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CLOSEDLOOP CONTROL OF FLOWINDUCED CAVITY OSCILLATIONS By QI SONG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 2008 Qi Song To my wife, Jingyan Wang; and my lovely son, Lawrence W. Song ACKNOWLEDGMENTS This study was performed while I was a member of the Interdisciplinary Microsystems Group (IMG) in the Department of Mechanical and Aerospace at the University of Florida in Gainesville, Florida, USA. First, I sincerely acknowledge my advisor, Dr. Lou Cattafesta, for providing me with this opportunity and giving me so much precious advice during my course time at UF. His guidance and encouragement always gave me sufficient confidence to conquer any difficulty. I thank all of my colleagues in the IMG group for their invaluable assistance. Finally, I appreciate my friends and my dear family for their tremendous consideration and unselfish support during my journey. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IS T O F T A B L E S .............................................................................. ............... 8 LIST OF FIGURES .................................. .. ..... ..... ................. .9 LIST OF ABBREVIATION S ......... ............................................ .... .................. 14 A B S T R A C T ................................ ............................................................ 16 CHAPTER 1 INTRODUCTON ............................... ... .. .... ..... ................. 18 L iteratu re R ev iew .............................................................................2 0 Physical M odels................................................... 21 PhysicsBased Models.................. .. ................................21 Num erical Simulations ......................... ........................ ......... 27 P O D T ype M odels................ .... .... ...... .................... ............ ........ .. .......... ...... 28 OnLine System ID and Active ClosedLoop Control Methodologies...........................28 U resolved T technical Issues ........................................ ............................................33 Technical Objectives ............................... .. .. ... .... ................... 33 Approach and Outline ............... ................. .............................. ....... 34 2 SYSTEM IDENTIFICATION ALGORITHMS ....................................... ............... 38 O v erv iew ................... ...................3...................8.......... SISO IIR F ilter A lgorithm s .......................................................................... ....................39 IIR O E A lgorithm ........................................................................ .......... .. 40 IIR EE Algorithm ..................................... .............................41 IIR SM Algorithm ..................................... .. .... ...... .. ............41 IIR C E A lgorithm ............... .... .... ........ .. .......................................... ................. 4 1 Recursive IIR Filters Simulation Results and Analyses............................................41 Accuracy comparison for sufficient system.......................................................43 Accuracy comparison for insufficient system............................. ....................43 Convergence rate ............................................... .. ......................... 44 C om putational com plexity ............................................... ............................ 44 C o n c lu sio n s ..............................................................................4 5 M IM O IIR F ilter A lgorithm ......................................................................... ....................46 3 GENERALIZED PREDICTIVE CONTROL ALGORIHTM ............................................62 Introduction ................... .......... ...... ...........................................62 M IM O A adaptive G P C M odel ........................................................................ ...................63 MIMO Adaptive GPC Cost Function ........... .. ........ ............................ 66 M IM O A adaptive G PC Law ..................................................... ................................... 66 M IM O A adaptive GPC Optim um Solution ........................................... .....................67 MIMO Adaptive GPC Recursive Solution................................... ...............68 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES ...........................................71 Schematic of the Vibration Beam Test Bed ............... .............. .....................71 System Identification Experimental Results........................................... .......................... 72 C om putational C om plexity ..................................................................... ..................72 Sy stem Id entification ............................................................................. .................... 7 3 Disturbance Effect ...................... .............. .... ................ 74 ClosedLoop Control Experiment Results.. .. ......................... .....................74 C om putational C om plexity ..................................................................... ..................74 ClosedLoop Results ......................................... .... .. ..... .............. .. 74 Estimated Order Effect ......... ... ........................ ....... 75 Predict Horizon Effect ......... ... .... ......... .. .. ......... ............. 76 Input W eight Effect ............. .... .... .. ................. ... .. .. ............. .. .............. 76 Disturbance Effect for Different SNR Levels During System ID..............................76 Su m m ary ......... .... ..... ....... ............ ....................................................77 5 WIND TUNNEL EXPERIMENTAL SETUP................................. 90 W ind T unnel F facility ................................................................................... ....................90 Test Section and C avity M odel.................................................... ............................... 91 Pressure/Temperature M easurement Systems ............................................. ............... 93 Facility Data Acquisition and Control Systems................................. ........................ 94 A ctu ato r S y ste m ................................................................................................................ 9 5 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION .............................113 B a c k g ro u n d ........................................................................................................................... 1 1 3 Data Analysis M ethods..................................... .................................... ........ 115 Noise Floor of Unsteady Pressure Transducers................................................................116 Effects of Structural Vibrations on Unsteady Pressure Transducers ..................................116 Baseline Experimental Results and Analysis ............................................... ............... 117 OpenLoop Experim ental Results and Analysis............................................................... 118 ClosedLoop Experimental Results and Analysis ..................................... .................120 7 SUMMARY AND FUTURE WORK ....................................................... ...................140 Sum m ary of C contributions .................................................... ................................... 140 F utu re W ork ......................................................14 1 APPENDIX A M A TR IX O PR A T IO N S ............................................................................ .....................143 6 Vector Derivatives ............... ................. ............ .................. ......... 143 D definition of V ectors ..................................... ........................ ...... ............... 143 Derivative of Scalar with Respect to Vector .....................................................143 Derivative of Vector with Respect to Vector ............................................................ 143 Second Derivative of Scalar With Respect to Vector (Hessian Matrix) .....................144 Table of Several Useful Vector Derivative Formulas ................................................144 Proof of the Formulas ....... .. ................................ ............ ................. 145 P roof (a) ...................................................................................................... ....... 145 P ro o f (b ) ...............................................................14 5 P ro o f (c ) ............................................................................................................ 1 4 6 P roof (d) ............... ......................146 The Chain Rule of the V ector Functions ............................................... .......... ..... 148 The Derivative of Scalar Functions Respect to a Matrix .................................................150 B CAVITY OSCILLATION MODELS .................................................155 R o ssiter M odel ...................................................................................................155 Linear Models of Cavity Flow Oscillations.................................. 156 Global Model for the Cavity Oscillations in Supersonic Flow ............................................159 Global Model for the Cavity Oscillations in Subsonic Flow.................... ..................163 C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS ...................... .......169 M IM O Sy stem Identification .......................................................................................... 169 Generalized Predictive Control Model ..................... ....... ...............171 D A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC R E SO N A N C E S .............................................................................179 M ason's R ule ............................................. ..... ................................... 179 Global Model for a Cavity Oscillation in Supersonic Flow ................................................180 Global Model for a Cavity Oscillation in Subsonic Flow ...................................................183 E CENTER VELOCITY OF ACTUATOR ARRAY .............................................................196 F PARAMETRIC STUDY FOR OPENLOOP CONTROL ..................................... 207 L IST O F R E F E R E N C E S ....................................................................................................... 2 19 BIO GR A PH ICA L SK ETCH ...............................................................227 LIST OF TABLES Table page 21 Summary of the IIR OE algorithm. .......................... ............. ...............................49 22 Sum m ary of the IIR EE algorithm ......... ................. .................................... ............... 50 23 Sum m ary of the IIR SM algorithm ........... ............... ............................. ............... 51 24 Sum m ary of the IIR CE algorithm ............................................................. .....................52 25 Simulation results of IIR algorithms for sufficient case .................................................53 26 Simulation results of IIR algorithms for insufficient case................................................54 27 Simulation conditions of IIR algorithms for sufficient case............................................55 28 Sum m ary of the IIR/LM S algorithm s.......................................................... ............... 56 41 Parameters selection of the vibration beam experiment ................ ......... .......................78 42 Summary of the results of the adaptive GPC algorithm.................... ................................ 79 51 Physical and piezoelectric properties of APC 850 device ...................................................100 52 Geometric properties and parameters for the actuator ....................................................... 101 53 Resonant frequencies with respective centerline velocities for each input voltage. ............102 A V sector derivative form ulas. ........................................................................ ...................154 D1 Components of the Mason's formula for supersonic case...................................................189 D2 Components of the Mason's formula for subsonic case.....................................................190 D3 Components of the Mason's formula for subsonic case.....................................................192 LIST OF FIGURES Figure page 11 Schematic illustrating flowinduced cavity resonance for an upstream turbulent boundary layer. .......................................................................................................... ......36 12 Tam and Block (1978) model of acoustic wave field inside and outside the rectangular cav ity ................... ............................................................ ................ 3 6 13 C classification of flow control. ...................................................................... ....................37 14 Block diagram of system ID and online control. ...................................... ............... 37 21 Linear timeinvariant (LTI) IIR Filter Structure..................................................................57 22 Simulation structure of the adaptive IIR filter..................................................................... 57 23 zplane of the test m odel ........... ... ................. ....... .. ...... .. .................. 58 24 3D plot of the MSOE performance surface of the insufficient order test system. .................58 25 Contour plot of the MSOE performance surface.................. ... ... ................ 59 26 Simulation results of weight track of the IIR algorithms for sufficient case..........................59 27 Simulation results of weight track of the IIR algorithms for insufficient case.....................60 28 Learning curve of IIR algorithms for sufficient case. ............. ........... .... ...........60 29 Computational complexity results from the experiment. ...................................................... 61 31 M odel predictive control strategy ................................................. .............................. 70 41 Schematic diagram of the vibration beam test bed ..................................................... 80 51 Schematic of the wind tunnel facility. ....................... ..................... ..........103 52 Schematic of the test section and the cavity model.......................................................... 103 53 Schematic of the control hardware setup ............. ................................ ... ............ 104 54 Bimorph bender disc actuator in parallel operation................................... ...............104 55 D designed ZN M F actuator array ......................... ....... ................................ ............... 105 56 Dimensions of the slot for designed actuator array. .................................. .................106 57 ZNMF actuator array mounted in wind tunnel.............................................................107 58 Bimorph 3 centerline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal...........................108 59 The comparison plot of the experiment and simulation result of the actuator design code for bim orph 3 ............ ......... ................ ........................................ ........................ .... 109 510 Current saturation effects of the amplifier .............................. ..............110 511 Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25, K=0.7).........111 512 Schematic of a single periodic cell of the actuator jets and the proposed interaction with the income ing boundary layer. ............................................................................112 61 Schematic of simplified wind tunnel and cavity regions acoustic resonances for sub sonic flow .......................................................................... 123 62 Noise floor level comparison at different discrete Mach numbers with acoustic treatment at trailing edge floor of the cavity with L/D=6 ............................ ..........123 63 x acceleration unsteady power spectrum (dB ref. g) for case with acoustic treatment and no cavity .............................................................................124 64 y acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic treatment and no cavity .............................................................................125 65 z acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic treatment and no cavity .............................................................................126 66 Spectrogram of pressure measurement (dB ref. 20e6 Pa) on the trailing edge floor of the cavity for the case with acoustic treatment and no cavity .............. .... ...............127 67 Spectrogram of pressure measurement (ref 20e6 Pa) on the trailing edge cavity floor without acoustic treatm ent at L/D=6. ........................................ .......................... 128 68 Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25, K=0.7).........129 69 Noise floor of the unsteady pressure level at the surface of the trailing edge of the cavity with and without the actuator turned on. .............................................. ............... 130 610 Openloop sinusoidal control results for flowinduced cavity oscillations at trailing edge floor of the cavity. ................................................... ...... .. ........ .... 131 611 Running error variance plot for the system identification algorithm. .............................133 612 ClosedLoop active control result for flowinduced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity. .......................................... ...............134 613 Input signal of the ClosedLoop active control result for flowinduced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity........................135 614 Sensitivity function (Equation 41) of the closedloop control for M=0.27 upstream flow condition .. ................................................................................ 136 615 Unsteady pressure level of the closedloop control for M=0.27, L/D=6 upstream flow condition w ith varying estim ated order.. .......... .................... ................................ 137 616 Unsteady pressure level of the closedloop control for M=0.27, L/D=6 upstream flow condition with varying predictive horizon s. ....................................... ............... 138 617 Unsteady pressure level comparison between the openloop control and closedloop control for M=0.27 upstream flow condition..... ..................... ............139 B Schem atic of R ossiter m odel. ...................... ....... ........ ...................................... 166 B2 Block diagram of the linear model of the flowinduced cavity oscillations....................... 166 B3 Block diagram of the reflection model. ........................................ ......................... 167 B4 Global model for the cavity oscillations in supersonic flow.......................................... 167 B5 Block diagram of the global model for a cavity oscillation in supersonic flow ................167 B6 Global model for a cavity oscillation in subsonic flow. ................... ............................. 168 B7 Block diagram of the global model for a cavity oscillation in subsonic flow. ....................168 D1 Global model for a cavity oscillation in supersonic flow ..................................................193 D2 Block diagram of the global model for a cavity oscillation in supersonic flow................193 D3 Signal flow graph of the global model for a cavity oscillation in supersonic flow............. 194 D4 Global model for a cavity oscillation in subsonic flow. ............................................. 194 D5 Block diagram of the global model for a cavity oscillation in subsonic flow...................195 D6 Signal flow graph of the global model for a cavity oscillation in subsonic flow..............195 E1 Hotwire measurement for actuator array slot la.................. .......................... 197 E2 Hotwire measurement for actuator array slot lb. .................................... .................198 E3 Hotwire measurement for actuator array slot 2a.............................................199 E4 Hotwire measurement for actuator array slot 2b.. ..................................... ............... 200 E5 Hotwire measurement for actuator array slot 3a..................................... .................201 E6 Hotwire measurement for actuator array slot 3b.. ..................................... ...............202 E7 Hotwire measurement for actuator array slot 4a ...... ......... ..............................203 E8 Hotwire measurement for actuator array slot 14b.. ............................... ...............204 E9 Hotwire measurement for actuator array slot 5a..................................... .................205 E10 Hotwire measurement for actuator array slot 5b.. ....................................................206 Fl OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 500 H z and 100 V pp voltage. ........................................ ............................................208 F2 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 500 H z and 150 V pp voltage. ........................................ ............................................208 F3 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 600 H z and 100 V pp voltage. ........................................ ............................................209 F4 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 600 H z and 150 V pp voltage. ........................................ ............................................209 F5 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 700 H z and 100 V pp voltage. ........................................ ............................................2 10 F6 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 700 H z and 150 V pp voltage. ........................................ ............................................2 10 F7 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 800 H z and 100 V pp voltage. ...................................................................... ..............2 11 F8 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 800 H z and 150 V pp voltage. ....................................................................... .........2 11 F9 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 900 H z and 100 V pp voltage. ........................................ ............................................2 12 F10 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 900 H z and 150 V pp voltage. ........................................ ............................................2 12 F11 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1000 H z and 100 V pp voltage. ............................................................. .....................2 13 F12 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1000 H z and 150 V pp voltage. ............................................................. .....................2 13 F13 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1100 H z and 100 V pp voltage. ...................................................................... ...........2 14 F14 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1100 H z and 150 V pp voltage. ...................................................................... ...........2 14 F15 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1200 H z and 100 V pp voltage. ............................................................. .....................2 15 F16 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1200 H z and 150 V pp voltage. ............................................................. .....................2 15 F17 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1300 H z and 100 V pp voltage. ............................................................. .....................2 16 F18 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1300 H z and 150 V pp voltage. ............................................................. .....................2 16 F19 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1400 H z and 100 V pp voltage. ............................................................. .....................217 F20 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1400 H z and 150 V pp voltage. ............................................................. .....................2 17 F21 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1500 H z and 100 V pp voltage. ............................................................. .....................2 18 F22 OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1500 H z and 150 V pp voltage. ............................................................. .....................2 18 LIST OF ABBREVIATIONS D Cavity depth L Cavity length M Freestream flow Mach number U. Freestream flow velocity a Mean sound speed inside the cavity m, Mode number (integer number 1,2...) Wo Natural frequency of second order system r Reflection coefficient 1 Damping ratio a Phase lag factor K Proportion of the vortices speed to the freestream speed Y Ratio of the specific heats 2, Spacing of the vortices ra Time delay inside the cavity T, Time delay inside the shear layer ADC Analog to digital converter ARMA Autoregressive and movingaverage CARIMA Autoregressive and integrated moving average CE Composite error DAC Digital to analog converter DNS Direct Numerical Simulations DSP Digital signal processing EE Equation error FFT Fast Fourier transform FIR Finite impulse response FRF Frequency response function GPC Generalized predictive control ID Identification IIR Infinite impulse response JTFA Jointtime frequency analysis LES Large Eddy Simulations LMS Least mean square LQG Linear quadratic Gaussian LTI Linear timeinvariant MIMO Multipleinput multipleoutput MPC Model predictive control MSOE Mean square output error OE Output error PDF Probability density function POD Proper orthogonal decomposition RANS Reynolds Averaged Navierstokes RLS Recursive least square SISO Singleinput singleoutput SM Steiglitz and McBride SNR Signal to noise ratio SPL Sound pressure level STR Selftuning regulator TITO Twoinput Twooutput Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CLOSEDLOOP CONTROL OF FLOWINDUCED CAVITY OSCILLATIONS By Qi Song May 2008 Chair: Louis Cattafesta Major: Aerospace Engineering Flowinduced cavity oscillations are a coupled flowacoustic problem in which the inherent closedloop system dynamics can lead to large unsteady pressure levels in and around the cavity, resulting in both broadband noise and discrete tones. This problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, and automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been working on this problem for more than fifty years. This is because not only is the physical nature of this problem rich and complex, but also it has become a standard test bed for controller deign and implementation in flow control. The ultimate goal of this research is to minimize the cavity acoustic tones and the broadband noise level over a range of freestream Mach numbers. Although openloop and closedloop control methodologies have been explored extensively in recent years, there are still some issues that need to be studied further. For example, a loworder theoretical model suitable for controller design does not exist. Most recent flowinduced cavity models are based either on Rossiter's semiexpirical formula or a proper orthogonal decomposition (POD) based models. These models cannot be implemented in adaptive controller design directly. In addition, closed loop control of high subsonic and supersonic flows remains an unexplored area. In order to achieve these objectives, an analytical system model is first developed in this research. This analytical model is a transfer function based model and it can be used as a potential model for controller design. Then, a MIMO system identification algorithm is derived and combined with the generalized prediction control (GPC) algorithm. The resultant integration of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics and reduce the flowinduced oscillations. A novel piezoelectricdriven synthetic jet actuator array is designed for this research. The resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. A validation vibration beam problem is used to demonstrate the combination of system ID and GPC algorithms. The result shows a 20 dB reduction at the single resonance peak and a 9 dB reduction of the integrated vibration levels. Both openloop control and closedloop control are applied to the flowinduced cavity oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced for both cases. The GPC controller can generate a series of control signals to drive the actuator array resulting in dB reduction for the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closedloop controller (i.e., the OASPL reduction is 3 dB). The relevant flow physics and active flow control actuators are examined and explained in this research. The limitations of the present setup are discussed. CHAPTER 1 INTRODUCTION Flowinduced cavity oscillations have been studied for more than fifty years, and the problem has attracted researchers in both fluid dynamics and controls. First, this problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, sunroofs and windows "buffeting" in automobiles, and junctions between structural and aerodynamic components in both (Kook et al. 1997). The flowacoustic coupling inherent in cavity resonance can lead to high unsteady pressure levels (both broadband noise and discrete tones), and can cause fatigue failure of the cavity and its contents. For example, the measured sound pressure levels in and around a weapons bay can exceed 170 dB ref 20 [aPa. For this reason, researchers are usually interested in suppression of flowinduced open cavity oscillations. Furthermore, this problem has become a standard test problem for designing, testing, and analyzing realtime feedback control systems. Although the standard rectangular cavity geometry is relatively simple, the physical nature of this problem is both rich and complex. Several good review articles on the flowinduced cavity oscillation problem are available in the literature (Rockwell, 1978, Komerath 1987, Colonius 2001, Cattafesta 2003). Figure 11 is a simplified schematic for two typical flow situations, corresponding to external (a) supersonic and (b) subsonic flow over a rectangular cavity with length, L, depth, D, and width, W. The cavity oscillation process can be summarized as follows. A (usually) turbulent boundary layer with thickness, 3, and momentum thickness, 0, separates at the upstream edge of the cavity. Both a turbulent boundary layer and laminar boundary layer generate the discrete tones caused by the external flow. However, a laminar boundary layer has been shown to produce louder tones, presumably because a turbulent flow generally results in a thicker shear layer with broadband disturbances, which leads to overall lower levels of oscillations (Tam and Block 1978; Colonius 2001). Following the description of Kerschen and Tumin (2003) and Alvarez et al.