met. For example, the moving distance is smaller than some tolerance value. Extremum Seeking Algorithm Artiyur and Krstic (2003) pres ent the theoretic details and some applications of the extremum-seeking algorithm. Simulations using the algorithm can be done following the block 45

PAGE 46

diagram in Figure 2-2 The simple proof that this algorithm will drive () f to its extremum is summarized below. First, assume that () f has a minimum f and can be approximated as the following form: 2 *() 2!f ff* (2.1) where is the optimal input and f is the local curvature of the cost function () f near Since it is assumed that () f has a minimum, f should be larger than zero for this case. Next, define the estimated error: (2.2) where is the estimated optimal input. From Figure 2-2 (2.3) sin()sin()awtawt Substituting equation (2.3) into equation (2.1) results in 2 *()sin() 2f yffawt (2.4) Expand equation (2.4) and apply 21cos(2) sin() 2 wt wt to obtain "2 "2 *2" "2"2 "2 *"sin()sin() 2 cos(2)sin() 44 2 fa f yf wtfawt fafa f fw t f a w 2 t (2.5) From Figure 2-2 this signal will pass through a high pass filter y hs sw Let 0hww then all the DC components in equation (2.5) will be removed while the oscillatory terms remain. 46

PAGE 47

"2 "cos(2)sin() 4fa wtfawt (2.6) Next, is multiplied by to give sin()awt "2 "cos(2)sin()sin() 4fa wtwtfawt 2 (2.7) Using the trigonometric identities 21cos(2) sin() 2 wt wt and sin(3)sin() cos(2)sin() 2 wtwt wtwt results in "2 ""2"2"sin(3)sin()1cos(2) 422 sin()sin(3)cos(2) 2882 fawtwt wt fa fafafafa wt wt wt (2.8) From Figure 2-2 this signal passes through a low pass filter. Let then all the high frequency terms will be remove d and only the DC term remains. 0lw w "2 f a (2.9) This signal then passes through an integrator 2 f ak s (2.10) This gives 2 f ak s (2.11) From equation (2.2) assuming f is fixed, then (2.12) From equations (2.11) and (2.12) one can obtain the first-order differential equation "2 f ak (2.13) with the solution 47

PAGE 48

"2 0 f ak te (2.14) Since f is assumed to be positive and and are positive constants, the estimated error a k will exponentially decay to zero. System Identification Algorithms System identification (ID) uses measured sign als (i.e., inputs and output s of the system) to identify (or estimate) the unknown parameters of an assumed dynamical systems model. It thus provides the necessary system information for control algorithms. System identification algorithms can be divided into two categories: offline (or batch) and online (or recursive). Offline algorithms first digitize a data record and then try to estimate the system using these data offline, usually via a least s quares method. Conversely, onlin e algorithms identify systems recursively while acquiring data in real-time. Online system identification is also known as adaptive filtering. In this research, three system ID algor ithms will be investig ated: ARMARKOV/LS, ARMARKOV/LS/ERA and recursiv e ARMARKOV/Toeplitz algorithms. They are all based on the ARMARKOV representation, which explicitly contains Ma rkov parameters (i.e., pulse response) of the system. The well known AR MA representation cont ains only one Markov parameter and is a special case of the ARMARK OV representation. The main advantage of these algorithms is their robustn ess with respect to low signa l-to-noise ratios (Akers and Bernstein 1997 A, B). The ARMARKOV/LS algor ithm is an offline algorithm and implements an overparameterized realization of the sy stem. The ARMARKOV/LS/ERA algorithm uses the same procedures to identify the system pa rameters as the ARMARKOV/LS algorithm, but implements a minimal realization of the system. The recursive ARMARKOV/Toeplitz 48

PAGE 49

algorithm is an online algorithm. The advantage of using an online algorithm is that it can adapt to the changing system. ARMARKOV/LS Algorithm Consider the discrete-time finite-dimensional linear time-invariant system: (1)()( ()()() ) x kAxkBu k y kCxkDuk (2.15) where and and are the number of inputs and outputs, respectively, of the system. For a single-input/single-out put (SISO) system, The algorithm is derived below for a SISO system. ,,,nnnilnliARBRCRDRil1il The Markov parameters are defined by jH (2.16) j=-1 j0j j jHD HCAB Next, define the ARMARKOV regressor vector 2()nkR : () (1 () () (1 yk ykn k uk ukn ) ) (2.17) where is the order of the system and n is the number of Markov parameters. Here, y and u denote measured input and output of the system described in equation (2.15) respectively. Next define the estimated output of the system () ykW (2.18) where the ARMARKOV weight matrix is defined by W 12 [ WHH ] (2.19) and 49

PAGE 50

1 ,1, 1 ,1,[] []n n n n R R (2.20) The expression of the weight matrix W is then determined to minimize the output error cost function defined below. First, define the output error kykyk (2.21) and the output mean squared error cost function 2 111111 ()()() 22NN T kkJk NN kk (2.22) where is the number of measurements. NSubstituting equations (2.17) (2.18) (2.19) and (2.21) into equation (2.22) results in 111 2N T kJykWkykWk N (2.23) 111 2N TTT kJykkWykW N k (2.24) 111 ( 2 )N TTT k TT TJykykkW N yk y kWkkWWkk (2.25) Because and TTkWyk T y kWk are transposes of each other and are also scalars, they are equal to each other. So, 111 2 2N TT TT T kJykykkWykkWWk N (2.26) To find the to minimize the output error cost function defined in equation W(2.22) we set the partial derivative of with respect to equal to zero. So, JW 0 T TJJ WW (2.27) From matrix calculus, we derive each term of in equation J(2.26) first, 50

PAGE 51

()()0 T Tykyk W (2.28) 222 T T T TT T T TkWykkykkyk W (2.29) () () 2 2T TT T T TT TT T TT TkWWk dWW kWWk WW W kkW Wkk Td W (2.30) Thus, 111 022 2T N TT T kJ kykWkk N W (2.31) 1111 NN TT kkWkkkyk NN (2.32) Finally, the expression of the weight matrix to minimize the output error cost function is given by W 1 1111 NN TT kkWk y kk NN k (2.33) After extracting the coefficients and from via equation jH W(2.19) we can obtain the system transfer func tion of the ARMARKOV represen tation, which is defined as follows 11 12 1 1 11 ,1 ,()nnn n nn nHzHzz Gz zz (2.34) This is called the ARMARKOV/LS identification algorithm and this algorithm assumes the numerator has the same order of the denom inator for simplicity. For the systems whose numerators and denominators do not have the same order, some parameters described in equation (2.34) will be identified to be approximately zero. 51

PAGE 52

The well-known ARMA represen tation only has one explicit Ma rkov parameter and it is a special form of the ARMARKOV representation with =1 in (2.34) 1 11,11 2 1 1,1 1,()nn n nn nHzz Gz zz, (2.35) For system identification problems, the orde r of the system is usually not known in advance, so we adjust and n to improve the performance of the system identification algorithm. ARMARKOV/LS/ERA Algorithm The ARMARKOV/LS/ERA algorithm obtains a minimal realization of the transfer function of the system from the Markov parame ters. It uses the same algorithm as the ARMARKOV/LS algorithm to obt ain the weight matrix by equation W(2.33) Then the Markov parameters can be extracted from equation jH (2.33) by using equation (2.19) Next, define the Markov block Hankel matrix for a SISO system: (2.36) ,, jj rsj jr jrsHH H HH s Qwhere are any positive integers. In this research, is set to be equal to for convenience. ,rsrsThen, we apply the singular value decomposition as describe in Akers and Bernstein (1997A) (2.37) ,,0, T rsrsHPS where and TTPPQQI,diagonal matrix of singular values.rsSFrom the Eigenvalue Realization Algor ithm (ERA), (Juang 1994, pp. 133-137) 52

PAGE 53

(2.38) 1/2 1/2 ,, 1, 1/2 1/2 1 T rs rsrs T rss T rrsASPHQS BSQE CEPS DH where The ERA also requires 11 0i iE ,1rsnThis arrives at the minimal state space realization of the system 1 3() GzCzAIBD (2.39) This is called the ARMARKOV/LS/ERA algorithm It is a minimal realization because the system order can be chosen as a minimal valu e when using the singular value decomposition in equation (2.37) However, an important drawback of this algorithm is that the singular value decomposition in equation (2.23) is very computational intensive. Theoretically, the rank of the matrix should be the rank of the system. However, in practical applications, the singular value decompos ition will return more singular values than the system order due to measurement noise, and so th e extra singular values should be small. So, only the largest n singular values obtained by the singular value decomposition will be used. ,rsSRecursive ARMARKOV/Toeplitz Algorithm First, define the ARMA RKOV regressor vector222()npkR : () (2 () () (2 yk ykpn k uk ukpn ) ) (2.40) 53

PAGE 54

where is the order of a system, n is the number of Markov parameters, and a new parameter p determines the averaging window of input-outpu t data that appears in the above regressor vector. It follows that () ykW (2.41) where the ARMARKOV/Toeplitz weight matrix (222)pnpWR is the block-Toeplitz matrix defined by 12 1200 00 00 00 0000 AHHB W AHH B (2.42) and 1 ,1,[]n n R 1 ,1,[]n n R and j H are the Markov parameters. As before, define the output error k and the output e rror cost function Jk (2.43) ()()() kYkYk 1 2TJkkk (2.44) Next, the gradient of with respect to can be calculated by Jk () Wk () ()() ()TJk Ukk Wk (2.45) where denotes the Hadamard product (i.e. element-wise matrix product) and (222) pnpUR is defined by (2.46) 11 ( ) 110000 00 00 0000nn nnII U II ( ) Finally, the recursive update law for the weight matrix is given by W 54

PAGE 55

() (1)()() () Jk WkWkk Wk (2.47) In equation (2.47) () k is the adaptive step size The optimal adaptive step size ()optk is defined as 2 2 2 2() () () ()optk k Jk Wk (2.48) where 2 denotes the spectral norm. The computationally efficient step size ()effk (namely, it is more computational efficient since it only needs to calculate the normal ARMARKOV regressor vector () k ) is defined as 2 21 () ()effk k (2.49) In order to assure convergence, () k should satisfy ()()optkk or ()()effkk where(0,2) After matrix is obtained by W(2.47) we can extract the coefficients and from jH (2.19) Then, we can obtain the system transf er function of the ARMARKOV representation form, which is defined in equation (2.34) Since the and coefficients are updated every iteration, this algorithm is called as the recursive ARMARKOV/To eplitz algorithm. jHAdaptive Disturbance Rejection Algorithms Disturbance rejection controll ers have been widely used in active noise control applications (Kuo and Morgan 1996). A block di agram of a standard disturbance rejection problems is shown in Figure 2-3 where w is the disturbance, is the control signal, u y is the reference signal, z is the performance signal and is the disturbance rejection controller. The cG 55

PAGE 56

goal for the controller is to generate a control signal u to minimize some cost function of the performance signal. The four transfer matrices, namely, the primary path the secondary path the reference path and the feedback path are standard terminology in the noise control literature (Kuo and Morgan 1996). The feedforward-type disturbance rejection algorithms, such as FXLMS and FXRLS, assume that zwGzuGywGyuGyuG 0 (no feedback path) and ywGI On the other hand, the ARMARKOV disturban ce rejection algorithm does not make these assumptions. All the disturbance rejection algor ithms require identifying the secondary path by online or offline system identification methods. zuGARMARKOV Disturbance Rejection Algorithm Consider the linear discrete time tw o-input/two-output system (shown in Figure 2-3 ) given by ()()zwzkGwk ( uk )zuG (2.50) ()()yw y kGwk (yu) Gu k (2.51) where the disturbance the control the reference () wk () uk () y k and the performance are in () zkwm R um R y l R and z l R respectively, and m and l denote the number of inputs and outputs, respectively. The system transfer matrices (primary path), (secondary path), (reference path) and (control path) are in zwGzuGywGyuG z wm l R z ulm R y wlm y ulm and R R respectively. The objective of the active noise or vibration c ontrol problems is to determine a controller uyml cGR that produces a control signal such that the performance measure is minimized (Sane et al. 2001). A measurement of is used to adapt ( uk )cG () yk () zk () zkcG The ARMARKOV form of (2.50) (2.51) is 56

PAGE 57

(2.52) ,2 111 ,2 11()(1)(1)(1 (1)(1)nn jz w jz w j jjj n zuj zuj jjzkzkjHwkjBwkj HukjBukj ) ) (2.53) ,2 111 ,2 11()(1)(1)(1 (1)(1)n n jy w jy w j jjj n yuj yuj jjykykjHwkjBwkj HukjBukj where j R ,,, z wlm zwjzwjBHR, ,,, z ulm zujzujBHR, ,,, y wlm ywjywjBHR ,,, y ulm yujyujBHR is the order of the system, and n is the number of the Markov parameters. Then, we define the extended performance vector () Z k the extended measurement vector and the extended control vector as () Yk () Uk ()()(1)TZkzkzkp (2.54) ()()(1)TYkykykp (2.55) ()()(1)T cUkukukp (2.56) where p is an averaging or windowing parameter and (1)cpnp The ARMARKOV regressor vectors zw and yw are defined by ()(2)()(2)T zwzkzkpnwkwkpn (2.57) and ()(2)()(2)T ywykykpnwkwkpn (2.58) Then (2.52) and (2.53) can be written as () ()zwzwzu Z kWBUk (2.59) () ()ywywyuYkWBUk (2.60) 57

PAGE 58

where zwWzu B and zwWyu B are the ARMARKOV weight matrices. Only zu B will be used in the control algorithm (shown later), and it will be obtained using the ARMARKOV/Toeplitz system identification algorithm. The ARMARKOV control matrix zu B is given by (2.61) ,1,2,1, ,1,2,100 0 0 0 zuzu zu zu zuzu zu zuzunlmlm lm zu lm lm zu zuzu zuHHBB B HHBB n where 0 z ulm is the zero matrix. Next, the ARMARKOV adaptive disturbance rej ection algorithm is derived. The control signal is given by () uk (2.62) ,, 2, 111()(1)(1)(1)cccn n cjc cj cjc jjjukukjHykjBykj where ,cj R and ,,,uyml cjcjHBR Similarly, the delayed versions are (2.63) ,, 1, 111(1)()()(cccnn cjc cj cjc jjjuk ukjHykjBykj )2) u y (2.64) 1 ,1 11(1)(2) (2)(c ccn cc jcc j n cj c cjcc jjukp ukjp HykjpBykjP Substituting all these equations in (2.56) and reordering gives (2.65) 1()(1)()cp ii iUkLkiRkwhere (2.66) ,1 ,0 ,2 ,1 ,()()()()()()()ucu c ccmcnmc c c cnkkIkIHkHkBkBk 58