(2004), when the turbulent boundary layer separates at the upstream edge of the cavity, the resulting high speed or "fast" acoustic wave, E,, and the low speed or "slow" acoustic wave, E,, propagate downstream in the supersonic flow case. In the subsonic flow case, only the socalled disturbance wave, E,, propagates downstream. In both cases, the shear layer instability, s, develops based upon its initial conditions coupled by the upstream traveling acoustic feedback wave, U, inside the cavity (and E, outside the cavity for subsonic flow). KelvinHelmholtztype (Tam and Block 1978) convective instability waves develop and amplify in the shear layer as they propagate downstream and finally saturate due to nonlinearity. In particular, the instability waves grow and form largescale vortical structures that convect downstream at a fraction of the freestream velocity. These structures then impinge near the trailing edge of "open" cavities (LID <10). The reattachment region acts as the primary acoustic source, which has been modeled as a monopole (Tam and Block 1978) or a dipole (Bilanin and Covert 1973) source. As a result, an upstream traveling acoustic wave, U, is generated inside the cavity. In subsonic flow, an additional acoustic wave, E,, propagates upstream outside the cavity. In this description, the acoustic feedback is modeled via acoustic waves that travel in the x direction. Finally, the loop is closed by a receptivity process, in which the upstream traveling waves are converted to downstream traveling instability and acoustic waves. The initial amplitude and phase of these waves are set by the incident acoustic disturbances through this receptivity process. Physically, some of the acoustic disturbance energy is converted to the instability waves at the upstream separation edge. Since the wavelength and the velocity of the instability waves and the acoustic disturbances differ, only those waves that are inphase ensure reinforcement of disturbances at that frequency. Therefore, this process is normally considered an introduction of a disturbance into the system, ultimately resulting in largeamplitude discrete tones inside and around the cavity. The measured broadband noise component is mainly due to the turbulent shear layer. The relevant dimensionless parameters are: L/D, L/W, L/O, and shape factor H = 6 /0 with the freestream flow parameters, Reynolds number Re, and Mach number Mn, all of which lead to tones (with Strouhal number St = jL/U, ) characterized by their strength as unsteady pressure normalized by the freestream dynamic pressure, p,,/q . In this study, three dimensional effects are not considered, since the cavity tones are generated by the interaction of the freestream flow and the longitudinal modes (coupled with vertical depth modes). Width modes are not relevant in this feedback loop if the width is small enough to prevent higherorder spanwise modes but large enough so that the mean flow over the cavity length is approximately two dimensional. Note that the width of the cavity does affect the amplitude of the cavity oscillations (Rossiter 1966) but is of secondary importance (Cain 1999). Therefore, a twodimensional model is reasonable from a physical perspective even though the unsteady turbulent motion is inevitably threedimensional (Bilanin and Covert 1973). Literature Review In this section, some published results related to the physics of flowinduced cavity oscillations are discussed. Since the ultimate goal of this research is to minimize the cavity acoustic tones and perhaps the broadband noise level, potential control methodologies and algorithms are also reviewed. A recent review paper by Cattafesta et al. (2003) gives a summary of the various passive and openloop cavity suppression studies. Physical Models In order to suppress the discrete tones and the broadband acoustic level of flowinduced open cavity resonance, an understanding of the physics is essential. From a control engineer's point of view, a simplified and loworder model is desirable in order to predict the resonant frequencies and amplitudes over a broad range of the governing dimensionless parameters. PhysicsBased Models Rossiter (1964) performed an extensive experimental study on the measurement of the unsteady pressure in and around a rectangular open cavity (2ft x 1.5ft) in a subsonic and transonic freestream air flow (0.5 unsteady acoustic tones generated in the cavity. For the deeper cavities (LID < 4 ), there was usually a single dominant tone, and the dominant frequency was observed to jump between different cavity tones. For the shallower cavities (LID > 4), two or more peaks were often observed and were approximately equal in magnitude. He proposed that the flow entering the cavity caused the external stream to accelerate, and then the flow decelerated near the reattachment region. As a result, pressure was lower near the separation region (leading edge) and higher near the reattachment region (trailing edge). As a result, he suggested that large eddies developed within the cavity due to this pressure gradient. He also used shadowgraphs to illustrate that the shear layer separates from the cavity leading edge, and the instability waves develop into discrete vortices that are shed at regular time internals from the front lip of the cavity (at Mach number 0.6 and with twodimensional cavity LID = 4 and a laminar boundary layer). He postulated that there were some connections between the vortex shedding and the acoustic feedback, and this phenomenon produced a series of periodic pressure fluctuations. When the frequency of one of these components is close to the natural frequency of the cavity, resonance occurs. In his study, Rossiter gave a semiempirical formula for predicting the resonant frequencies of these peaks at a specific Mach number. The derivation of the Rossiter equation is given in Appendix B, and the resulting formula for the dimensionless Strouhal number is S L _(m, a) St fmL (m ) (11) U 1 where f, is the resonance frequency for integer mode m,, L is the length of the cavity, U. is the freestream velocity, a is the phase lag factor (in fractions of a wavelength), K is the ratio of the vortex propagation speed to the freestream velocity, and MAJ is the freestream Mach number. Empirical constant values of K = 0.57 and a = 0.25 are shown to best fit the measured frequencies of resonances over a wide range of the Mach numbers for his experiment. These experimental constants account for the phase shift associated with the coupling between the shear layer and acoustic waves at the two ends of the cavity, and this phase shift is approximately independent of frequency. The phase speed cKU of the vortices is a weak function of MA,, L/O and D/O (Colonius 2001). Different integer values m, give different frequencies, commonly referred to as "shear layer" or "Rossiter" modes. In conclusion, Rossiter's formula is based on an integer number of 2r phase shifts, 2k r, around a resonant feedback loop consisting of a downstream unstable shear layer disturbance and an upstream feedback acoustic wave inside the cavity. This phase shift is a necessary condition for selfsustaining oscillations (Cattafesta et al. 1999a). However, Rossiter's expression does not account for the depth or width of the cavity and only successfully predicts the longitudinal cavity resonant frequencies at moderatetohigh Mach numbers. It also does not predict the amplitude of the oscillations. Heller and Bliss (1975) corrected the Rossiter equation for the higher sound speed in the cavity, in which the static temperature in the cavity was assumed to be the stagnation temperature of the upstream. The modified Rossiter formula is f,L (m a) St f LU' ) (12) 1 2 1+ M 2 where y is defined as the ratio of specific heats. They gave a discussion on the physical mechanisms of the oscillation process based on water table visualization experiments. They suggested that the unsteady motion of the shear layer leads to a periodic mass addition and removal at the cavity trailing edge, leading to subsequent modeling efforts that employ an acoustic monopole source. In addition, the wave motion of the shear layer and the wave structure within the cavity were strongly coupled. Bilanin and Covert (1973) modeled the cavity problem by splitting the domain into two parts outside and inside the cavity. These two flow fields were separated by a thin mixing layer, which was approximated by a vortex sheet, and the flow was assumed to be inviscid. The dominant pressure oscillations at the trailing edge were modeled by a single periodic acoustic monopole. They also assumed that the pressure field from the trailing edge source had no effect on the vortex sheet itself. Hence, the main disturbance was introduced at the leading edge of the shear layer. Kegerise et al. (2004) illustrated the agreement between the disturbance sensitivity function defined in control systems and the performance measurement of output disturbances. Their analysis confirmed the notion that the disturbances were mainly introduced into the cavity at the cavity leading edge. Tam and Block (1978) carried out extensive experimental investigations at low subsonic Mach numbers (M < 0.4) and postulated that vortex shedding was probably not the main factor for cavity resonance over the entire Mach number range. They made two key assumptions, namely that the rectangular cavity flow was twodimensional, and the mean flow velocity inside the cavity was zero. These two assumptions were based on experimental evidence of little correlation between the mean flow and the acoustic feedback inside the cavity. Tam and Block proposed a process of flowinduced cavity oscillations as follows. The shear layer oscillated up and down at the trailing edge of the cavity. The upward movement was uncorrelated with the generation of the acoustic waves, because if the shear layer covered the trailing edge, then the external flow passes over the trailing edge without impingement. They argued that only the downward motion of the shear layer into the cavity caused significant generation of pressure waves and subsequent radiation of acoustic waves in all directions (Figure 12). For example, some of the waves radiating into the external flow (e.g., wave A) were argued to have minor effects on the oscillations inside the cavity. However, the effect of the waves propagating inside of the cavity was deemed more significant. The resulting acoustic waves included the upstream propagating waves (e.g., wave C) and the reflected waves from the floor (e.g., wave F) and the upstream wall (e.g., wave E). Subsequent reflections of the acoustic waves by the walls, the cavity, or the shear layer were deemed negligible. They concluded that the directly radiated wave and the first reflected waves by the floor and upstream end wall of the cavity provided the energy to excite the instability waves of the shear layer. These disturbances within the shear layer were then amplified as the instability waves propagate downstream. When the disturbances amplitudes became large, nonlinear effects were important and ultimately established the amplitude of the discrete tones. A mathematical model of the cavity oscillation and acoustic field were developed. In order to calculate the phases and waves generated at the trailing edge, a periodic line source was simulated at the trailing edge of the cavity. In addition, the reflections of the acoustic waves by the cavity walls were modeled by periodic line image sources about the cavity walls. Their model accounts for the finite shear layer thickness effects and produces a more accurate estimation of the resonance frequencies than Rossiter's model. However, their resulting model is complicated and difficult to employ for control law design. Rowley et al. (2002 b, 2003, 2006) provided an alternative viewpoint for understanding flowinduced cavity oscillations. They showed that selfsustained oscillations existed only under certain conditions. The resonant frequencies were due to the instabilities in the shear layer interacting with the flow and acoustic fields. The amplitude of the oscillations was determined by nonlinear saturation. However, at other conditions, the cavity oscillations could be represented as a lightly damped but stable linear system. The oscillations were caused by the amplification of external disturbances via the closedloop dynamics of the cavity. The amplitude of each mode was determined by the amplitude of the external forcing disturbances and some frequencydependent gain of the system. They modeled the dynamics of the shear layer as a secondorder system and the acoustic propagation process via a onedimensional, standingwave model. The impingement and receptivity procedures were simply modeled as a constant unity gain. Finally, the Rossiter formula was derived under some specific conditions. The derivation of this model is provided in Appendix B. They also used Gaussian white noise as input and examined the probability density function (PDF) and the phase portrait of the output pressure signal at different Mach numbers. Their results showed that under some conditions, the selfsustained regime of Rossiter modes was valid. However, at other conditions, called the forced regime, open cavity oscillations may be represented as lightly damped stable linear systems. External random forces drove the finite amplitude cavity oscillations, which implies they will disappear if the external forces were removed. This physical linear model was also proposed as a potential model for controller design. Kerschen and Tumin. (2003) and Alvarez et al. (2004) provided a promising global model to describe the flowinduced cavity oscillation problem for two different flow patterns (Figure 1 1). Their model combined scattering analyses for the two ends of the cavity and a propagation analysis of the cavity shear layer, internal region of the cavity, and acoustic nearfield. They solved a matrix eigenvalue problem to identify the resonant frequencies of the cavity oscillation. From their resulting characteristic functions, four and twelve closed loops could be identified for the supersonic flow and subsonic flow cases, respectively. One more feedback loop makes the subsonic flow much more complex than the supersonic flow. For example, some of these closed loops were major loops, such as closed loop s, U and U, D (Figure 11), while the other closed loops were considered minor loops. The combined effects of these loops caused the cavity resonances in the cavity flow. Besides these closed loops, the forward propagation paths, such as S, D, Ed, E,, and E,, also have critical effects on the amplitude of the oscillations. This global model provides more insights for controller design. A detailed derivation of this model is provided in Appendix B. Clearly, the physicsbased models described above provide physical insight concerning flowinduced cavity oscillations. However, the original Rossiter model and the global model derived by Kerschen and Tumin. (2003) can only estimate the resonance frequencies of the cavity flow. The linear model derived by Rowley et al. (2002b, 2003, 2006) is transfer function based model but is not sufficiently accurate to design a control system. A transfer function based model, which is an extension of Kerschen et al.'s model, of cavity acoustic resonances is derived and given in Appendix D. For this approach, a signal flow graph is first constructed from the block diagram of the Kerschen et al's physical model, and then Mason's rule (Nise 2004) is applied to obtain the transfer function from the disturbance input to the selected system output. This method can give predictions for both the resonant frequencies of the flowinduced cavity oscillations and the amplitude of the cavity tones. In addition, this method also provides a linear estimate for the system transfer function from the disturbance input at the leading edge and the pressure sensor output within the cavity walls. Therefore, this model is a potential global model for controller design in this research. Numerical Simulations Some computational fluid dynamic (CFD) methods, such as Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) and Reynolds Averaged Navierstokes (RANS), provide useful information for understanding the issues of physical modeling of cavity oscillations. A review paper by Colonius (2001) gives a summary of issues related to each of these topics. More recent research on these topics can be found by Rizzetta et al. (2002, 2003) and Gloerfelt (2004). The Detached Eddy Simulation (DES) method, which is a involves a hybrid turbulence modeling methodology, has also been used to calculate the flow and acoustic fields of the cavity (Allen and Mendonca 2004; Hamed et al. 2003, 2004). Another hybrid RANSLES turbulence modeling approach is presented by Arunajatesan and Sinha (2001, 2003). They model the upstream boundary layer flow field and the shear layer region via RANS and LES models, respectively. All of these computational methods provide, at a minimum, good flow visualization and physical insight, and, at a maximum, quantitative information on the details of flow dynamics. PODType Models The previous analytical physical models are not accurate enough to design a control system. Furthermore, CFD methods are far too computationally intensive at the present time to provide a reasonable framework to design and test potential controllers. This translates into the need for new methods to develop more accurate reducedorder models. Therefore, simulation and experimental data based models were proposed and later used for the controller design. Rowley et al. (2001) introduced a nonlinear dynamical model for flowinduced rectangular cavity oscillations, which was based on the method of vectorvalued proper orthogonal decomposition (POD) and Galerkin projection. The POD method obtains lowdimensional descriptions of a highorder system (Chatterjee 2000). For the cavity flow problem, data resulting from the temporalspatial evolution of the numerical simulations or experiments is used to construct a loworder subspace system that captures the main features (coherent structures) of the cavity flow. A more detailed explanation of POD methods for cavity flow are given by Rowley et al. (2000, 2001, 2002c, 2003a). Some of the control methodologies discussed in next section can be constructed based on the resultant model obtained by POD (Caraballo et al. 2003, 2004, 2005; Samimy et al. 2003, 2004; Yuan et al. 2005). Instead, we turn our attention to an alternative experimentalbased modeling approach that employs system identification techniques. Here, the nonlinear infinitedimensional governing equations are modeled by a reduced set of differential (in continuous) or difference (in discrete time) equations. This method is the focus of this study and is discussed in the following section. OnLine System ID and Active ClosedLoop Control Methodologies Previous studies aimed at suppression of the flowinduced cavity tones have employed mainly passive or openloop active flow control methodologies. The standard classification of the flow control techniques is shown in Figure 13. The review paper by Cattafesta et al. (2003) provides a detailed overview of various passive and openloop control methodologies. However, passive and openloop approaches are only effective for a limited range of flow conditions. Active feedback flow control has recently been applied to the flowinduced cavity oscillation problems. The closedloop control approaches have advantages of reduced energy consumption (Cattafesta et al. 1997), no additional drag penalty, and robustness to parameter changes and modeling uncertainties. In general, closedloop flow control measures and feeds back pressure fluctuations at the surface of the cavity walls (or floor) to an actuator at the cavity leading edge to suppress the cavity oscillations in a closedloop fashion. In general, past active control strategies have taken one of two approaches for the purpose of reducing cavity resonance. First, they can thicken the boundary layer in order to reduce the growth of the instabilities in the shear layer. Alternatively, they can be used to break the internal feedback loop of the cavity dynamics. Most closedloop schemes exploit the latter approach. Early closedloop control applications used manual tuning of the gain and delay of simple feedback loops to suppress resonance (Gharib et al. 1987; Williams et al. 2000a,b). Mongeau et al. (1998) and Kook et al. (2002) used an active spoiler driven at the leading edge and a loop shaping algorithm to obtain significant attenuation with small actuation effort. Debiasi et al. (2003, 2004) and Samimy et al. (2003) proposed a simple logicbased controller for closedloop cavity flow control. Loworder modelbased controllers with different bandwidths, gains and time delays have also been designed and implemented (Rowley et al. 2002, 2003, Williams et al. 2002, Micheau et al. 2004, Debiasi et al. 2004). Linear optimal controllers (Cattafesta et al. 1997, Cabell et al. 2002, Debiasi et al. 2004, Samimy et al. 2004, Caraballo et al. 2005) have been successfully designed for operation at a single flow condition. These models are all based on reducedorder system models, and most of these controller design methods are based on model forms of the frequency response function, rational discrete/continuous transfer function, or statespace form. However, the coefficients of these model forms are assumed to be constant, and this assumption requires that the system is time invariant or at least a quasistatic system with a fixed Mach number. Although the physical models of flowinduced cavity oscillations have been explored extensively, they are not convenient for control realization. This is because these models are highly dependent on the accuracy of the estimated internal states of the cavity system. In addition, cavity flow is known to be quite sensitive to slight changes in flow parameters. So a small change in Mach number can deteriorate the performance of a singlepoint designed controller (Rowley and Williams 2003). Therefore, adaptive control is certainly a reasonable approach to consider for reducing oscillations in the flow past a cavity. Adaptive control methodology combines a general control strategy and system identification (ID) algorithms. This method is thus potentially able to adapt to the changes of the cavity dimension and flow conditions. It updates the controller parameters for optimum performance automatically. The structure of this method is illustrated in Figure 14. Two distinct loops can be observed in the controller. The outer loop is a standard feedback control system comprised of the process block and the controller block. The controller operates at a sample rate that is suitable for the discrete process under control. The inner loop consists of a parameter estimator block and a controller design block. An ID algorithm and a specified cost function are then used to design a controller that will minimize the output. The steps for realtime flow control include: (i) Use a broadband system ID input from the actuator(s) and the measured pressure fluctuation output(s) on the walls of the cavity to estimate the system (plant and disturbance) parameters. (ii) Design a controller based on the estimated system parameters. (iii) Control the whole system to minimize the effects of the disturbance, measured noise, and the uncertainties in the plant. Based on this adaptive control methodology, some adaptive algorithms adjust the controller design parameters to track dynamic changes in the system. However, only a few researchers have demonstrated the online adaptive closedloop control of flowinduced cavity oscillations. Cattafesta et al. (1999 a, b) applied an adaptive disturbance rejection algorithm, which was based upon the ARMARKOV/Toeplitz models (Akers and Bernstein 1997; Venugopal and Bernstein 2000, 2001), to identify and control a cavity flow at Mach 0.74 and achieved 10 dB suppression of a single Rossiter model. Other modes in the cavity spectrum were unaffected. Insufficient actuator bandwidth and authority limited the control performance to a single mode. Williams and Morrow (2001) applied an adaptive filteredX LMS algorithm to the cavity problem and demonstrated multiple cavity tone suppression at Mach number up to 0.48. However, this was accompanied by simultaneous amplification of other cavity tones. Numerical simulations using the least mean squares (LMS) algorithm were shown by Kestens and Nicoud (1998) to minimize the output of a single error sensor. The reduction was associated with a single Rossiter mode, but only within a small spatial region around the error sensor. Kegerise et al. (2002) implemented adaptive system ID algorithms in an experimental cavity flow at a single Mach number of 0.275. They also summarized the typical finiteimpulse response (FIR) and infiniteimpulse response (IIR) based system ID algorithms. They concluded that the FIR filters used to represent the flowinduced cavity process were unsuitable. On the other hand, IIR models were able to model the dynamics of the cavity system. LMS adaptive algorithm was more suitable for realtime control than the recursiveleast square (RLS) adaptive algorithm due to its reduced computational complexity. Recently, more advanced controllers, such as direct and indirect synthesis of the neural architectures for both system ID and control (Efe et al. 2005) and the generalized predictive control (GPC) algorithm (Kegerise et al. 2004), have been implemented on the cavity problems. From a physical point of view, the closedloop controllers have no effect on the mean velocity profile (Cattafesta et al. 1997). However, they significantly affect streamwise velocity fluctuation profiles. This control effect eliminates the strength of the pressure fluctuations related to flow impingement on the trailing edge of the cavity. Although closedloop control has provided promising results, the peaking (i.e., generation of new oscillation frequencies), peak splitting (i.e., a controlled peak splits into two sidebands) and mode switching phenomena (i.e., nonlinear interaction between two different Rossiter frequencies) often appear in active closed loop control experiments (Cattafesta et al. 1997, 1999 b; Williams et al. 2000; Rowley et al. 2002 b, 2003; Cabell et al. 2002; Kegerise et al. 2002, 2004a). Explanations of these phenomena are provided by Rowley et al. (2002b, 2006), Banaszuk et al. (1999), Hong and Bernstein (1998), and Kegerise et al. (2004). Rowley et al. (2002b, 2003) concluded that if the viewpoint of a linear model was correct, a closedloop controller could not reduce the amplitude of oscillations at all frequencies as a consequence of the Bode integral constraint. Banaszuk et al. (1999) gave explanations of the peaksplitting phenomenon. They claimed that the peak splitting effect was caused by a large delay and a relatively low damping coefficient of the openloop plant. Cabell et al. (2002) explained these phenomena by the combination of inaccuracies in the identified plant model, high gain controllers, large time delays and uncertainty in system dynamics. In addition, narrowbandwidth actuators and controllers may also lead to a peaksplitting phenomenon (Rowley et al. 2006). Hong and Bernstein defined the closedloop system disturbance amplification (peaking) phenomenon as spillover. They illustrated that the spillover problem was caused by the collocation of disturbance source and control signal or the collocation of the performance and measurement sensors. For this reason, the reduction of broadband pressure oscillations was not possible if the control input was collocated with the disturbance signal at the leading edge of cavity. Therefore, Kegerise et al. (2004) suggested a zero spillover controller which utilized actuators at both the leading and trailing edges of the cavity for closedloop flow control. Unresolved Technical Issues Although the flowinduced cavity oscillation problem has been explored extensively, there are still some unresolved issues that need to be studied further. * A suitable theoretical model does not exist that estimates both the discrete frequencies as well as the amplitude of the peaks. * A feedback controller that reduces both broadband and tonal noise over a wide range of Mach numbers has not been achieved. An adaptive zero spillover control algorithm may reduce both the tones and broadband acoustic noise associated with cavity oscillations. * The necessity for a highorder system model is a critical problem for controller design and implementation, because this highorder system results in significant computational complexity for application in digital signal processing (DSP) hardware. As such, the convective delays between the control inputs and the pressure sensor outputs must be specifically addressed in the control architecture. * Closedloop control of high subsonic and supersonic flows is an unexplored area. Technical Objectives According to those unresolved technical issues, the ultimate goals of this dissertation are summarized as follows. * A feedback control methodology will be developed for reducing flowinduced cavity oscillation and broadband pressure fluctuations. * Adaptive system ID and control algorithms will be combined and implemented in real time. * The relevant flow physics and the design of appropriate active flow control actuators will be examined in this research. * The performance, adaptability, costs (computational and energy), and limitations of the algorithms (spillover, etc.) will be investigated. Approach and Outline In order to achieve these objectives, some design and application approaches warrant additional consideration. First, a potential theoretical model of cavity acoustic resonances is derived based on the global model of Kerschen and Tumin. (2003). This model (derived in Appendix D) provides the framework to estimate the amplitudes and frequencies of the cavity tones. This model has a low system order and also accounts for the convective delay between the disturbance input and the output pressure measurement. Second, during the controller design, the controlled system is a continuous system; therefore, all the sensors measurements and the actuators inputs are analog signals. However, for the present realtime application, the control algorithms are implemented using a DSP. For this reason, additional hardware, such as analogtodigital converters (ADC), digitaltoanalog converters (DAC), antialiasing filters, and power amplifier, must be included in the whole control design procedure. Finally, multiple actuators and multiple sensors are employed in this study in order to design an adaptive zero spillover control algorithm to explore the possibility of achieving broadband acoustic noise reduction in addition to suppression of the cavity tones themselves. This active control method development procedure can be summarized as the following stages according to Elliott (2001). * Study the simplified analytical system model and understand the fundamental physical limitations of the proposed control strategy. * Obtain the sensor output and derive the states or coefficients from the system ID algorithms using offline or online methods. * Calculate the optimum performance using different control strategies and find the control law for realization. * Simulate the different control strategies and tune the candidate controller for different operating conditions. * Implement the candidate controller in realtime experiments. The thesis is organized as follows. Several SISO IIR system ID algorithms and a more general MIMO system ID algorithm are derived and discussed in the next Chapter. Then the MIMO adaptive GPC algorithm is described in Chapter 3. This is followed by a description of the sample experimental setup and the discussion of preliminary experimental results. Chapter 5 describes the wind tunnel facilities and the data processing methods. Wind tunnel experimental results for both openloop (baseline) and closedloop are then presented and discussed in Chapter 6. Finally, the conclusions and future work are presented in Chapter 7. Turbulent Boundary Layer M>1 ES E X S D D < L A Turbulent ' Boundary Layer M <1 V S ........................ D D L > B Figure 11. Schematic illustrating flowinduced cavity resonance for an upstream turbulent boundary layer. A) In supersonic flow.B) In subsonic flow. y A U. Simulated Line Source E F D 4< L > Figure 12. Tam and Block (1978) model of acoustic wave field inside and outside the rectangular cavity. Flow Control Approaches Figure 13. Classification of flow control. (after Cattafesta et al. 2003) Uncertainties Figure 14. Block diagram of system ID and online control. CHAPTER 2 SYSTEM IDENTIFICATION ALGORITHMS This chapter provides a detail discussion of the system identification algorithms. Several typical adaptive SISO IIR structure filters are chosen as the candidate digital filters. These algorithms are applied to an example from Johnson and Larimore (1977) for simulation analysis. Then, four interested aspects of these filters, accuracy, convergence, computational complexity, and robustness, are examined and summarized. Finally, a more general MIMO system ID algorithm is derived from one of the promising SISO system ID algorithms. The resulting model is used to combine with the MIMO adaptive GPC model which is discussed in next chapter. Overview As discussed in the first chapter, IIR structure filter is an applicable mathematical model to capture the cavity dynamics. Furthermore, this kind of structure can be a starting point and easily combined with many controller design strategies. Therefore, in this Chapter, several system ID algorithms based on the IIR filter structure are examined. The ideal of the system ID is to construct a predefined IIR structure filter, which has the similar frequency response of the actual dynamic system, using the information from the previous and present input and output time series data of the dynamic system. In general, the system ID algorithms fall into two big categories, the batch method and the recursive method. The batch method directly identifies the final system parameters in onetime calculation using a block data from the input and a block data from the output. Nevertheless, the recursive method updates the estimated system parameters within each sampling period using the latest input and output data in time domain. At each iteration of calculation, the system parameters may not be the optimal values. However, these estimated parameters will finally converge to the true values of the system internal states. Successful identifying the system internal states depends on two major assumptions. First, the input signal and the output signal must have a good correlation. Then, the system ID model has the same structure of that of the estimated system model. The recursive method is more attractive for present experiment, because this updating method is more suitable for online implementation and it can also track the change of the system dynamics. Furthermore, the computational complexity of recursive method is much lower than the batch method. SISO IIR Filter Algorithms Netto and Diniz (1995) give a summary of some popular adaptive IIR filter algorithms. In this section, the Output Error (OE), Equation Error (EE), Steiglitz and McBride (SM), and Composite Error (CE) algorithms are selected and illustrated. The general structure of an IIR filter is shown in Figure 21. The filter output may be expressed as S(k)= aZ(k i) bx(k j) =\1 J=0 (21) = i(k)0(k) where represents the 'estimation' values. a and b/ are the adjustable coefficients of the model, while ha and hb is the estimated order of the feedback loop and forward path, T respectively. o(k)=[f(ki) x(kj)]T, 0(k)= a b] and i ,...,,;j= 0,1,..., b. This IIR filter structure, Equation 21, is the same as the autoregressive and moving average (ARMA) model (Haykin 2002). Based on different error, the value difference between the filter output and the system output, definitions, quite a few IIR filter algorithms have been presented by Netto and Diniz (1995). In their simulations, they use an "insufficient" model, which models a secondorder system using a firstorder system to test each algorithm. The results from their paper show that the Modified Output Error (MOE) algorithm may converge to a meaningless stationary point. The same result is also shown by Johnson and Larimore (1977). The Simple Hyperstable Algorithm for Recursive Filters (SHARF) algorithm, the modified SHARF algorithm, and the Bias Remedy LeastMeanSquare Equation Error algorithm (BRLE) also show poor convergence rates. The Composite Regressor (CR) algorithm has similar problems as the MOE algorithm, since this algorithm combines the EE and MOE methods. Therefore, in this section, tests of these poor performing algorithms are not discussed. Fundamentally, there are two approaches for an adaptive IIR filter, the OE algorithm and the EE algorithm, which have been derived by Haykin (2002) and Larimore et al. (2001), respectively. Many other adaptive IIR filter algorithms are mainly derived from these algorithms, or combine some good features from the OE and the EE filters. Therefore, a summary of each of these two algorithms is provided in the following section. Two other algorithms, the SteiglitzMcBride algorithm (SM) and the Composite Error algorithm (CE), are also introduced, because both these algorithms also show good performance in our Simulink simulations. IIR OE Algorithm The IIR OE algorithm is summarized in Table 21. To ensure the stability of the algorithm, generally, the upper bound of step size / is set to 2/ where Amax is the / max maximum eigenvalue of the autocorrelation matrix of the regress vector OE (k). The step sizes of the following algorithms are also satisfying this criterion. Furthermore, in order to guarantee the convergent approximation of a, and /f, this algorithm requires slow adaptation rates for small values of n, and hb (Haykin 2002). IIR EE Algorithm The IIR EE algorithm is summarized in Table 22. Since the desired response is the supervisory signal supplied by the actual output of plant during the training period, the EE algorithm may lead to faster convergence rate of the adaptive filter (Haykin 2002). IIR SM Algorithm The IIR SM algorithm is summarized in Table 23. Since the EE algorithm and the OE algorithm possess their own advantages as well as drawbacks (discuss later), the motivation of the SM algorithm is to combine the desirable characteristics of the OE and the EE methods. IIR CE Algorithm This algorithm tries to combine both the EE algorithm and the OE algorithm in another way. As shown in Table 24, a parameter / is used to switch this algorithm between the EE algorithm and the OE algorithm. Recursive IIR Filters Simulation Results and Analyses In adaptive control experiments, the accuracy, the convergent rate, the computational complexity and the robustness are the main issues of the system ID algorithms. Here, computer simulations are examined in order to compare these aspects of the four system ID algorithms. The setup for the following Simulink simulation is shown in Figure 22. A Gaussian broad band white noise with zero mean and unity variance is chosen as the reference input signal. The prototype test model is a secondorder dynamical system (Johnson and Larimore 1977) with the transfer function 0.05 0.4z H(z1')= (22) 11.1314z 1 +0.25z 2 From the zplane plot (Figure 23), it clearly shows that this test model is a stable and nonminimum phase system, which has two real poles at z = 0.3011 and z = 0.8303, and two zeros at z = 0 and z = 8. In the following simulations, a sufficient order identification problem is firstly examined, which means a secondorder system model with the transfer function b, (k) + b,(k): H(z1, k) = o (k) + is used to estimate the test model. Then, an 1 a (k)z a2(k)z insufficient order identification problem is investigated. This approach uses a firstorder bo(k) system model with the transfer function H(z k) (k) to estimate the test 1a (k)z1 model. The mean square output error (MSOE) surface of this insufficient order dynamical system is obtained by Shynk (1989) 2 ^ b MSOE= cr2 2b0H(c,)+ b (23) 1\a1 A 3D surface plot and a contour plot of the MSOE performance surface are shown in Figure 24 and Figure 25, respectively. The plots show that the MSOE surface of the test model is bimodal with a global minimum (denoted by "*") at(b,a ) = (0.311,0.906), which yieldsMSOE* = 0.277, and a local minimum (marked by "+") at (b, a) = (0.114,0.519), which corresponds toMSOE+ = 0.976. The input x(k) and the test model output y(k) (with or without disturbance v(k)) are introduced to the adaptive IIR filter algorithms at the same time. The adaptive IIR filter algorithms calculate the error signal and update the weights at each iteration. Accuracy comparison for sufficient system Table 25 and Figure 26 show the simulation results and weight tracks of the four IIR algorithms for the sufficient case, respectively. For the sufficient case, the algorithms minimize the mean square error between the system output and the filter output, and the estimated weights converge to the original coefficients of the test model. Accuracy comparison for insufficient system The simulation results and weight tracks of the IIR algorithms for the insufficient case are shown in Table 26 and Figure 27, respectively. The OE algorithm starts from two different initial conditions. One point is closer to the global minimum, and the other one is closer to the local minimum. This method adjusts its weights via stochastic gradient estimation to the closest stationary point of the initial condition. Similarly, two initial conditions are selected for EE algorithm. One of them is close to the global minimum, and the other one is much closer to the local minimum. This algorithm can avoid the local minimum and adjust its weights to let the final mean square error value arrive at the area near the global minimum. However, for this insufficient order situation, the final solution exhibits bias compared to the optimum solution. The SM algorithm combines the advantages of the OE algorithm and the EE algorithm. This algorithm avoids the local minima and converges to the global minimum with different initial points, which is like the EE algorithm. At the same time, the final solution for this algorithm is very close to the optimum solution. As addressed above, the CE algorithm is a combination of the OE algorithm and the EE algorithm. It uses a weighting parameter / to switch and weight between the OE algorithm and the EE algorithm. For this insufficient identification problem, this method performs well. If the weighting parameter / is close to 0, this algorithm is more like the OE algorithm, and the interesting feature of this algorithm shows that it converges to the global minimum in the MSOE surface. However, when / is close to 1, the biased characteristic of the EE algorithm is apparent in the results. Convergence rate For convergence rate comparison, the same step size and number of iterations are chosen for simulations. The simulation conditions and the learning cures of the IIR algorithms for the sufficient case are shown in Table 27 and Figure 28, respectively. Obviously, the EE and SM algorithms converge faster than the OE and CE algorithms. Computational complexity In order to apply the ID algorithm on an adaptive control algorithm for realtime implementation, the computational complexity for one iteration of the ID algorithm have to be less than the sampling time of the DPS processor used for realtime experiment. Four algorithms are compared for computational complexity by the turnaround time with the increase of the number of unknown for each algorithm (Figure 29). The hardware used for experiment is PowerPC 750 (480MHz) microprocessor (12.6 SPECfp95). The experimental results are shown in Figure 29. The computational complexity of all of the IIR algorithms is approximately linear. And the CE algorithm needs more computational time for each iteration than time requirements for the other three algorithms. Conclusions Varies of IIR adaptive filters are examined in this Chapter, the objective of these digital filters is to identify the system coefficients (internal states) from the input and output signals. The OE algorithm and the EE algorithm are two basic structures of an adaptive IIR filter. Beyond that, two other algorithms, the SM algorithm and the CE algorithm, are also examined. Simulation results show that the mean square error value calculated by the OE algorithm converges to the optimum solution for both the sufficient case and the insufficient case if the "proper" initial condition is chosen. This means that the OE algorithm may converge to local minima in the MSOE surface. Furthermore, this algorithm does not guarantee that the poles of the ARMA model always lie inside the unit circle in the zplane. Thus, the OE method may become unstable (Haykin 2002) during the experiment. Therefore, a small enough step size and stability monitoring are required to ensure the convergence of the algorithm. However, the optimum step size is unknown, and the stability monitoring highly increases the computational complexity. These are the main drawbacks that should be considered in applications. The mean square error value calculated by the EE algorithm avoids the local minima and converges to the global minimum in the MSOE surface. The convergent rate and the computational complexity are good for realtime implementation. Unfortunately, the final solution is biased when the test model uses a lowerorder system to model a higherorder system (Shynk 1989, Netto and Diniz 1995). Both the SM algorithm and the CE algorithm can find the global minimum in the MSOE surface. However, the good performance of the SM algorithm does not occur in general and, in fact, cannot be assured in practice (Netto and Diniz 1992). Moreover, the CE algorithm produces good results when 0.04 algorithm is unimodal, and the bias is negligible (Netto and Diniz 1992). However, the stability of the CE algorithm must still be monitored, and the computational complexity is also high for this algorithm. A summary of the four algorithms is given in Table 28. The robustness results of each ID algorithms come from the experiment discussed in Chapter 4. As the results, the EE algorithm is the best algorithm comparing to the other three ID algorithms. Therefore, in next step, a MIMO IIR filter is going to be derived based on this algorithm. MIMO IIR Filter Algorithm In this section, a MIMO system ID algorithm is developed based on the SISO IIR EE algorithm. First, a linear system model is summed with the r inputs [u]rl and the m outputs [y] For simplification, the order p of the feedback loop is assumed the same as the order of the forward path. At specific time index k, the system can be expressed as y(k) = ay(k 1) + ay(k 2) + .. + a y(k p) (24) + 0u(k)+ P+u(k 1)+ f2u(k 2) + ... + f (k p) where u, (k) y, (k) u ,,(k) y,(k) u(k) = [(k)], = ( ,y(k) [y(k)]m 1 Ur(k)J rl y, (k) ta = [al]mmI.' C2 = [ak 2,mm< ap Lap]mM fo =['8,0 mx 'l=[AL,, '. =[ ,]P1gI Define the observer Markov parameters o(k) = a  a I )60 P ]m[m.pr(p+i)] and the regression vector y(k 1) y(k p) (p(k) = u(k) u(k ) [m*p+r*(p+l)] substituting Equation 26 and Equation 27 into Equation 24 yields a matrix equation for the filter outputs [y(k)]ml = [L(k) [(pr(p))] (k)][(m pr(p ))] Furthermore, the errors are defined as [ )(k)]l = [j(k)]m [y(k)]m Finally, the observer Markov parameters 26 can be identified recursively by 0(k + 1) = (k)/ p6(k)'T (k) In order to automatically update the step size, choose C + PL2 (25) (26) (27) (28) (29) (210) (211) where c is a small number to avoid the singularity when ( 2 is zero. The main steps of the MIMO identification for one iteration are summarized as follows Step 1: Initialize [\(k)] m[(mp+r*(p+l))] = [0]. Step 2: Construct regression vector [(p(k)][m*p+r (p1)] 1 according to Equation 27. Step 3: Calculate the output error [E(k)]mx1 according to Equation 29. Step 4: Calculate the step size according to Equation 211. Step 5: Update the observer Markov parameters matrix [9(k)] m[m*p+r(p+l)] according to Equation 210. Then, the calculation for the next iteration goes back to step 2. The detail derivation for this MIMO ID algorithm is given in Appendix B. And the experimental results of the algorithm, the computational complexity, and the disturbance effects will be discussed in later Chapters. The calculation result of this MIMO ID gives an estimated model of the system with the form of Equation 24. In the following Chapter, a MIMO control algorithm is developed based on this MIMO ID model. Table 21. Summary of the IIR OE algorithm. Initialization: 0(k) = [a (0) b (0)]r = 0, where i Computation: For k = 1,2,... OE(k)= Z a E(k i) + bx(k j) =) y ( = eOE (k) = y(k) 0E (k) a, (k)& E(k i)+ Z^,r(k (), for Define: 1= ,(k) & x(k j)+ a a, (k ), for =1 OE (k)= [a, (k) 1 (k)]T 0(k + 1) (k) + /eo (k)oo (k) where / is the step size. In practice: i 1,..., , j =0,1,...,b yfoE(k i) yoE(k i)+akyfoE(k i l), for i = ,...,h Define: 1 n, tfOE(k j) =x(k j)+ ak fE(k jl),for J = 0, .b 1=1 OEf(k) [f E(k i) I oE(k j)]l (k + 1) (k) +/e, (k)O (k) 1,...,ha,j = o,1,...,IA Table 22. Summary of the IIR EE algorithm. Initialization: 0(k) [ a (0) b (0) 0 where i= 1,...,n,j = 0,1,...,Ib Computation: For k = 1,2,... iY(k)= C^y(k i) + x(k j) t=1 ]=0 e (k)= y(k) Y(k) 0__ (k) = [y(k i) I x(k j)]T +(k + 1) = 0(k)+ +/e_ (k)EE, (k) where /u is the step size. Table 23. Summary of the IIR SM algorithm. Initialization: 0(k) [ a (0) b (0)] 0 where i= 1,..., 0a,j = 0,1,...,1* b Computation: For k = 1,2,... YiE(k))= C^Zy(k i) + x(k j) t=1 ]=0 es (k)= 1 e, (k) Y=1 ha y,(k i)= j(k i)+ takyj (k i ),for i= ,..., Define: f s(kj)= x(kj)+ ,akiSM(kj1),for j=0,1,...,hb /=1 Osu (k) ['s (k i)I fM (k j)] 0(k + 1) = 0(k) +ues (k)osM (k) Table 24. Summary of the IIR CE algorithm. Initialization: 0(k) [, (0) b(0)] 0 where i= 1,..., ,, j = 0,1,..., hb Computation: For k = 1,2,... Step 1: OE (k) = Z a oE(k i) + Z x(k j) eoE(k) = y(k) YoE(k) foE (k ) = OE(k i) + E (k l),for i = 1,..., Define: / IxE(k j)= x(k j)+ E(kj1), for = 0,1,. .,b 1=1 ~OEf (k)= [LfE(ki) OfE(k j)]' Step 2. y(k)= Z ,y(k i) + x(k j) eEE(k)= y(k) E(k) EE(k) = [y(k i) I x(k j)]T Step 3: eCE (k) = fle (k) + (1 )eOE (k) wCE (k) = PEE (k) + (1 P)E (k) O(k + 1) = (k) + pec (k)CE (k) where 0 < < 1 Table 25. Simulation results of IIR algorithms for sufficient case. Initial Point Final Point Adaptive Algorithm (0) (0) Numberof () (n) Structure Parameters iterations () ( =0000 004 [0.05 ,0.4007] , OE = 0.005 12000 .,0.4 ] ,0 0 [1.13,0.2484] EE U = 0.01 8000 [0.05 0.4] L0 0] [1.131 0.2493] SM u 0.005 K 011000 [0.05022,0.4006] F 007 [0.050,0.4006] u = 0.005,/ = 0.04 12000 [1.13,0.2485] u 0.01, 0.60 08000 [0.04997,0.400] L0 0] [1.131,0.2496] Table 26. Simulation results of IIR algorithms for insufficient case. Initial Point [O (0),"a,(0)] [0.5,0.1] [0.5,0.2] [0.5,0.1] [0.11,0.52] [0.5,0.1] [0.5,0.2] [0.11,0.52] [0.5,0.1] [0.5,0.2] Number of iterations 5500 12000 7000 7000 3000 4500 5500 5000 19000 Global Min. [0.311,0.906] Final Point [0.3098,0.8998] [0.0928,0.4896] [0.04577,0.8755] [0.05003,0.8719] [0.3132,0.9039] [0.3062,0.8992] [0.2967,0.9031] [0.3037,0.9112] [0.315,0.9181] Adaptive Structure Algorithm Parameters OE EE SM CE 0.001 0.003 0.001 0.001 0.0005 0.0005 0.0005 0.001, 8 0.003, 8 0.04 0.04 Table 27. Simulation conditions of IIR algorithms for sufficient case. Initial Point SAlgorithm hO (0) (0) Number of Adaptive Structure Paramet(0) (0) iterations OE / = 0.005 o 12000 10 01 EE / = 0.005 o 12000 10 01 SM o = 0.005 o 12000 CE 0 CE u = 0.005,/8= 0.04 0 0 12000 Table 28. Summary of the IIR/LMS algorithms. Rank Order: A(High or Good) > B > C > D(Low or Bad) Computational Convergent Rate Complit Complexity Robustness Accuracy SM CE Output y(k) Figure 21. Linear timeinvariant (LTI) IIR Filter Structure. Additive White Noise v(k) Test Model Output y(k) Figure 22. Simulation structure of the adaptive IIR filter. Input x(k) c(k) E 1  _ 0 "C m zplane of Selected System     1 0 1 2 3 4 5 6 7 8 Real Part Figure 23. zplane of the test model. 18 16 14 12 10 8 6 4 2 66 a 0.4 6.8 Figure 24. 3D plot of the MSOE performance surface of the insufficient order test system. 0.8 0.4 0.2 S0 0.2 0.4 0.6 0.8 Local Minimum 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 b Figure 25. Contour plot of the MSOE performance surface. IIR OE method 5 1 ) 2000 4000 6000 8000 10000 number of iterations IIR SM method 12000 IIR EE method 0.5 0.5 1 1.5 0 2000 4000 6000 8000 10000 120 number of iterations IIR CE method 00 0.5 0.5 1 1 1.5 5 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 number of iterations number of iterations Figure 26. Simulation results of weight track of the IIR algorithms for sufficient case. IIR OE method IIR SM method 0 02 04 06 b IIR CE method Figure 27. Simulation results of weight track of the IIR algorithms for insufficient case. Learning Curve LIuJ 04 C 03 02 0 1 0 2000 4000 6000 8000 10000 12000 number of iterations Figure 28. Learning curve of IIR algorithms for sufficient case. IIR EE method 100 70 50 1 10 20 30 40 50 64 Number of Unknown Figure 29. Computational complexity results from the experiment. CHAPTER 3 GENERALIZED PREDICTIVE CONTROL ALGORIHTM This Chapter describes the background of the generalized predictive control (GPC) algorithm. Then, the GPC algorithm is developed based on the MIMO system ID model (discussed in Chapter 2). Both batch method and recursive version are given and discussed. Introduction The generalized predictive control (GPC) algorithm belongs to a family of the most popular model predictive control (MPC). The MPC algorithm is a feedback control method, different choices of dynamic models, cost functions and constraints can generate different MPC algorithms. It was conceived near the end of the 1970s and has been widely used in industrial process control. The methodology of MPC is represented in Figure 31, where k is the time index number, u(k) are the input sequences, and y(k) are the actual output sequences. The y(k) and y, (k) are estimated output and reference signals, respectively. Two comments are made here to describe all MPC algorithms. First, at each time step, a specific cost function is constructed by a series of future control signals up to u(k+s ) and a series of future error signals, which are the differences between the estimated output signals y(k+j) and the reference signals y,(k+j). Second, a series of future inputs u(k+j) are calculated by minimizing this cost function, and only the first input signal is provided to the system. At the next sampling interval, new values of the output signals are obtained, and the future control inputs are calculated again according to the new cost function. The same computations are repeated. Some important MPC algorithms, such as model algorithmic control (MAC), dynamic matrix control (DMC) and GPC, have become popular in industry. MAC explicitly uses an impulse response model and DMC applies the step response process model in order to predict the future control signals (Camacho 1995). The GPC method, which is inherited from generalized minimum variance (GMV) (Clarke 1979), was proposed and explained by Clarke (1987 a, b). The GPC algorithm is an effective selftuning predictive control method (Clarke 1988). It uses controlled autoregressive and integrated moving average (CARIMA) model to derive a control law and can be used in real time applications. Juang et al. (1997, 2001) give the derivation of the adaptive MIMO GPC algorithm. This algorithm is an effective control method for systems with problems of nonminimum phase, open loop unstable plants or lightly damped systems. It is also characterized by good control performance and high robustness. Furthermore, the GPC algorithm can deal with the multidimension case and can easily be combined with adaptive algorithms for selftuning realtime applications. The problem of flowinduced open cavity oscillations exhibit several theses issues, therefore, the GPC is considered as a potential candidate controller. Two modifications are made for this algorithm. First, a input weight matrix is integrated into the cost function, this control matrix can put the penalty for each control input signal and further to tune the performance for each input channels. Second, a recursive version of GPC is developed for realtime control application. MIMO Adaptive GPC Model In this section, a MIMO model, which has the same form of the MIMO ID algorithm, is considered. A linear and time invariant system with r inputs [u]rl and m outputs [y]l. at the time index k can be expressed as y(k)= ay(k 1) + ay(k 2).. + ay(k p) + u(k) 3u(k 1) + u(k 2) + u(k (31) + nu(k)+ Pzu(k 1)+ Pu(k 2)+... + Pu(k p) where Su (k) y, (k) u(k)= [u(k)] y(k) [y(k) ur (k)J rl ym(k)Jmi fla = [,8,0 l, = ,[ ]x,,, *., p [p p , {a=[oA]mm=[a2 [2]mm,<,ap=[ [a9] Shifting j step ahead from the Equation 31, the output vector y(k + j) can be derived as y(k+ j)= aJ) y(k 1) +... + cr y(k p + 1) +a, Jy(k p) + fu(k + j)+ ,1)u(k + j 1) +. +jfl )u(k) + AP, )u(k 1) + Jp + Pu(k p) where [al(j lmxm [a2(J)m]m [ap(j) ]mXm [Cl!;m 1)a,2 + a0 1) 1) Ot p 1) a,1 p [fo(j' ]m r (al(J )fi + PA(J1) [fi(I) ]mrM, (a1(J 1) A + l2(j 1)) [,A", (a,(J )f,1 + fiA 01) ^"'L"r ". a1" )8 and with initial [a(O)]mxm [2Jm]m (35) LLap 1()Jmm a , LAp()m, 1 j , [ (O)]m am [ rp(O]mx__ The quantities 8,(k) (k = 0,1,..) are the impulse response sequence of the system. Defining the following the vector form (32) (33) (34) Su(kp) u(k) u(k p + 1) u(k + 1) up(kp)= k ,u(k)= u(k u(k ) ,,x u(k + j) r.( )x u(k 1 (rp) 1 k + 1/ (r*(j+))xl (36) (36) (y (k p) yp(k p)= y(k p+) Sy(k 1) 9(m*p)x1 the predictive index j = 0,1, 2, .q, q +1, s 1, and usk) u(k + 1) ,y(k) y(k1) (37) \u(k + s 1) ( y(k + s 1), (.m* Finally, the predictive model for future outputs, ys, is obtained, this future outputs consists of a weighted summation of future inputs, us, previous inputs, up, and previous outputs, yp y (k)= Tu (k) + Bup (k p)+ Ayp (k p) (38) where p o0 ... 0 T (1) fl ... 0(39) o(s1) () (s2) f. 0 (m*s)(r*s) B p() Bp (1) f il) ( 1 B .= (310) p(s1) p l(s\1) ... 