PAGE 59

()()( 2)()( 1)T uy c ccc ccckukuknpykyknp (2.67) (1) ()0 0uu u cuuimm im pimmLI (2.68) and (2.69) 11 1 111 2 1 22 1 222 2 2(1) ()(1) () (1) ()(1) ()00000 00000uc uy uc u yqimqqqpimqilqqqpil i qimqqqpimqilqqqpilI R I c y c y)ywith and 1cuqnm2(1ccqnl Thus from (2.59) and (2.65) we obtain 1()()(1)()()()cp zwzwzui iuy zwzwzu i Z kWkBLkiRkWkBUk (2.70) Next, evaluate the performan ce of the current value of ()k based upon the behavior of the system during the previous p steps to result in the definition of the estimated performance () Z k by 1 ()()()()cp zwzwzuiiuy i Z kWkBLkRk (2.71) Substituting (2.70) into (2.71) we obtain the estimated perf ormance in terms of known and measured variables 1 ()()()()()cp zu iiuy i Z kZkBUkLkRk (2.72) Using (2.72) we define the estimated performance cost function 1 ()()() 2TJkZkZk (2.73) The purpose for the ARMARKOV adaptive controller is to obtain the controller parameters ()k such that the performance cost function is minimized. Using matrix derivative formulae, the gradient of with respect to ()Jk()k()Jk is given by 59

PAGE 60

1() ()() ()cp TTTT izuuyi iJk LBZkkR k (2.74) The gradient is used in the update law () (1)()() () Jk kkk k (2.75) where () k is the adaptive step size. An implementable adaptive step size ()impk is used 2 2 21 () ()imp czuuy Fk p Bk (2.76) where F and 2 denote the Frobenius norm and th e spectral norm (Golub and Van Loan 1996), respectively. The steps involved in implementing the ARMARKOV adaptive disturbance rejection algorithm are summarized as follows: Obtain the matrix zu B (eqn. (2.61) ) by using the recursive ARMARKOV/Toeplitz system identification algorithm (eqn. (2.42) ) or the offline ARMARKOV/LS (eqn. (2.33) ). Calculate the control signal from the controller parameter matrix () uk () k and the vector (eqn. ()uyk (2.65) ). Use the signals ()uk() z k and to update the vectors ()yk () Z k (eqn. (2.72) ) and ()uyk (eqn. (2.67) ). Calculate the gradient () () Jk k (eqn. (2.74) ). Calculate the implementable adaptive step size ()impk (eqn. (2.76) ). Update the controller parameter matrix ()(1)kk (eqn (2.75) ). 60

PAGE 61

Figure 2-1. Flow chart of downhill simplex algorithm. 61

PAGE 62

62 Figure 2-2. Block diagram for the extremum seeking control. Figure 2-3. Block diagram of disturbance rejection control.

PAGE 63

CHAPTER 3 SIMULATION AND VALI DATION EXPERIMENTS Before the algorithms are used for closed -loop separation control in the wind tunnel experiments, they are tested by using Matlab/Sim ulink simulations or validation experiments. The purpose of this chapter is to ensure that the algorithms work as desired. Optimization Simulations Downhill Simplex Simulation Results The downhill simplex algorithm is programme d in Matlab. The performance of the algorithm is illustrated by a 1-dimensional and a 2-dimensional simulation cases. This algorithm can be easily extended to higher dimensions. In the 1-dimensional case, the cost function () f x is chosen to be an 8th order polynomial function of x which has a local minimum at 14.2 x and a global minimum at as shown in 67.3 x Figure 3-1 Two initial conditions are selected to demonstrate that this algorithm can be trapped by a local minimum. The first initial condit ion is at about 45 x One should notice that for this 1-dimensional problem, there should be two independent points (a simplex) as the initial condition. As shown in Figure 3-1 the downhill simplex algorithm crawls down to the global minimum (red trace). On the other hand, the second initial condition is at about 30x which leads the algorithm to the local minimum (b lue trace). This is di ctated by the inherent downhill nature of the algorithm. Another example is to demonstrate how the downhill simplex algorithm works in twodimensional space. The cost function is obtaine d in Matlab by the peaks command. The formula for the cost function is as follows: 22223 5(1 Z = 3(1-x)(1)10() 532 2 21 ) x xyxyx eyxyee (3.1) 63

PAGE 64

This function has two local minima and a global minimum as shown in Figure 3-2 Similar as the 1-dimensoinal case, the optimization algorit hm converges to either a local minimum (blue trace) or the global minimum (red trace) de pending on the initial condition. Although each iteration of the algorithm requires several steps (Cha pter 2), it only takes 9 iterations to find the global minimum. This result is encouraging and suggests that it can be fast for some cases. One can also adjust the termination to lerance to control the time consum ption. On the other hand, the time consumption of the separation control experiments is also de pendent on other factors, such as data acquisition. This will be discussed further in Chapter 5. Extremum Seeking Simulation Results The extremum seeking algorithm is implemented in Simulink. Figure 3-3 shows the simulation block diagram for the extremum seeki ng control. In the simulation, the algorithm seeks a maximum instead of a minimum. One can easily modify the program to search for a minimum by adding a negative sign to the cost func tion. Two numerical m odels are tested. The first model is a quadratic function 2 *ff* which has a single maximum f at as shown in Figure 3-4 In this case, f is set to 10 and is set to 5. The second model is a double hump model, which is fitted by a 8th order polynomial function, which is the same as the model shown in Figure 3-1 with a opposite sign. It has a local maximum and a global maximum as shown in Figure 3-5 Table 3-1 summarizes all the parameters that are us ed in the simulations. Recall that the detailed derivation of algorithm is given in Chapter 2. Note that the parameters and are the main factors that affect the convergence rate a nd stability. Thus, they are varied in the simulations to understand how they affect the performance of the algorithm. a w 64

PAGE 65

Figure 3-6 demonstrates how affects the conve rgence rate while is fixed to be 50 Hz. Clearly, the convergence rate increases when decreases. This is consistent with the analytical solution shown in eqn. a wa(2.14) for 21 k a where the convergence rate ( "2 f ak ) is dependant on 1a. However, when is too small, the algorithm becomes unstable. a wFigure 3-7 shows how affects the convergence rate while a is fixed to be 0.001. Apparently, the convergence rate increases with When is too large, the algorithm again becomes unstable. w wFigure 3-8 and Figure 3-9 show the results of the double hump model. Clearly, the extremum seeking algorithm drives the cost function f to the local minimum. Vibration Control Testbed Setup Figure 3-10 shows a detailed sketch of the whole vi bration control testbed setup. A thin aluminum cantilever beam with one piezocerami c plate bonded to each side is fixed on a block base and connected to the ground. The two piezo ceramic plates are used to excite the beam by applying electrical field across their thickness. The piezoceramic plate bonded to the upper side of the beam is called the "distu rbance piezoceramic" because it is used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the lower side of the beam is called the "control piezoceramic" because it is supplied with the controller output signal to counteract the disturbance actuator. The beam system has a natural frequency of about 97 Hz. The goal of the disturbance rejection controller is to mitigate the vibration of the aluminum beam generated by an unknown disturbance signal. The controller tries to generate a signal to counteract the vibration of the aluminum beam generated by the disturb ance piezoceramic. The performance (or the residue) signal of the cont roller is measured at the center of the tip of the beam by a laser-optical displacement sensor. The performance signal is filtered by a high 65

PAGE 66

pass filter with 1cz f H to filter out the dc offset of the displacement sensor and then amplified by an amplifier with a gain of 10. The disturbance and control signals are generated by our dSPACE (Model DS1005) DSP system with 466MHz PowerPC CPU and amplified by two separate channels of an amplifier by a same gain of 50. The types and conditions of the signals are discussed in details in the next section. The dSPACE system has a 16-bit A/ D and a 16-bit D/A board. The computer can acquire data using Mlib/Mtrace programs in MATLAB through the dSPACE system. The whole system was a two-input/two-output system. One input was the control signal and the other input was the unmeasured distur bance signal. The two outputs are termed a reference output and a perfo rmance output. For this vali dation test, the reference and performance outputs were identica l. The disturbance rejection algorithm was implemented in the Simulink environment and compiled and downloade d to the dSPACE system. The disturbance signal was band-limited white noise with frequency of 0-150 Hz. The disturbance rejection algorith m runs in one of the followi ng two modes: 1) ID, then control (shown in Figure 3-11 ): the system (control model) is identified by the ARMARKOV/Toeplitz system ID algorithm and the identified system weight matrix zu B is transferred to the ARMARKOV co ntrol algorithm; then the controller is turned on and the control signal is switched to the controller output. 2) ID and control (shown in Figure 3-12 ): the ID and control ar e turned on simultaneously. The input ( ) to the system for ID can be either band-limited white noise or a repetitive linear chirp signal. The controller uses the identified system to achieve maximum suppressi on of the vibration of the beam, subject to constraints on the maximum allowable actuator si gnal. The ID and control mode is better when the system is a time variant system becau se this mode updates the system information IDu 66

PAGE 67

during every iteration. However, the ID and control mode adds an additional signal to the control signal all the time and this certainly aff ects the performance of the disturbance rejection controller. The tradeoff between the adaptation ability and effects on the performance should be kept in mind. IDuResults of the Vibr ation Control Tests Computational Tests For real-time control applications, the turnaroun d time (defined as the time for the program to execute one iteration) is required to be le ss than the sampling time. Complex algorithms are computationally intensive and have large turnaround time, which requires choosing a corresponding larger sampling time (or a smaller sampling frequency s f ). From the Shannon sampling theorem, the sampling frequency must be larger than twice th e highest frequency of interest to avoid aliasing. Thus, algorithms with high computational complexity may not be feasible in flow control applications. The tradeoff between choosing a large s f to satisfy the sampling theorem and choosing a small s f to allow a large turnaround time must be considered. This section analyzes the effect s of varying the parameters of the ID and control algorithms on the computational intensity. This serves as a reference for choosing the parameters with regard to the computational intensity. The sampling frequency was 1024 Hz fo r the computationa l tests. In Figure 3-13 and Figure 3-14 the turnaround time of the system ID algorithm by varying either or n p is plotted, while the other two parameters are fixed at unity. It is shown th at the turnaround time is approximately linearly proportional to both and while the slope for is approximately twice of that for n n The dependence of the turnaround time on p is approximately quadratic. Clearly the averaging window number p has much more significant impact on the 67

PAGE 68

computational intensity than the other two parameters. Figure 3-15 investigates the effects of varying p on the computational intensity with respect to n As shown, the computational intensity is proportional to p From these results, it is suggested to hold p to be a small number and increase to improve the system ID performance. Figure 3-16 shows the effects of varying or cnc on the computational intensity of the ARMARKOV disturbance rejection algorithm. The results are si milar to those of the system ID algorithm. The turnaround time is a pproximately linearly proportional to and cnc while the slope for is approximately twice that for cnc System Identification The ARMARKOV/Toeplitz system ID algorithm requires the following three parameters: the order of the system the number of Markov parameters n and a parameter p that determines the size of the averag ing window. The SNR is also a parameter that can affect the performance of the system ID algorithm. When the number of Markov parameters is unity, the ARMARKOV model reduces to an ARMA model. The values of the parameters are limited by the requirement that the turnaround time must be less the sampling time. As shown in the last section, has the smallest effect on the turnaround time; thus in this sectio n the performance of the system ID algorithm with varying is compared. The offline non-parametric fit of the freque ncy response of the beam system is also implemented as a comparison and shown as green dot lines in Figure 3-20 Figure 3-23. The non-parametric fit uses the invfreqz comma nd in MATLAB and implements a second order approximation. The invfreqz command returns the system matrices and A B of the state space representation. The zero-pole map of the non-parametric fitted system is shown in Figure 3-17. As shown, the beam system is a low damp ing system because it has two poles that are 68

PAGE 69

very close to the unit circle. The controllability matrix is 11.655;01BAB (for a second order system), which has full rank 2. This means that the system is controllable. The sampling frequency was 1024 Hz. The input signal used for the system ID was a periodic chirp signal. The frequency response shown in Figure 3-20 Figure 3-23 was implemented with NFFT=1024, no overl ap, and a rectangular window. Figure 3-18 shows a very good match between the measured and fitted outputs of the system with the system ID parameters of 2pn 10 and SNR=20 dB. Meanwhile, as shown in Figure 3-19 the weight tracks of the system ID converge at about 0.5 seconds. Figure 3-20 to Figure 3-23 show the comparison between the measured and fitted frequency response with the system ID parameters of 2pn SNR=20 dB and varying A significant improvement of the system ID is obtained when changes from 1 (ARMA case) to 10. Figure 3-24 compares the mean square error (MSE ) verse time of the system ID with varying Surprisingly, it is found that the case with 10 has the best performance. This indicates that for the beam system, increasi ng the number of Markov parameters does not necessarily improve the system ID performance. This also suggests that for a certain system, there may exist an optimal number of Ma rkov parameters. It is also shown in Figure 3-24 that for the case with 40 the convergence rate is sl ower than the case with 1 although the final MSE is better. This indicates that too many Markov paramete rs may be detrimental to the performance of the ARMARKOV/ Toeplitz system ID. Figure 3-25 compares the mean square error (MSE ) verse time of the system ID with varying SNR. The SNR is computed by using the formula: 22 1010log() s nSNR where 2 s is the variance of the control signal and 2n is the variance of the disturbance signal. It is clear that 69

PAGE 70

the system ID performs better with a higher SNR. This suggests that when the system ID is conducted with unknown disturbance, it is better to apply a large system input within the maximum allowable range. Adaptive Disturbance Rejection The ARMARKOV disturbance rejection al gorithm requires the following three parameters: the order of the controller the number of Markov parameters of the controller cnc and the adaptive step size constant that controls the convergence rate of the controller. The controller uses the system informa tion identified by th e ARMARKOV/Toeplitz system ID with the parameters of 2 pn 10 and SNR=20 dB (shown in Figure 3-21 ). All cases of the controller design use the same identified system. Band-limited white noise with frequency of 0 150 Hz and variance of 0.09 is used to excite the disturbance piezoceramics. Figure 3-26 and Figure 3-27 show the time data of the performance and control signals with the parameters of 2cn 20c and The system ID is off for the whole period and the controller is off initially and turned on at t = 20 sec. In 1 Figure 3-28 the power spectra of the performance signa l with control off and on are compared. The power spectra were calculated by using the time data of 20-second duration with NFFT=1024, 50% overlap and a hanning window. The control on case is taken after the controller is turned on for 30 seconds. The performance of suppression is calculated by 22 1010logoffon and this case gives 11.7 dB s uppression. Interestingly, in Figure 3-28 the power around 70 Hz and 120 Hz of the control on case is higher than that of the control off case. This is generally defined as spill over (Hong and Bernstein 1998). Hong and Bernstein (1998) used the Bode integral c onstraint to analyze th e spillover problem a nd concluded that the spillover is inevitable if the reference and perfor mance signals are collocated or the disturbance 70