1) (m*s)x(r*p) ap ap1 a, = : .. : (311) (s1) (s1) (s1) a a,^ ... a, SP P 1 1 (m*s)x(m*p) The detail derivation of the GPC model is given in Appendix B. MIMO Adaptive GPC Cost Function Assume the control inputs (present input and future inputs) depend on the previous inputs and output and can be expressed as up(k p) (s*r) 1 (s*r) [p*(m+r)] y(k p)12) Lp P][p*(m+r)] 1 Two potential cost functions are list below. The first one consists terms of future outputs and a trace of the feedback gain matrix J(k)= yf,(k)Qy,(k)+ 7tr(H'H) (313) and the second definition of cost function based on the total energy of future outputs as well as the inputs J(k) = (y (k)Qy, (k) + uf (k)Ru, (k)) (314) The output weight matrix Q, input weight matrix R and the control horizon s are important parameters for tuning the controller. The horizon s is usually chosen to be several times longer than the rise time of the plant in order to ensure a stable feedback controller (Gibbs et al. 2004). Also, if the predict horizon range is from zero to infinity, the resulting controller approaches the steadystate linear quadratic regulator (Phan et al. 1998). MIMO Adaptive GPC Law In order to minimize the cost function, three approaches are considered as follows. * Based on Equation 313, the control coefficients can be update using adaptive gradient algorithm. * Based on Equation 314, the optimum solution can be derived. However, this method requires the calculation of a matrix inverse, so the computational complex is higher. * Based on Equation 314, the control coefficients can be updated using an adaptive gradient algorithm. The first approach is examined by Kegerise et al. (2004). In the next section, the latter two approaches are derived. MIMO Adaptive GPC Optimum Solution Based on the cost function 314, the goal is to find [H](s*r)x[p*(m+r)] or [u (k)](sr)a 1 to minimize the cost function. We will show that both minimizing the cost function 314 respect to control matrix [H](s*r)x[p*(m+r)] and input vector [u (k)](s r)l will provide the same result. To simplify the expression, let's define = Up (k p) (315) P [p*(m+r)]x1 y (k p) v L"p P l[p*(m+r)]x1 Substituting the predictive model 38 and control law 312 into the cost function 314 gives J(k) = (y, (k)Qy,(k) + u (k)Ru, (k)) =I(Tu +[B A][v ])Q(Tu++[B A][vp]) (316) + ([H][1]) R([H][v) with some algebraic manipulation, the gradient of cost function respect to the control matrix [H](s*r)[p* (m+ r) can be obtained. The optimum solution is obtained when the gradient equal to zero. J(k)= T T, [V P]'T +R([H][H v )[v] =TTQ((Tu+[B A][vp ]) [VP]T +Ru [Iv ] (317) =(TQT+R)u [v] +T TQ[B A][vvP][ V =0 thus, u = (TTQT+R) TTQ[B A][v] (318) Alternatively, from Equation 316, setting the gradient of the cost function with respect to the input vector [u (k)](s ,) to zero gives J(k) =(Tu, +[B A][v ) QT+ R (319) 9u (319) =0 thus, u =(T TQT+R) TTQ[B A][v] (320) A comparison of Equation 320 to Equation 318 shows that these two approaches yield the same result. It is easy to apply the optimal solution of the Equation 320 on the cavity problem. However, the matrix inversion calculation has high computational complexity. Only if the model order is low enough, the optimal input can be used in realtime application. MIMO Adaptive GPC Recursive Solution To avoid calculating the inverse of the matrix in Equation 320, the stochastic gradient descend method can be used to update the control matrix H using the following algorithm "J(k) H(k + 1) = H(k) u (321) H (k) Substituting Equation 312 into Equation 317 gives OJ(k) (TTQT+R)[H[] vp ] [VP ]T OH(k) +TTQ[B A][vp][vP ] (322) ={(TQT+R)[H] +TQ[B A]}[v ]v [V therefore, the recursive solution is given by H(k 1) =H(k) (TTQT+R)H(k)+TTQ[B A]}[vp][v]T (323) Since only present r controls [u(k)] i are applied to the system, only the first r rows in Equation 323 are used [h(k+1)]=[h(k)] u(T'QT+R)[H(k)]+TQ[B A][v][v ] (324) first r rows In next chapter, this adaptive feedback controller, which is the combination of the MIMO system ID (discussed in Chapter 2) and the GPC algorithm, is implementation on a vibration beam test bed. The output weight matrix Q, input weight matrix R and the control horizon s are tuning for testing their effects to the control performance. Horizon k2 k1 k k+l k+2 k+3 Figure 31. Model predictive control strategy. k+j k+s CHAPTER 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES In this Chapter, the MIMO system ID (discussed in Chapter 2) algorithm and the GPC algorithm (discussed in Chapter 3) are implemented on a vibration beam test bed. Since the objective idea of this sample experiment is similar to the flowinduced cavity oscillation, which is the disturbance rejection problem, the results of this vibration beam experiment will give us some insights to guide the later flow control applications of using this realtime adaptive control mythology. First, computational complexity of ID algorithm, ID results in time domain and frequency domain, and the disturbance effect for ID algorithm are examined. Then, the output and input weight matrices as well as the control horizon are tuned for testing the control performance with varies of these parameters. Schematic of the Vibration Beam Test Bed Figure 41 shows a detailed sketch of the whole vibration control testbed setup. A thin aluminum cantilever beam with one piezoceramic (PZT5H) plate bonded to each side is mounted on a block base and connected to an electrical ground. The two piezoceramic plates are used to excite the beam by applying an electrical field across their thickness. The piezoceramic plate bonded to the right side of the beam is called the "disturbance piezoceramic" because it is used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the left side of the beam is called the "control piezoceramic" because it is supplied with the controller output signal to counteract the unknown disturbance actuator. The beam system has a natural frequency of about 97 Hz. The goal of the disturbance rejection controller is to mitigate the tip deflection of the aluminum beam generated by an external unknown disturbance signal. The controller tries to generate a signal to counteract the vibration of the aluminum beam generated by the "disturbance piezoceramic". The performance signal and the feedback signal of the controller are collocated, which is measured at the center of the tip of the beam by a laseroptical displacement sensor (Model MicroEpsilon OptoNCDT 2000). This device gives an output sensitivity of 1 V/mm with a resolution of 0.5/um and a sample rate of 10kHz. The performance signal is filtered by a high pass filter (Model Kemo VBF 35) with f, = 1 HZ to filter out the dc offset of the displacement sensor and then amplified by a highvoltage amplifier (Model Trek 50/750) with a gain of 10. The disturbance and control signals are generated by dSPACE (Model DS1005) DSP system with 466MHz Motorola PowerPC microprocessor and amplified by two separate channels of the power amplifier by the same gain of 50. The types and conditions of the signals are discussed in detail in the next section. The dSPACE system has a 5channel 16bit ADC (DS2001) and a 6channel 16bit DAC (DS2102) board. The signals are acquired using Mlib/Mtrace programs in MATLAB through the dSPACE system. The block diagram of the vibration beam test bed is shown in Figure 42. System Identification Experimental Results Computational Complexity During the realtime adaptive control of flowinduced cavity oscillations, computational complexity is an important issue. Kegerise et al. (2004) use 80 order estimated model for the system ID and 240 prediction horizon for the recursive GPC algorithm to capture the dynamics of the cavity system. Therefore, the computational complexity of the online adaptive controller have to reach or beyond these lengths of parameters. Figure 43 shows the changes of turn around time with the increasing estimated system order of the MIMO system ID algorithm. It is clear that the computational complexity of this algorithm is approximately linear, and the time requirement to estimate the same system order for the two inputs and two outputs (TITO) system is approximately three times longer than the requirement of the SISO system. And both cases have enough turn around time for subsonic cavity experiment. System Identification Before identifying the parameters of the system, the system order has to be estimated. Using the ARMARKOV/LS/ERA algorithm (Akers et al. 1997; Ljing 1998), the eigenvalues of the triangle matrix calculated from singular value decomposition (SVD) of the vibration beam system is shown in Figure 44. This plot shows that the minimum reasonable estimated order of the system is 2. For different ID algorithms, the estimated system order may be different in practice. Therefore, the resulting identified system transfer function should be checked in order to match the experimental data shape in both time domain and frequency domain. A periodic swept sine signal (Figure 45) is chosen as the ID input signal (without external disturbance). The sampling frequency is 1024 Hz, the sweep frequency produced by dSPACE system is from 0 Hz to 150 Hz, and the amplitude of the sweep sine signal is 0.25 volt. The performance of the system ID algorithm improves with increasing estimated system order. For this case, the estimated order of system ID block is set to 10. Figure 45 shows that the output of the system ID algorithm matches the system output very well in the time domain. The coherence function (Figure 46) shows good correlation between the input and output signals. The zeropole location and the transfer function between the input and sensor output are shown in Figure 47 and Figure 48 respectively. Three system ID methods are used for comparison. Two batch methods calculate transfer function in frequency domain using the experimental data and the FRF method to fit the frequency domain data. One recursive method updates the system coefficients in realtime. Notice that all of these three methods give the similar shape in frequency domain and capture the two dominant poles of the system (Figure 4 7). However, the FRF fit batch method gives a lower order model than the recursive method. Disturbance Effect External disturbance degrades the performance of the system ID algorithm. Figure 49 shows the vibration shim experiment system ID result with different external disturbance levels. A larger SNR (lower external disturbance level) in the input signal generally give more accurate identified system models. However, although the lower SNR input signal may result in a suboptimal system model, the closedloop control implementation based on this model still works well. The results are shown later. ClosedLoop Control Experiment Results Computational Complexity The controller design block is the most time consuming blocks in the entire adaptive control implementation. Estimated model order and the length of the predict horizon are two main parameters effecting the computational complexity. Figure 410 illustrates the computational complexity of the main C code Sfunction block which maps the observer Markov parameters (discussed in Chapter 2) to predict model coefficients (discussed in Chapter 3). The result shows that the turnaround time increases more quickly with increasing estimated system order than increasing of the prediction horizon. ClosedLoop Results The experimental parameters for the closedloop control are list in Table 41, and some result plots are presented here. Figure 411 shows the sensor output signal in time domain, in which the control signal is initiated at time 0. Power spectra of open loop (base line) vs. closedloop sensor output and the closedloop sensitivity are shown in Figure 412 and Figure 413, respectively. The sensitivity function is define as S= (f (41) Equation 41 provides a scalar measurement of disturbance rejection. A value less than one (negative log magnitude) indicates disturbance attenuation, while a value greater than one (positive log magnitude) indicates disturbance amplification. Although the resonance of the open loop system can be mitigated by the closedloop controller, a spillover phenomenon is also observed in Figure 413. As discussed in Chapter 1, the spillover problem is generated because, for this special case, the performance sensor output and the measurement sensor output (feedback signal) are collocated. Next, the effects of the adaptive GPC parameters are examined. Figure 414 shows the effect of the changes of the estimated model order. Figure 415 shows the effect of the changes of the predict horizon. Figure 416 shows the effect of the changes of the input weight. Figure 418 shows the effect of the different level of disturbance (SNR) during the system ID. The results for each case are discussed below. Estimated Order Effect In general, increasing the estimated order of the GPC, up to a certain point, can improve the performance of the closedloop control (Figure 414). The experimental result shows that when the estimated model order is greater than 4, the closedloop controller can not improve the performance any more. Predict Horizon Effect It is clearly see that increasing the predict horizon can improve the performance of the closedloop controller (Figure 415). Input Weight Effect The input weight penalizes the magnitude of the input signal. For this experiment, +0.75 volt saturation is given to the input signal to avoid the damage of the actuator. In order to restrict the input signal within the limits of the saturation, the input weight should be carefully tuned to obtain a realizable GPC. Although a smaller input weight improve performance of the closedloop controller (Figure 416), it also generates a larger control signal (Figure 417). Therefore, the tuning idea is to decrease the input weight as low as possible under the input saturation constraints. Disturbance Effect for Different SNR Levels During System ID As mentioned above, the level of the external disturbance signal (different SNR) is an important issue for the accuracy of the system ID (Figure 49). However, the adaptive closed loop controller gives the surprising results (Figure 418). Three cases are examined and compared in this section. First, the open loop (base line) case is the power spectrum of the output measurement of laser sensor without any control input. Second, the external disturbance is turned off during the system ID. Finally, the external disturbance is turned on with some level during the system ID. The result shows that the higher disturbance level (low SNR) does not have a detrimental effect on the performance of the closedloop system. In fact, the performance of the closedloop controller with low SNR is improved slightly. Summary Table 42 gives a summary of the experimental results of adaptive GPC algorithm. It can be seen that the GPC algorithm gives the better control performance with the larger estimated system order, the higher prediction horizon and the lower input weight. In Chapter 6, the similar control approach combining the system ID algorithm and the GPC algorithm will be implemented on the flowinduced cavity oscillations problem. Since the control ideas for both the vibration beam problem and the cavity oscillations problem are disturbance rejection, the successful implementation of the system ID algorithm and GPC algorithm to the vibration beam test bed may give the guidance to the flow control. Table 41. Parameters selection of the vibration beam experiment. Fs Disturbance GPC Low Pass Filter White Noise (IIR Butterworth, 4th order) 1024Hz Var = 0.09 fc = 150Hz Input Weight Prediction Estimated H Horizon Order Table 42. Summary of the results of the adaptive GPC algorithm. Input Weight Prediction Horizon Integrated Reduction (In dB) 7.0 9.2 8.4 2.8 Reduction at Resonance (In dB) 11.2 20.1 13.4 4.2 Estimated Order Figure 41. Schematic diagram of the vibration beam test bed. Figure 42. Block diagram of the vibration beam test bed. Analog Part Digital Part x 10a dt d1 II 0.8 .E 0.6 0 C i < 0.4 0.2 C" Computational Complexity of MIMO ID 0 100 200 300 400 500 600 700 800 900 1000 Estimated System Order Figure 43. Computational complexity of the MIMO system ID. Eigenvalues of the System 1.5 ** ++ ++ + 0 5 10 15 20 25 Number of Eigenvalues Figure 44. Eigenvalues of the triangle matrix obtained by the SVD method of the vibration beam system. (Calculated by ARMARKOV/LS/ERA algorithms with 50 Markov parameters and estimated order of the denominator is 10). 81 / TITO I     TI             I         I    T  I    r  1 0.4 024  S 0 01 02 0.3 System 0 05    1 r (0.5 1 0 0.1 0.2 0.3 Figure 45. (bottom). Input 0.4 0.5 0.6 0.7 0.8 0.9 n Output and ID Output 0.4 0.5 0.6 0.7 0.8 0.9 1 Times) Input time series (top), system output and system ID algorithm output time series Coherence V V;2_ .... ....  .. .. .    .____ ______ __ ________________________ __ _ _ _  _   .   L  ,, ,,          1 0.9 0.8 07 E LU 0.6 0.5 L.L i 0.4  0.3 0 0.2 0.1 n Frequency (Hz) Figure 46. Coherence function of system input and system output. Zplane (FRF:top, ID: bottom)     I I I [ I I I 3 2 1 0 1 2 3 Real Part   I Oy 4 3 2 1 0 Real Part 1 2 3 4 Figure 47. Zeropole location of FRF (top) and system ID algorithm. The estimated order of FRF fit function is 2 and the estimated order of system ID algorithm is 10. TF Magnitude 40 S* experiment OD 20 r C f f   0 FRF fit r0 2    60 L 0 50 1 0 50 100 150 200 (; 100 (D 0 c 200 300 0 Frequency (Hz) S experiment    FRF fit ID 'r S.  _.  5 1  50 100 1 50 100 15 Frequency (Hz) Figure 48. Identified transfer function using the experiment data by frequency response function (experiment), frequency response function fit(FRF fit) and time domain system ID algorithm (ID). The estimated order of FRF fit function is 2 and the estimated order of system ID algorithm is 10. 83 0.5 0 0.5 1ik Learning Curve (Fs=1024Hz) 10 ro 104 10 10 L 0 51 5 10 Times) Figure 49. Learning curve of system ID with different input SNR. The estimated system order is 10, sampling frequency is 1024 Hz. dt 1 0.8 x 103 Computational Complexity of Controller Design (SISO) 0 5 10 15 20 25 30 35 40 Prediction Horizon A Figure 410. Computational complexity of the main controller design C Sfunction. A) For SISO case. B) For TITO case. 84  2nd order * 3rd order a 5th order / S 10th order e 16th order IEstim ated Order BEstimatedt Order   ^j /  T    T     T    x 10 Computational Complexity of Controller Design (TITO) dt 1  2nd order  3rd order  5th order e 10th order  e 12th order .E 0     _0 6      ^.  ^ F Estimated Order < 0.4 L    1 _ 0 .2   ... ...........   0.2 0 5 10 15 20 25 30 Prediction Horizon B Figure 410. Continued System Output w/o:control with :c ntrol 0.6  0.4 2! 0.2 0.4  0.6  15 10 5 0 5 10 15 Time(s) Figure 411. System output time series data. The control signal is introduced at 0 second, the estimated order is 10, predict horizon is 10, and the input weight is 1. 85 Power Spectrum of performance 10 i I open loop (w/o control) 20 losedloop (with control) 30^[   IL E 40    0      70 80 0 50 100 150 Frequency (Hz) Figure 412. Power spectrum of output signal with control and without control signal. The estimated order is 10, predict horizon is 10, and the input weight is 1. Sensitivity 0.2 0.    O 0.2  0 Cl) 0 0.2 0.4 (.9 0 S0.6 ry 0.8 1 0 50 100 150 Frequency (Hz) Figure 413. Sensitivity function of the system. The estimated order is 10, predict horizon is 10, and the input weight is 1. Power Spectrum of performance 10  open loop 4th order 20  6thorder  0 6th order 10th order 30    40 S50 a 60  70  70  80 0 50 100 150 Frequency (Hz) Figure 414. Power spectrum of output signals for different estimated order. Predict horizon is 10, and the input weight is 1. Power Spectrum of performance 10 ' open loop 20  predict horizon is 4.. predict horizon is 10 30 E 0 60   70 80 0 50 100 150 Frequency (Hz) Figure 415. Power spectrum of output signals for different predict horizon. The estimated order is 4, and the input weight is 1. 87 Power Spectrum of performance 10  open loop 20 inputweightis 1 .. input weight is 10 30  .  70 j 40  50    8  C    80 0 50 100 150 Frequency (Hz) Figure 416. Power spectrum of output signals for different input weight. The estimated order is 10, predict horizon is 10. Control Signal (input weight 1) 0.2 0.1 0 .0 1    Control Signal (input weight 10) 0.04 > 0.00 4       o 0 0.02   0.04 0 5 10 15 Times) Figure 417. Control signals for different input weight. The estimated order is 10, predict horizon is 10. 88 Power Spectrum of performance 10  open loop  IDw/o dist. 20 ID with dist. (SNR=0) 30  40 a_ 60  70 50  80 0 50 100 150 Frequency (Hz) Figure 418. Power spectrum of output signals for different system ID disturbance conditions. The estimated order is 10, predict horizon is 10, and the input weight is 1. CHAPTER 5 WIND TUNNEL EXPERIMENTAL SETUP The experimental facilities and instruments used in this study are described in detail in this Chapter. These devices consist of a blowdown wind tunnel with a test section and cavity model, unsteady pressure transducers, data acquisition systems, and a DSP realtime control system. Finally, the actuator used in this study is described. Wind Tunnel Facility The compressible flow control experiments are conducted in the University of Florida Experimental Fluid Dynamics Laboratory. A schematic of the supply portion of the compressible flow facility is shown in Figure 51. This facility is a pressuredriven blowdown wind tunnel, which allows for control of the upstream stagnation pressure but without temperature control. The compressed air is generated by a Quincy screw compressor (250 psi maximum pressure, Model 5C447TTDN7039BB). A desiccant dryer (ZEKS Model 730HPS90MG) is used to remove the moisture and residual oil in the compressed air. The flow conditioning is accomplished first by a settling chamber. The stagnation chamber consists of a 254 mm diameter cast iron pipe supplied with the clean, dry compressed air. A computer controlled control valve (Fischer Controls with body type ET and Acuator Type 667) is situated approximately 6 meters upstream of the stagnation chamber with a 76.2 mm diameter pipe connecting the two. A flexible rubber coupler is located at the entrance of to the stagnation chamber to minimize transmitted vibrations from the supply line. The stagnation chamber is mounted on rubber vibration isolation mounts. A honeycomb and two flow screens are located at the exit of the settling chamber and the start of the contraction section, respectively. The honeycomb is 76.2 mm in width (the cell is 76.2 mm long) with a cell size of 0.35 mm. Two antiturbulence screens spaced 25.4 mm apart are used; these screens have 62% open area and use 0.1 mm diameter stainless steel wire. For the current experiment, the facility was fitted with a subsonic nozzle that transitions from the 254 mm diameter circular crosssection to a 50.8 mm x 50.8 mm square crosssection linearly over a distance of 355.6 mm. The profile designed for this contraction found in previous work provides good flow quality downstream of the contraction (Carroll et al. 2004). The overall area contraction ratio from the settling chamber to the test section is 19.6:1. For the present subsonic setup, the freestream Mach number can be altered from approximately 0.1 to 0.7, and the facility run times are approximately 10 minutes at the maximum flow rate due to the limited size of the two storage tanks, each with volume of 3800 gallons. Test Section and Cavity Model A schematic of the test section with an integrated cavity model is shown in Figure 52. The origin of the Cartesian coordinate system is situated at the leading edge of the cavity in the midplane. The test section connects the subsonic nozzle exit and the exhaust pipe with 431.8 mm long duct with a 50.8 mm x 50.8 mm square cross section. The cavity model is contained inside this duct and is a canonical rectangular cavity with a fixed length of L = 152.4 mm and width of W = 50.8 mm and is installed along the floor of the test section. The depth of the cavity model, D, can be adjusted continuously from 0 to 50.8 mm. This mechanism provides a range of cavity lengthtodepth ratios, L/ D, from 3 to infinity. The cavity model spans the width of the test section W. However, a small cavity width is not desirable, because the side wall boundary layer growth introduces threedimensional effects in the aft region of the cavity. As a result, the growth of the sidewall boundary layers in the test section may result in modest flow acceleration. The boundary layers have not been characterized in this study. Nevertheless, the cavity geometry applied in this study is consistent with previous efforts in the literature (Kegerise et al. 2007a,b) considered to be shallow and narrow, so twodimensional longitudinal modes will be dominant (Heller and Bliss 1975). Removable, optical quality plexiglas windows with 25.4 mm thickness bound either side of the cavity model to provide a full view of the cavity and the flow above it. The floor of the cavity is also made of 14 mm thick plexiglas for optical access. Two different wind tunnel cavity ceiling configurations are available. The first one is an aluminum plate with 25.4 mm thickness that can be considered a rigidwall boundary condition. This boundary condition helps excite the cavity vertical modes and the "cuton" frequencies of the cavity/duct configuration (Rowley and Williams 2006). The performance of this ceiling is discussed in the next chapter. In order to simulate an unbounded cavity flow encountered in practical bombbay configurations, a flushmounted acoustic treatment is constructed to replace the rigid ceiling plate. The new cavity ceiling modifies the boundary conditions of the previous sound hard ceiling. This acoustic treatment consists of a porous metal laminate (MKI BWM series, Dynapore P/N 408020) backed by 50.8 mm thick bulk pink fiberglass insulation (Figure 52). This acoustic treatment covers the whole cavity mouth and extends 1 inch upstream and downstream of the leading edge and trailing edge, respectively. This kind of acoustic treatment reduces reflections of acoustic waves. The performance of this treatment is assessed in the next chapter. The exhaust flow is dumped to atmosphere via a 5 angle diffuser attached to the rear of the cavity model for pressure recovery. A custom rectangulartoround transition piece is used to connect the rectangular diffuser to the 6 inch diameter exhaust pipe. Three structural supports are used to reduce tunnel vibrations (Carroll et al. 2004). Two of these structural supports attach to both sides of the test section inlet flange, and the additional structural support is installed to support the iron exhaust pipe (Figure 52). Pressure/Temperature Measurement Systems Stagnation pressure and temperature are monitored during each wind tunnel run and converted to Mach number via the standard isentropic relations with an estimated uncertainty of 0.01. The reference tube of the pressure transducer is connected to static pressure port (shown in Figure 52) using 0.254 mm ID vinyl tubing to measure the upstream static pressure of the cavity. The stagnation and static pressures are measured separately with Druck Model DPI145 pressure transducers (with a quoted measurement precision of 0.05% of reading). The stagnation temperature is measured by an OMEGA thermocouple (Model DP80 Series, with 0.10C nominal resolution). Two pressure transducers are located in the test section to measure the pressure fluctuations. The first transducer is a flushmounted unsteady Kulite dynamic pressure transducer (Model XT19050A) and is an absolute transducer with a measured sensitivity (2.64 0.06) x 10 7 V/Pa with a nominal 500 kHz natural frequency, 3.447 x 105 Pa (50 psia) max pressure, and is 5 mm in diameter. This pressure transducer is located on the cavity floor (y = D) 0.6 inch upstream from the cavity real wall (x = L), and 8.89 mm (z = 8.89 mm) away from the midplane. This position allows optical access from the midplane of cavity floor for flow visualization and avoids the possibility of coinciding with a pressure node along the cavity floor (Rossiter 1964). The second pressure transducer is also an Kulite absolute transducer (with measured sensitivity (5.13 + 0.03) x10 7 V/Pa and nominal 400 kHz natural frequency, 1.724 x105 Pa (25 psia) max pressure, 5 mm in diameter), and it is flush mounted in the tunnel side wall 63.5 mm downstream of the cavity as shown in Figure 52. From a series of vibration impact tests performed in a previous study (Carroll et al. 2004), the results indicated that the pressure transducer outputs are not affected by the vibration of the structure. An experiment to validate this hypothesis is discussed in the next Chapter. Due to a modification of the experimental setup, the second pressure sensor is moved to the cavity floor (Figure 52) for both openloop control and closedloop control. A PC monitors the upstream Mach number, stagnation pressure, and stagnation temperature, as well as the static pressure. This computer is also used for remote pressure valve control (Figure 51) in order to control the freestream Mach number using a PID controller. In addition, an Agilent E1433A 8channel, 16bit dynamic data acquisition system with builtin antialiasing filters acquires the unsteady pressure signals and communicates with the wind tunnel control computer via TCP/IP for synchronization. The code for both data acquisition and remote pressure control output generation are programmed in LabVIEW. The pressure sensor timeseries data are also collected for both the baseline and controlled cavity flows for posttest analysis. Facility Data Acquisition and Control Systems The schematic of the control hardware setup is shown in Figure 53. For the realtime digital control system, the voltage signals from the dynamic pressure transducers are first pre amplified and lowpass filtered using Kemo Model VBF 35. This filter has a cutoff range 0.1 Hz to 102 kHz, and three filter shapes can be used. Option 41 with nearly constant group delay (linear phase) in the pass band and 40dB/octave rolloff rate is chosen. The cutoff frequency is 4 kHz for a sampling frequency of 10.24 kHz. The signal is then sampled with a 5channel, 16bit, simultaneous sampling ADC (dSPACE Model DS2001). The control algorithms are coded in SIMULINK and C code Sfunctions and are compiled via Matlab/RealTime Workshop (RTW). These codes are uploaded and run on a floatingpoint DSP (dSPACE DS1006 card with AMD OpteronTM Processor 3.0GHz) digital control system. The DSP was also used to collect input and output data from the DS2001 ADC boards as well as computing the control signal once per time step. At each iteration, the computed control effort is converted to an analog signal accomplished using a 6channel 16bit DAC (DS2102). This signal is passed to a reconstruction filter (Kemo Model VBF 35 with identical settings to the antialias filters) to smooth the zeroorder hold signal from the DAC. The output from this filter is then sent to a highvoltage amplifier (PCB Model 790A06) to produce the input signal for the actuator. The computer is also able to access the data with the dSPACE system via the Matlab mlib software provided by dSPACE Inc. Actuator System In order to achieve effective closedloop flow control, high bandwidth and powerful (high output) actuators are required. The following issues should be considered for selecting the actuators (Schaeffler et al. 2002). * The selected actuators must produce an output consisting of multiple frequencies at any one instant in time. * The bandwidth of the actuators should enable control of all significant Rossiter modes of interest. * The control authority must be large enough to counteract the natural disturbances present in the shear layer. According to Cattafesta et al. (2003), one kind of actuator called "Type A" has these desirable properties. Such actuators include piezoelectric flaps and have successfully been used for active control of flowinduced cavity oscillations by Cattafesta (1997) and Kegerise et al. (2002). Their results show that the external flow has no significant influence of the actuator dynamic response over the range of flow conditions. Their later work (Kegerise et al. 2004; 2007a,b) also shows that one bimorph piezoelectric flap actuator is capable of suppressing multiple discrete tones of the cavity flow if the modes lie within the bandwidth of the actuator. Therefore, the piezoelectric bimorph actuator is a potential candidate for the present cavity oscillation problem. Another candidate actuator is the synthetic or zeronet massflux jet (Williams et al. 2000; Cabell et al. 2002; Rowley et al. 2003, 2006; Caraballo et al. 2003, 2004, 2005; Debiasi et al. 2003, 2004; Samimy et al. 2003, 2004; Yuan et al. 2005). This actuator can be used to force the flow via zeronetmass flux perturbations through a slot in the upstream wall of the cavity. Although the actuator injects zeronetmass through the slot during one cycle, a nonzero net momentum flux is induced by vortices generated via periodic blowing and suction through the slot. In this research, a piezoelectricdriven synthetic jet actuator array is designed. This type of synthetic jet based actuators normally gives a larger bandwidth than the piezoelectric flap type of actuators. A typical commercial parallel operation bimorph piezoelectric disc (APC Inc., PZT5J, Part Number: P412013TJB) is used for this design. The physical and piezoelectric properties of the actuator material are listed in Table 51. The composite plate is a bimorph piezoelectric actuator, which includes two piezoelectric patches on upper and lower sides of a brass shim in parallel operation (Figure 54). The final design of the actuator array consists of 5 single actuator units. Each actuator unit contains one composite plate and two rectangle orifices shown in Figure 55. The designed slot geometries for the actuator array are shown in Figure 56. Another advantage of this design is that it avoids the pressure imbalance problem on the two sides of the diaphragm during the experiment. Since the two cavities on either side of a single actuator unit are vented to the local static pressure, the diaphragm is not statically deflected when the tunnel static pressure deviates from atmosphere. The challenge is whether these actuators can provide strong enough jets to alter the shear layer instabilities in a broad Mach number range and also whether the actuators produce a coherent signal that is sufficient for effective system identification and control. A lumped element actuator design code (Gallas et al. 2003) was used together with an experimental trialanderror method to design the single actuator unit. The final designed geometric properties and parameters of the single actuator unit are listed in Table 52. To calibrate this compact actuator array, the centerline jet velocities from each slot are measured using constanttemperature hotwire anemometry (Dantec CTA module 90C10 with straight general purpose 1D probe model 55p 1 and straight short 1D probe support model 55h20). A Parker 3axis traverse system is used to position the probe at the center of actuator slots. The sinusoidal excitation signal from the Agilent 33120A function generator is fed to the 790A06 PCB power amplifier with a constant gain of 50 V/V. The piezoceremic discs are driven at three input voltage levels: 50 Vpp, 100 Vpp, and 150 Vpp, respectively, over a range of sinusoidal frequencies from 50 Hz to 2000 Hz in steps of 50 Hz. Each bimorph disc serves as a wall between two cavities labeled side A and side B. The notation used to identify each bimorph and its corresponding slots is shown in Figure 57. The rms velocities of the slots 3A and 3B located in the centerline of the cavity are shown in Figure 58 as an example. The maximum centerline velocities measured at the three excitation voltages for each slot are listed in Table 53. A summary of the measurements of the centerline velocities and currents to the actuator array for each slot are provided in Appendix E. The piezoelectric plate is tested over a range of frequencies and amplitudes to determine the current saturation associated with the amplifier. Figure 59 shows the simulation result calculated by the LEM actuator design code and is superposed on the experimental result, Figure 58. The results show that, the LEM actuator design code provides a pretty accurate rms velocity estimation of synthetic jet over a large frequency range between 50 Hz and 2000 Hz. Finally, the measure input current level to the actuator array after the amplifier is measured. The results are shown in Figure 510 and indicate that the input current will saturate above 136mApp, which means if the input voltage is larger than 100 Vpp, the current to the actuator will keep a constant value. During the closedloop experiments, an upper limit of 150 Vpp is used since the current probe is unavailable. Figure 511 shows the spectrogram of the pressure measurement on the cavity floor with acoustic treatment. The Rossiter modes (Equation 12) with ca=0.25, K=0.7 are superimposed on this figure. The experimental details are explained in the next chapter. For this dissertation, the lower portion of the Mach number range (from 0.2 to 0.35) is our control target as an extension to previous work by Kegerise et al. (2007a,b). The desired bandwidth of the designed actuator should cover the dominant peaks of Rossiter mode 2, 3 and 4, which is between 500 Hz and 1500 Hz. (Rossiter mode 1 is usually weaker compared to Rossiter modes 2, 3 and 4.) Over this frequency range, the designed actuator can generate large disturbances. In addition, the array produces normal oscillating jets that seek to penetrate the boundary layer, resulting in streamwise vortical structures. In essence, it acts like a virtual vortex generator. A simple schematic of the actuator jets interacting with the flow vortical structures is shown in Figure 512. The approach boundary layer contains spanwise vorticity in the xy plane (the coordinate is shown in Figure 5 2). By interacting with the ZNMF actuator jets, the 2D shape of the vortical structures transform to a 3D shape with spanwise vortical structures. These streamwise vortical disturbances seek to destroy the spanwise coherence of the shear layer, and the corresponding Rossiter modes are disrupted (Arunajatesan et al. 2003). Alternatively, the introduced disturbances may modify the stability characteristics of the mean flow, so that the main resonance peaks may not be amplified (Ukeiley et al. 2003). Unfortunately, the flow interaction was not characterized in this dissertation and will be addressed in future work. Instead of using one specific amplitude and one frequency in openloop control, a closed loop control algorithm is used in this study to exam the effects of the disturbance with multiple amplitudes and multiple frequencies. Thus, the present actuator represents a hybrid control approach, in which we seek to reduce both the Rossiter tones and the broadband spectral level. Table 51. Physical and piezoelectric properties of APC 850 device. Shim (Brass) Piezoceramic Bond Elastic Modulus (Pa) 8.963 x 1010 5 x1010 3.98 x108 Poisson's Ratio 0.324 0.31 0.3446 Density (kg /m3) 8700 7400 1060 Relative Dielectric Const. 