PAGE 71

and control actuators are collocated. For th is vibration control test, the reference and performance signals are collocated, t hus the spillover is unavoidable. Figure 3-29 Figure 3-30 and Figure 3-32 show the performance of the disturbance rejection algorithm with varying cnc and respectively, while other parameters are held constant. From Table 3-2 and Table 3-3 it is interesting to find that there is not much difference of the suppression performance for varying and cnc However, the step size parameter does play a significant role. Larger gives much faster convergence and better performance. However, if is too large, it is possible for the controller to become unstable. This tradeoff should be kept in mind when choosing The disturbance rejection controller can also be run at the ID and control mode. In this mode, the band-limited white noise with frequency of 0 150 Hz and variance of 0.09 is used to excite the disturbance piezoceramics. Meanwhil e, the band-limited white noise with frequency of 0 150 Hz and variance of 0.01 is added to th e controller output. The c ontrol signal is shown in Figure 3-33 and the performance signal is shown in Figure 3-34 Comparing with Figure 3-27, the control signal of the ID and control case is significantly larg er at the beginning because of the additive signal and the evolution of the controller output is buried under it. IDu Figure 3-35 compares the power spectra of the performance signal of the two different modes. It is surprising that th e ID and control mode results in lower power around the natural frequency of the beam. This is hard to see in Figure 3-34 because the ID, then control mode seems much better. However, it is not surprising that the ID and control mode results in higher power at other frequencies than the natura l frequency because of the additive signal In addition, IDu 71

PAGE 72

Table 3-5 shows that the ID, then control m ode gives better suppres sion performance of the overall power. Unfortunately, for this setup, it is not feasible to test the adaptability of the two modes. However, this will be done in the wind tunnel experiments. As a summary, the computational tests are condu cted first to determine how the parameters affect the computational complexity of the system ID and control algorithm s. It is shown that the averaging window number p has much more significant impact on the computational intensity than the other two parameters for th e system ID algorithm. The dependence of the computational complexity vs. is approximately twice of that for n Similarly it is found that the turnaround time of the c ontrol algorithm is approximate ly linearly proportional to and cnc while the slope for is approximately twice of that for cnc The ARMARKOV/Toeplitz syst em ID algorithm successfu lly identifies the system (control model) and results in very good fre quency response approximations. A significant improvement of the performance of the AR MARKOV system ID over the ARMA (when 1 ) system ID is found. However, too many Markov parameters of the ARMARKOV system ID may be detrimental to the performance. Higher SNR improves the performance, thus when the system ID is conducted with unknown noise, the input signal should be chosen as large as possible within the maximum allowable level. The order of the controller and the number of Markov parameters cnc do not play significant roles on the performance of the ARMA RKOV controller for th is vibration control test. However, this conclusion may vary with different systems and remains to be investigated. The step size constant significantly affects the converg ence rate of the controller. should be chosen as large as possible before it makes th e controller unstable. Th e spillover effect is 72

PAGE 73

identified in this vibration control test. This effect is unavoidable becau se the reference signal and performance signal are colloc ated (Hong and Bernstein 1998). Table 3-1. Parameters for the simulations. Fs (Hz) 500 Perturbation amplitude a = 0.001, 0.002, 0.005 Adaptation gain 21 ka Perturbation frequency ( ) Hzw = 30, 40, 50 High pass filter cutoff frequency ( ) Hz 10hw Low pass filter cutoff frequency ( ) Hz10lw Table 3-2. Suppression performance of the disturbance rejecti on algorithm with 20c and varying 0.1 cn 20c 0.1 1 5cn cn 10cn Suppression (dB) 8.9 8.6 9.0 Table 3-3. Suppression performance of the disturbance rejecti on algorithm with 2cn 0.1 and varying c 2cn 0.1 1c 20c 40c Suppression (dB) 8.4 8.7 7.9 Table 3-4. Suppression performance of the disturbance rejecti on algorithm with 2cn 20c and varying 2cn ,2c0 0.01 0.1 1 Suppression (dB) 3.9 8.9 11.7 73

PAGE 74

Table 3-5. Suppression performance of the dist urbance rejection algorithm at different modes with 2cn 20c and 0.1 2cn ,2 0c and 0.1 ID, then control ID and control Suppression (dB) 7.1525 4.9994 0 10 20 30 40 50 60 70 80 90 100 -100 -80 -60 -40 -20 0 20 xCost function Two different initial conditions f (x) Figure 3-1. One-dimensional exampl e of the downhill simplex algorithm. 74

PAGE 75

Figure 3-2. Two-dimensional exampl e of the downhill simplex algorithm. Figure 3-3. Simulation block diagra m for extremum seeking control. 75

PAGE 76

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 ff = f* = 10 when = = 5 Figure 3-4. Single glob al maximum test model: 2 *ff* where and *10 f *5 0 10 20 30 40 50 60 70 80 90 100 -20 0 20 40 60 80 100 f Figure 3-5. Double hump model with one lo cal maximum and one global maximum. The function is fitted by a polynomial: -128-107-76-55 -34-23-12-1.2e-1.6e+1.4e-2.3e +1.5e-4.4e+3.8e2.35.4 f 76

PAGE 77

0 10 20 30 40 50 60 70 80 90 100 -5 0 5 10 t (sec) a=0.001 a=0.002 a=0.005 0 10 20 30 40 50 60 70 80 90 100 -20 -10 0 10 t (sec)f a=0.001 a=0.002 a=0.005 Figure 3-6. converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while *f 10 a 50 w 0 10 20 30 40 50 60 70 80 90 100 -10 -5 0 5 10 t (sec) w=30 Hz w=40 Hz w=50 Hz 0 10 20 30 40 50 60 70 80 90 100 -150 -100 -50 0 50 t (sec)f w=30 Hz w=40 Hz w=50 Hz Figure 3-7. converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while *f 10 w 0.001 a 77

PAGE 78

0 50 100 150 200 250 300 350 400 -2 0 2 4 6 8 10 12 14 t (sec) Figure 3-8. converges to the local optimal input (see Figure 3-5 ) *14 ( ). 0.001,50 aw 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 40 45 t (sec)f Figure 3-9. f converges to the local maximum (see Figure 3-5 ) 40 f ( ). 0.001,50 aw 78

PAGE 79

Figure 3-10. Vibrati on Control Testbed. Figure 3-11. Block diagram of vibration control with ID, then control. 79

PAGE 80

Figure 3-12. Block diagram of vibra tion control with ID and control. Figure 3-13. Effects of varying or n p on the computational intensity of the ARMARKOV/Toeplitz system ID. 80

PAGE 81

Figure 3-14. Effects of varying p on the computational intens ity of the ARMARKOV/Toeplitz system ID. Figure 3-15. Effects of varying p on the growth rate of the co mputational intensity of the ARMARKOV/Toeplitz system ID with respect to n 81

PAGE 82

Figure 3-16. Effects of varying or cnc on the computational intensity of the ARMARKOV disturbance rejection. -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real PartImaginary Part Figure 3-17. Zero-pole map of the non-parametric fit of the frequency response. 82

PAGE 83

0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 -1.5 -1 -0.5 0 0.5 1 1.5 t (sec) Measured ARMARKOV/Toeplitz ID Figure 3-18. Measured output a nd fitted output by the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. 0 1 2 3 4 5 6 7 8 9 10 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t (sec)ARMARKOV/Toeplitz ID weight tracks Figure 3-19. Weight tracks of the ARMARKOV/Toeplitz system ID with 2pn10 and SNR=20 dB. 83

PAGE 84

0 50 100 150 0 10 20 Magnitude Frequency response Non-parametric fit ARMARKOV/Toeplitz ID 0 50 100 150 -200 0 200 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 3-20. Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2pn 1 and SNR=20 dB. 0 50 100 150 0 10 20 Magnitude Frequency response Non-parametric fit ARMARKOV/Toeplitz ID 0 50 100 150 -200 0 200 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 3-21. Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2pn 10 and SNR=20 dB. 84

PAGE 85

0 50 100 150 0 10 20 Magnitude Frequency response Non-parametric fit ARMARKOV/Toeplitz ID 0 50 100 150 -500 0 500 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 3-22. Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2pn 20 and SNR=20 dB. 0 50 100 150 0 10 20 Magnitude Frequency response Non-parametric fit ARMARKOV/Toeplitz ID 0 50 100 150 -200 0 200 400 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 3-23. Comparison of measured frequency response, non-parametric fit and the ARMARKOV/Toeplitz system ID with 2pn 30 and SNR=20 dB. 85

PAGE 86

0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 x 10-3 t (sec)MSE =1=10=20=30 Figure 3-24. Comparison of MSE of th e ARMARKOV/Toeplitz system ID with SNR=20 dB with different 2pn 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 x 10-3 t (sec)MSE SNR=40dB SNR=20dB SNR=0dB Figure 3-25. Comparison of MSE of th e ARMARKOV/Toeplitz system ID with 2pn20 with different SNR. 86

PAGE 87

0 5 10 15 20 25 30 35 40 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 t (sec)Performance signal Control off Control on Figure 3-26. Performance signal of the ARMA RKOV disturbance rejection to band-limited white noise (0-150 Hz) with 2cn 20c and 1 0 5 10 15 20 25 30 35 40 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 t (sec)Control signal Figure 3-27. Control signal of the AR MARKOV disturbance rejection with 2cn 20c and 1 87

PAGE 88

0 50 100 150 -100 -80 -60 -40 -20 0 20 Frequency (Hz)Power spectrum (dB) Control off Control on Figure 3-28. Power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 20c and 1 0 50 100 150 -100 -80 -60 -40 -20 0 20 Frequency (Hz)Power spectrum Control off Control on nc=1 Control on nc=5 Control on nc=10 Figure 3-29. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection with 0.1 20c and different cn 88

PAGE 89

0 50 100 150 -100 -80 -60 -40 -20 0 20 Frequency (Hz)Power spectrum Control off Control on c=1 Control on c=20 Control on c=40 Figure 3-30. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 0.1 and different c 0 5 10 15 20 25 30 35 40 -1 0 1 =0.01 0 5 10 15 20 25 30 35 40 -1 0 1 Performance signal =0.1 0 5 10 15 20 25 30 35 40 -1 0 1 t (sec) =1 Figure 3-31. Comparison of convergence of the ARMARKOV disturbance rejection with 2cn 20c and different 89

PAGE 90

0 50 100 150 -100 -80 -60 -40 -20 0 20 Frequency (Hz)Power spectrum Control off Control on =0.01 Control on =0.1 Control on =1 Figure 3-32. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 20c and different 0 5 10 15 20 25 30 35 40 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t (sec)Control signal Figure 3-33. Control signal of the ARMARKOV di sturbance rejection at ID and control mode with 2cn 20c and 0.1 90

PAGE 91

91 0 5 10 15 20 25 30 35 40 -1 -0.5 0 0.5 1 ID, then control 0 5 10 15 20 25 30 35 40 -1 -0.5 0 0.5 1 t (sec)Performance signal ID and control Figure 3-34. Comparison of convergence of the ARMARKOV disturbance re jection at different modes with 2cn 20c and 0.1 0 50 100 150 -100 -80 -60 -40 -20 0 20 Frequency (Hz)Power spectrum (dB) Control off ID, then control ID and control Figure 3-35. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection at different modes with 2cn 20c and 0.1

PAGE 92

CHAPTER 4 EXPERIMENTAL SETUP AND DATA ANALYSIS METHOD The separation control experime nts are conducted in an open -return low-speed wind tunnel with a 30.48 cm (1 ft) by 30.48 cm test secti on. The wind tunnel has two anti-turbulence screens, an aluminum honeycomb and a 9:1 contr action ratio. The airspeed is controlled by the variable frequency of the motor fan. A two-dimensional NACA 0025 airfoil that is e quipped with synthetic jet actuators, Kulite dynamic pressure transducers and a lift/drag balance is used as the test model. A Particle Image Velocimetry (PIV) system is used for flow visualization and quantitative flow field measurements. A Dantec CTA hot wire system is used to measure instantaneous velocity. This chapter describes each part of the experime ntal setup in detail. A brief description of the Higher Order Statistical Analys is (HOSA) is also presented in this chapter because it may be used for nonlinear flow instability analysis. NACA 0025 Airfoil Model A two-dimensional NACA 0025 airfoil with chord le ngth of 15.24 cm (6 in.) is built as a test bed for flow separation control ( Figure 4-1 ). The span of the airf oil model is 29.21 cm (11.5 in.), which allows for a slight gap on either end to accommodate a sidewall-mounted straingauge sting balance. The boundary layer is tripped at the lead ing edge region using No. 60 sand grit. Two pairs of synthetic jets are embedded in the airfoil at approxim ately 3% chord and 30% chord, respectively. Six ports near the rear of the airfoil at th e mid-span location are available for dynamic pressure transducers. The six ports are located at approx imately 44.0%, 52.5%, 61.0%, 69.5%, 77.9% and 86.4% chord. A pre-amplifier PCB board for the dynamic pressure transducers can be also installed in the airfoil. The detailed side view of the airfoil is shown in Figure 4-1 92

PAGE 93

Synthetic Jet Actuators The airfoil is fitted with two pairs of synthetic jet arrays (each with 0.5 mm wide slots separated by 2.4 mm), which ar e located in the central 1/3rd spanwise region of the airfoil (see Figure 4-1 ). The first pair is located near the leading edge of the airfoil, at approximately 3% chord, while the second is placed near the point of maximum thickness at about 30% chord. The first array is fixed, while the second array can be translated be tween 25% chord and 37% chord. The detailed design procedures of the synthetic jet actuators that ar e used in this research can be found in Gallas et al. (2003) and Gallas (2005). The primary goal of the design is to maximize the magnitude of the volume flow rate th rough the orifice per a pplied voltage (i.e. outacQV, where is the volume flow rate and is the applied ac voltage) over a frequency range of while the size of the syntheti c jet actuators are limited by th e geometry of the airfoil. As mentioned, the frequency response of the synt hetic jet actuators is another important design criterion. The frequency response of the actuators must be chosen appropriately to effectively control (via amplitude and burst modulation te chniques) the flow sepa ration over a range of frequencies, ranging from the lo w frequency shedding in the wake to the high frequency shear layer instability. The side and top views of the synthetic jet actuators are shown in outQHzacVOkFigure 4-3 The cavity is 151 mm long, 28 mm high and 2 mm wi de. Five piezoceramic disks are attached to one side of the cavity. They are driven in phase using a single amplified drive signal to achieve maximum flow rate. A thin slot (0.5 mm wide by 101.6 mm l ong) at the top of the cavity permits oscillatory fluid flow. Two closely spaced synthetic jets are obtained by introducing a rigid wall to se parate them, as shown in Figure 4-3 The two synthetic jet actuators are nominally identical. See the detailed characterization of the actuators in CHAPTER 5 93

PAGE 94

Experimental Methods Flow Visualization Flow field velocity data over the surface of th e airfoil and in the wake are acquired using Particle Image Velocimetry (PIV). The PIV system consists of a pair of New Wave Minilase 15 Hz, 50 mJ per pulse, Nd:YAG lasers with appropriate light sheet opt ics. The width of the light sheet is approximately 1-2 mm at the plane of measurement. A TSI model 630157 Powerview Plus 2MP 10-bit CCD camera is used to acquire images. This camera contains 1600 x 1200, 7.4 m square pixels. A series of Nikon lenses (60 mm, 75240 mm, 200 mm) are available. The flow is seeded with water-based fog fluid by a LeMaitre G150 seeder and the seeding density is adjusted to insure uniform seeding density. The laser pulse generator and the camera are synchronized by a TSI Model 610032 Synchronizer which is configured to acquire a pair of images using TSI INSIGHT Software version 6.1.1. The computation of the velocity fi eld begins by dividing the image into a grid of interrogation windows overlapped in space by 50%. These windows typically range from 32 x 32 pixels to 64 x 64 pixels. The velocity is determined by the know n distance that a particle is displaced during the known time dT. The INSIGHT Software utilizes an FFT cross-correlation process in conjunction with a Gaussian peak search algorithm to cal culate the average velocity of the particles in the interrogation widow. A numb er of validation schemes are available in the software, such as range outlier rejection and median filtering. Lift/Drag Balance A strain-gauge balance is designe d to measure lift and drag forces of the airfoil test bed. The detailed design procedure can be found in Griffin (2003). Two pairs of strain gauges are attached to the cantilever that supports the airfoil to measure th e normal and axial forces on the airfoil, respectively ( Figure 4-4 ). The layout of the strain gauges and Wheatstone bridge 94

PAGE 95

configuration are shown in Figure 4-5 and Figure 4-6 The output of the Wheatstone can be calculated by the following equation outVR V R (4.1) From the above equation, we can see that the output is linearly depe ndant on the change of resistance R The output of the strain gauges is measur ed by a high-resolution HP34970A DAQ system and is averaged over 2 power line cycles to el iminate 60 Hz noise. The lift and drag are calculated from the normal and axial forces together with the angle of attack ( Figure 4-4 ) via the following equations: (4.2) cos()sin() LNAOAAAOA (4.3) sin()cos() DNAOAAAOA where and L D stand for lift and drag, respectively, while and N A stand for normal and axial force, respectively. Before the balance can be used for the wind t unnel experiments, it is calibrated by adding known weights on the balance and measuring the output from the normal and axial strain gauges. Figure 4-7 and Figure 4-8 show typical normal and axial for ce calibrations vs. balance output. Very good linear relationships between the bala nce output and the forces on the balance are achieved. The coefficients of the linear equations are used to back out the forces on the airfoil from the voltage output of the strain gauges. The balance is also validated by comparing with the lift and pressure drag coefficients measured by integrating the static pressure around the airfoil. The pressure taps are located at the center span of the airfoil and the static pressure is measured via a Heise static pressure gauge. Figure 4-9 shows the static pressure distributions on the airfoil su rface at different AOAs when 95

PAGE 96

Re150,00020. From Figure 4-9 it can been identified that the flow is separated at AOA=1 and The suction zones on the upper surface shrink dramatically. This is generally referred to as pressure loss due to flow separation and is primarily res ponsible for deteriorating lift to drag ratio. 5The lift and drag coefficients are calculated by integrating the static pre ssure around the airfoil surface, assuming surface friction is negligible compared with pressure forces and the flow is two-dimensional. Figure 4-10 and Figure 4-11 show the comparison of the lift and drag coefficients calculated by the two different methods at Re100,000 and R The uncertainty was calculated at %95 confidence interval (i.e. e150,000 2 uncertainty N where is standard deviation and is number of measurements). As shown, they agree reasonably well considering measurement uncertainties. This vali dates that the balance works as desired. The main reason for the differences is the three-dimens ional effect as the pressure taps only measure at the center span. NDynamic Pressure Transducers To measure the pressure fluctuat ion on the airfoil surface, it is required that the pressure sensors must be compact so that they can be instal led within the limited space in the airfoil. It is also desired that they have large enough bandwidth to capture the characteristics of the oscillations of the flow above th e airfoil, and their response is lin ear with respect to the pressure load within the range of interest. For these r easons, a number of commercially available MEMS Kulite LQ125-5A dynamic pressure transducers ( Figure 4-12 ) are used to obtain dynamic pressure response on the upper surface of the airfoil. The transducers can be flush mounted in the six available locations on the upper surface. A pre-amplifier/filter board for the transducers 96

PAGE 97

is designed to eliminate dc response ( 1.5cutoff f Hz ) and amplify the outputs by a gain of 100. The pre-amplifier/filter board can be installed insi de the airfoil so that the airfoil acts like an electronic enclosure. Before the transducers can be used in the experiments, they are dynamically calibrated in a 2.54 cm (1 in.) by 2. 54 cm plane wave tube (PWT). A speaker was used as a source, and a Brel & Kjr (Mode l 4318) microphone was used as a reference transducer. Figure 4-14 shows the linear response of a typica l Kulite sensor that is obtained by fixing the frequency and increasing the input amp litude of the speaker. The frequency response is measured using a periodic chirp signal ( Figure 4-14 ). As shown, the frequency response does not vary up to approximately 3000 Hz, whic h is sufficient for this research. Hot Wire Anemometry A Dantec constant-temperature hot wire anem ometry system (CTA module 90C10) is used to measure time-resolved velocity in the unsepar ated flow above the airfoil. The CTA system includes A/D converter and all the signal conditioners needed. Befo re the measurements, a static calibration is performed by the calibration m odule and the flow unit (90H01 and 90H02). A typical calibration cu rve is shown in Figure 4-15 Since the output of the hot wire system usually drifts due to temperature changes, connections, etc, the calibration should be done before each measurement. Two algorithms are commonly used for curve fitting. One is a polynomial that is used here, and the other is Kings law (power law): 1 2 nUEAB where is the voltage output of the hot wire and U is the flow velocity (Jrgensen 1996). The difference between the temperatures at calibration and measurements should also be corrected by means of E 0 1 1 w corr wTT EE TT where is the wire temperature, is the temperature at calibration, is the raw wire voltage, is the temperature during measurement and is the corrected wT0T1E1TcorrE 97

PAGE 98

voltage (Jrgensen 1996). During experiments, the hot wire probe (55P11) is mounted on a 2dimensional Velmex traversing system wh ich has spatial resolution of about 1.6 / mstep in both directions. Control System Hardware and Software The control systems for the separation contro l experiments are implemented by a dSPACE (Model DS1005) DSP system with a 466MHz PowerPC CPU. The dSPACE system has a 5channel 16-bit A/D board (DS2001) and a 6-chan nel 16-bit D/A board (DS2102) as the data acquisition equipments. The range of the data acq uisition boards can only be -10 to +10 V, 0 to 10 V or -5 to +5V. The control algorithms are first programmed in Matlab/Simulink and C programs (c-mex sfunction) and then compiled and downloaded to the dSPACE system. The compiled programs together with the data acquisition boards are able to run the experiments in real time. The computer is also able to acquire data into Matlabs workspace through the dSPACE system via the m-lib pr ograms provided by the dSPACE. Higher Order Statistical Analysis (HOSA) Higher order spectral anal ysis is used to uncover the nonlinea r interactions in signals or to identify nonlinear systems (Nikias and Mendel 1993). As discussed in CHAPTER 1 there are three characteristic frequencies a nd nonlinear interactions between them are inherent in separated flow. Unfortunately, the power spectrum alone is incapable of providing a ny conclusive proof of the nonlinear interactions. The power spectrum only provides proof of presence of power at certain frequencies. On the other hand, highe r-order spectral method can quantify quadratic coupling between frequency pairs. For exampl e, it can provide the information that the generation of power at a certain frequency is the result of quadrat ic coupling of other frequencies. The auto-bispectrum uses third order cumulants and is defined as 98

PAGE 99

*1 ,limxxxij ijij T B ffEXfXfXff T (4.4) and the auto-bicohere nce is defined as 2 2, ,xxxij xxxij x xixxjxxijBff bff PfPfPff (4.5) where X f denotes the Fourier transform of x t, denotes the complex conjugate and denotes the auto-spectrum of xxPf x t. The auto-bicoherence is bounded by zero and un ity. Disturbances with frequencies i f j f and ij f f are quadratically coupled if 2,ijbff 1 not quadratically coupled if bfijf2,0 and partially coupled if 20,ijbff 1 Just as the auto-spectrum has the cross-spectrum as its counterpart for signals x t and y t, the auto-bispectrum has the cross-bispectrum as its counterpart, which is defined as: *1 ,limxxyij ijij T B ffXfXfYff T (4.6) Similarly, the cross-coherence is obtained by normalizing the cross-bispectrum and defined as follows: 2 2, ,xxyij xxyij x xixxjyyijBff bff PfPfPff (4.7) Some examples that illustrate the applications of the HOSA are given in the APPENDIX 99

PAGE 100

Figure 4-1. NACA 0025 airfoil model with actuators and pressure tran sducers installed. (Adapted from Holman et al. 2003) Instantaneous Time-averaged Figure 4-2. Schematic of a synthetic jet actuator. 100

PAGE 101

text text text text text side view top view piezoceramic disks splitter plate (2.4 mm thick) slots (0.5 mm wide) Figure 4-3. Synthetic jet array. (Adapted from Holman et al. 2003) Figure 4-4. Forces on NACA0025 airfoil. 101

PAGE 102

Figure 4-5. Closer view of the strain gauges. Figure 4-6. Wheatstone bridge configuration of the balance. 102

PAGE 103

Normal Force ( N ) Figure 4-7. Normal fo rce vs. balance output. Axial Force ( N ) Figure 4-8. Axial fo rce vs. balance output. 103

PAGE 104

0 10 20 30 40 50 60 70 80 90 100 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x/c (%)CpAOA = 0 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 -1.5 -1 -0.5 0 0.5 1 x/c (%)CpAOA = 5 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 x/c (%)CpAOA = 10 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x/c (%)CpAOA = 15 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 -1.5 -1 -0.5 0 0.5 1 x/c (%)CpAOA = 20 Upper surface Lower surface Figure 4-9. Static pressure distributions on the airfoil surf ace at different AOA at Re150,000 104

PAGE 105

-5 0 5 10 15 20 25 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 AOA (degree)CL CL from pressure CL from balance -5 0 5 10 15 20 25 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 AOA (degree)CD CD from pressure CD from balance Figure 4-10. Comparison of lift and drag coeffici ents measured by the static pressure and the balance at Re100,000 105

PAGE 106

-5 0 5 10 15 20 25 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 AOA (degree)CL CL from pressure CL from balance -5 0 5 10 15 20 25 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 AOA (degree)CD CD from pressure CD from balance Figure 4-11. Comparison of lift and drag coeffici ents measured by the static pressure and the balance at Re150,000 106

PAGE 107

Figure 4-12. Picture of a Kulite transducer. Sensor out p ut () mV Figure 4-13. Linear response (at 500Hz ) of a typical Kulite transducer. 107

PAGE 108

108 Fre q uenc y res p onse Figure 4-14. Frequency response of a typical Kulite transducer. Figure 4-15. Hot wire calibration curve.

PAGE 109

CHAPTER 5 RESULTS AND DISCUSSION The experimental results of the dynamic feedback control an d nonlinear control approaches are presented in two parts. Dynamic Feedback Control Experimental Configuration Figure 5-1 shows the complete experimental confi guration. The system ID and control algorithms run on the dSPACE controller system in real-time (~4 KHz). The controller generates control signal that is amplified by an amplifier. Th e pressure fluctuation signals measured by Kulite sensors are amplified and filter ed before sending to the dSPACE controller. System Identification Coherent flow structures Unlike POD-based approaches (Holmes et al 1998; Tadmor et al. 2007; Ausseur et al. 2007), the unsteady surface pressure si gnals are used exclusive of th e velocity field for feedback in this research. Although system ID and POD-based met hods of modeling the flow are different, they all attempt to capture the signat ure of the separated flow the coherent flow structures. In this section, we show that th e surface pressure signals indeed represent the footprint of the cohere nt flow structures. The following experiment is devised to show this. A continuous pulse train (repetition rate at 1 Hz and amplitude at 50 V) is fed to the ac tuator A1 at Re=120,000. The pressure signals are phase averaged relative to th e pulse signal. Four thousand av erages are taken to obtain the statistically converged pressure signal profiles. As shown in Figure 5-2 a vortex (produced by the actuator pulse) propagates downstream indica ted by the surface pressure fluctuations. The vortex reaches the sensor S1 first and then S2 S6. After the vortex passes by, the averaged 109

PAGE 110

surface pressure fluctuating approaches zero. Th is is because the random pressure fluctuations that are not correlated with th e pulse input from the actuato r possess random phase and are averaged out. The convective velocity is much slower than the free stream velocity, with a nominal lifetime of more than 5 airfoil chord lengths. With the link between the flow structure and the surface pressure clearly esta blished, the goal of the system ID approach is to correlate the actuator input and the correspondin g surface pressure fluctuations and utilize the re lationship to model the coherent flow structure wi th linear dynamical equations. Linear prediction The control approach is based on the assumption that the coherent flow structures may be modeled by linear dynamical equations. This sec tion demonstrates that the linear model is capable of predicting the downstr eam evolution of the flow dynami cs (measured by the pressure sensors) subject to the actuati on upstream (provided by the ZNMF actuators). In fact, previous studies in turbulent boundary layer (Rathnasingham and Breuer 2003) and cavity flows (Cattafesta et al. 1999) have suggested that linear approximations can reasonably predict inherently nonlinear flow structures. The system ID algorithm is applied first to demonstrate this. The computational requirements are demanding for both the system ID and control algorithms. When implementing the algorithms, the algorithm parameters were chos en due to hardware limitations and were not optimized. The sampling frequency is chosen to be 4096 Hz. In Figure 5-3 z denotes the pressure signal measured by th e #6 pressure sensor shown in Figure 5-1 subject to a band-limited random input provided to actuator A1. Using the input and z, one can fit a model to represent the flow structures using the a pproach described in earlier sect ions. One should be aware that this model actually includes actuator and sensor dynamics and other hardware in the loop, e.g. amplifiers and filters. Here, in z Figure 5-3 is the estimated using the model mentioned z 110

PAGE 111

above, and the data show a reasonable match ove r a range of time or frequency scales. The errors are primarily due to the random turbulent st ructures that are uncorrelated with the actuator input, and they cannot be modeled by the system ID algorithm. An indication of the convergence of the system ID to the actual flow model is the expected value of the Mean Squared Erro r (MSE), which is defined as 2 ()() MSEEzkzk As shown in Figure 5-4 at the beginning the MSE has a relatively large value. This is because the model parameters are initialized to zero. Then the model parameters are trained by the ARMARKOV system ID algorithm targeting th e objective of minimizing the error. Frequency response and the performance of system ID To further evaluate the performance of syst em ID approach, we compare the frequency response of the flow system determined using conventional FFT methods for single-input/singleoutput systems described in Chapter 6 of Bendat and Piersol (2000) with that determined using the converged ARMARKOV system ID model parameters. Wh en computing the frequency response, the parameters are s f =4096 Hz, NFFT=1024, 75% overlap, a Hanning window and 320 effective averages. Figure 5-5 shows that the system ID does not perf ectly match the frequency response, but it does capture the essential characte ristics over a broad frequency range It is also clear that the coherence between the input and out put is close to zero at frequenc ies less than 600 Hz, which is a characteristic of the present piezoelectri c zero-net mass flux actuators, which posses a resonance near 1200 Hz (Holman et al. 2003). The low coherence renders the FFT-based frequency response estimate uncertain and highlight s the difficulty of desi gning a control system using classical frequency domain approaches. 111