2400 d31 (m/V) 200 x1012 200 Vpp/mm for 0.15 mm Maximum Voltage Loading r 0.15 thickness is 30 Vpp Resonant Resistance (Q) 200 Electrostatic Capacitance (pF) 210, 000 30% Operating Temp. (C) 2070 Table 52. Geometric properties and parameters for the actuator. Geometric Properties of the Diaphragm APC PZT5J, P412013TJB Piezo. Configuration Bimorph Disc Bender Shim Diameter (mm) 41 Clamped Diameter (mm) 37 Shim Thickness (mm) 0.1 Piezoceramic Diameter (mm) 30 Piezoceramic Thickness (mm) 0.15 Ag Electrode Diameter (mm) 29 Total Bond Thickness (mm) 0.03 ( 0.015 on each side) Radius a0 (mm) 1/2 Length of the Orifice L (mm) 1 Width of the Orifice wd (mm) 3 Volume in side A (mm3) 4064 Volume in side B (mm3) 2989 Damping 4 0.04 Table 53. Resonant frequencies with respective centerline velocities for each input voltage. Input Voltage 50Vpp 100 Vpp 150Vpp Slot Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) 1A 1150 35. 1100 58 1150 62 1B 1150 44 1100 69 1150 74 2A 1150 27 1100 47 1150 52 2B 1150 37 1100 60 1150 66 3A 1150 27 1100 45 1150 49 3B 1150 38 1100 60 1150 65 4A 1150 36 1150 58 1200 62 4B 1150 44 1150 68 1200 73 5A 1150 40 1150 60 1150 65 5B 1150 49 1100 72 1150 78 Pressure Valve Manual Valve Honeycomb Settling Chamber Subsonic Nozzle Connect to Test Section Screens Figure 51. Schematic of the wind tunnel facility. Fiberglass Exhaust Structural Support L Dynamic Pressure Sensor P2 Static Pressure Port Cavity Model Dynamic Pressure Sensor P1 Unit: Inch J Figure 52. Schematic of the test section and the cavity model. Dimensions are inches. Inlet Structural Supports Perforated Metal Plate Reconstruction Filter Antialiasing Filter Fc = 4 kHz Power Amplifier Gain = 50x Actuator Array Cavity Flow Fc = 4 kHz Figure 53. Schematic of the control hardware setup. o Piezoceramic Figure 54. Bimorph bender disc actuator in parallel operation. The physical and geometric properties are shown in Table 51 and Table 52. 16bit Cont e 16bit c F Controller k s ADC ADC Fs=10.240 kHz dSPACE System R i S Side A Side B A B C D Figure 55. Designed ZNMF actuator array. A) Operation plot. B) Assembly diagram of single unit. C) Singe unit of the actuator. D) Actuator array. 57.15(4irnch) I  iirrur nwardprcrn u wrsirncnwr ."". Steel &u r~lur Ii~r .L  Unit: mm TITLE SEE D'WC. N<. REV A cover SCALE 1'1 WEI HT SHEET1 0F1 Figure 56. Dimensions of the slot for designed actuator array. 106 A I "'"'**'C'*'*'L "*~'*E ~~~"""'~~'"" '*r c~.r**r.rr.. ** .*r E~. rl**l. rr. ~r~ r 4ir Figure 57. ZNMF actuator array mounted in wind tunnel. 50V PP 100 V PP 150V PP S30 0 2 I 20 500 1000 Frequency [Hz] 1500 2000 50V PP 100 V PP 150V PP 40 o 30 (U ;> 0 500 1000 1500 2000 Frequency [Hz] B Figure 58. Bimorph 3 centerline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal. A) For side A. B) For side B. Sim 50 Vpp 70 Exp. 50 Vpp Sim 100 Vpp 60 0 Exp. 100 Vpp E 50  >o 40 > 30 20 10 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Figure 59. The comparison plot of the experiment and simulation result of the actuator design code for bimorph 3. The output is the centerline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal for side B. 160 50 Vpp 100 Vpp 140 _150 Vpp 120 g 100 80 60 40 20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Figure 510. Current saturation effects of the amplifier. 5000 150 4500 140 4000 ..:... 3500 130 N .. 3000 120 (D2500 110 10 0 0 .. . Figure 511. Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with 500 7080 070 0.1 0.2 0.3 0.4 0.5 0.6 Mach number Figure 511. Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25, K=0.7). Leading Edge of the Cavity Figure 512. Schematic of a single periodic cell of the actuator jets and the proposed interaction with the incoming boundary layer. CHAPTER 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION Experimental results for the baseline uncontrolled and controlled cavity flows are presented in this chapter. First, the effects of structural vibrations on the unsteady pressure transducers are illustrated. Then, a joint timefrequency analysis of the unsteady pressure measurement for an uncontrolled cavity flow is shown. Flowacoustic features are deduced from the results. An improved test environment is established by replacing the original hardwall ceiling of the wind tunnel with an acoustic liner. This new test section minimizes the effects of the vertical acoustic modes. Finally, the results of both openloop and adaptive closedloop control experiments using the ZNMF actuator array is presented and discussed in detail. The ability of the actuator to alter both broadband and tonal content of the unsteady pressure spectra is demonstrated at low Mach numbers. Background As discussed in the first Chapter, flowinduced cavity oscillations are often analyzed via unsteady pressure measurements in and around the cavity. However, these measurements are often contaminated by other dynamics associated with the specific characteristics of the wind tunnel test section. As a result, the unsteady pressure spectrum may be due to the cavity oscillations or other phenomena. The experimental results of Cattafesta et al. (1999), Debiasi and Samimy (2004), and Rowley et al. (2005) show that some of the resonant frequencies measured within the cavity track or lock on to vertical acoustic duct modes at some test conditions. This effect can be reduced by adding acoustic treatment at the ceiling above the mouth of the cavity (Cattafesta et al. 1998; Williams et al. 2000; Ukeiley et al. 2003; Rowley et al. 2005). This acoustic treatment modifies the soundhard boundary condition and thus mitigates the contribution of the cavity vertical resonance modes to the unsteady pressure measurements. Consequently, the modified cavity model will ideally exhibit the behavior of an unbounded cavity flow and be dominated by Rossiter modes. Alvarez et al. (2005) developed a theoretical prediction method and showed that the wind tunnel walls lead to a significant increase in the growth rate of a resonant mode for frequencies near the cuton frequency of a crossstream mode. In the present baseline (i.e., uncontrolled) experimental study, flowacoustic resonances in the test section region and in the cavity region are examined. A schematic of the simplified wind tunnel model and the cavity region of the experimental setup are shown in Figure 61. Using the same nomenclature of Alvarez et al. (2005), the domain is divided into three regions: an upstream tunnel region (x < 0), a cavity region (0 < x (x > L). The Rossiter modes (R,, i = 0,1,...) are the combined result of a receptivity process at x = 0, instability growth in the unstable shear layer, sound generation due to impingement of the shear layer at x = L, and upstream and downstream propagating acoustic waves within the cavity region. The resulting flow oscillations are interesting targets for fluid dynamics and control researchers to analyze and mitigate. Additional vertical cavity acoustic modes (V,, i = 0,1,...) and cavity cuton modes (C,, i = 0,1,...) can also be present, as discussed above. These acoustic modes are generated by the reflections from the ceiling and area changes of the cavity model. During the wind tunnel experiments, the vertical modes (V) are undesirable and should be reduced in order to mimic the unbounded bomb bay problem more accurately. As explained in Alvarez et al. (2005), the upstream region can support duct cuton modes DI (i = 0,1,...) and upstream propagating duct modes (T', i = 0,1,...) due to the acoustic 114 scattering process. Similarly, the downstream region can support duct cuton modes Dd (i = 0,1,...) and downstream propagating tunnel modes ( Td,i = 0,1,...) due to the scattering process. Here, we focus our attention on the propagating modes in the cavity and the downstream tunnel regions. Data Analysis Methods A schematic of unsteady pressure transducer locations for this study was presented in Chapter 5 (Figure 52). P] and P, measure the unsteady pressure fluctuations in the cavity and downstream regions, respectively. The cavity and wind tunnel acoustic modes can be obtained experimentally using two approaches. One way is to measure the output of each unsteady dynamic pressure sensor for different fixed freestream Mach numbers and then find the spectral peaks for each discrete Mach number. However, with this method it is difficult to track the gradual frequency changes with Mach number. The other choice is to record each unsteady pressure sensor output continuously as the Mach number is increased gradually over the desired range. Then, ajointtime frequency analysis (JTFA) (Qian and Chen 1996) is applied to these recorded pressure time series data. JTFA provides information on the measurement in both the time and frequency domains. Finally, the time axis is converted to Mach number via synchronized measurements of the Mach number versus time. Similar analysis methods can be found in Cattafesta et al. (1998), Kegerise et al. (2004), and Rowley et al. (2005). In this study, the sampling frequency for experimental data collection is 10.24 kHz and the frequency resolution is 5 Hz. The cutoff frequency of the antialiasing filter is 4 kHz, and 500 continuous blocks of time series data are used in the analysis. During the experiment, Mach number sweeps from 0.1 to 0.7 in about 100 seconds. Noise Floor of Unsteady Pressure Transducers The effective insitu noise floor of the two unsteady pressure transducers is presented in Figure 62. Each noise floor measurement is compared with the spectra obtained at different discrete Mach numbers for the acoustically treated L/D=6 cavity. Within the tested frequency range, the signaltonoise ratio (SNR) is in excess of 30 dB, which demonstrates adequate resolution of unsteady pressure transducers for the present experiments despite their large full scale pressure ranges. Effects of Structural Vibrations on Unsteady Pressure Transducers A series of initial impulse impact tests are performed before the baseline experiments. As discussed in Chapter 5, with the wind tunnel turned off, the pressure transducer outputs are not affected by hammer or shakerinduced structural vibrations. A simple test is described here to investigate the effects of structural vibrations while the wind tunnel is running. To avoid confounding cavity oscillations, the cavity floor is mounted flush with the tunnel floor (D = 0). A piezoceramic accelerometer (PCB Piezotronics Model 356A16) is used to measure the structural vibrations. It is attached to the test section outer wall using wax at the location indicated in Figure 52, which is close to one of the pressure transducers (P2). This piezoceramic accelerometer is connected to a multichannel signal conditioner (PCB Piezotronics Model 481A01). Three channels of the piezoceramic accelerometer corresponding to the x, y, and z directions are measured. The coordinate directions of x and y are shown in Figure 52, and z is the corresponding lateral direction using the righthand rule. The accelerometer is calibrated with a reference shaker (PCB Piezotronics, Model: 394C06) that provides 1 g (rms) at 1000 rad/s (159.2 Hz). 116 The JTFA results (Figure 63 to Figure 65) for all components of the accelerometer measurements show that the power of the structural vibration spreads is broadband with a few spectral peaks. A modest peak at 1000 Hz is present in the lateral (z) and vertical (y) directions. In addition, some higher frequency peaks (i.e., 2450 Hz in the x or flow direction, 1880 Hz in the lateral direction and 3200 Hz in the vertical direction) can also be detected. However, the JTFA results of P, (Figure 66) and P2 do not display any of these resonances. These results confirm that the unsteady pressure transducers are not affected by structural vibrations. Baseline Experimental Results and Analysis The rigid ceiling plate (no acoustic treatment) above the mouth of the cavity is considered first. JTFA results of the unsteady pressure transducer measurement for this case are shown in Figure 67. Numerous flowacoustic resonances can be observed in the plots. For easy reference in the subsequent discussion, these features are numbered 1 and 2. The final goal of the baseline experiment is to simulate the unbounded weapon bay using the cavity model in the test section. Therefore, the active flow control scheme targets the Rossiter modes (feature 1 in Figure 67). The other unknown acoustic features 2 in Figure 67 are undesirable features that we wish to eliminate. These acoustic modes come from the bounded wind tunnel walls, the mismatched acoustic impedance due to area change, and the leading and trailing edges of the cavity. In order to better mimic an unbounded cavity flow in a closed wind tunnel, the boundary condition of the cavity ceiling must be altered to eliminate the unexpected modes within the cavity region. A flushmounted acoustic treatment (discussed in Chapter 5) is fabricated to replace the previous solid tunnel ceiling. The new cavity ceiling modifies the zero normal velocity boundary condition of the previous sound hard top plate. The unsteady pressure transducer JTFA measurement for the trailing edge floor of the cavity is shown in Figure 68. The results illustrate a very clean flow field below Mach 0.6. The acoustic features 24 in Figure 67 are eliminated within the cavity region. Therefore, the experimental Rossiter modes R, shown in JTFA plot (Figure 68) now follow the estimated Rossiter curves. At higher upstream Mach numbers (M > 0.6), the experimental Rossiter modes deviate slightly from the expected Rossiter curves. This is partly because the estimated curves use the upstream static temperature to calculate the speed of sound. This estimation does not account for the expected significant static temperature drop due to the large flow acceleration near the aft cavity region seen by Zhuang et al. (2003). Another possible reason for these deviations of the flowacoustic resonance comes from the structural vibration coupling with the Rossiter modes. At high Mach numbers above 0.6, the structural vibrations may cause a lockon phenomenon with the Rossiter modes. For this study, all experiments are thus performed below M= 0.6. In conclusion, the observed flowacoustic behavior of the acoustically treated cavity model behaves as expected below M = 0.6 and is therefore suitable for application of openloop and closedloop flow control. OpenLoop Experimental Results and Analysis The openloop and closedloop experimental results using the designed actuator array are shown in this section. Before the control experiments, measurements of the pressure sensor at the surface of the trailing edge of the cavity with the without the actuator turned on are shown in Figure 69. Without the upcoming flow, the noise floor shows a significant peak at 660 Hz and a small peak at 2000 Hz. The pressure sensor can also sense the acoustic disturbances associated with the excitation frequency and its harmonics, and the measured unsteady pressure level can reach 115120 dB. The extent to which the measured levels deviate from theses values with flow on (considered below) indicates the relative impact of the actuator on the unsteady flow. First, openloop active control is explored. The purpose of the openloop experiments is to verify if the synthetic jets generated from the designed actuator array can affect and control the cavity flow. A parametric study for the openloop control is explored first. A sinusoidal signal is chosen as the excitation input with the frequency swept from 500 Hz to 1500 Hz. The open loop experimental results are shown in Appendix F. The opencontrol performance is best over the frequency range 1000 Hz to 1500 Hz, which corresponds to the resonance frequencies of the actuator array. Since at the resonance frequency, the actuator array can generate larger velocity jet, and the blow coefficient Be = l/(pUoAc ,) increases. As a result, the control effect increases. For these openloop tests, the upstream flow Mach number is varied from 0.1 to 0.4. For illustration purposes, results are examined here for two sinusoidal signals with 200 Vpp and excitation frequencies at either 1.05 kHz or 1.5 kHz to drive the actuator array. The 1.05 kHz excitation frequency is close to the resonance frequency of the actuator, while the 1.5 kHz frequency lies between the second and third Rossiter modes. The experimental results shown in Figure 610 illustrate that this actuator array can successfully reduce multiple Rossiter modes, particularly at Mach number 0.2 and 0.3. In addition, the pressure fluctuation is mitigated at the broadband level on the surface of the cavity floor for all the tested flow conditions. However, new peaks are generated by the excitation frequencies and their harmonics, especially at low Mach number 0.1. With increasing upstream Mach number, the unsteady pressure level also increases and the effect of the control is reduced. Note the synthetic jets introduce temporal and spatial disturbances to modify the mean flow instabilities and destroy the coherence structure in 119 spanwise, respectively. The effectiveness of the actuator scales with the momentum coefficient, which is inversely proportional to the square of the freestream velocity. So, as the upstream Mach number increases, the synthetic jets are eventually not strong enough to penetrate the boundary layer and the control effect is reduced. Future work should perform detailed measurements to validate this hypothesis. The results of the openloop control suggest that this kind of actuator array can generate significant disturbances not only along the flow propagation direction but also in the spanwise direction of the cavity. The combination of these effects disrupts the KelvinHelmholtz type of convective instability waves, which are the source of the Rossiter modes. As a result, multiple resonances are reduced via active control. The experimental results also show the limitation of the openloop control. ClosedLoop Experimental Results and Analysis The openloop control results suggest that this compact actuator array may be effective for adaptive closedloop control. As discussed above, the synthetic jets add disturbances to disrupt the spanwise coherence structure of the shear layer and result in a broadband reduction of the oscillations. However, at the same time, the coherence between the drive signal and the unsteady pressure transducer will be reduced. High coherence is considered essential for accurate system identification methods. To exam the accuracy of the system ID algorithm with the change of the estimated order, an offline system ID analysis is first performed. The nominal flow condition is chosen at M = 0.275 (to match that of Kegerise et al. 2007a,b) with a L/D=6 cavity, and two system ID signals, one with white noise (broadband frequency and amplitude 0.29 Vrms ) and the other with a chirp signal (amplitude 0.86 Vrms and fL = 25 Hz to fH = 2500 Hz in T = 0.05 sec), are used as a broadband excitation source to identify the system. The running error variances the system ID are shown in Figure 611. It is clear that the larger the estimated order, 120 p, the more accurate is the system ID algorithm. However, due to the limitations of the DSP hardware, we cannot choose very large values of the estimated order for system ID algorithm on line. One potential advantage of the closedloop adaptive control algorithm is that it does not rely exclusively on accurate system ID. Figure 612 shows the result of the closedloop realtime adaptive system ID together with the GPC control algorithm for an upstream Mach number 0.27. Based on the above system ID results, due to the DSP hardware limitation, the estimated GPC order and the predictive horizon are chosen as 14 and 6, respectively. The breakdown voltage of the actuator array restricts the excitation voltage level; therefore, the diagonal element of the input weight penalty matrix R (Equation 314) is chosen as 0.1. This research represents an extension of Kegerise et al. (2007b) where the system ID algorithm and the closedloop controller design algorithm are used simultaneously in a realtime application. It is important to note that only the system ID white noise or chirp signal is used to identify the openloop dynamics, and the feedback signal is not used for this purpose. Clearly, the results show that the GPC controller can generate a series of control signals to drive the actuator array resulting in significant reductions for the second, third, and fourth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closedloop controller; the OASPL reduction is 3 dB. The input signal is shown in Figure 613. The sensitivity function discussed in Chapter 4 (Equation 41) is shown in Figure 614. A negative amplitude value indicates disturbance attenuation, while a positive value indicates disturbance amplification. The results show that all the points are negative, which indicates the closedloop controller reduces the pressure fluctuation power at all frequencies. The spillover phenomenon (Rowley et al. 2006) is not observed in Figure 614. As discussed in Chapter 1, the spillover problem is generated because either the disturbance source and control signal or the performance sensor output and the measurement sensor output (feedback signal) are collocated. The Bode's integral formula is shown in 61. flog S(ico) do = r Re(pk) (61) k where Pk are the unstable poles of the loop gain of the closedloop system. So, for a stable system, any negative area at the left hand side of the Equation 61 must be balanced by an equal positive area at the left hand side of the Equation 61. However, for present closedloop control study, the left hand side of the Bode's integral formula is 38 rad/sec, which shows that Bode's integral formula does not hold here. Since this formula is valid for a linear controller, the combination of the adaptive system ID and controller is apparently nonlinear. A more detailed study is required in the future to validate this hyothesis. A parametric study of the GPC is then studied by varying the estimated order and the predictive horizon. Figure 615 and Figure 616 show that the control effects improve with increasing order and predictive horizon. This trend matches the simulation results shown in Chapter 4. The comparison between the openloop and closedloop results is shown in Figure 617 for the same flow condition. Notice that the baseline measurement for a same flow condition can vary a little from case to case. The openloop control uses a sinusoidal input signal at 1150 Hz forcing and 150 Vpp and the rms value of the input is 53V. The closedloop control uses the estimated order 14, the predictive horizon 6, and the input weight 0.1, and the input rms value is 43 V. Upstream Region . Cavity Region  Downstream Region Figure 61. Schematic of simplified wind tunnel and cavity regions acoustic resonances for subsonic flow. 0.4 0.5 0.55  0.58 0.6 0.65 0.69 SNoise Floor 3000 frequency [Hz] 4000 5000 6000 Figure 62. Noise floor level comparison at different discrete Mach numbers with acoustic treatment at trailing edge floor of the cavity with L/D=6. 150 140  t, I 130 '. 120 ' .I r SNoise Floor 1000 2000 ;IliLa ~d~ 5000 100 4500 4000 9 3500 80 4 3000 2500 ) 70 2000 1500 60 1000 50 500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Mach number Figure 63. x acceleration unsteady power spectrum (dB ref Ig) for case with acoustic treatment and no cavity. 124 5000 100 4500 90 4000 9 3500. 80 S3000 70 = 2500 60 2000 60 1500 50 1000 40 500  0. 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Mach number Figure 64. y acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic treatment and no cavity. 125 I~ ~ ~~. I;~ ~~~;~~~~ 0.1 0.2 0.3 0.4 Mach number Figure 65. z acceleration unsteady power spectrum (dB ref. Ig) for case with acoustic treatment and no cavity. 126 5000 4500 4000 3500 T 3000 S2500 L 2000 LI. 1500o 1000 500 0 U.b U.6 0.7 ~ .s, ;. r .. . ".;r 'S ~~;.; .Rr~~P~ 11~1 5000 4500140 140 4000 3500 130 T 3000 S.120 = 2500 2000 :'" :. 'i:.. 1 10 1500 1000 . 100 500 90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Mach number Figure 66. Spectrogram of pressure measurement (dB ref. 20e6 Pa) on the trailing edge floor of the cavity for the case with acoustic treatment and no cavity. Noise spike near 600 Hz is electronic noise. 5000 .: :: ' 4500 .. .. 2 1 150 4500, 150 4000 . 140 3500 N 3000 130 2500 120 2000 1500 1.110 1000 00 100 0.1 0.2 0.3 0.4 0.5 0.6 Mach number Figure 67. Spectrogram of pressure measurement (ref 20e6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D=6. Unknown acoustics features are denoted as "2," while the Rossiter modes are denoted as "1." 5000 150 4500 140 4000 ..:... 3500 130 N .. 3000 120 (D2500 110 10 0 0 .. . Figure 68. Spectrogram of pressure measurement (ref20e6 Pa) on the cavity floor with 500 7080 070 0.1 0.2 0.3 0.4 0.5 0.6 Mach number Figure 68. Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with a=0.25, K=0.7). 129  TE noisefloor TE 1050Hz150Vpp 1000 2000 3000 Frequency [Hz] 4000 5000 6000  TE noisefloor TE 1500Hz and 150Vpp 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure 69. Noise floor of the unsteady pressure level at the surface of the trailing edge of the cavity with and without the actuator turned on. A) The exciting sinusoidal input has frequency 1050 Hz and amplitude 150 Vpp. B) The exciting sinusoidal input has frequency 1500 Hz and amplitude 150 Vpp. The peaks near 600 Hz and 2000 Hz are electronic noise. 130 115 110 105 100 95 90 85 80 75 0 500 1000 1500 2000 2500 frequency [Hz] 125 120 115 110 105 100 95 0 500 1000 1500 2000 25 frequency [Hz] 3000 3500 4000 Mach 0.2, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp 00 3000 3500 4000 00 3000 3500 4000 Figure 610. Openloop sinusoidal control results for flowinduced cavity oscillations at trailing edge floor of the cavity. A) At Mach number 0.1. B) At Mach number 0.2. C) At Mach number 0.3. D) At Mach number 0.4. The cavity model with 6 inch long and L/D=6.  Mach 0.1, Baseline  1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp Mach 0.3, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp frequency [Hz] C Mach 0.4, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp 0 500 1000 1500 2000 2500 frequency [Hz] 3000 3500 4000 Figure 610. Continued "' r i p=2 ........... p = 4 p=6 p=8 p=14 p =50 p= 100 Time(s) A x 107 1.4 1.2 p=2 .......... p = 4 p=6 p=8 p=14 p =50 p= 100 1.5 Time(s) Figure 611. Running error variance plot for the system identification algorithm. A) With chirp signal as input. B) With white noise signal as input. Upstream Mach number is 0.275, L/D=6. x 107 1.4 0.8 0) ._ 0.6 0.4 0.2 TE Baseline, M=0.27 TE ClosedLoop, p=14, s=6 130 125 120 115 110 105 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency [Hz] Figure 612. ClosedLoop active control result for flowinduced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity. The control algorithm uses an estimated order of 14 for both the system ID and GPC algorithms, and the predictive horizon is chosen as 6. A chirp signal is used as the system ID excitation source. 134 10/ V 0 \ S10 20 30 40 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency [Hz] Figure 613. Input signal of the ClosedLoop active control result for flowinduced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity. The control algorithm uses an estimated order of 14 for both the system ID and GPC algorithms, and the predictive horizon is chosen as 6. A chirp signal is used as the system ID excitation source. Disturbance Amplification 0 0.1 UO 0) j M 0.2 U) SD isturban 0.3 0.4 0 500 1000 1500 2000 2500 Frequency [Hz] ce Attenuation 3000 3500 4000 Figure 614. Sensitivity function (Equation 41) of the closedloop control for M=0.27 upstream flow condition. The estimated order is 14, prediction horizon is 6, and the input weight R is 0.1. This sensitivity is calculated based on Figure 612. 136  TE Baseline, M=0.27 TE ClosedLoop, p=2, s=6 TE ClosedLoop, p=14, s=6 0 500 1000 1500 2000 2500 Frequency [Hz] 3000 3500 4000 Figure 615. Unsteady pressure level of the closedloop control for M=0.27, L/D=6 upstream flow condition with varying estimated order. The prediction horizon is 6, and the input weight is 0.1. The excitation source for the system ID is a swept sine signal. 135 130 125 120 115 110 105 135 S TE Baseline, M=0.27 TE ClosedLoop, p=14, s=2 130 TE ClosedLoop, p=14, s=6 125 120 115 110 105 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency [Hz] Figure 616. Unsteady pressure level of the closedloop control for M=0.27, L/D=6 upstream flow condition with varying predictive horizon s. The estimated order of the system is 6, and the input weight is 0.1. The excitation source for the system ID is a swept sine signal. 