PAGE 112

Acoustic contamination In all of the discussions above we have ignored an important potential issue related to acoustic contamination. It is well known that zero-net mass flux actuators can produce significant sound. In the present control problem, the pressure sensors are intended to capture the hydrodynamics of the coherent flow st ructures. However, as demonstrated by Figure 5-6 the pressure sensors do not discrimi nate between acoustic and hydrodyna mic pressure fluctuations. Since the pressure measurements contain both components, the disturba nce rejection control algorithm will try to suppress th e acoustic power as well as the hydrodynamic power, possibly resulting in an undesirable reduction in the actu ator amplitude. Furthermore, from a control standpoint, the acoustic and hydrodynamic paths have significantly different propagation speeds, leading to significant phase lag differences and an unstable controller. One way to address this problem is to es timate a frequency-wavenumber spectrum using Fourier-based methods, but such a method is not amenable to a real-time control system. A second approach, adopted here, incorporates a di gital filter to predict and remove the acoustic signal. We design this digital filter using the same system ID method described in the earlier section, i.e. using the system ID method to pred ict the acoustic signal with the actuators on and the flow off and then subtracting the com puted acoustic component from the sensor measurement with both the actuators and flow on (see Figure 5-6 ). The ID parameters were p=1, n=100, =1. Note this filter has much higher order than that used in Figure 5-3 However, this will not add significant computational intensity b ecause after the filer is designed by the system ID algorithm the filter parameters are fixed during the closed-loop control. Figure 5-7 shows a comparison between the actual m easured acoustic noise with wi nd tunnel off and the predicted acoustic noise by the digital filter. Good agreement is achieved. To te st this digital filter further, 112

PAGE 113

we used the same digital filter in Figure 5-7 but reduced the actuator amplitude by 50%. Figure 5-8 shows the digital filter works well when the input signal is changed, indicating the linear behavior of the acoustic signal produced by the actu ator for typical excitation levels used in the experiment. Next, the digital filter is applied to the m easurements with the wind tunnel running. Figure 5-9 shows the power spectra comparison of the pressure measurements before and after applying the digital filter for a system ID cas e. The power spectrum with the digital filter applied clearly shows lower power at the frequency band 500 Hz to 1500 Hz, where the piezoelectric actuator generates most of its acous tic noise due to the ac tuator resonance as indicated in Figure 5-6 The digital filter is thus able to mitigate the acoustic noise component and is used for all results presented below. Disturbance Rejection Closed-loop control As described earlier, the dist urbance rejection algorithm re quires both reference (used for feedback) and performance signal measurements. As described ear lier, the goal of the algorithm is to minimize the fluctuations of the pe rformance signal. A ccording to Venugopal and Bernstein (2000), the reference and performance signals can be the same. Herein, different reference and performance transducer combinations are tested for comparison. Since S1 is the closest to the leading edge and S6 is the closest to the trailing edge, it is reasonable to investigate the extremes shown in Table 5-1 The disturbance rejection algorith m is applied to all four ca ses and the ID and controller parameters are summarized in Table 5-2 Note that higher values increase the nu mber of adjustable parameters in the system ID and the controller, which may increase pe rformance. However, th is is not guaranteed. 113

PAGE 114

First, to make sure the control is inducing a global effect, we examine the lift/drag performance that is the aerodynamic objective of the separation control system. The control objective is to attach the sepa rated flow and thereby reduce th e fluctuating pressure spectrum associated with the convection of the vertical stru ctures over the airfoil su rface. The lift and drag are measured after the closed-l oop control algorithm converges by the balance, which is only capable of providing mean or time-averaged data. The lift-to-drag ratios fo r all the cases are also summarized in Table 5-1 and include uncertainty estimates that account for calibration and random errors (Tian 2007). All four closed-l oop control cases give similar L/D improvement, L/D of the uncontrolled baseline case. Note that the lift is increased while the drag is decreased. Close inspection via tu ft and smoke flow visualization reveals that, in all 4 cases, the controller is able to full y attach the separated flow. Since th e four closed-loop control cases give the same L/D within experimental uncertainty, th is indicates that the ch oice of the performance and reference sensor locations does not have a significant imp act on the integrated lift/drag performance for this flow condition. In the fo llowing sections, we choose to study case #2 using S6 more closely. ~7Effect of control on surface pressure signals Recall that we use the actuator and surface pr essure signals to model the plant dynamics, and the disturbance rejection algorithm attempts to suppress the surface pressure spectra. Figure 5-10 shows the time traces of the performance surface pressure (measured by S6) and control input signals for case #2 before and after the contro l is turned on. The re sults clearly show that before the closed-loop control is initiated, the perfor mance pressure signal has relatively large amplitude. After the ID and control is initiated from a zero initial condition for all parameters, the performance pressure signal starts to decr ease driven by the control input from the ZNMF actuators generated by the disturbance reject ion algorithm. The entire process takes 114

PAGE 115

approximately 79 convective time scales to lear n the dynamics and optimize the controller. After a steady state is achieved, the pressure sign al stays at the lower level corresponding to an attached low oscillation flow, as will be shown in the next section. Figure 5-11 shows the comparison of the power spect ra for baseline and case #2 measured by the performance transducer S6. For the clos ed-loop control case, the power spectra are based on the surface pressure signal af ter the disturbance rejection algorithm converges. The noise floor of the transducer is also plotted for comparison to verify th at the pressure signals for all cases are well above the noise floor. Figure 5-11 clearly shows that th e disturbance rejection algorithm is able to lower the sp ectrum of the surface pressure si gnal compared with the baseline case at all frequencies. Quantitative flow visualization Preliminary experimentation shows that A1 and A2 give similar results but A3 and A4 are ineffective because they are located downstream of the separation location. Thus, the results with A1 are studied closed. Normalized stream wise velocity and vorti city contours (obtained using 500 PIV image pairs) over th e airfoil for the base line and closed-loop control case #2 are shown in Figure 5-12 and Figure 5-13 For the closed-loop control case, the images are taken after the disturbance rejection al gorithm converges. The actuator A1 and pressure sensors S1 S6 are shown as circles on the airfoi l surface. For the baseline case in Figure 5-12 (a), the flow separates from the leading edge just downstream of actuator A1 and all six pressure sensors are located inside the separated region. Instantane ous PIV data reveal th at the separated flow features large coherent vortices sweeping over the airfoil upper surface, which results in highly unsteady pressure signals on th e airfoil upper surface. The di sturbance rejection algorithm senses the pressure fluctuations and genera tes actuation signals to negate the pressure fluctuations. This process ultimately organizes th e unsteady flow into an attached turbulent flow 115

PAGE 116

in a closed-loop (smart) fashion. As shown in Figure 5-12 (b), the flow is fully attached for the closed-loop control case. This ma y explain why the four cases in Table 5-1 result in similar lift/drag performance, i.e. they share similar information about the flow before and after the closed-loop control is initiated. Control input In order to gain physical insight into the actuator control signal that the disturbance rejection algorithm generates to attach the flow, we examine the input voltage power as well as the corresponding input electrical power to the actuator A1. The first quantity has units of V2, while the second quantity has an SI unit of W. Note that these two quantities are different because piezoelectric actuators are capacitive devi ces that draw higher current as frequency is increased by virtue of the derivative operation, iCdVdt where denotes the capacitance of the piezoelectric actuators. The electrical pow er is calculated by multiplying the input voltage by the current of the actuator. The current is measured by a stand-alone current probe (Tektronix TCP A300). CThe voltage and rms electrical power spectra are shown in Figure 5-14 The total rms electrical power sums up to 12.7 mW. It is clear from Figure 5-14 (a) that the disturbance rejection algorithm generates a broadband contro l input to the actuator with spectral peaks in both the low (20 Hz 80 Hz) and the high (around 1 kHz) frequency ranges. The emphasis at low frequencies is due to the larger scale cohere nt flow structures described earlier, while the emphasis at higher frequencies corresponds to the smaller scale shear layer structures. More detailed discussions on the two types of flow stru ctures can be found in Ti an et al. (2006) and Wu et al. (1998). Since the flow is most recep tive at these inherent frequency scales, the 116

PAGE 117

disturbance rejection algorithm attempts to util ize the two characterist ic frequency scales by energy addition at these frequencies. On the other hand, actuator characterization e xperiments reveal that the actuator produces very small output at low frequencies (< 500 Hz ) (Tian et al. 2006). This is clear from Figure 5-5 and may be deduced from Figure 5-14 (b). The electrical power is concentrated at higher frequencies near the actuator resonance (than th e voltage signal power) while remaining almost flat at low frequencies. This implies that although the contro ller attempts to control the low and high frequency instabilities associated with the wake and shear layer, respectively, the dynamic response of the actuator significantly influences the control system dynamics. It is important to recall that the plant in cludes the actuator. Inspection of Figure 5-14 also brings into question whet her the voltage power should be used as a penalty function. The electrical power may be a better choice for the penalty function since it reflects the actual power consumption by the actuator. Such questions must await future studies. Discussion To examine how the adaptive controller pe rforms under different flow conditions, the angle of attack is varied continuously from 12 to 20, with a fixed free stream Re of 120,000. The lift-to-drag ratios for the baseline and controlled cases are plotted in Figure 5-15 Clearly, the baseline flow separates starti ng from 12. The adaptive contro ller gives the best performance at AoA=12. The performance deteriorates as AoA increases. The improvement in lift-to-drag ratio becomes very small at AoA=20, which means the controller is ineffective at this AoA. The comparison of the power spectra for baseline and case #2 measured by the performance transducer S6 is shown in Figure 5-16 It is clear that unlike the 12 counterpart (shown in Figure 5-11 ), the power spectrum of the closed-loop control case is higher than that of 117

PAGE 118

the baseline case. This defeats the purpose of the closed-loop c ontroller and ther efore makes the controller ineffective. The key question is why the pressu re spectral characteristics look conversely different at different AoAs. Our hypothesis is that the flow can only be partially attached at higher AoA in the mean sense. Inst antaneously the partially attached flow contains highly unsteady flow vortices that cause the increase in the pressure spectra. Possible solutions include usi ng alternate surface sensors, such as MEMS-based direct shear stress sensors for feedback instead of pres sure sensors. This can also solve the acoustic contamination issue. However, we believe that the key limitation of the present scheme, as evidenced by its failure at higher angles of at tack, include the assumption of linearity in the system identification and disturbance rejection algorithm. The nonlinearity, especially in the partially attached unsteady flow, has strong impact on the performance of the closed-loop controller. Future improvements to the current approach include exploration of nonlinear control methods, which is the subject of th e next section. By using the modulated signals as input (Tian et al. 2006), the nonlinear interactions in the unsteady flow is promoted. On the other hand, nonlinear dynamical approaches are desirable co mpared to the quasi-static approach in our companion study. The nonlinear system iden tification algorithms are demonstrated in simulations by Pillarisetti and Ca ttafesta (2001). Future di rection includes implementing a nonlinear dynamical system identifi cation and control scheme in the wind tunnel experiments. Nonlinear Control Experimental Configuration As shown in Figure 5-17 dual-timing control loops are configured to implement the optimization algorithms (described in the next section). The first loop synchronously controls the actuators and measures the low-pass filtered and amplified balance signal, while the second 118

PAGE 119

loop averages the balance output and performs optimization in an asynchronous fashion. The second loop acts as a supervisory cont roller that updates the control parameters in the first loop. The sampling rate of the first loop is 40 kHz, while the second loop runs on a host PC at O(Hz). The optimization algorithm is programmed in Matlab and communicates directly with the dSPACE system to adjust the actuator signal parameters. Flow Instabilities In the present flow conditions (AoA=2 and 0Re120,000c ), the baseline uncontrolled flow is massively separated and doe s not reattached before the trai ling edge (i.e. post stall). As mentioned earlier, this type of post-stall flow is characterized by leading-edge shear layer rollup and vortex shedding in the wake (Wu et al. 1998). These two types of flow structures are clearly visible in the instan taneous snapshot of the flow shown in Figure 5-18 The shear layer rollup structures in the left figure have a much smalle r length scale than the vo rtex shedding structures in the wake shown in the right figure. To characterize the velocity fluctuations in the two flow structures a hot-wire anemometer is used. The hot-wire is traversed vertically ac ross the shear layer (near the separation point) and the wake vortices (1 chord aft of the trailing edge). The maximal location in the attached sub-regions is then determined at the two streamwise locations. rmsu Figure 5-19 shows a plot of the power spectral density (PSD) of the wake and the shear layer at the respective peak rms locations. The PSD was estimated using a 4096 point FFT, a Hanning window with 75% overlap, and 320 effective blocks. The plots zoom in on two intere sting regions. The left plot clearly shows the dominant wake frequency wake f ~ 40 Hz. The right plot shows the much higher shear layer frequency at SL f ~2040 Hz. The plot also provid es evidence for the nonlinear coupling between the shear layer and wake inst abilities via the presen ce of the shear layer 119

PAGE 120

frequency SL f and the sum/difference frequencies SLwake f f Note that SL f is much higher than wake f in accordance with classical scaling arguments that ~SL sepfU and ~wake wakefUW. Based on the definition FfcU SL f f gives and ~( FO30)wa ke f f give This evidence supports our hypothesis th at more than a single characteristic frequency exists, perhaps explai ning the wide range of effectiv e forcing frequencies reported in the literature. ~( FO 0.6) To study our hypothesis about the nonlinear quadr atic coupling between the instabilities, higher-order spectral analysis of the same velocity data in Figure 5-19 is performed (Nikias and Petropulu 1993). The auto-bicoherence contour plot shown in Figure 5-20 is bound between 0 and 1 and is only nonzero due to nonlinear quadr atic phase coupling (lock-on). The autobicoherence thus quantifies the fraction of power in a random signal as a function of triad between two frequency components and their sum or difference 1 f 2 f 12 f f A close inspection of the contour plot reveals distinct features at (in particular near 2 w f f 2 ,SLSL,1.5SL f ff), 2 SL f f and 2 SL w f ff (in particular between SL f and 1.5SL f ), and especially along the lines 12 SL f f f and 12 SL w f ff f. These data conclusively show the presence of nonlinear quadratic coup ling between the Kelvin-Helmholtz and wake instabilities. Actuator Calibration Frequency response ZNMF actuator dynamics is a critical issue for the control of a separated flow. A typical ZNMF device contains a cavity a nd a vibrating diaphragm to dr ive oscillatory flow through a small orifice on the cavity. The synthetic jet represents a coupled electro-mechanical-acoustic system with frequency dependent properties de termined by device dimensions and material 120

PAGE 121

properties (Gallas et al. 2003). Gallas et al. use the lumped element modeling approach to model these actuators. In this research, although modeling our ZNMF devices is not needed, we do need to characterize the frequency response of the actuator A1 (that is used in this research). Since the ZNMF actuator is an inherently nonlinear device, the traditional approach that uses a swept sine as an input signal is not appropriate. Instead, a single sine wave is used as input, and the anemometer signal is recorded. The frequency is then in creased in a loop, while the forcing amplitude is held constant. Figure 5-21 shows the rms velocity per input voltage in the frequency band from 500 Hz to 2500 Hz. Three different input leve ls are used. The peak output occurs at approximately 1200 Hz, and significant output is apparent ly limited to a bandwidth of 500-1500 Hz. The output level is very low for fr equencies less than 500 Hz and larger than 1500 Hz. This precludes the possibility of directly forcing either the low-frequency (~40 Hz) wake or high frequency (~2020 Hz) shear layer instabilities via sinusoidal excitation. It also highlights the preferential output of the actuator near its resonance frequency. Furthermore, the nonlinear nature of the actuators is revealed, since the frequency response function is not independent of the input voltage. (If the actuator were linear, th ese curves would collapse.) However, that is this nonlinear behavior that is leveraged to en able forcing at low a nd high frequencies, as explained below. Types of actuation waveforms Three typical multi-modal waveforms are studied to take advantage of the multiple instabilities of a separated flow. They are shown in Figure 5-22 : (a) amplitude, (b) burst and (c) pulse modulation. In (a) and (b ), the lower plot in the figure is the result of a point-by-point product of the top two waveforms. Such forcing is, in general, a modula tion of a (usually) high frequency carrier signal, (e.g., a sine wave with frequency c f ) by a low frequency modulation 121

PAGE 122

signal (either a sine wave or square pulse with frequency m f ). In addition, a parameter is multiplied to determine the amplitude. For the BM signal, there is an additional parameter, the duty cycle, which determines how many sine wave periods occur in each burst In this research, the duty cycle is adjusted such that only one peri od occurs in each burst. The last case in part (c) is a pulse train, which can be interpreted as th e modulation of a constant signal by square pulse. Similar to the BM signal, the duty cycle can be an additional parameter. In this research, it is kept to be the shortest achievable width on th e dSPACE control, i.e. one discrete sample At In this case, there is only one waveform parameter m f to vary. As one moves from amplitude to burst to pulse modulation, the modulation process re sults in an increasingly rich signal spectrum with broader spectral content, which improves the likelihood that the excitation waveform will excite an inherent instability. Furthermore, the required actuator power reduces as will be shown in the following section. Next we study the output of the actuator subject to the AM waveform excitation as an example. Figure 5-23 shows the velocity output of the ZNMF actuator A1 measured by the hotwire anemometry subject to an AM excitation, where A 50 Vpp (peak-to-peak voltage), Hz and Hz. The wire was placed at a sufficient distance (~1mm) above the actuator slot so that there is no reverse flow occurs and, hence, no signal rectificat ion is needed. The temporal record shown in (a) clearly shows the low frequency oscillations at 50 Hz and the high frequency oscillations at 1180 Hz. The corresponding power spect ral density of the velocity signal shown in (b) reveals that the high power is at the low modulation frequency 50 Hz while the second highest power is at the hi gh carrier frequency, with additional harmonic distortion peaks. Recall that th e response to the sinusoidal exc itation at frequencies less than 500 Hz is very low, as shown in 50mf 1180cf Figure 5-21 The modulation enables the actuator to generate high 122

PAGE 123

power signals at low frequencies, while the si nusoidal response is limited by the actuator dynamics. This allows an actuator operating at or near its resonant freque ncy via a carrier signal at c f to generate significant disturbances at charac teristic frequencies of the flow that are far from the natural frequency of the device. This char acteristic is attributed to the nonlinear nature of the actuator system. Similar beha vior is observed for BM and PM. and electrical power calibration C Figure 5-21 showed that the actuator response varies significantly with actuation frequency when subjected to sinusoidal excitation. This is typical for a ZNMF actuator. When the performance of separation control is studied, the actuation frequenc y is always of prime importance. However, when the actuation frequency is varied, the actuator response is also varied even for constant amplitude. As menti oned earlier, it has been shown that sinusoidal authority varies monotonically with rmsu /JVU up to some maximum value (Seifert et al. 1993, 1996, 1999; Glezer and Amitay 2002; Mittal and Ra mpunggoon 2002). In other words, when the actuation frequency is varied, th e two performance-determining parameters (namely frequency and amplitude) are varied simultaneously. Unfortuna tely, this can lead to misleading results. For example, one may find that the performance is the best at the peak response frequency of the actuator, which could be simply because the actuator is providing higher output as opposed to the flow being more receptive as that frequency. The same problem above still exists even when multi-modal waveforms are used; the actuator response varies when Am f and c f are varied. To separate amplitude forcing effects, the following calibration is performed. For th e AM and BM signals, a two-dimensional grid in the ,mc f f space is generated, and C and rms electrical power c onsumed by the actuator are measured for various excitation amplitudes. This time-consuming task takes days to complete 123

PAGE 124

and verify repeatability. The results are shown in Figure 5-24 and Figure 5-25 The profiles for five amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are recorded and shown. The lowest profile is for the 30 Vpp case and the highest profile is for the 50 Vpp case. For the PM signal, there is only one frequency parameter m f ; thus a contour plot in the ,mAf space is shown in Figure 5-26 The amplitudes are again 30 Vpp to 50 Vpp with 5 Vpp increment. There are several important observations in th e results. For the AM signal, the response ( C and electrical power) is approximately independent of the modulation frequency m f but strongly dependent on c f The C at a fixed m f is similar to the sinu soidal velocity response shown in Figure 5-21 In addition, the shape of the el ectrical power surf aces are somewhat different than the shapes of the C and the peak frequencies are different too. This means that when the actuator provides the maximum velocity output, the electrical power consumption is not maximized. For the BM signal, the C and electrical power resp onses are dependent on both m f and c f They monotonically increase with m f as expected. As with the AM signal, the shapes of the C and electrical power surfaces are different and possess different peak frequencies. The profiles are simpler for the PM signal since there is only one frequency parameter. Both the C and electrical power responses increase with m f and as expected. Furthermore, typical levels of AC are and for the AM, BM and PM signals, respectively. 410510610With this information on C and electrical power, additional constraint functionality is added to the adaptive optimization program to hold C or electrical power constant, while m f and c f are varied. This is done by adjusting in accordance with AFigure 5-24 Figure 5-25 and 124

PAGE 125

Figure 5-26 in each iteration of the optim ization algorithm. Thus, it is possible that at certain ,mc f f combinations, constant C or electrical power cannot be achieved. In this case, m f and c f are set to the values at th e boundary such that the moment um or power constraint is maintained. This will hopefully become clearer when the optimization results are discussed in the next section. Adaptive Control Results First, the constrained optimiza tion is carried out at AoA=12 and Re=120,000. At this AoA, the dynamic control approach in our compani on study works well. The constraint is set at = 7.15 x 10-6. This is a relatively low level, which can be achieved in most portions of the C ,mc f f space for the AM and BM signals and all values of m f for the PM signal. To protect the actuator from physical damage, the maximum amplitude is limited to 50 Vpp. The results are summarized in Table 5-3 The lift-to-drag ratios are comp arable with the results using the dynamic control approach (within uncertainties). This is not surprising because the flow is completely attached by both approaches at AoA=12. On the other hand, when the AoA is 20 where the dynamic control a pproach fails to attach the flow, the present nonlinear control shows its clear performance improvement. Therefore, a more detailed study is performed at this AoA. In addition to the constrained cases for C = 7.15 x 10-6, the constrained optimization is also carried out with constant electrical power of 0.0005 W. The constrained optimization results are summarized in Table 5-4 Typical search paths are plotted in Figure 5-27 to Figure 5-29. The shaded areas in Figure 5-27 and Figure 5-28 denote the achievable parameter space, inside which the optimization is c onstrained. It is clear that th e achievable areas cover most of the parameter space for the AM cases while lim ited space for the BM cases. Since there are two 125

PAGE 126

parameters: m f and c f three initial points are needed in each optimization run as mentioned in earlier sections. The symbols denote the search paths fo llowing each initial point during the optimization. On the other hand, for the PM signal, only two initial points are needed. Different sets of initial points are chosen to cover most of the opera tional space to achieve a global optimum. In Table 5-4 the first column indicates if the constraint is active in the optimization process. The last three columns summarize th e optimal values for converged results. The experimental uncertainties are also added to the LD values. By simply observing the L/D values, we find that the performance for the AM case (a) gives the best performance. The AM and BM signal are superior to the PM signal with constant C = 7.15 x 10-6. For the AM and BM cases (a), note that the converged c f is near the shear layer frequency SL f On the other hand, the modulation frequency m f assumes a value near the wake frequency wake f and its superharmonics. Most importantly, effective separation control is achieved by using the multimodal waveforms with C of at least an order-of-magnitude smaller than typical values reported in the literature (Table 2 in Gree nblatt and Wygnanski 2000). Selected sinusoidal excitations are also tested on our airfoil model to compare with the results using the multimodal waveforms. The voltage amplitude for these tests is held at 50 Vpp. The results are summarized in Table 5-5 It is clear that sinusoida l control (even with higher C ) gives poorer performance than the AM and BM excitations. In addition, except the PM signal, the nonlinea r control is able to achieve much better performance than the dynamic contro l in our companion study at AoA=20. The nonlinear 126

PAGE 127

control approach benefits from the nonlinear co upling of the instabilities and the integrated performance (L/D) measurements instead of local unsteady pressure measurements. Table 5-1. Case descriptions and performance for disturbance re jection experiments. (AoA=12 and Re=120,000) Case # Reference transducer yPerformance transducer z L C D C LD Baseline 0.21 0.02 0.21 0.09 1.01 0.08 1 S1 S1 0.84 0.01 0.12 0.01 6.97 0.37 2 S6 S6 0.83 0.01 0.12 0.01 7.21 0.46 3 S1 S6 0.84 0.01 0.12 0.01 7.11 0.40 4 S6 S1 0.84 0.01 0.12 0.01 7.09 0.43 Table 5-2. Summary of parameters in disturbance re jection algorithm. p n cpcnc 1 2 10 1 pn 2 20 Table 5-3. Constrained optimi zation results using the AM, BM and PM signals. (Baseline L/D=1.01 at AoA=12 and Re=120,000) Signal type Constraint Constraint Active? Converged m f Converged c f Converged LD AM C = 7.15 x 10-6 No 74 1667 7.47 0.45 BM C = 7.15 x 10-6 Yes 48 1305 7.63 0.36 PM C = 7.15 x 10-6 No 16 NA 7.14 0.25 127

PAGE 128

Table 5-4. Constrained optimi zation results using the AM, BM and PM signals. (Baseline L/D=1.1 at AoA=20 and Re=120,000) Signal type Constraint Constraint Active? Converged m f Converged c f Converged LD AM: Case (a) C = 7.15 x 10-6 Yes 61 2405 2.18 0.07 AM : Case (b) Power=0.0005 No 202 1005 1.77 0.05 BM : Case (a) C = 7.15 x 10-6 Yes 55 1979 1.95 0.05 BM : Case (b) Power=0.0005 Yes 56 1318 1.52 0.04 PM : Case (a) C = 7.15 x 10-6 No 16 NA 1.49 0.04 PM : Case (b) Power=0.005 No 29 NA 1.48 0.04 Table 5-5. Results using sinusoidal excitations. /sepF f LU C LD 0.5 ~0 1.14 0.02 15 3.16 x 10-4 1.76 0.03 26 7.79 x 10-6 1.77 0.03 128

PAGE 129

Figure 5-1. NACA 0025 airfoil m odel with actuators, sensors a nd closed-loop control system. Figure 5-2. Phase averaged pulse response m easured by six pressure sensors. The slow propagation velocity of the coherent flow structures is clearly visible. 129

PAGE 130

Figure 5-3. Comparison between measured signal from the pressure sensor (#6 in Figure 5-1 ) and the fitted output by ARMARKOV system ID algorithm for long and short time intervals. Results show a reasonable match at low frequencies between measured and fitted outputs. For ARMARKOV ID: p=1, n=2, =10. Figure 5-4. Mean Squared Error (Running MSE) between measured and fitted outputs. Results show that the ARMARKOV ID algorithm converges, i.e. error being minimized. 130

PAGE 131

Figure 5-5. Comparison between frequency re sponse (FR) and fitted response by ARMARKOV ID algorithm. Parameters for FR: s f =4096 Hz, NFFT=1024, 75% overlap and Hanning window. For ARMARKOV system ID: p=1, n=2, =10. Figure 5-6. Dual signal paths from the actuator to the pr essure sensor (acoustic and hydrodynamic). A digital filter is introduced to remove the acoustic component by turning off the flow to isolate the acoustic path. Figure 5-7. Actual measured and predicted acoustic noise using a band-limited random signal to the actuator. Figure 5-8. Actual measured and predicted acoustic noise using the same filter as in Figure 5-7 but with one half of the input amplitude. 131

PAGE 132

Figure 5-9. Power spectra of the sensor signa ls (with wind tunnel runn ing) before and after applying acoustic filter. Figure 5-10. Performance surface pressure (S1) and control input signals (in Volt) before and after the ID and control is initiated for case #2. Control is established within 1 second or <100 convective time scales. Figure 5-11. Power spectra of the pressure transducer output for the baseline and the closed-loop control cases measured by S6 (performance). 132

PAGE 133

Figure 5-12. Contours of streamwise velocity uU for (a) baseline and (b) closed-loop control case #2 at AoA = 12 and Rec=120,000. Figure 5-13. Contours of vorticity for (a) baselin e and (b) closed-loop control case #2 at AoA = 12 and Rec=120,000. Figure 5-14. (a) Voltage and (b) electrical power spectra of the actuator A1 input signal for the closed-loop control case. 133

PAGE 134

Figure 5-15. Performance comparison at different AoA. Figure 5-16. Power spectra of the pressure transducer output for the baseline and the closed-loop control cases measured by S6 (performance) at AoA = 20 and Rec=120,000. Figure 5-17. NACA 0025 airfoil model with actuators, sensors and closed-loop control system. 134

PAGE 135

135 Figure 5-18. Flow structures in separated flow. Figure 5-19. Wake (1 chord aft of TE) and shea r layer (near separation) power spectral density functions at peak rms location.

PAGE 136

Figure 5-20. Auto-bicoherence of the same velocity signal analyzed in Figure 5-19 using the same parameter settings. The auto-bicohe rence is zero except where nonlinear phase quadratic phase coupling occurs due to inte ractions between the shear layer and wake instabilities. Figure 5-21. Frequency response of ZNMF actuator A1. 136

PAGE 137

()sin2sin2cmetAftft (a) amplitude modulation (AM) ()sin2square pulsecetAft (b) burst modulation (BM) () square pulse etA (c) pulse modulation (PM) Figure 5-22. Various waveforms of unit amplitude 1A that can be used to excite multiple instabilities or modes in a separated flow. Figure 5-23. Velocity response (a ) and its power spectral density (b ) subject to an AM excitation for the ZNMF actuator A1. A 50 Vpp 50mf Hz and 1180cf Hz. Measurements were made outside the region of reverse flow. Figure 5-24. C (a) and electrical power (b) profile s ubject to AM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. 137

PAGE 138

Figure 5-25. C (a) and electrical power (b) profile s ubject to BM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. Figure 5-26. C (a) and electrical power (b) profile s ubject to PM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. 138

PAGE 139

139 Figure 5-27. Constrained search using AM: Ca ses (a) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. Figure 5-28. Constrained search using BM: Ca ses (a) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. Figure 5-29. Constrained search using PM: Cases (a) and (b). AOA = 20 deg. and Re = 120,000.