135 S TE OpenLoop 130 TE Baseline, M=0.27 TE ClosedLoop 125 ' 120 e 0 115  110  1050T 100 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency [Hz] Figure 617. Unsteady pressure level comparison between the openloop control and closedloop control for M=0.27 upstream flow condition. The openloop control uses a sinusoidal input signal at 1150 Hz forcing and 150 Vpp and the rms value is 53 V. The closed loop control uses an estimated order 14, predictive horizon 6, input weight 0.1, and the input rms value is 43 V. 139 CHAPTER 7 SUMMARY AND FUTURE WORK This chapter summarizes the previously discussed work and presents contributions from this study. Future work is summarized that addresses detailed measurements of the actuator system, a systematic experimental analysis of the flow using various flow diagnostics, and a more detailed parametric study of openloop and closedloop control. Summary of Contributions The contributions of this research are summarized here. First, a global model of flow induced cavity oscillation is derived that provides insight into the required structure for a plant model used for subsequent control. When simplified, this model matches the Rossiter model. Second, a novel piezoelectricdriven synthetic jet actuator array is designed for this research. The resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. This actuator array produces normal zeronet mass flux jets that seek to penetrate the boundary layer, resulting in streamwise vortical structures. These streamwise vortical disturbances destroy the spanwise coherence of the shear layer, and the corresponding Rossiter modes are disrupted. Alternatively, the introduced disturbances modify the stability characteristics of the mean flow, so that the main resonance peaks may not be amplified. Next, a MIMO system ID IIRbased algorithm is developed based on the structure inferred from the global model. This system ID algorithm combined with a GPC algorithm is applied to a validation vibration beam problem to demonstrate its capabilities. The control achieves 20 dB reduction at the single resonance peak and 9 dB reduction of the integrated vibration levels. Finally, this control methodology is extended and applied to subsonic cavity oscillations for on line adaptive identification and control. Openloop active control uses a sinusoidal signal with 200 Vpp and an excitation frequency of either 1.05 kHz or 1.5 kHz, which are detuned from the Rossiter frequencies, to drive the actuator array. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced. However, when the upstream Mach number increases (greater than Mach number 0.4), the effects of the synthetic jets from this actuator are gradually reduced. Adaptive closedloop control is then applied for an upstream Mach number of 0.27; the estimated GPC order is 14 and the predictive horizon is 6. To avoid saturation in the control signal, the input weight penalty is chosen as 0.1. The GPC controller can generate a series of control signals to drive the actuator array resulting in dB reduction for the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closedloop controller (i.e., the OASPL reduction is 3 dB). However, unlike previously reported closedloop cavity results, a spillover phenomenon is not observed in the closedloop control result. As discussed in Chapter 1, the spillover problem is generated by a linear controller because the disturbance source and control signal or the performance sensor output and the measurement sensor output (feedback signal) are collocated. The nonlinear nature of the adaptive system may be responsible for this effect. Future Work Recommended future work consists of the following items. * The phaselocked centerline velocity of the actuator array should be measured using, at least, by hotwire anemometry. * The actuator system for the active flow control needs to be explored in detail. Since the size of the rectangular slots are small (1 mm by 3 mm), the present size of the hotwire (1 mm) cannot provide spatiallyresolved measurements. The hot wire is also not suitable to decipher the 3D velocity field resulting from the interaction of the jets with the boundary layer. Laser Doppler Velocimetry (LDV) or stereo Particle Image Velocimetry (PIV) measurements can provide good spatial resolution of the 3D, turbulent velocity field that results from the interaction between the ZNMF jets and the grazing boundary layer. * The turbulent boundary layer characteristics (e.g., incoming boundary layer thickness) at the leading edge of the cavity and the mean flowfield of the baseline uncontrolled case should be measured. * The impedance of the ceiling liner should be measured using an acoustic impedance tube. * Another potential system order estimation algorithm from information theory of empirical Bayesian linear regression by Stoica (1989, 1997) should be applied to this problem. * Parametric studies are recommended to analyze the performance, adaptability, cost function, and limitations (spillover, etc.) of the adaptive MIMO control algorithms over and above that of the present SISO experiments. * The effectiveness of the adaptive closedloop controller should be evaluated with changes in the upstream Mach number. APPENDIX A MATRIX OPERATIONS Vector Derivatives In this appendix, finite length vectors derivatives are illustrated. During most optimization method development, they are the fundamental tools to find the optimum value. Only real numbers are considered in this appendix. Definition of Vectors Define the vectors u and y as following u1 Y, (u) y, (ul, u") u2 y,(u) (u,,u,) u = ,y y= = (Al) u n1 Y.m(") M A ym(u,"'n)u x For special case, if n 1 or m = 1, the vector u or y is reduced to scalar, respectively. Derivative of Scalar with Respect to Vector au = [ u I(A 2) Note the derivative of scalar is a row vector. Derivative of Vector with Respect to Vector a y ly OYl y ... OY2 y= 0u = 0U2 n (A3) du Ym aym 2Ym aY, 1 2 n mxn Second Derivative of Scalar With Respect to Vector ay 82y a2y 0 ay 'y2 a2 OU2 uu 1uaouy _^. 8u, (Hessian Matrix) a2y a2y Ou1 cU2 c1 nSu" a2y a2y cOu2 Ou2 u28un aUJaU2 On"O _ Example 1 Given: Y3 1 L 2 1 2 u= u2, y= 1 22 2x1 32 +3u2 3 3A Find: l u Solution: From the Equation A3, the derivative of matrix is computed as Qu 1i 1 1 1, ay, y _yu yu,1 u2 93 anu u k u 3 3 (A6) 1 2 3 2I3 2u, 1 0 0. 3 2u3 23 Table of Several Useful Vector Derivative Formulas Table Ai lists the most common vector derivative results, these results are very useful for MIMO controller design. If [A, ,] is symmetric, the last formulas in Table Ai can be expressed as (A4) (A5) '= 2([A7) [A x] Proof of the Formulas Proof (a) Ynxl y1 Y2  n nxl [A,,,,][u,,,1 a11 a12 a1 n U1 a21 a22 a2n a 2 a a a * _ n1 n2 nn _nx n nxl a11+a1 12u2 + .. +ain a21,u +a2, u + +a,2Un an^lH +anu 2 U 2 + + annUn n (A8) According to Equation A3 a11 a21 [A nl az1 < aK J (A9) . aln n  nn nxn yn,, =[u,, n]T [Anxn] all S[ i ia2 =[ 1u 2 U "]L x an =[auI +a7212 + +a lnln a12 aIn a22 a2n an2 ". nn nxn ... alnu" +a2nu2 +' +"+annUnnu According to Equation A3 Proof (b) (A10) (A7) FaI a21 anl1 au i i i (A11) Aln a2n nn n,,n = [A nx ] Proof (c) Yix = [ ].Ti [". x] =[ 2 (A12) [1 .2 ""n n 1 LUn Jnxl = [2 +U22 +... + Un21] According to Equation A2 =[2u, 2u2 2u pu " =2[u1 u2 ]l, (A13) =2[u n1 ] Proof (d) Y>xi = [uxll]T [AI,][u x] 1 1 1 lxn " 12 1 = [al 21 + a.U + + t.n).n ... 2ln.l +ni7 +... + tl 2n2 n LiE i 1 = [(aul, +a2lu2 + +aunlUn)u 1 + ( + (aUl +a 2nU2 + 1+ann,,un) n ] According to Equation A2 y (~1 +au +a 21u2 +a + lun)+(a1ual +a12u2 + + +alnun) " u (a, u, +a2 + +a, )+ (a,,,u +a + + +a,,]n) i, = [aI,+u +C L21U +a+C un anuI + +72A + +a+annUn, n + [a11, +a21U2 +. +au, ... a,,,u +anu + +annUn ] n [ +ai2n2+'"+a" ... an+an2u2+...+annun] a11 a12 aln a a 2a2 a2 =[u U2 .. 21 u22 (A15) ani an2 "" ann al 1 a21 n +[u1 22 u]n a12 a 22 a2 + u 2 ... *** n an a2n ann nxn =[un ]T [Ann + [u nx ]T [An ]T Example 2 Given: uz 1 1 0 u= u2 ,A= 1 0 1 3 3I0 0 1 (A16) Y 1x [31s [A3x3[1[3 1] Find: Bu Solution: Y = [U 31 ]T [A 33 ][u3x1] 1 1 0 u I = u2, 2 13 1 0 1 u2 (A17) u12 2 3x 2 = 112 + U 23 + U 2 According to Equation A2, the derivative of is computed as 8u = [2u1 u3 u2 +2u3 ]1 au Now, calculate the gradient of using the Table A1 Qu =[uI ]T [A 3]+[uI]T [A 3] au [1 1 1" 02 1 1 0 =[u1 u u1 u2 ] 3]0 1 +[ 2 [ 1u u3 3]1 0 0 0 0 1 0 1 13x3 =[2u, u3 u +2u3 ]3 The result of Equation A19 is the same as that in Equation A18. The Chain Rule of the Vector Functions Define the vectors u, y, z and w as following u1 y,(u) u, y2(u) u= ,y= n l y (")l zI (y) (z) z,(y) w2(z) Z= ,W= ,(z Lzr(y).rl w ,(z From Equation A3 (A18) (A19) (A20) ouZ Each element of Equation A21 may be expanded using chain rule as Oz1 OzC 1 Oz1 0y2 Oz 0Ym Oz, Oz, y, + z, y, + Oz, C Si &Y, Y j k=l C Uk i, 2,...,r where 1, 2,,n j=Substitute Equati,2,on An Substitute Equation A22 into Equation A21 1 Oz k=ml k au1 k=k kz1 2Yk U m OZ, I az az Ol OY2 ay, ay2 Oz, Oz, ay, ay2 k1 k=l @k az, k=lk m O!r kI k=l k &l az2 aYm "L rxm Lu ]mxn k=l @k 1n k=I iyk 1n 5$z 2 k y kn k=I Ok 1n m aZ ar ay1 4 1 21 Oy1 11/2 chi^ 8 i rxm [L 1 2 (A21) (A22) LIz_~ ]rn^ aYm au" (A23) Jmxn Similarly, for more vectors, the chain rule just builds the new derivatives to the right. Iu = z ] ~ m u(A24) Qu ,,, z y Qu a The Derivative of Scalar Functions Respect to a Matrix Define a matrix 1 42 ... 4n h, h,, h, H= 22 (A25) hl hm2, hmn n and a scalar function J = f(H). (A26) The gradient of J with respect to H represented by 8J aJ 8J aJ 8J 8J 8J h= h h ah22 (A27) OH 8J aJ 8J ah,, ah, aknh Example 3 Find the gradient matrix, if J is the trace of a square matrix H. J tr(H) = f h (A28) According to Equation A27 the gradient of J with respect to H is SJ F 0 1 ... 0 O 8H ~ = i (A29) 0 0 1  1 Kxn = [I]ln n Example 4 Find the gradient matrix, if J is the trace of a square matrix H'H, where H is defined by Equation A25 and need not be square. The scalar function J can be expressed as J = tr(HH) t h1 ,, 21 h h,11_ h12 ... l ,n h r 12 h22 hm 2 21 h 22 ... h2n = tr i i : i . h,, h2,, h ,,,,, n h,, h 2,, h ,,,,,n n = (h2 +h22, +...+h2,l)+(h21 +h22, +...+h2m2)+... (A30) + (h +2h 2 + h +h 2) n . =zh2 J 1 1 and the gradient of J is 24, 22 ... 2h 8J 2kh, 2h2 .. 2h H .,, : i mI (A31) 22h, 2hm2 2 h ,n 2[H] n Example 5 Define the vectors A, B, and H as following a, b, A= a ,B= b2 a b nImxl nxl H.1 _, ,1 f I hl h,, ... h/) Al k2 mn.my., Find the derivatives of a scalar function J respect to matrix H. Express the scalar function as J = AHB hI1 h1, ... h,n b, h,, h22,, hn b2 =[al a2^ a 4H= K Z hl hm2 ... hmn .. b bi = (alhl +a2h2 +...+ahl) ... (alhn +a2h2n +...+ahal)] b2 bn m m =Zakhklbl +...+ Zakhhb k=l k=1 n m =ZZakhk,b, l= k=l So, the derivative of J respect to the matrix H is a,b, a, aamb, am a, Sa2 [b = [A]ml B a bn ab a2bn,, a1mbn mn b b, ,b2 T lxn (A32) (A33) (A34) 1 b .. b, ]x Note that since J is a scalar function, J = J' = BTHTA, and the following equation is also hold. =a O (BTHT A)J LH lOH [ (A35) =[A],[BT Table A1. Vector derivative formulas. [Anxn] [A ,jn+u]T 2[unxI [nx ]T [A nxn I + [u nx ]T [A nxn ]T nxI, = [A nx, [u nxl ] yIxn = [unx,]T [Ann] IxI, = [uxi]T [unxI] ixI, = [I"nx,] [A nxn ] [unxI APPENDIX B CAVITY OSCILLATION MODELS Rossiter Model The derivation of the Rossiter Model depends on the following assumptions (Rossiter 1964) * Frequencies of the acoustic radiation are the same as the vortex shedding frequency. * There are mV complete wave length of the vertex motion at the time t = to + t'. (e.g. m =1,2,...) The schematic of the Rossiter model is illustrated in Figure Bl, and the symbols in the plot are listed as follows L,D Cavity length and depth U, Free stream velocity a Mean sound speed inside the cavity m, Mode number (integer number 1,2...) a Phase lag factor between impact of the large scale structure on the trailing edge and the generation of the acoustic wave. K Proportion of the convective vortices speed to the free stream speed A, Spacing of the vortices At specific initial time t = to, an acoustic wave forms at the trailing edge with the distance aAi (Figure B1). Appropriately choose the time t', such that, the propagating wave front just reaches the cavity leading edge. By the assumption, the mode number m, is an integer number. Then, L = at' (Bl) L = mnA, aAl cKUj ' And the frequencies of the oscillations are related the phase speed and the wavelength of the vortical disturbance. f U (B3) Substituting the Equation Bl in to Equation B2, L =m, a, KU L a (B4) 1+KUjL =(m,a)A, a Then, combining the Equation B3 and Equation B4 resulting 1+aUj L=(m a) KU. a f fL (m, a) (B5) a The Rossiter model is defined by St f U. IK a (m a St = ( m :=1,2,... +M Linear Models of Cavity Flow Oscillations The block diagram of the linear model of the flowinduced oscillations is show in Figure B2 (Rowley et al. 2002). And the closedloop cavity transfer function can be expressed as (B2) G(s)S(s)A(s) P(s) = (B7) 1 G(s)S(s)A(s)R(s) Furthermore, the shear layer model is considered as a secondorder system with a time delay G(s)= Go0(s)e w2+2w+ s2 + 2wo + o2 L KU~ where wo Natural frequency of second order system Damping ratio r, Time delay inside the shear layer The acoustics model A(s) can be represented as a reflection model (Figure B3) where Ta Time delay inside the cavity r Reflection coefficient The closedloop transfer function of the reflection model can be written as es, A(s)= 2 1 re 2S and L a = a To recover the Rossiter formula, additional assumptions are required (B8) (B9) (B10) (B1) * Impingement model S(s) and receptivity model R(s) are unit gains. * No reflections in acoustic model in B10, r = 0. * The shear layer model is only a constant phase delay Go(s) =e 2 (B12) Depending on these assumption, combine the Equation B7, Equation B10 and Equation B12, G(s)A(s) 1 G(s)A(s) (B13) (B13) e12~a s(r+r,) 1e e12 2Tes(z+z" ) In order to find the resonant of the system, substitute the poles locations s = iw into the characteristic function of the Equation B13, and then combine the Equation B9 and Equation B11 resulting 1 e 2Tae w(e+z,) = 0 e12n, 2 e12~a w(r+r,) L L 2.(m, a)= w(+ ) (B14) wL m, a 2.crUo U+ 1 + a K Define St f U. (m a (B15) m m 1,B2,.. The linear model results Equation B15 matches the Rossiter formula Equation B6. Global Model for the Cavity Oscillations in Supersonic Flow The global model for cavity oscillations in supersonic flow is shown in Figure B4. To be consistent with the notation of Kerschen et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude S, upstream propagating cavity acoustic modes U and downstream propagating cavity acoustic modes D, and 'fast' modes Ef and 'slow' modes E nearfield acoustic waves in the supersonic stream. The local amplitudes of all quantities at the leading edge are denoted by the decoration while quantities at the trailing edge do not have the decoration. The scattering processing at the leading edge is modeled by S CSU b CDU^ SC U (B16) _E _CEU Csu, CDU, CE and CEsU are the four scattering coefficients for the upstream end of the cavity. The first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at the trailing edge is modeled by S U=[cus CUD C CUE E (B17) Cus, CUD, CU and CuE are the four scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by S = SelaL,D= D eedL 3 3 (B18) Ef =L 2Efe zL1L,Es =L 2Ee l2 where a and Td are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. M, and M2 are the wavenumbers of the 'fast' and 'slow' downstream propagating nearfield acoustic waves, respectively. Finally, the upstream propagating acoustic cavity mode is U= UezlL (B19) where r, is the complex wavenumber of this mode. The global model can be expressed in a block diagram (Figure B5). Substituting Equation B19 into Equation B17 results le L = U = CusS + C UDD + CUE +C, E U S (B20) [e'"L CUs CUD CUE CU D =0 Ef Also, substituting Equation B18 into Equation B16, the following formula can be obtained Se 'L C De dL = 3 L C= U (B21) E, L 2Efe CEfU S3 C _E_ L 2 Ee2L C EHu These four equations can be written in matrix form CsU CDU C CE,U EsU e laL 0 0 0 e ldL 0 0 L 2e lML 0 0 0 0 L 2e 2L (B22) Now, combine Equation B20 and Equation B22 yields Define e,L _CU _CUD Cs e 0 C. 0 e"rL CUEf 0 0 3 0 0 L 2 eL1L CEU A= X= CU 0 0 0 3 0 0 0 L 2 e 2L eL _Us CUD _CUE CSU e a 0 0 CDU 0 e dL 0 3 C,,, 0 0 L 2 e L CEsU U S D Ef E A. U S D Ef E c^ C 0 0 0 3 0 0 0 L 2e M2L (B23) (B24) Therefore, Equation B23 can be written as AX = 0 (B25) Notice that the quantities of X are the incident waves on the two ends of the cavity. The global mode has to satisfy Equation B25 which corresponds to the condition det(A) = 0. Calculating the determinant and simplifying, 3 L 2 (CE fUCUEfe MlL CEU CUEs 2L (B26) (B26) +L (e + CsuCuseU U + CDUCUDe L) 0 Assume a simple case where only Cus 0, Cs, # 0, and all other scattering coefficients are zeros. Therefore, Equation B26 can be simplified to e ' L + CSUC Se aL 0 Csu Cse (a+)L = 1 (B27) (Csue'L (CuseL) I L' Enforce the phase criterion and notice that the length is normalized by L = and U Equation B27 results CsuCse(a+,)L= 1 (B28) Arg(Csu )+ Arg(Cus)+ Re [(a + ,,)]L = 27rm or L' 2z7m mArg(Cs) Arg(Cus) L U Re[a + (B29) (B29) oL' m (Arg(Cs)+ Arg(Cus)) /2~ 2;rU Re[a +r,] Consider the normalized wave number Re[a']= / L JKU/U^, K (B30) Re [r, 0 = M c / U and define St oL' 2;r U (B31) a = (Arg(Cs )+ Arg(Cs ))/ 2r Therefore, Equation B29 can be written as oL'= L = 2rm Arg(Csu ) Arg(Cus) L U Re [a + r, L' m a (B32) St= ,m = 1, 2,... 2.r U 1 +M This matches the Rossiter formula Equation B6. Global Model for the Cavity Oscillations in Subsonic Flow The global model for cavity oscillations in subsonic flow is shown in Figure B6. To be consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude S, upstream modes U and downstream modes D propagating acoustic modes in the cavity, and upstream modes E^ and downstream modes Ed propagating nearfields acoustic waves in the subsonic flow. The local amplitudes of all quantities at the leading edge are denoted by the decoration ^), while quantities at the trailing edge do not have the decoration. The scattering processing at the leading edge is modeled by SSU SE," D = CDU CDE,, (B33) d EdU Ed E, Csu CDu, CEdu, CSE, C DE and CEdE are scattering coefficients for the upstream end of the cavity. Again, the first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at the trailing edge is modeled by FU C CUD CUE d =CS CD C D (B34) Cus, CUD CUE > CEus CEuD and CEEd are scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by 3 S = Sea, D = De L, ,Ed = L 2Ed edL (B35) where a and Td are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. Md is the complex wavenumber of the downstream propagating nearfield acoustic wave. Finally, the upstream propagating acoustic cavity modes are 3 U= Ue^,E =L 2EUeL (B36) where zr and M are the complex wavenumbers of the upstream traveling cavity mode and the upstream traveling acoustic nearfield mode, respectively. The global model can be represented in a block diagram (Figure B7). Again, substituting Equation B35 into Equation B34and Equation B36 into Equation B33, and combine the results, a matrix equation AX = 0 can be obtained, where 1 0 useaL CUD eL L 2CU eLdL 3 U 0 1 CESe C edL L 2C edL E EE,D E,,Ed A= eL L2C eL 1 0 0 X= S (B37) 3 1D CDU eL L 2CDE, e ML 0 1 0 3 C edUL L2C elL 0 0 1 EdEdE, The global mode has to satisfy AX = 0 which corresponds to the condition det(A) = 0. Calculating the determinant and simplifying, 3 CSU se (a+US )L CDUCUDe(d )L 'L CEd CUEd (d e )L 3 3 L 2CSE' C ESe +M)L L CDE, CE, e'(d+M,)L +L 3CE CE,Ed' e(Md+M,)L +LC C C C C C C (a+d+M+,)L SU ES DE, UD DU ED SEUS +Le ' CSUCUSCDE, E,,D CDUCUDCSE ES (B38) (C C` C +C C Cs CUS L 3 e(a+Md ,+M )L CSU ES EdE, UEd EdU EEd SEHUS C SUCUS CEd CE CEd CSEd CSE,S +Lc c c c +3C( c c+ c L3 e'(dd+M,+M )L CDUCED EdE UEd EdU E,E DE CUD CDU UD EdE E,,Ed EdU UEd DE E,,D Assume a simple case where only Cus # 0, Csu # 0, and all other scattering coefficients are zero. Therefore, Equation B38 can be simplified to CsuCuse (a+z,)L = (B39) (Cssue ) (Cuse L)= This matches the supersonic case results Equation B27 'a ~ ta' a, + KU t  ) ( ) ( t= t+t' Figure Bl. Schematic of Rossiter model. Shear Layer Actuator Input (Trailing Edge) Impingment Figure B2. Block diagram of the linear model of the flowinduced cavity oscillations. Acoustics Feedback Sensor ~I 3 3 Figure B3. Block diagram of the reflection model. Turbulent Boundary Layer M > 1 9Ts D D I( < L Figure B4. Global model for the cavity oscillations in supersonic flow. CUS D CUEf CUEs Figure B5. Block diagram of the global model for a cavity oscillation in supersonic flow. y X S I )D U Turbulent Boundary Layer ..M <.1 ,X S ..< L Figure B6. Global model for a cavity oscillation in subsonic flow. Figure B7. Block diagram of the global model for a cavity oscillation in subsonic flow. D D I( APPENDIX C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS MIMO System Identification Assume a linear and time invariant system, with the r inputs [u]r1l and the m outputs [y]m,, at the time k, the system can be expressed as y(k) = ay(k 1)+ay(k 2)+. +ay(k p) (C1) +f,u(k)+ APu(k 1) + ,u(k 2) + ...+ ,u(k p) where u, (k) y, (k) S ,((k) )=[y(k), y2(k) u(k) = [u(k)] = k) ,y(k) = [y(k)]= Yk) S(,k). y,(k) (C2) {a [a I...mIa2 =[a'2I...mI<... lap =[La P mXm Ao, [fAo,,,,A= [AL,8,,= ,p =[p'~m Define B(k)=[a,  a, P,  f] (C3) (k) 1 o P lmx[m*p+r*(p+l)] (C3) and y(k 1) y(k p) (P(k) = (C4) u(k) u(k p) [m*p+r*(p+l)]xl and these yield the filter outputs [j)(k)]m = [L (k)]m[(m, p+,(p+l))] [o(k)][ p+r(,,,))] (C5) Therefore, the error between the two outputs is defined as [s(k)]l = [j(k)].1 [Y(k)]m, (C6) (C6) = (k)q(k) y(k) and the scalar error cost function is defined by J(k)= e (k)e(k) (C7) 2 To identify the observer Markov parameters C3, the following equations based on the gradient descend method is developed OJ(k) (k + 1) = (k) / ^ (C8) a0(k) where / is the step size. Substituting Equation C6 into Equation C7 J(k) = (k)e(k) 2 = l[(k)p(k) y(k)] [j(k)p(k) y(k)] (C9) = [O p'O y yOp + yT y] and the gradient of error cost function is aJ(k) 1 T T T 80(k) 21 Y ( 1= O[p9P + (pO' 2yCpT 2= LO[rp + pp'T 2ypT] (C10) =(0op,)opT [F (k)][sub g T (k) E [m*p+* (p+ l) Finally, substituting Equation C10 into Equation C8 and yielding +(k +1) = 0(k) (k)q' (k) (C11) In order to automatically update the step size, choose P = (C12) cr+ k2 where c is a small number to avoid the infinity number when qp 2 is zero. Here the main steps of the MIMO identification are given as follows Step 1: Initialize [\(k) lm[mp+rpl))] =[01. Step 2: Construct regression vector [q,(k) ][mp+r*p+l)]1 according to Equation C4. Step 3: Calculate the output error [e(k)]m1 according to Equation C6. Step 4: Calculate the step size according to Equation C12 Step 5: Update the observer Markov parameters matrix [O(k)] m[m pr(p)] according to Equation C11. And then go back to step 2 for next iteration. Generalized Predictive Control Model In this section, a MIMO model, which is the same as the model of the MIMO ID Cl, is considered. Assume a linear and time invariant system, with the r inputs [u]r and m outputs [y]mm', at the time index k, the system can be expressed as y(k)= ay(k 1)+ay(k 2)+..+ay(k p) (C13) +flu(k)+ Pu(k 1)+ fPu(k 2)+. + flpu(k p) where u, (k) y, (k) u (k) y (k) u(k) = [(k)]= ( y(k) [y(k) Yk) ur(k)Jrl y,(k)j (C14) {tI = [al]ma2 = [a2]m... ap lap ]mXm fa = [1, ],, l a = [a,, ],, ..., p [= p, L, Shifting Equation C13 one time step ahead and can be expressed as y(k +1) = acy(k) + azy(k 1) + + apy(k p +1) (C15) +P,u(k + 1) + P,u(k) + P,u(k 1) + + fpu(k p + 1) Substituting Equation C13 into Equation C15 y(k +1) = alaty(k 1) + aazy(k 2) +.. + alapy(k p) +a2y(k 1) + ay(k 2) +. + ay(k p +1) +apfu(k) + a1Pu(k 1) +.. + atpu(k p) +P,u(k + 1) + P,u(k) + P,u(k 1) + + Pu(k p+ 1) = (a + a) y(k )+(aa, + a) y(k 2)+.. +(alap + ac) y(k p +1) + (acla) y(k p)+ flu(k +1) +(aP0 + P1)u(k)+(c~a, + P2)u(k 1) +... (C16) +(a,3p + 3,)u(k p+ 1)+(aflp)u(k p) = [a x y(k 1)+ a2 ] y(k2)+ [a(k p +1)+ [a]x y(k p) + [o u(k + 1)+ lo') ]m, u(k)+ [PA(')I u(k 1) +.. + ,< u')]mr,(k p + 1)+ [p(')]mx, uk p) where [a(2] : [ Jm] m L< m a1a, + a aa2 +a3 a ^ap + a = a,a, [ o() mxr [A(')Jmxr (all l fl2) :(W+A) (C17) [1)] = a +fm The output vector y(k + 1) is the linear combination of the past outputs, the past inputs and future inputs. By induction, the output vector y(k + j) can be derived as y(k + j) = [a(J ] 4 y(k 1)+ [a2) ] x y(k 2)+ .. + [a,0 ] x y(k p +1) + [ap)m] y(k p)+[Pfo ] u(k + j)+[f,: ]m u(k +j .1)+... [ ](C18) +[lomw] u(k)+[P)] m u(k1)+ [Pl)]mr (k2)+ p+ [ f fJxr fx u(k p) where [ apl()] m [ p() ]m m 1) +a~'2J1) 1) a, +a a 1) 1)api + ap01) a,0 ( [,6I]mxr (a 1( l)flo + fij1)) (C19) [/ 1(j ) 1 x = (a1(Jl1)'# 1 + 18( 01) 58p (j)Lmxr a"(.^" [1aO ]mxr [Pl mxr [f 1(0')]m [f(O()] rmxrl The quantities [fpA(k) (k = 0,1, ) are the impulse response sequences of the system. with initial [ai(0) ]m [a2 ]m a, 1i0l ]m [a (0) ]m a= m m x. = apl x. = ap (C20) Define the following the vector form u,(k p)= [u (k p)], ) [U(k+)],: [u)(k P1)] 1 [u(k + j)]ri [y(k p)] x Substituting Equation C21 into Equation C18 and express it in matrix form as [y(k+ j)]mxl [" [m xr)m (r*(j 1)) l(k)](r*(j+l))x1 S( l r 2 mr r mx(r*p) I (k p) 1()x (C22) + p([a ] ... [a2] 1 [a,]mxm ) qy (kp)](m*p)x1 Now, let the predictive index = 0,c1,2, n, q +1, , s 1, and define d bye /I [u (k)L rx l us,(k) [u(k +1)Ix],x \[u(k+ s 1 ) ] rx1 (r*s)x1 (C23) [y(k)]mxl Ys(k)= [y(k + 1)]mxl A[y(k + pm m can, b(m)xl A predictive model can be expressed by ([a mxr [O]mxr ([fo(1) mxr [0 lmxr [af (1)]m [/0(s2) ]mxr [, 2) ]. L mxr [Y(k)]m 1 [y(k + 1)]mx x, L[y(k+s 1)]mx J(m*s)x1 =I + II+III [(k)],"/ L'" [O]mxr )mx(r .s) [O]x l [O rxl ( .(r .)xl [O]m 0lr m(r*s) [u(k)Jrxl [u(k + 1)]rx1 [O]rx1 (r*s)xl S [u(k)]r 1 [ mxr )mx(r (k +1)]rxI [u(k+s 1)]Irx s [0]mr [O]mxr ... [0]  0imx ' w e_ [u(k)], . [u(k +)]r 1 (m*s)(rs) (k+1) or for simplification I [T](ms)x(r*s) [u (k)](rs)xl where (C24) (['0(s1) ]m [ ]mxr [Olmxr [a(1) [ ]mxr [P: ,: (C25) (C26) [ y. (k) ](m.s) [Tf1 (m*sx(r*s [Lf lmxr [OLmxr [l(s1) [m o(s2) mxr ... [0]L ... [L0] SJOmxr 0 mxr J The matrix [T] is called Toeplitz matrix. And the second part of Equation C24 is [u(kP)]L [u(k p+)], [u(k +l)] S[u(k 1)],)L (r)1 (5s1) Ir I'flp1(S1) Ir [P mxr mxr ,_ fLP mxr pl1 mxr fp (1) mxr 'fp1 (1) mxr 5p(s1) mr [p1(s1) mr W ^~s!]~ [apl(ljm [u(kP))L [u(k P +1)]r [u(k 1)]r1 ) (r*p) (m*) [u(k p)] [u(k P+)]L (C28) [u(k 1)] /Jr (rp)xl or for simplification II = [B](ms)x(r*p) [u(k p)]()xl 1 (C27) [PLlsOmr ) mx(r/n p) [(1(1) ]m 1(s1) mxr (m*sx(r* where (C29) (m*s)x(r*s) /Il r (1) M1(1) [( m mx [ I mxr mx(rlp) [B](m*s)x(r*p) [ ]mxr c1(1) ]mx [f1 (1) ]m L Im, (C30) LfP Imxr Lp1 mxr _lp) m, J fp1 (1) Im, [fp(s1) mr p1 (s1) mr, and the third part of Equation C24Is r [y(k p)],x mam [y(k p +1)]r [y(k 1)]rx1 )(m*p)xl [y(k p)]i x ) m () [y(k p+1)] I[y(k1)]rl (m*p)x1 (l i 1) mm [ap(s1) Im [iaP mxm lm P1 \mxm lap(1) mxm [p1(1) Imm [ap (1) ,m [ap(s1) Im [a mxm [a(1)mxm [y(kp)],r1 [y(kp+1)]l S[y(k p)],r [y(k +1)]r [y(k 1)]rxi J(m.p)x1 (m*s)x(m*p) or for simplification III [A](ms)(m*p) [yp(k p)](m*p)l where m*s)xl (C31) (C32) (m*s)x(r*p) (ap,, p almxm ([a<'L],,, [cu,."'] S(s1) mxmmx(m*p) [A] (m*s)x(n*p) [P mxm [a1 /mm [a) [ p1)]( Im _ (s1) mxm L p1s 1)mxm ... [a ,]xi . [ (1) ] m L (s1) mxmI (C33) Combine Equation C26, Equation C29 and Equation C32 in to Equation C24, [ys (k)](m.s) = [T],(ms)(r]s) [u (k)](rs) 1 +[B](m*s)(r*p) [IP(k p)](.)x1 +[A](m*sx(m*p) [y(k p)](m*p) 1 (C34) l(m*s)x(m*p) APPENDIX D A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES In this section, a potential theoretical model of cavity acoustic resonance is derived based on the model ofKerschen et al. (2003). The model combines scattering analyses for the two ends of the cavity and the propagation analyses of the cavity shear layer, internal region of the cavity, and acoustic nearfield. Kerschen et al. solve a matrix eigenvalue problem to identify the frequencies of the cavity oscillation. A different approach for characterizing the same model is illustrated in this section. A signal flow graph is first constructed from a block diagram of the physical model, and then Mason's rule (Nise 2004) is applied to obtain the transfer function from the disturbance input to the selected system output. This method gives a prediction for the resonant frequencies of the flowinduced cavity oscillations. In addition, this method also provides a linear estimate for the system transfer function. Mason's Rule Mason's rule reduces a signal flow graph to a transfer function between of any two nodes in the network. A signal flow graph connects "nodes," used to represent variables, by line segments, called "branches." First, some definitions are given as follows * Input node: a node that has only outgoing branches. * Output node: a node that has only incoming branches. * Path: a string of connected branches and nodes. It contains the same branch and node only once. * Forward path: a path that traverses from the input node to the output node of the signal flow graph in the direction of signal flow. It touches the same node only once. * Forward path gain: the product of gains found by traversing the path from the input node to the output node of the signal flow graph in the direction of signal flow. * Loop: a path that starts at a node and ends at the same node without passing through any other node more than once and follows the direction of the signal flow. * Loop gain: the product of branch gains in a loop. * Touching: two loops, a path and a loop, or two paths that have at least one common node. * Nontouching loop: Loops that do not have any nodes in common. * Nontouching loop gain: The product of loop gains from nontouching loops. The closed loop transfer function, T(s), of a linear dynamic system represented by a signal flow graph is (Nise 2004) N PkAk T(s)= k (Dl) where N: number of forward paths pk: the kth forward path gain A : 1 E(loop gains)+ E(nontouching loop gains taken two at a time)  (nontouching loop gains taken three at a time)+ E(nontouching loop gains taken four at a time) ... Ak : A (loop gain terms in A that touch the kth forward path). In other words, Ak is found by eliminating from A those loop gains that touch the kth forward path. Global Model for a Cavity Oscillation in Supersonic Flow The global model for cavity oscillations in supersonic flow is shown in Figure D1. To be consistent with the notation of Kerschen et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude S, upstream modes U and downstream modes D propagating acoustic modes in the cavity, and 'fast' modes Ef and 'slow' modes E nearfield acoustic waves in the supersonic stream. The local amplitudes of all quantities at the leading edge are denoted by the decoration (), while quantities at the trailing edge do not have the decoration. The scattering processing at the leading edge is modeled by S CSU S CDU C U (D2) _E_ CEU Csu, CDU CEU and CE,. are the four scattering coefficients for the upstream end of the cavity. The first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at the trailing edge is modeled by S U =[Cs CD Cr CuE D (D3) E CUs, CUD, CU and CUE are the four scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by S = eL, D=bedL 3 3 (D4) Ef =L 2Efe zLL,E =L2Ee llL where a and Td are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. M, and M2 are the wavenumbers of the 'fast' and 'slow' downstream propagating nearfield acoustic waves, respectively. Finally, the upstream propagating acoustic cavity mode is U = UeL (D5) where r, is the complex wavenumber of this mode. The global