PAGE 140

CHAPTER 6 SUMMARY AND FUTURE WORK An adaptive system identification and feedb ack control algorithm is applied to the separation control problem for a NACA 0025 airf oil at nominal angles of attack of 12 and and a chord Reynolds number of 120,000 with a tr ipped boundary layer, corresponding to control of a massively leading-edge se parated flow. In particular, a recursive ARMARKOV system ID algorithm is used to model the flow dynamics a nd provide the informati on required to implement the disturbance rejection algorithm in real time with no prio r knowledge of the system dynamics. Phase-locked PIV and fluctuating surface pressure measurements provided evidence of the link between the separated flow vertical structures and the surface pressure fluctuations. The chosen control objective was thus to suppress the airfoil surface pressure fluctuations. The disturbance rejection algorithm was able to automatically generate contro l input to the ZNMF actuator, emphasizing low (i.e., wake) and high (i.e., shea r layer) characteristic frequencies of the separated flow. The effect of the control is to enhance near-w all mixing and suppress the highly unsteady flow structures. This adaptive control sc heme is able to completely reattach the flow using low (~12.7 mW) power to a single piezoel ectric synthetic jet actuator. The closed-loop control results show ~ 7 x improvements in th e lift/drag ratio, with a corresponding increase in lift and reduced drag and concomitant reductions in the fluctuating surface pressure spectra. The present results are, to the best of our knowledge, the first expe rimental demonstration of an adaptive dynamic feedback contro l of a separated flow. The results reveal the tremendous potential of closed-loop fl ow control to real aircraft applic ations but also reveal key issues worthy of further study. 20First, in terms of positives, the adaptive cl osed-loop control scheme has several attractive features. It is quite general, and no prior knowledge of the system dynamics is required. The 140

PAGE 141

system identification and disturbance rejecti on algorithms are integrated, and the system dynamics can be obtained with minimal a priori user knowledge. The cont roller is implemented using DSP hardware and can be easily incorporated in hardware-in-the-loop applications. It can be applied to not only flow separation control pr oblems, but also, for example, cavity oscillation control and turbulent bounda ry layer control. Second, in terms of unresolved technical issu es, there remain many concerning the actuator and sensor dynamics. Clearly, better actuators with not just higher out put but flatter dynamic response over wider frequency range are desirabl e. While measuring un steady surface pressure is relatively straightforward, the potential ac oustic contamination issue was highlighted. This difficulty was mitigated with an acoustic digita l filter here but at the cost of additional computational complexity from an already limited DSP. This issue suggests the use of alternate surface sensors, such as MEMS-based direct shear stress sensors for feedback instead of pressure sensors. Or perhaps thermal sensors may be suffici ent despite their sensitivity to more than just shear stress. Ultimately, it is believed that th e key limitation of the present scheme, as evidenced by its failure at higher angles of attack, incl ude the assumption of linearity in the system identification and disturbance re jection algorithm. The second pa rt explores nonlinear control methods. In the post-stall separated flow where th e flow does not reattach, there are two characteristic instabilities: the shear layer Kelvin -Helmholtz instability and wake instability. Our experiments have the evidence for such instabili ties. In addition, the second order spectral analysis has quantified the quadratic phase coupling between the two instabilities, which indicates the separated flow is a complex multi-frequency system. 141

PAGE 142

Three multi-modal waveforms (namely amplitude modulation, burst modulation and pulse modulation) are used targeting excitation of the multi-frequency separated flow system. A simplex optimization approach for controlling the separated flow has been developed to search for the optimal actuation parameters of the thre e waveforms using the ZNMF devices. It is typical for the C response to vary when the waveform frequency parameters vary for the ZNMF devices. This can potentially lead to misleading results about the optimal forcing frequency. The actuator dynamics is taken into consideration in the optim ization approach. To offset the actuator dynamics implications, a special routine is devised to hold the C at constant during the optimization process utilizing the pre-calibrated actuator response profiles. This is specifically done by varying the actuation voltages according to the response profiles keep C to the at constant levels. The constrained optimization results seeking to maximize LD are promising and reveal the importance of forcing nonlinear interactions between the shear layer and wake instabilities. Effective separation control is achieved by using oscillat ory momentum coefficients which is more than an order-of-magnit ude smaller than test cases on our model with sinusoidal excitations and typical values reported in the literature (see summary in Greenblatt and Wygnanski 2000). Specifica lly, the optimized carrier frequency 651010 Oc f targets the shear layer frequency while the optimized modulation frequency m f targets the wake frequency and its super-harmonics. The nonlinear control is able to achieve similar performance at AoA=12 and much better performance than the dynamic control in our companion study at AoA=20. The nonlinear control approach benefits from the nonlinear coupling of the flow 142

PAGE 143

143 instabilities and the integrated performance (L/D) measuremen ts instead of local unsteady pressure measurements. Future direction includes exploring nonlinear dynamic control strategies and implementing feasible penalties in the cost functions.

PAGE 144

APPENDIX HIGHER ORDER SPECTRUM EXAMPLES This chapter gives an illustration of the appl ication of higher order spectral analysis on systems with quadratic a nd cubic nonlinearities. Example 1: Quadratic System Second order systems are tested as examples to analyze nonlinear couplings using the bispectral methods. The two systems that are tested are 2 y xxn and 2 y xn where x is input, y is output and is additive white noise. nCase 1 The quadratic nonlinear system in Figure A-1 is used as the test model. In the model () x t is input, () y t is output and is uncorrelated additive white noi se. Note there is a linear part in this model. () nt Input is a pure 15 Hz sinusoid signal: ()sin(215) x tAt (4.8) where is one. A The parameters used to compute the auto-spectrum are: Fs=128, NFFT=1024, Hanning window, 75% overlap. The parameters used to compute the auto-bicoherence and crossbicoherence are: Fs=128, NFFT=256, Hanni ng window, 75% overlap. Averaging using 100 blocks is used for all cases. The power spec trum of the output y, auto-bicoherence of y and cross-bicoherence between x a nd y are plotted below. Based on trigonometric identities, the output can be expressed as follows: 144

PAGE 145

2sin(215)sin(215) 1cos(2215) sin(215) 2 11 sin(215)cos(230) 22yttn t t tt n n (4.9) Based on this, there should be peaks at 15 and 30 Hz. The power spectrum plot in Figure A-2 confirms this. Figure A-3 shows the auto-bicohere nce of y. There is a strong peak at (15, 15), which means that the 15 Hz signal interacts with itself to generate the 30 Hz signal that is shown in the power spectrum plot. The cross-bicoherence shown in Figure A-4 also shows this interaction. Furthermore, the cr oss-bicoherence recovers that the 30 Hz signal also interacts with the original signal to generate additional 15 Hz signal (i.e. 1215 f fH z ). Case 2 The nonlinear model in Figure A-5 is also tested. Compared with the case 1, there is no linear part in this model. The input and additive white noise are the same as the last case, as are the parameters for spectral analysis. The plot of power spectrum is as expected, w ith dc and 30 Hz components. On the other hand, the auto-bicoherence of y is close to zero at all points ( Figure A-7 ). This is reasonable since there is only one frequency com ponent in the output y and from the definition, the auto-bicoherence represents quadratic coupling between two frequency components. This is also the case for the cross-bicoherence as shown in Figure A-8 Example 2: Cubic System From the definitions in Chapter 4, the bispectr al analysis calculates quadratic coupling. This works as expected with quadratic systems as shown above. A question naturally arises. Does it work with cubic or higher-order nonlineariti es? This section uses the third order system in Figure A-9 as the test model. 145

PAGE 146

All the signals and processing para meters are the same as for the previous cases. Ttrigonometric identities as follows: 3sin(15*2)sin(15*2) 31 sin(15*2)sin45*2 44 ytt tt n n (4.11) The power spectrum of y verifies that ther e are two peaks at 15 Hz and 45 Hz, which agrees with the above equation. It is also clear that the auto-bicoherence of y and crossbicoherence between x and y are very close to zero ( Figure A-11 and Figure A-12 ). This result shows that the bi-spectral analysis is not able to reveal cu bic nonlinearities. Example 3: x x The system in Figure A-13 is also tested. By means of least mean square estimation, this system can be approximated by a cubic system: 31632 1515 A y xxxx A (4.12) Substituting eqn. (7.1) into eqn. (7.5) y takes the form as follows: 81 sin(215)sin245 35 y t t (4.13) All the signals and processing parameters are th e same as for the previous cases. In the plot of the power spectrum, there are several peaks. This indicates that the system is much more complicated than the previous cases. The auto -bicoherence and the cros s-bicoherence are very close to zero ( Figure A-15 and Figure A-16 ), which indicates that the bispectral analysis does not work for this system. This is expected as the system acts like a cubic system according to eqn. (7.6) 146

PAGE 147

Figure A-1. Test model #1 for higher order spectrum analysis. 0 10 20 30 40 50 60 70 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power Spectrum of y f (Hz) Figure A-2. Power spectrum of y for test model #1. 147

PAGE 148

0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)Auto-Bicoherence of y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure A-3. Auto-bicoherence of y for test model #1. f1 (Hz)f2 (Hz)Cross-Bicoherence between x and y 0 10 20 30 40 50 60 -60 -40 -20 0 20 40 60 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure A-4. Cross-bico herence between x and y for test model #1. 148

PAGE 149

Figure A-5. Test model #2 for higher order spectrum analysis. 0 10 20 30 40 50 60 70 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power Spectrum of y f (Hz) Figure A-6. Power spectrum of y for test model #2. 149

PAGE 150

0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)Auto-Bicoherence of y 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Figure A-7. Auto-bicoherence of y for test model #2. f1 (Hz)f2 (Hz)Cross-Bicoherence between x and y 0 10 20 30 40 50 60 -60 -40 -20 0 20 40 60 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 Figure A-8. Cross-bico herence between x and y for test model #2. 150

PAGE 151

Figure A-9. Test model #3 for higher order spectrum analysis. 0 10 20 30 40 50 60 70 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power Spectrum of y f (Hz) Figure A-10. Power spectrum of y for test model #3. 151

PAGE 152

0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)Auto-Bicoherence of y 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Figure A-11. Auto-bicoherence of y for test model #3. f1 (Hz)f2 (Hz)Cross-Bicoherence between x and y 0 10 20 30 40 50 60 -60 -40 -20 0 20 40 60 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 Figure A-12. Cross-bicoherence be tween x and y for test model #3. 152

PAGE 153

Figure A-13. Test model #4 for hi gher order spectrum analysis. 0 10 20 30 40 50 60 70 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power Spectrum of y f (Hz) Figure A-14. Power spectrum of y for test model #4. 153

PAGE 154

154 0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)Auto-Bicoherence of y 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Figure A-15. Auto-bicoherence of y for test model #4. f1 (Hz)f2 (Hz)Cross-Bicoherence between x and y 0 10 20 30 40 50 60 -60 -40 -20 0 20 40 60 0.02 0.03 0.04 0.05 0.06 0.07 Figure A-16. Cross-bicoherence betw een x and y for test model #4.

PAGE 155

LIST OF REFERENCES Akers, J. C. and Bernstein, D. S., ARMARKOV Least-Squares Identification, Proceeding of the American Control Conference, pp. 191-195, New Mexico, June 1997A. Akers, J. C. and Bernstein, D. S., TimeDomain Identification Using ARMARKOV/Toeplitz Models, Proceeding of the American Control Conference, pp. 186-190, New Mexico, June 1997B. Allan, B., Juang, J., Seifert A., Pack L. and Brown, D, Closed-loop Separation Control Using Oscillatory Flow, ICASE Report No. 2000-32. Amitay, M, and Glezer, A., Controlled transients of flow reattachment over stalled airfoils, Heat and Fluid Flow vol 23, pp 690 699, 2002. Amitay, M., Smith, D., Kibens, V., Rarekh, D. and Glezer A., Aerodynamic Flow Control over an Unconventional Airfoil Using Synthetic Jet Actuators, AIAA Journal, Vol. 39 No.3, pp 361-370, March 2001. Artiyur, K. B. and Krstic, M ., Real-Time Optimization by Extremum-Seeking Control, WileyInterscience, 2003. Ausseur, J.M., Pinier, J.T., Gl auser, M.N and Higuchi, H., Controller Development for ClosedLoop Feedback Control of Flows 35th AIAA Fluid Dynamics Conference and Exhibit, AIAA-2005-5264, Toronto, Canada, 2005. Ausseur, J., Pinier, J. and Glauser, M, Flow Separation Control Using a Convection Based POD Approach, 3rd AIAA Flow Control Conference, AIAA-2006-3017 San Francisco, California, 2006. Banaszuk, A., Narayanan S. and Zhang Y., Adaptive Control of Flow Se paration in a Planar Diffuser, AIAA paper 2003-0617, 2003. Bendat, J.S. and Piersol, A.G., Random Da ta: Analysis & Measurement Procedures, 3rd ed., Wiley-Interscience, 2000. Becker, R., King, R., Petz, R. and Nitsche, W., Adaptive Closed-Loop Separation Control on a High-Lift Configuration us ing Extremum Seeking, AIAA 2006-3493, June 2006. Camacho, E.F. and Bordons, C., Model Predicti ve Control in the Process Industry, SpringerVerlag Berlin Heidelberg New York, 1995. Cattafesta, L.N., Garg, S., Choudhari, M. and Li, F., Active Control of Flow Induced Cavity Resonance, AIAA 1804, 1997. Cattafesta, L. N., Shulkla, D., Garg, S. and Ro ss, J.A., Development of an Adaptive WeaponsBay Suppression System, AIAA99 1901, 1999. 155

PAGE 156

Cattafesta, L.N., Garg, S. and Shukla, D., Dev elopment of Piezoelectr ic Actuators for Active Flow Control, AIAA Journal Vol. 39, No. 8, Aug. 2001. Clarke, D.W., D. Phil, M. A., Gawthrop, P. J. and D. Phil, B. A., Self-Tuning Control, IEEE Proc ., Vol. 126, No. 6, June 1979. Clarke, D.W., Mhtadi C. and Tuffs, P.S., Gener alized Predictive Cont rol-Part I. The Basic Algorithm, Automatica Vol. 23, No. 2, pp. 137-148, 1987A. Clarke, D.W., Mhtadi C. and Tuffs, P.S., Generali zed Predictive Control-Part II. Extensions and Interpretations, Automatica Vol. 23, No. 2, pp. 149-160, 1987B. Clarke, D.W.,Application of Generalized Predictive control to industrial Processes, IEEE Control Systems Magazine pp. 49-55, April 1988. Coller, B., Noack, B., Narayanan, S., Banaszuk, A. and Khibnik, A., Reduced-Basis Model for Active Separation Control in a Planar Diffuser Flow, AIAA 2000-2562, 2000. Crook, A., Sadri, A. M. and Wood, N. J., The Development and Implementation of Synthetic Jets for the Control of Separa ted Flow, AIAA-99, 1999. Cutler, C.R, and Ramaker, B.L., Dynamic Matr ix ControlA Computer Control Algorithm, Proc. JACC San Francisco, WP5-B, 1980. Gad-el-Hak M., Flow Control: Passive, Active, and Reactiv e Flow Management, Cambridge, 2000. Gallas, Q., Holman, R., Nishida, T., Carroll, B., Sheplak, M., and Cattafesta, L., Lumped Element Modeling of Piezoelectric-Driven Synthetic Jet Actuators, AIAA Journal Vol. 41, No. 2, pp. 240-247, 2003. Gallas, Q., Ph.D Thesis, University of Florida, 2005. Golub, G. H. and Van Loan, C. F., Matrix Computations, 3rd ed., Baltimore, MD: Johns Hopkins, 1996. Greenblatt, D and Wygnanski, I, Dynamics Stal l Control by Periodic Exc itation, Part 1: NACA 0015 Parametric Study, Journal of Aircraft Vol. 38, No. 3, MayJune 2001A. Greenblatt, D and Wygnanski, I, Dynamics St all Control by Periodic Excitation, Part 1: Mechanisms, Journal of Aircraft Vol. 38, No. 3, MayJune 2001B. Greenblatt, D and Wygnanski, I, Effect of Leading-Edge Curvature on Airfoil Separation Control, Journal of Aircraft Vol. 40, No. 3, May-June 2003. Greenblatt, D and Wygnanski, I, The contro l of flow separation by periodic excitation, Progress in Aerospace Sciences Elsevier Science Ltd, 36:487-545, 2000. 156

PAGE 157

Greenblatt, D., Nishri, B., Dara bi A. and Wygnanski, I. Some Factors Affecting Stall Control with Particular Emphasis on Dynamic Stall, AIAA-99-3504 June-July 1999. Griffin, B., Senior Thesis, 2003. Haftka, R.T. and Grdal, Z., Elements of Struct ural Optimization, 3rd edition, Kluwer, 1992. Halfon, E., Nishri, B., Seifer t, A. and Wygnanski, I., Eff ects of Elevated Free-Stream Turbulence on Actively Controlled Separation Bubble, Journal of Fluids Engineering, Vol. 126, P 1015 1024, Nov. 2004. Hajj, M. R., Miksad, R. W. and Powers, E.J., Perspective: Measurements and Analyses of Nonlinear Wave Interactions with Higher-Order Spectral Moments, Journal of Fluids Engineering, v119, Mar. 1997, pp 3-13. Haykin, S., Adaptive Filter Theory, 4th edition, Prentice Hall, 2002. Ho, C. M. and Huerre, P, Perturbed Free Shear Layers, Annual Review Fluid Mechanics 16: 365-424, 1984. Holland, J., Adaptation of Natural and Artificial Systems, The University of Michigan Press, Ann Arbor, MI, 1975. Holman, R. and Gallas, Q., Carrol B. and Cattafe stta, L.N., Interaction of Adjacent Synthetic Jets in An Airfoil Separation Application, AIAA paper 2003-3709, June 2003. Holmes, P., Lumley, J.L. and Berkooz, G., Turbul ence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Univer sity Press, Cambridge, 1998. Hong, J. and Bernstein, D., Bode Integral C onstraints, collocation, and Spillover in Active Noise and Vibration control, IEEE Transactions on Cont rol Systems Technology vol. 6, No.1, January, 1998. Honohan, A. M., Amitay, M., and Glezer, A., Aerodynamic Control Using Synthetic Jets, AIAA-2000-2401, 2000. Hsiao, F.-B., Liu, C.F., and Shyu, J.-Y., Control of Wall-Separated Flow by Internal Acoustic Excitation, AIAA Journal Vol. 28, No. 8, pp 1440-1446, 1990. Huang, L.S., Maestrello, L. and Bryant, T.D., S eparation Control over an Airfoil at High Angle of Attack by Sound Emanating from the Surface, AIAA Paper 87-1261 1987. Huerre, P. and Monkewitz, P., Local and Global Instabilities in Spatia lly Developing Flows, Annual Review Fluid Mechanics, 22: pp 473-537, 1990. Johnson, C. R., and Larimore, M. G., "Comments on and Additions to 'An Adaptive Recursive LMS Filter'", Proc. of the IEEE Vol. 65, No. 9, pp 1399-1402, Sep. 1977. 157

PAGE 158

Jrgensen, F. E., The Computer-Controlled Constant-Temperature Anemometer. Aspects of Set-up, Probe Calibration, Data Acquisition and Data Conversion, Meas. Sci. Technol. 7 pp 1378-1387, 1996. Juang, J., Applied System Identificati on, Prentice-hall, New Jersey, 1994. Juang, J.N. and Phan, M.Q., Deadbeat Predictive Controllers, NASA TM 112862, May, 1997. Juang, J.N. and Phan, M.Q., Identification a nd Control of Mechanical Systems, Cambridge University Press. Kegerise, M. A., Spina, E. F., Garg, S. a nd Cattafesta, L., Mode-Switching and Nonlinear Effects in Compressible Flow over a Cavity, Physics of Fluids, v16 n3, Mar. 2004, pp 678 687. Kegerise, M.A., Cattafesta L.N. and Ha, C., A daptive Identification and Control of FlowInduced Cavity Oscillations, AIAA Paper 2002-3158, June 2002. Kennedy, J., The particle swarm: social adaptation of knowledge, Proc. Intl. Conf. on Evolutionary Computation Indianapolis, IN, 303308. Piscataway, NJ: IEEE Service Center, 1997. Kumar, V. and Alvi, F. S., Efficient Contro l of Separation Using Mi crojets, AIAA Paper 20054879, June 2005. Kuo, S.M. and Morgan, D.R., Active Noise Control: A Tutorial Review, Proceedings of the IEEE, Vol. 87, No. 6, June 1999. Kuo, S.M. and Morgan, D.R., Active Noise Control Systems New York: Wiley, 1996. Kuo, S.M., Kong X. and Gan W. S., Application of Adaptive Feedback Noise Control System, IEEE Transactions on Control Systems Technology, Vol. 11, No. 2, March 2003. Larimore, M. G., Johnson, C. R., and Shah, N., Theory and Design of Adaptive Filters, Pearson Education, Mar. 2001. Ljung, L., System Identification: Theory for th e User, Prentice Hall PTR, Upper Saddle River, New Jersey, 1987. Lee, K.H., Cortelezzi, L., Kim J. and Speyer J ., Application of reduced -order controller to turbulent flows for drag reduction, Physic s of Fluids, Vol. 13, NO. 5, 2001. Margalit, S., Greenblatt, D., Seifert, A., and Wy gnanski, I., Active Flow Control of a Delta Wing at High Incidence using Segmented Piezoelectric Actuators, AIAA-2002-3270, 2002. 158

PAGE 159

Mittal, R., Kotapati, B. and Cattafesta, L., Num erical Study of Resonant Interactions and Flow Control in a Canonical Separated Flow, AIAA paper 2005-1261 Reno, Nevada, Jan. 2005. Mittal, R. and Rampunggoon, P., On the Virtual Aero-Shaping Effect of Synthetic Jets, Phys. Fluids vol. 14, no. 4., pp. 1533-1536, 2002. Morari, M. and Ricker, N.L., M odel Predictive Control Toolbox, Users Guide The MathWorks, Version 1, October, 1998. Narayanan, S. and Banaszuk A., Experimental Study of a Novel Active Separation Control Approach, AIAA paper 2003-60, Reno, Nevada, Jan. 2003. Netto, S. L., and Diniz, P. S. R, "Composite Algorithms for Adaptive IIR Filtering", Electronics Letters Vol. 28 No. 9, p. 886-888, 23rd Apr. 1992. Netto, S. L., and Diniz, P. S. R., "Adaptive IIR Filtering Algorithm for System Identification: A General Framework", IEEE Transactions on Education Vol. 38, No. 1, Feb. 1995. Nishri, B. and Wygnanski I., E ffects of Periodic Excitation on Turbulent Flow Separation from a Flap, AIAA Journal Vol. 36, No.4, April 1998. Nikias, C.L. and Mendel, J.M., Signal Processing with Higher-Order Spectra, IEEE Signal Processing Magazine, 1053-5888, July 1993. Nikias, C. L. and Petropulu, A. P., Higher-O rder Spectra Analysis A Nonlinear Signal Processing Framework, Prentice-Hall Upper Saddle River, NJ, 1993. Ol, M., McCauliffe, B., Hanff, E., Scholz, U ., and Kaehler, C. C omparison of Laminar Separation Bubble Measurements on a Low Reynolds Number Airfoil in Three Facilities AIAA-2005-5149, June 2005. Pack, L., Schaeffler, N., Yao, C., and Seifert, A., Active Control of Flow Separation from the Slat Shoulder of a Supercritical Airfoil, AIAA-2002-3156, 2002. Pinier, J.T., Ausseur, J.M., Glauser, M.N and Higuchi, H., Proportional Closed-loop Feedback Control of Flow Separation, AIAA Journal, in review. Pillarisetti, A. and Cattafesta, L. N., Adaptiv e Identification of Fluid-Dynamic Systems, AIAA Paper 2001-2978, June 2001. Prandtl, L. Proceedings of Third Internationa l Mathematical Congress, Heidelberg, pp 484-491, 1904. Prasad, A. and Williamson, C., The Instability of the Separated Shear Layer from a Bluff Body, Physics of Fluids, Vol. 8, No.6, pp 1347-1349, June 1996. 159

PAGE 160

Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., Numerical Recipes in Fortran, 2nd edition (January 15, 1992), Cambridge University Press. Raju, R., Mittal, R., and Cattafesta, L., Towards Physics Based Strategies for Separation Control over an Airfoil using Synthetic Jets, AIAA Paper 2007-1421, January 2007. Rathnasingham, R. and Breuer K., Active Control of Turbulent Boundary Layers, Journal of Fluid Mechanics, Vol 495, pp. 209-233, 2003. Rathnasingham, R. and Breuer K., System Iden tification and Control of Turbulent Boundary Layer, Phys. Fluids, pp1867-1869, July 1997. Roshko, A. On the Development of Turbulen t Wakes from Vortex Streets, NACA Report 1191, 1954. Rouhani, R., and Mehra, R.K., Model Algor ithmic Control (MAC), Basic Theoretical Properties, Automatica Vol. 18, No. 1, pp.401-414, 1982. Sane, H.S., Venugopal, R. and Bernstein, D.S ., Disturbance Rejection Using Self-Tuning ARMARKOV Adaptive Control with Simultaneous Identification, IEEE Transactions on Control Systems Technology, Vol. 9, No. 1, January 2001. Schubauer, G.B. and Skramstad H.K., Laminar boundary layer oscillations and transition on a flat plate, NASA Report 909, 1948. Seifert, A., Darabi, A. and Wygnanski, I., D elay of Airfoil Stall by Periodic Excitation, Journal of Aircraft Vol. 33, No. 4, JulyAugust 1996. Seifert, A. and Pack, L. G., Compressibility and Excitation Location Effects on High Reynolds Numbers Active Separation Control, Journal of Aircraft Vol. 40, No. 1, Jan. Feb. 2003 A. Seifert, A. and Pack, L. G., Effects of Sweep on Active Sepa ration Control at High Reynolds Numbers, Journal of Aircraft, Vol. 40, No. 1, Jan. Feb. 2003 B. Seifert, A., Pack, L. G., Active Flow Separation Control on Wall-Mounted Hump at High Reynolds Numbers, AIAA Journal Vol. 40, No.7, July 2002. Seifert, A., Pack, L. G., Oscillatory Cont rol of Shock-Induced Separation, Journal of Aircraft Vol. 38, No. 3, May June 2001. Seifert, A., Pack, L. G., Separ ation Control at Flight Reynolds Numbers: Lessons Learned and Future Directions, AIAA paper 2000-2542 June 2000. Seifert, A., Pack, L. G., O scillatory Control of Separation at High Reynolds Numbers, AIAA Journal, Vol. 37, No.9, pp 1062-1071, Sep. 1999. 160

PAGE 161

Sheplak, M., Cattafesta, L., and Tian, Y. Micro machined Shear Stress Sensors for Flow Control Applications, IUTAM Symposium on Fl ow Control and MEMS, London, September 2006. Shynk, J. J., "Adaptive IIR Filtering", IEEE ASSP Magazine p. 4-21, Apr. 1989. Sinha, S.K., Flow Separation Control w ith Microflexural Wall Vibrations, Journal of Aircraft Vol. 38, No.3, May-June 2001. Soderstrom, T. and Stoica, P., System Iden tification, Prentice-Hall, New York, 1989. Song, Q., Tian, Y. and Cattafesta, L., MIMO Feedback Control of Flow Separation, AIAA Paper 2007-0109, Jan. 2007. Tadmor, G., Centuori, M., Noack, B., Luchtenburg, M., Legmann, O. and Morzy ski, M., Low Order Galerkin Models for the Actuated Flow Around 2-D Airfoils, AIAA 2007-1313, January, 2007. Tian, Y., Cattafesta, L., and R. Mittal, Ada ptive Control of Separated Flow, AIAA Paper 2006-1401, Jan. 2006. Tian, Y., Song, Q., and Cattafesta, L., "Adaptiv e Feedback Control of Flow Separation," 3rd Flow Control Conference, San Fran cisco, CA, AIAA-2006-3016, June 2006. Venugopal, R. and Berstein D.S., Adaptive Di sturbance Rejection Using ARMARKOV System Representation, Proceeding of the 36th Conference on Decision & Control pp1884-1889, Dec. 1997. Wang, Y., Haller, G., Banaszuk, A. and Tadm or, G., Closed-Loop Lagrangian Separation Control in a Bluff B ody Share Flow Model, Physics of Fluids, Vol. 15, No. 8, P 2251 2266, August, 2003. White, F. M., Viscous Fluid Flow, 2nd edition, McGraw-Hill, 1991. Wiltse, J. N. and Glezer, A., Manipulation of Free Shear Flows Using Piezoelectric Actuators, Journal of Fluid Mechanics, vol. 249 pp. 261-285, 1993. Wu, J. -Z., Lu, X. -Y., Denny, A. G., Fan, M., and Wu, J. -M. Post-Stall Flow Control on an Airfoil by Local Unsteady Forcing, J. Fluid Mech. vol. 371, pp. 21-58, September 1998. Wygnanski, I., Some New Observations Affec ting the Control of Separation by Periodic Forcing,AIAA-2000-2314, 2000. Zhou, Kemin, Essentials of robust contro l, Prentice-Hall, New Jersey, 1998. 161

PAGE 162

162 BIOGRAPHICAL SKETCH Ye Tian was born in 1978 in Chengdu, China. After graduating from Jintang high school in 1997, he went to the Beijing University of Aeronautics and Astronautics, where he got his Bachelor of Engineering specializing in auto matic control in 2001. Then he came to the University of Florida as a Ph.D. candidate in the Department of Mechanical and Aerospace Engineering. Under the guidance of Dr. Louis Cattafesta, he was engaged in research in flow control applications funded by NASA and AFOSR. He obtained his Master of Science in 2004 and his Ph.D. in 2007.