model can be represented in a block diagram (Figure D2) or a signal flow graph (Figure D3). From the signal flow graph (Figure D3), the transfer function between the disturbance input N and the upstream propagating cavity acoustic mode U' can be found. First, identify the components of Equation Dl. The results are listed in Table Dl. The characteristic function of the system can be identified from Equation Dl 4 A= k k=l1 S 3 (D6) =1 Csuuse L "+CDUCUDe(+ + L 2 (CEUCUE e(M)L +E CEuC, e M2 The numerator of the transfer function can be derived from D1. The Ak terms are formed by eliminating from A those loop gains that touch the kth forward path. 4 3 SPk Ak=CsuCuse + DUCUDe +L EUUE ML +CEUUEe 2 (D7) k=l Finally, the transfer function between the disturbance input N and the upstream traveling wave U' in the cavity is 4 PkAk TU'N(S) k=l CC3L C L (CEfL C eI (D8) CsuCuse' +CDuCuDe +L 2 CEfUUE f eL u uCEU C UE e2 1 CsuCuse( +CDuCuDe (+)L + L 2 E U E, Hf L E+ u CuEe e('M2 The characteristic function A in Equation D6 is the same as the eigenvalue relation derived by Kerschen et al. (2003). And the transfer function in Equation D8 gives more information concerning the physical model. For example, this model can predict the resonant frequencies as well as the nodall" regions or zeros of the flow field. Global Model for a Cavity Oscillation in Subsonic Flow The global model for cavity oscillations in subsonic flow is shown in Figure D4. To be consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude S, upstream modes U and downstream modes D propagating acoustic modes in the cavity, and upstream modes E and downstream modes Ed propagating nearfields acoustic waves in the subsonic flow. The local amplitudes of all quantities at the leading edge are denoted by the decoration ^), while quantities at the trailing edge do not have the decoration. The scattering processing at the leading edge is modeled by s CSU CSEu D = CDU CDE (D9) E. Ed EdU EdEj Csu Dv,DU > CE, Cs E, CDE and CEdE, are scattering coefficients for the upstream end of the cavity. Again, the first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at the trailing edge is modeled by L L ECS D CUE, C D (D10) Cus, CUD, CE CES CEuD and CEE are scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by 3 S = SeL, D = )De" Ed = L EddL (D11) where a and Td are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. Md is the complex wavenumber of the downstream propagating nearfield acoustic wave. Finally, the upstream propagating acoustic cavity modes are 3 U = UeL =L E eeA iL (D12) where ,, and M are the complex wavenumbers of the upstream traveling cavity mode and the upstream traveling acoustic nearfield mode, respectively. The global model can be represented in a block diagram (Figure D5) or a signal flow graph (Figure D6). From the signal flow graph (Figure D6), the transfer functions between the disturbance input N and the upstream propagating cavity acoustic modes, U' and E ', can be found. The transfer function between the disturbance input N and the upstream propagating cavity acoustic mode U' is calculated by first identifying the components of Equation D1. The results are listed in Table D2. The characteristic function of the system can be identified from Equation D1 12 6 A= 1/k 2k k=l k=l C,,C,,e"' ) +C DUC (+UD )L 3 3 +L CE EU UE(Md + L 2SUC CEHSCDE CUDI1 3 +L3CSUCES CEdE CUUEd 2 + L 2CDU CED Cs CUSI1 +L3CDU CEC CE C UEd 3CL U CEd C C US2 3 +L 3CU CEE CDE CUD 3 +L 2CS C se+M)L 3 +L 2CD C D(e +M,)L L 3CE CE e(Md +Mu)L L 2CCDEu CED L CE E Ed L 2 SU US D^u EuD^ SU US Edu EuEd 2 3 + +L 2CDUCUDCSE CESI1 L+, 3CDU CUD CEdE CEE 13 +L3CEU CEUC SCE + L3CEI U CUE DE CEDI3 CsuCuse + CDU CeiUD +)L 3 3 = 1 +L2CE uUEde(Md +L L 2CSEu CES e(+M)L 3 +L 2CD C (e (d+Mu)L + L3C CE e'(Md+,)L DEu EuD EdEu EuEd CSUCES DECUD DU ED SE, US (C C C C 3C C C (D13) DSU ED EdE, UEd EdU EEd SE, US C C C C CC E, SU US EdE, ^ E,'Ed EdU C CCES) (CDUCECE CUE +CUC C C (D13) L3I 3LC +jED ECC EE E^ EDE, UD CDU UD EdE, EEd EdU UEd DE ED 1 where I, = e'(a+Td,+r,)L ,2 = e(a+Md+Mu+z )L ,3 e'(z"+Md+M,"" +)L (D14) The numerator of the transfer function can be derived from D1 9 SPkAk k=l = CsuCuse ) 1L CDE, EDe EdEL ECEd C de 3 "(CDUCUD dL L 2CSE C e(a+M,)L L C CEEd 3 3 3 +L E C UE Su ES (a+M)L 3DE E +M)L S SUCES DE, UD + ES EdE" C UEd d 3 lDUEuDSE rUSez(+Mu+a)L 3 DU u EdEud +Md)L "+( 3CE_ CEuEd SEuUS (Md mu + (L3EdU EuEd CDu D +Mu dL 3 Li3 (ed )KC+C C C C C SCsuCuseL +CDUCUe ZdL + L 2C UEd uE dL + CSUCUSCDE, ED DUCUD SE, E2S c c c c +c cc c  +L3e (a+Md+Mu)L SUESEdEUEd CEdUCEEdSECUS (D15) CCC C C C C CS CC c 3 e(d+Md+Mc)L c c c L3 ,(+Md+M,)L DU ED EdECUEd EdU EuEd DE, UD CDU UD EdE, EEd EdU JUEd DE, ED Finally, the transfer function between the disturbance input N and the upstream traveling wave U' in the cavity is T7' (s) 9 YPkAk k=l A 3 Cs uCse'L +C C DeLL + L 2CEu C eMdL + +L2 e(a+d+M)L SUCESCDEu UD DU E,uD SE US CsuCu DE, ED DU UD SEuEuS CsuCUSCEEuC d Ed d E EUCUEd SEES (C C C C+C C C C L3 (+Md+M)L CDU ED EdEu UEd EdU E,Ed DE, CUD CdDU CEUDE CEEEEd CEdU CUEd CDE CEuD 3 L C su C use a + C D UM UD ( L +C D )L C L _)2 EdU U L CEd C E e ( +L 2CSe (a+Mu )L L 2 C L +M)L LE 3 C M(MdL _L e(a+zd+M,+,)L CSUCESCDECUD+ CDUC E.DC SE CUS e'_CsuCusC DE CHED DU UD CSE CES (D16) 3 (a+Md +M,+,)L SU ES EdE UEd E+C EdE dSE US 3 ,(d+Md+Mu,)L CDU EuD EdEUE d EdUd EuEd DEu UD CDU UD EdE, E,,Ed EdU UECdDEu EH D The characteristic function in Equation D13 is the same as the eigenvalue relation derived by Alvarez et al. (2003), but the transfer function in Equation D16 gives more information concerning the physical model. For example, this model can predict the resonant frequencies as well as the nodall" regions or zeros of the flow field. Then, the transfer function between the disturbance input N and the upstream propagating acoustic mode E', is calculated. Similarly, first, identify the components of Equation D1. The results are listed in Table D3. Because the loop gains and the nontouching loop gains are the same as before (Table D2), only the forward path gains are listed. The characteristic function of the system is the same as Equation D13. And the numerator of the transfer function can be derived from D1 3 3 Z pkAk SUCEse L + DU CEDe +L 2 ECEE edL (D17) k=l Finally, the transfer function between the disturbance input N and the upstream traveling wave E', in the cavity is TEN(S) 3 PkAk k=l A 3 CsuCESeL +CDU CE1D eZ L +L 2CEd CE edL +L CSECESe (+L DE CEDe L C3 E CEE, 2E( M (Md+L)L 1 CSUCuse" +CDU UD e+L L 2EdU UE d i +L 2C (a+ML L 2 2 r '(qrd +Mu)L +L3 jr r (Md+M )L ' SEuEuS DEu EuD EdEu CuCC _L e(a+d+,+,)L CSUCESCDE CUD+ DU ED SE, US CSUCU CD CD DUC C (D18) 3 e(a+MdM )L SU ES EE UEd EdU EUEd SE US CSUCUSCEdE, ECEd EdUUEd SECE5S _L3 ,(,+M,+M,+ )L DUED EEUEE d EdU EuEd DECUD DU UD EdE EC EdC CUEd DE ED ^ ^DU^UD^ ,C,* E^ CEdU UE^D IE CED ) Table D1. Components of the Mason's formula for supersonic case. Index Forward Paths Forward Path Gains 1 0 1 2 6 10 11 p, = CsuCse 2 0 1 3 7 10 11 2 = CDUCUDeLL 0 1 44 810 11 0>1 >54 >8 >10> 11 0 1 5 9 10 11 Loops 1 2 3 6 3 10 1 13 7 101 1 4 38 10 1 1 5 9 10 1 3= L 2CEU CU e IML 3 4 = L 2 C C e 2L Loop Gains 1I = Csu~se a)L /12 = CDU UDe( d)L 3 ,/1 3L 2C H CuL e Ml 13=L ^EfU UEf 3 14= L2 Cu e (,M2)L Index 1 Table D2. Components of the Mason's formula for subsonic case. Index Forward Paths Forward Path Gains 1 0 1 3 6 9 11 p = CsuCuseaL 2 0 1 4 7 ~9 11 P2 =CDUCUDedL 3 3 0 1 5 8 9 11 3= L 2 CE Ee dL 3 4 0 1 3 36 10 2 4 7 9 11 p4 =L 2CSUCECDE'UD +,+)L 5 01>3 >610 >2>5>8>9 >11 ps =L 3CSUESC CUE, E a+ d)L 3 6 01 >4>7102>2> 6911 p6 = LCDCE DC Cue+M"+)L 7 0>1>4>7>10>2>5>8>9>11 p7=L CDUC C E uE(M, Md)L 7 0 1 447 10 2 5 8 911 p = 3DUCDciE E (q,+A"+Ac)L 8 0o1>5>810>2>3>6>911 p8 =3 CEU E CE CSE_ USC(Mdu++a)L 9 0>1>5 >8>10>2>4>7>9>11 p9 = L3CCE ECDE CuCDeM(d+,+d)L Index Loops Loop Gains 1 1 3 6 9 1 1/ = CCsuuse (+)L 2 14>479>7 1 12 CCDUCDe(",)L 3 3 1 5 8 591 /13 L2 EC UE (Md+ ,)L 3 4 13 6 10 2 4 7 9 1 14 = U~CEUCS DE, C UDe'a+ ,+d+,)L 5 1 3 6 10 ~4 2 5 8 9 1 15 3 (+M+d+ )L 6 1>4>7>10>23>>91 /1 =L C, C CU e' ,)L 66 L" 2CDU CE,,DC SE CUS 7 1 4>7 10 2 5 87 = 310 9C C5 8DCEIE9 C UE( _,+A'+M,+Md+ r,)L 8 1 5 8 10 2 3691 18 = L3 CEU CEc _SCECSE CCS(+,+a+ )L 9 15>85 10 2 4>4791 /19 = L 3CUC C C,, CuDe (Md"M"+,++,)L 3 10 2> 3> 6 > 10 >2 /11 L 2 Cs,,(+M)L 10 SE E, S^^ + 3 11 24>47102 111 L2 (DE ECEz( )L S11 CD, CDE1, De 12 2 5 8102 12 3= C_3 EC CE, et(d +eM+,, )L Index Nontouching Loops Nontouching Loop Gains 3 1 Loop 1 and Loop 11 /21 L 2CSU USCDE CE"D (+" ++M)L 2 Loop 1 and Loop 12 122 = 3C SU CUSCEE, dEEd e( +l, 3 3 Loop 2 and Loop 10 /23 L 2CDUCUDSE C C, ~(++c+M)L Loop 2 and Loop 12 Loop 3 and Loop 10 Loop 3 and Loop 11 124 3C DU UD C E C E (d e'd d ,)L /2 = Lr3C C CU C Sel(Md+r,+a+M )L 125 EdU UEd SE ES 126 L3C EdUE DE EuDe(Md ur Td M Components of the Mason's formula for subsonic case. Forward Paths Forward Path gains 0 1 3 6 10 12 p, = CsCCEseaL 0 1 4 7 10 12 P2 = CDUCDe dL 0>1 >5 >8 >10 >12 p3 = L 2CEU CEe_ edL Table D3. Index 1 2 Turbulent Boundary Layer M>1 19T x ES L Figure D1. Global model for a cavity oscillation in supersonic flow. Figure D2. Block diagram of the global model for a cavity oscillation in supersonic flow. D D Ir ~t~ E,. tALE Figure D3. Signal flow graph of the global model for a cavity oscillation in supersonic flow. Turbulent Y Boundary Layer S L Figure D4. Global model for a cavity oscillation in subsonic flow. Figure D5. Block diagram of the global model for a cavity oscillation in subsonic flow. S ,L S Figure D6. Signal flow graph of the global model for a cavity oscillation in subsonic flow. APPENDIX E CENTER VELOCITY OF ACTUATOR ARRAY In this appendix, the center velocity of each slot (notation see Chapter 5) and corresponding current measurement of the actuator array are shown. slot la 50V PP 100 V PP 150V PP 40 o 30 20 10 500 1000 Frequency [Hz] A slot la 0.06 0.0 o 50 V PP 0.05 + 100 Vpp + 150 Vp 0.04 0.03 0.02 0.01 0 I 0 500 1000 Frequency [Hz] B Figure E1. Hotwire measurement for actuator array slot la. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 1500 2000 slot lb 50V PP 100 V PP 150V PP .X 40 20 20 1000 Frequency [Hz] 0.06 o 50V pp 0.05 +100 Vpp 150V + PP 0.04 + 0.03 + 0.02 + 0.01 ++ o000 0 0 500 slot lb + + o O 00 OO o0 1000 Frequency [Hz] B Figure E2. Hotwire measurement for actuator array slot lb. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 500 1500 2000 a a, (U slot 2a 1000 Frequency [Hz] A slot 2a 0 1 I 0 500 1000 Frequency [Hz] B Figure E3. Hotwire measurement for actuator array slot 2a. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) S40 30 0 > 20 50V PP 100 V PP 150V PP 500 1500 2000 50V PP 100 V pp PP 150V PP 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 slot 2b 500 1000 Frequency [Hz] A slot 2b o 50V pp 100 V PP + 150V PP 0 1 I 0 500 1000 Frequency [Hz] B Figure E4. Hotwire measurement for actuator array slot 2b. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 200 50 S40 o 30 50V PP 100 V PP 150V PP 1500 2000 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 :++++++ slot 3a 1000 Frequency [Hz] A slot 3a 0 1 I 0 500 1000 Frequency [Hz] B Figure E5. Hotwire measurement for actuator array slot 3a. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 40 30 20 , 20 50V PP 100 V PP 150V PP 500 1500 2000 50V PP 100V PP 150V PP 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 slot 3b 1000 Frequency [Hz] A slot 3b 0 1 I 0 500 1000 Frequency [Hz] B Figure E6. Hotwire measurement for actuator array slot 3b. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 202 50 S40 o 30 50V PP 100 V PP 150V PP 500 1500 2000 50V PP 100V PP 150V pp 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 slot 4a 60 50 40 o 30 > 0 1 I 0 500 1000 Frequency [Hz] B Figure E7. Hotwire measurement for actuator array slot 4a. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 203 ++ 000 o 0 1000 Frequency [Hz] A slot 4a 50V PP 100 V PP 150V PP 500 1500 2000 50V PP 100 V PP 150V pp 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 slot 4b 50V PP 100 V PP 150V PP . 40 0 1000 Frequency [Hz] A slot 4b 0 1 I 0 500 1000 Frequency [Hz] B Figure E8. Hotwire measurement for actuator array slot Current measurement of the actuator array. 1500 2000 14b. A) Center RMS velocity. B) 204 500 1500 2000 50V PP 100V PP 150V PP 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 60 50 40 o 30 20 10 500 1000 Frequency [Hz] A slot 5a 50V PP 100V PP 150V PP 0 I 0 500 1000 Frequency [Hz] B Figure E9. Hotwire measurement for actuator array slot 5a. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 205 slot 5a 50V PP 100 V PP 150V PP 1500 2000 0.06 0.05 S0.04 < 0.03 S0.02 0.01 500 1000 Frequency [Hz] A o 50V pp +100 V 150V PP 0 500 1000 Frequency [Hz] B Figure E10. Hotwire measurement for actuator array slot 5b. Current measurement of the actuator array. 1500 2000 A) Center RMS velocity. B) 206 slot 5b 60 . 40 0 50V PP 100 V PP 150V PP 1500 2000 slot 5b +t**t 0.06 0.05 S0.04 < 0.03 0.02 U 0.02 0.01 APPENDIX F PARAMETRIC STUDY FOR OPENLOOP CONTROL In this appendix, a parametric study results for openloop control are shown. To illustrate the openloop control, a fixed flow condition (M=0.31) is chosen for all experimental cases. The frequencies of the excitation input signals to the actuator array are varied from 500 Hz to 1500 Hz, and for each frequency, two excitation voltage levels, 100 Vpp and 150 Vpp, are chosen. 207 TE Baseline TE Open Loop      'Ir 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F1. OpenLoop control result for M=0.31 and excitation sinusoidal 500 Hz and 100 Vpp voltage. 140 130 r 120 110 100 input with frequency TE Baseline TE Open Loop 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F2. OpenLoop control result for M=0.31 and excitation sinusoidal 500 Hz and 150 Vpp voltage. input with frequency 208 130 I I S100 II .. TE Baseline TE Open Loop 100 S100 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F3. OpenLoop control result for M=0.31 and excitation sinusoidal 600 Hz and 100 Vpp voltage. 140 130  120 110 100 90 input with frequency TE Baseline TE Open Loop 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F4. OpenLoop control result for M=0.31 and excitation sinusoidal 600 Hz and 150 Vpp voltage. input with frequency 209 :"i TE Baseline TE Open Loop 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F5. OpenLoop control result for M=0.31 and excitation sinusoidal 700 Hz and 100 Vpp voltage. 140 130  120 110 100 input with frequency TE Baseline TE Open Loop 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F6. OpenLoop control result for M=0.31 and excitation sinusoidal 700 Hz and 150 Vpp voltage. input with frequency 210 130 120 '; 100 90 80 70 TE Baseline TE Open Loop     ' ^A~rWil  r   I I^ :: : : : : : : : : : : : I : : :: : 100 90 80 70 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F7. OpenLoop control result for M=0.31 and excitation sinusoidal 800 Hz and 100 Vpp voltage. 140 130  120 110 100 input with frequency TE Baseline TE Open Loop 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F8. OpenLoop control result for M=0.31 and excitation sinusoidal 800 Hz and 150 Vpp voltage. input with frequency 1 TE Baseline TE Open Loop 100 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F9. OpenLoop control result for M=0.31 and excitation sinusoidal 900 Hz and 100 Vpp voltage. input with frequency 130 120 TE Baseline TE Open Loop K 110 100 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F10. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 900 Hz and 150 Vpp voltage. 212 "(: TE Baseline TE Open Loop 110 o 100 90 80 70 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F11. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1000 Hz and 100 Vpp voltage. 140 130 , TE Baseline TE Open Loop 120 100 90 80 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F12. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1000 Hz and 150 Vpp voltage. 213 '' :'~ TE Baseline TE Open Loop     1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F13. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1100 Hz and 100 Vpp voltage. TE Baseline TE Open Loop 100 90 80 70 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F14. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1100 Hz and 150 Vpp voltage. 214 I JUV 120 110 100O STE Baseline TE Open Loop     I I I 130 120 110 0 100 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F15. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1200 Hz and 100 Vpp voltage. 130 120 110 100 90 80 70 TE Baseline TE Open Loop 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F16. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1200 Hz and 150 Vpp voltage. 215  TE Baseline  TE Open Loop 7    S100 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F17. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1300 Hz and 100 Vpp voltage. 140 130  120 '' TE Baseline TE Open Loop 100 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F18. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1300 Hz and 150 Vpp voltage. 216 ' I v~va TE Baseline TE Open Loop 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F19. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1400 Hz and 100 Vpp voltage. 140 130 , TE Baseline TE Open Loop 120 110 100 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F20. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1400 Hz and 150 Vpp voltage. 217 100  , I_ _ I4 I 1~ TE Baseline TE Open Loop 100 1000 2000 3000 Frequency [Hz] 4000 5000 6000 Figure F21. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1500 Hz and 100 Vpp voltage. TE Baseline TE Open Loop 100 90 0 1000 2000 3000 4000 5000 6000 Frequency [Hz] Figure F22. OpenLoop control result for M=0.31 and excitation sinusoidal input with frequency 1500 Hz and 150 Vpp voltage. 218 LIST OF REFERENCES Akers, J. C., and Bernstein, D. S. 1997(a) ARMARKOV LeastSquares Identification. Proceedings of the American Control Conference, Albuquerque, New Mexico. Akers, J. C., and Bernstein, D. S. 1997(b) TimeDomain Identification Using ARMARKOV/Toeplitz Models. Proceedings of the American Control Conference, Albuquerque, New Mexicl, B. Allen, R. and Mendonca, F. 2004 DES Validations of Cavity Acoustics over the Subsonic to Supersonic Range. 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In May 2004, he earned a Master of Science degree in electrical engineering. He received his Ph.D. degree with an aerospace engineering major in May 2008 at the University of Florida. His research focuses on the active closedloop control of the cavity oscillations. 227 PAGE 1 CLOSEDLOOP CONTROL OF FLOWINDUCED CAVITY OSCILLATIONS By QI SONG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1 PAGE 2 2008 Qi Song 2 PAGE 3 To my wife, Jingyan Wang; and my lovely son, Lawrence W. Song 3 PAGE 4 ACKNOWLEDGMENTS This study was performed while I was a member of the Interdiscip linary Microsystems Group (IMG) in the Department of Mechanical a nd Aerospace at the University of Florida in Gainesville, Florida, USA. First, I sincerely acknowledge my advisor, Dr. Lou Cattafesta, for providing me with this opportuni ty and giving me so much precious advice during my course time at UF. His guidance and encouragement al ways gave me sufficient confidence to conquer any difficulty. I thank all of my colleagues in the IMG group for their invaluable assistance. Finally, I appreciate my friends and my dear family for their tremendous consideration and unselfish support during my journey. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 LIST OF ABBREVIATIONS........................................................................................................14 ABSTRACT...................................................................................................................................16 CHAPTER 1 INTRODUCTON.................................................................................................................. .18 Literature Review.............................................................................................................. .....20 Physical Models...............................................................................................................2 1 PhysicsBased Models.....................................................................................................21 Numerical Si mulations....................................................................................................27 PODType Models...........................................................................................................28 OnLine System ID and Active Cl osedLoop Control Methodologies...........................28 Unresolved Technical Issues...........................................................................................33 Technical Objectives........................................................................................................... ...33 Approach and Outline.............................................................................................................34 2 SYSTEM IDENTIFICATION ALGORITHMS....................................................................38 Overview....................................................................................................................... ..........38 SISO IIR Filter Algorithms....................................................................................................39 IIR OE Algorithm............................................................................................................40 IIR EE Algorithm............................................................................................................41 IIR SM Algorithm...........................................................................................................41 IIR CE Algorithm............................................................................................................41 Recursive IIR Filters Simulation Results and Analyses..................................................41 Accuracy comparison for sufficient system.............................................................43 Accuracy comparison for insufficient system..........................................................43 Convergence rate......................................................................................................44 Computational complexity.......................................................................................44 Conclusions.....................................................................................................................45 MIMO IIR Filter Algorithm...................................................................................................46 3 GENERALIZED PREDICTI VE CONTROL ALGORIHTM...............................................62 Introduction................................................................................................................... ..........62 MIMO Adaptive GPC Model.................................................................................................63 5 PAGE 6 MIMO Adaptive GPC Cost Function.....................................................................................66 MIMO Adaptive GPC Law....................................................................................................66 MIMO Adaptive GPC Optimum Solution......................................................................67 MIMO Adaptive GPC Recursive Solution......................................................................68 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES..............................................71 Schematic of the Vibration Beam Test Bed...........................................................................71 System Identification Experimental Results...........................................................................72 Computational Complexity.............................................................................................72 System Identification.......................................................................................................73 Disturbance Effect...........................................................................................................74 ClosedLoop Control Experiment Results..............................................................................74 Computational Complexity.............................................................................................74 ClosedLoop Results.......................................................................................................74 Estimated Order Effect....................................................................................................75 Predict Horizon Effect.....................................................................................................76 Input Weight Effect.........................................................................................................76 Disturbance Effect for Different SNR Levels During System ID...................................76 Summary.................................................................................................................................77 5 WIND TUNNEL EXPERIMENTAL SETUP........................................................................90 Wind Tunnel Facility........................................................................................................... ...90 Test Section and Cavity Model...............................................................................................91 Pressure/Temperature Measurement Systems........................................................................93 Facility Data Acquisition and Control Systems......................................................................94 Actuator System................................................................................................................ ......95 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION..............................113 Background...........................................................................................................................113 Data Analysis Methods.........................................................................................................115 Noise Floor of Unsteady Pressure Transducers....................................................................116 Effects of Structural Vibrations on Unsteady Pressure Transducers....................................116 Baseline Experimental Results and Analysis.......................................................................117 OpenLoop Experimental Results and Analysis...................................................................118 ClosedLoop Experimental Results and Analysis................................................................120 7 SUMMARY AND FUTURE WORK..................................................................................140 Summary of Contributions...................................................................................................140 Future Work..........................................................................................................................141 APPENDIX A MATRIX OPRATIONS.......................................................................................................143 6 PAGE 7 Vector Derivatives................................................................................................................143 Definition of Vectors.....................................................................................................143 Derivative of Scalar w ith Respect to Vector.................................................................143 Derivative of Vector with Respect to Vector................................................................143 Second Derivative of Scalar With Resp ect to Vector (Hessian Matrix).......................144 Table of Several Useful Vector Derivative Formulas...................................................144 Proof of the Formulas....................................................................................................145 Proof (a)..................................................................................................................145 Proof (b).................................................................................................................145 Proof (c)..................................................................................................................146 Proof (d).................................................................................................................146 The Chain Rule of the Vector Functions..............................................................................148 The Derivative of Scalar Func tions Respect to a Matrix......................................................150 B CAVITY OSCILLATION MODELS..................................................................................155 Rossiter Model................................................................................................................. .....155 Linear Models of Cavity Flow Oscillations..........................................................................156 Global Model for the Cavity Oscillations in Supersonic Flow.............................................159 Global Model for the Cavity Oscillations in Subsonic Flow................................................163 C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS............................................169 MIMO System Identification................................................................................................169 Generalized Predictive Control Model.................................................................................171 D A POTENTIAL THEOR ETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES....................................................................................................................179 Masons Rule........................................................................................................................179 Global Model for a Cavity Oscillation in Supersonic Flow.................................................180 Global Model for a Cavity Oscillation in Subsonic Flow....................................................183 E CENTER VELOCITY OF ACTUATOR ARRAY..............................................................196 F PARAMETRIC STUDY FO R OPENLOOP CONTROL..................................................207 LIST OF REFERENCES.............................................................................................................219 BIOGRAPHICAL SKETCH.......................................................................................................227 7 PAGE 8 LIST OF TABLES Table page 21 Summary of the IIR OE algorithm.........................................................................................49 22 Summary of the IIR EE algorithm..........................................................................................50 23 Summary of the IIR SM algorithm.........................................................................................51 24 Summary of the IIR CE algorithm.........................................................................................52 25 Simulation results of IIR algorithms for sufficient case.........................................................53 26 Simulation results of IIR algorithms for insufficient case......................................................54 27 Simulation conditions of IIR algorithms for sufficient case...................................................55 28 Summary of the IIR/LMS algorithms.....................................................................................56 41 Parameters selection of the vibration beam experiment.........................................................78 42 Summary of the results of the adaptive GPC algorithm.........................................................79 51 Physical and piezoelectric properties of APC 850 device....................................................100 52 Geometric properties and parameters for the actuator..........................................................101 53 Resonant frequencies with respective cen terline velocities for each input voltage.............102 A1 Vector derivative formulas................................................................................................ ..154 D1 Components of the Mason s formula for supersonic case...................................................189 D2 Components of the Masons formula for subsonic case......................................................190 D3 Components of the Masons formula for subsonic case......................................................192 8 PAGE 9 LIST OF FIGURES Figure page 11 Schematic illustrating flowinduced cav ity resonance for an upstream turbulent boundary layer...................................................................................................................36 12 Tam and Block (1978) model of acoustic wave field inside and outside the rectangular cavity......................................................................................................................... .........36 13 Classification of flow control............................................................................................ .....37 14 Block diagram of syst em ID and online control...................................................................37 21 Linear timeinvariant (LTI) IIR Filte r Structure.....................................................................57 22 Simulation structure of the adaptive IIR filter........................................................................57 23 zplane of the test model................................................................................................. ........58 24 3D plot of the MSOE performance surf ace of the insufficient order test system..................58 25 Contour plot of the MSOE performance surface....................................................................59 26 Simulation results of weight track of the IIR algorithms for sufficient case..........................59 27 Simulation results of weight track of the IIR algorithms for insufficient case.......................60 28 Learning curve of IIR algorithms for sufficient case.............................................................60 29 Computational complexity results from the experiment........................................................61 31 Model predictive control strategy......................................................................................... ..70 41 Schematic diagram of the vibration beam test bed.................................................................80 51 Schematic of the wind tunnel facility...................................................................................10 3 52 Schematic of the test section and the cavity model..............................................................103 53 Schematic of the control hardware setup..............................................................................104 54 Bimorph bender disc actua tor in parallel operation..............................................................104 55 Designed ZNMF actuator array............................................................................................10 5 56 Dimensions of the slot for designed actuator array..............................................................106 57 ZNMF actuator array mounted in wind tunnel.....................................................................107 9 PAGE 10 58 Bimorph 3 centerline rms velocities of the single unit piezoele ctric based synthetic actuator with different excita tion sinusoid input signal...................................................108 59 The comparison plot of the experiment a nd simulation result of the actuator design code for bimorph 3.................................................................................................................. .109 510 Current saturation effects of the amplifier..........................................................................110 511 Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7).........111 512 Schematic of a single periodic cell of th e actuator jets and th e proposed interaction with the incoming boundary layer...................................................................................112 61 Schematic of simplified wind tunnel a nd cavity regions acoustic resonances for subsonic flow...................................................................................................................123 62 Noise floor level comparison at differe nt discrete Mach nu mbers with acoustic treatment at trailing edge floor of the cavity with L/D=6................................................123 63 x acceleration unsteady power spectrum (dB ref. 1g) for case with acoustic treatment and no cavity....................................................................................................................124 64 y acceleration unsteady power spectrum (dB re f. 1g) for case with acoustic treatment and no cavity....................................................................................................................125 65 z acceleration unsteady power spectrum (dB re f. 1g) for case with acoustic treatment and no cavity....................................................................................................................126 66 Spectrogram of pressure measurement (dB ref. 20e6 Pa) on the trailing edge floor of the cavity for the case with ac oustic treatment and no cavity..........................................127 67 Spectrogram of pressure measurement (re f 20e6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D=6...............................................................................128 68 Spectrogram of pressure measurement (re f 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7).........129 69 Noise floor of the unsteady pressure level at the surface of the trailin g edge of the cavity with and without the actuator turned on..........................................................................130 610 Openloop sinusoidal control results for flowinduced cavity oscillations at trailing edge floor of the cavity....................................................................................................131 611 Running error variance plot for the system identification algorithm.................................133 612 ClosedLoop active control result for flowinduced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity.....................................................................134 10 PAGE 11 613 Input signal of the ClosedLoop activ e control result for flowinduced cavity oscillations at Mach 0.27 at the trai ling edge floor of the L/D =6 cavity........................135 614 Sensitivity function (Equation 41) of the closedloop control for M=0.27 upstream flow condition..................................................................................................................136 615 Unsteady pressure level of the clos edloop control for M=0.27, L/D=6 upstream flow condition with varying estimated order...........................................................................137 616 Unsteady pressure level of the clos edloop control for M=0.27, L/D=6 upstream flow condition with varying predictive horizon s....................................................................138 617 Unsteady pressure level comparison be tween the openloop c ontrol and closedloop control for M=0.27 upstream flow condition...................................................................139 B1 Schematic of Rossiter model............................................................................................... 166 B2 Block diagram of the linear model of the flowinduced cavity oscillations........................166 B3 Block diagram of the reflection model................................................................................167 B4 Global model for the cavity os cillations in supersonic flow................................................167 B5 Block diagram of the global model for a cavity oscillation in supersonic flow..................167 B6 Global model for a cavity os cillation in subsonic flow.......................................................168 B7 Block diagram of the global model for a cavity oscillation in subsonic flow.....................168 D1 Global model for a cavity os cillation in supersonic flow....................................................193 D2 Block diagram of the global model for a cavity oscillation in supersonic flow..................193 D3 Signal flow graph of th e global model for a cavity osci llation in supersonic flow.............194 D4 Global model for a cavity os cillation in subsonic flow.......................................................194 D5 Block diagram of the global model for a cavity oscillation in subsonic flow.....................195 D6 Signal flow graph of the global model for a cavity oscillation in subsonic flow................195 E1 Hotwire measurement for actuator array slot 1a.................................................................197 E2 Hotwire measurement for actuator array slot 1b................................................................198 E3 Hotwire measurement for actuator array slot 2a.................................................................199 E4 Hotwire measurement for actuator array slot 2b................................................................200 11 PAGE 12 E5 Hotwire measurement for actuator array slot 3a.................................................................201 E6 Hotwire measurement for actuator array slot 3b................................................................202 E7 Hotwire measurement for actuator array slot 4a.................................................................203 E8 Hotwire measurement for actuator array slot 14b..............................................................204 E9 Hotwire measurement for actuator array slot 5a.................................................................205 E10 Hotwire measurement for actuator array slot 5b..............................................................206 F1 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 500 Hz and 100 Vpp voltage...........................................................................................208 F2 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 500 Hz and 150 Vpp voltage...........................................................................................208 F3 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 600 Hz and 100 Vpp voltage...........................................................................................209 F4 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 600 Hz and 150 Vpp voltage...........................................................................................209 F5 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 700 Hz and 100 Vpp voltage...........................................................................................210 F6 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 700 Hz and 150 Vpp voltage...........................................................................................210 F7 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 800 Hz and 100 Vpp voltage...........................................................................................211 F8 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 800 Hz and 150 Vpp voltage...........................................................................................211 F9 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 900 Hz and 100 Vpp voltage...........................................................................................212 F10 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 900 Hz and 150 Vpp voltage...........................................................................................212 F11 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1000 Hz and 100 Vpp voltage.........................................................................................213 F12 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1000 Hz and 150 Vpp voltage.........................................................................................213 12 PAGE 13 F13 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1100 Hz and 100 Vpp voltage.........................................................................................214 F14 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1100 Hz and 150 Vpp voltage.........................................................................................214 F15 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1200 Hz and 100 Vpp voltage.........................................................................................215 F16 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1200 Hz and 150 Vpp voltage.........................................................................................215 F17 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1300 Hz and 100 Vpp voltage.........................................................................................216 F18 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1300 Hz and 150 Vpp voltage.........................................................................................216 F19 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1400 Hz and 100 Vpp voltage.........................................................................................217 F20 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1400 Hz and 150 Vpp voltage.........................................................................................217 F21 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1500 Hz and 100 Vpp voltage.........................................................................................218 F22 OpenLoop control result for M=0.31 and ex citation sinusoidal input with frequency 1500 Hz and 150 Vpp voltage.........................................................................................218 13 PAGE 14 LIST OF ABBREVIATIONS D Cavity depth L Cavity length M Freestream flow Mach number U Freestream flow velocity a Mean sound speed inside the cavity vm Mode number (integer number 1,2) 0w Natural frequency of second order system r Reflection coefficient Damping ratio Phase lag factor Proportion of the vortices speed to the freestream speed Ratio of the specific heats v Spacing of the vortices a Time delay inside the cavity s Time delay inside the shear layer ADC Analog to digital converter ARMA Autoregressive and movingaverage CARIMA Autoregressive and integrated moving average CE Composite error DAC Digital to analog converter DNS Direct Numerical Simulations DSP Digital signal processing EE Equation error 14 PAGE 15 FFT Fast Fourier transform FIR Finite impulse response FRF Frequency response function GPC Generalized predictive control ID Identification IIR Infinite impulse response JTFA Jointtime frequency analysis LES Large Eddy Simulations LMS Least mean square LQG Linear quadratic Gaussian LTI Linear timeinvariant MIMO Multipleinput multipleoutput MPC Model predictive control MSOE Mean square output error OE Output error PDF Probability density function POD Proper orthogonal decomposition RANS Reynolds Averaged Navierstokes RLS Recursive least square SISO Singleinput singleoutput SM Steiglitz and McBride SNR Signal to noise ratio SPL Sound pressure level STR Selftuning regulator TITO Twoinput Twooutput 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 CLOSEDLOOP CONTROL OF FLOWINDUCED CAVITY OSCILLATIONS By Qi Song May 2008 Chair: Louis Cattafesta Major: Aerospace Engineering Flowinduced cavity oscillations are a coupl ed flowacoustic problem in which the inherent closedloop system dynamics can lead to large unsteady pressure levels in and around the cavity, resulting in both broadband noise and di screte tones. This problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, and automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been working on this problem for more than fifty year s. This is because not only is the physical nature of this problem rich and complex, but also it has become a standard test bed for controller deign and implementation in flow control. The ultimate goal of this research is to minimize the cavity acoustic tones and the broadband noise level over a range of freestr eam Mach numbers. Although openloop and closedloop control methodologies have been explored extensively in recent years, there are still some issues that need to be studied further. For example, a loworder theoretical model suitable for controller design does not exist. Most recen t flowinduced cavity models are based either on Rossiters semiexpirical formula or a proper orthogonal decomposition (POD) based models. These models cannot be implemented in adaptive controller design directl y. In addition, closedloop control of high subsonic and supers onic flows remains an unexplored area. 16 PAGE 17 17 In order to achieve these objectives, an analytical system model is first developed in this research. This analytical model is a transfer function based model and it can be used as a potential model for controller design. Then, a MIMO system identificat ion algorithm is derived and combined with the generalized prediction control (G PC) algorithm. The resultant integration of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics and reduce the flowinduced oscillations. A novel piezoelectricdriven synt hetic jet actuator arra y is designed for this research. The resulting actuator produces high ve locities (above 70 m/s) at the cen ter of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. A validation vibration beam problem is used to demonstrate the combin ation of system ID and GPC algorithms. The result shows a ~20 dB re duction at the single resonance peak and a ~9 dB reduction of the integr ated vibration levels. Both openloop control and cl osedloop control ar e applied to the flowinduced cavity oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced for both cases. The GPC controlle r can generate a series of control signals to drive the act uator array resulting in dB re duction for the second, third, and fouth Rossiter modes by 2 dB, 4 dB, and 5 dB respectively. In a ddition, the broadband background noise is also reduced by this closedloop cont roller (i.e., the OASPL reduction is 3 dB). The relevant flow physics and active flow control actuators are exam ined and explained in this research. The limitations of the present setup are discussed. PAGE 18 CHAPTER 1 INTRODUCTON Equation Section 1 Flowinduced cavity oscillations have been st udied for more than fifty years, and the problem has attracted researchers in both fluid dynamics and controls. Firs t, this problem exists in many practical environments, such as land ing gear bays and weapon delivery systems on aircraft, sunroofs and windows buffeting in auto mobiles, and junctions between structural and aerodynamic components in both (Kook et al. 1997). The flowacoustic coupling inherent in cavity resonance can lead to high unsteady pressu re levels (both broadba nd noise and discrete tones), and can cause fatigue failure of the cavity and its contents. For example, the measured sound pressure levels in and around a weapons bay can exceed 170 dB ref 20 For this reason, researchers are usually inte rested in suppression of flowinduced open cavity oscillations. Furthermore, this problem has become a sta ndard test problem for designing, testing, and analyzing realtime feedback control systems. Although the standard rectangular cavity geometry is relatively simple, the physical nature of this problem is both rich and complex. Several good review articles on the flowinduced cavity oscillati on problem are available in the literature (Rockwell, 1978, Komerath 1987, Colonius 2001, Cattafesta 2003). Pa Figure 11 is a simplified schematic for two t ypical flow situations, corresponding to external (a) supersonic and (b) subsonic fl ow over a rectangular cavity with length, depth, and width, W. The cavity oscillation process can be summarized as follows. A (usually) turbulent boundary layer with thickness, L D and momentum thickness, separates at the upstream edge of the cavity. Both a turbulent boundary layer and laminar boundary layer generate the discrete tones cau sed by the external flow. Howe ver, a laminar boundary layer has been shown to produce louder tones, presumably b ecause a turbulent flow generally results in a 18 PAGE 19 thicker shear layer with broadband disturban ces, which leads to overall lower levels of oscillations (Tam and Bl ock 1978; Colonius 2001). Following the description of Kerschen and Tumin (2003) and Alvarez et al.(2004), when the turbulent boundary layer separates at the upstream edge of the cavity, the resulting high speed or fast acoustic wave, and the low speed or slow acoustic wave, fE s E, propagate downstream in the supersonic flow case. In the subsonic flow case, only the socalled disturbance wave, propagates downstream. In both cas es, the shear layer instability, S, develops based upon its initial conditions coupled by the upstream traveling acoustic feedback wave, U, inside the cavity (and outside the cavity for subsonic flow). KelvinHelmholtztype (Tam and Block 1978) convective instability wave s develop and amplify in the shear layer as they propagate downstream and finally saturate due to nonlinearity. In part icular, the instability waves grow and form largescale vortical structur es that convect downstream at a fraction of the freestream velocity. These structures then impi nge near the trailing edge of open cavities ( ). The reattachment region acts as the primary acoustic source, which has been modeled as a monopole (Tam and Block 1978) or a dipole (Bilanin and Covert 1973) source. dEuE/1 LD 0 As a result, an upstream traveling acoustic wave, U, is generated inside the cavity. In subsonic flow, an additional acoustic wave, propagates upstream outsi de the cavity. In this description, the acoustic fee dback is modeled via acoustic waves that travel in the uE x direction. Finally, the loop is closed by a receptivity proces s, in which the upstream traveling waves are converted to downstream traveling instability an d acoustic waves. The initial amplitude and phase of these waves are set by the incident acoustic disturbances through this receptivity process. Physically, some of the acoustic dist urbance energy is converted to the instability waves at the upstream separation edge. Since the wavelength and the veloc ity of the instability 19 PAGE 20 waves and the acoustic disturba nces differ, only those waves that are inphase ensure reinforcement of disturbances at that frequency. Therefore, this process is normally considered an introduction of a disturbance into the system, ultimately resulting in largeamplitude discrete tones inside and around the cavity. The measured broadband noise component is mainly due to the turbulent shear layer. The relevant dimensionless parameters are: LDLW, L and shape factor *H with the freestream flow parameters, Reynolds number Re and Mach number M all of which lead to tones (with Strouhal number StfLU ) characterized by their strength as unsteady pressure normalized by the freestream dynamic pressure, rmspq In this study, three dimensional effects are not considered, since the cavity tones are generated by the interaction of the freestream flow and the lo ngitudinal modes (coupled with vertical depth modes). Width mode s are not relevant in this feedback loop if the width is small enough to prevent higherorder spanwise modes but large enough so that the mean flow over the cavity length is approximately two dimensional. No te that the width of the cavity does affect the amplitude of the cavity oscillations (Rossiter 1966) but is of secondary importance (Cain 1999). Therefore, a twodimensional model is reasonable from a physical perspective even though the unsteady turbulent motion is inevitably thr eedimensional (Bilanin and Covert 1973). Literature Review In this section, some publis hed results related to the physics of flowinduced cavity oscillations are discussed. Since the ultimate go al of this research is to minimize the cavity acoustic tones and perhaps the broadband noise level, potential control methodologies and algorithms are also reviewed. A recent review pa per by Cattafesta et al. (2003) gives a summary of the various passive and ope nloop cavity suppression studies. 20 PAGE 21 Physical Models In order to suppress the disc rete tones and the broadband aco ustic level of flowinduced open cavity resonance, an understanding of the phys ics is essential. From a control engineers point of view, a simplified and loworder model is desirable in order to predict the resonant frequencies and amplitudes over a broad range of the governing dimensionless parameters. PhysicsBased Models Rossiter (1964) performed an extensive expe rimental study on the measurement of the unsteady pressure in and around a rectangular open cavity ( 21.5 f tft ) in a subsonic and transonic freestream air flow (0.). He observed broadband noise and a series of unsteady acoustic tones generated in th e cavity. For the deeper cavities ( ), there was usually a single dominant tone, and the domin ant frequency was observed to jump between different cavity tones. For the shallower cavities ( ), two or more peaks were often observed and were approximately equal in magnit ude. He proposed that the flow entering the cavity caused the external stream to accelerate, and then the flow decelerated near the reattachment region. As a result, pressure wa s lower near the separa tion region (leading edge) and higher near the reattachment region (trailing edge). As a re sult, he suggested that large eddies developed within the cavity due to this pre ssure gradient. He also used shadowgraphs to illustrate that the shear layer separates from th e cavity leading edge, and the instability waves develop into discrete vortices that are shed at regular time internals from the front lip of the cavity (at Mach number 0.6 and with twodimensional cavity 51.2M /LD 4/ LD 4/4LD and a laminar boundary layer). He postulated that there were some connections between the vortex shedding and the acoustic feedback, and this phenomenon produced a se ries of periodic pre ssure fluctuations. 21 PAGE 22 When the frequency of one of these components is close to the natural fr equency of the cavity, resonance occurs. In his study, Rossiter gave a semiempiric al formula for predicting the resonant frequencies of these peaks at a specific Mach nu mber. The derivation of the Rossiter equation is given in Appendix B, and the resulting form ula for the dimensionless Strouhal number is 1v mm fL St U M (11) where m f is the resonance frequency for integer mode is the length of the cavity, U is the freestream velocity, vmL is the phase lag factor (i n fractions of a wavelength), is the ratio of the vortex propagation speed to the freestream velocity, and M is the freestream M number. Empirical constant values of ach 0.57 and 0.25 are shown to best fit the measured frequencies of resonances over a wide range of the Mach numbers for his experiment. These experimental constants account for the phase sh ift associated with the coupling between the shear layer and acoustic waves at the two ends of the cavity, and this phase shift is approximately independent of frequency. The phase speed U of the vortices is a weak function of M L and D (Colonius 2001). Different integer values give different frequencies, commonly referred to as shear layer or Rossiter mode s. In conclusion, Rossiters formula is based on an integer number of vm2 phase shifts, 2 k around a resonant feedback loop consisting of a downstream unstable shear layer di sturbance and an upstream feedb ack acoustic wave inside the cavity. This phase shift is a necessary condition fo r selfsustaining oscilla tions (Cattafesta et al. 1999a). However, Rossiters expression does not account for the depth or width of the cavity and only successfully predicts the longitudinal ca vity resonant frequencies at moderatetohigh Mach numbers. It also does not predic t the amplitude of the oscillations. 22 PAGE 23 Heller and Bliss (1975) corrected the Rossiter equation for the higher sound speed in the cavity, in which the static temperature in the cavity was assumed to be the stagnation temperature of the upstream. Th e modified Rossiter formula is 21 1 1 2v mm fL St U M M (12) where is defined as the ratio of specific heat s. They gave a discussion on the physical mechanisms of the oscillation process based on wa ter table visualization experiments. They suggested that the unsteady motion of the shear layer leads to a periodic mass addition and removal at the cavity trailing edge, leading to subsequent modeling efforts that employ an acoustic monopole source. In addition, the wave motion of the shear layer and the wave structure within the cavity were strongly coupled. Bilanin and Covert (1973) modeled the cavit y problem by splitting th e domain into two parts outside and inside the cavity. These two flow fields were separated by a thin mixing layer, which was approximated by a vortex sheet, and th e flow was assumed to be inviscid. The dominant pressure oscillations at the trailing edge were modele d by a single periodic acoustic monopole. They also assumed that the pressure field from the trailing ed ge source had no effect on the vortex sheet itself. Hence, the main distur bance was introduced at the leading edge of the shear layer. Kegerise et al. (2004) illustrated the agreement be tween the disturbance sensitivity function defined in control systems and the perf ormance measurement of output disturbances. Their analysis confirmed the notion that the disturbances were mainly introduced into the cavity at the cavity leading edge. 23 PAGE 24 Tam and Block (1978) carried out extensive expe rimental investigations at low subsonic Mach numbers ( ) and postulated that vortex sheddi ng was probably not the main factor for cavity resonance over the en tire Mach number range. They made two key assumptions, namely that the rectangular cavity flow was twodimensional, and the mean flow velocity inside the cavity was zero. These two assumptions were based on experimental evidence of little correlation between the mean flow and the acoustic feedback inside the cavity. Tam and Block proposed a process of flowinduced cavity oscillations as follows. The shear layer oscillated up and down at the trailing edge of the cavity. The upward movement was uncorrelated with the generation of the acoustic waves, because if the shear layer covered the trailing edge, then the external flow passes over the trailing edge without impingeme nt. They argued that only the downward motion of the shear la yer into the cavity caused signi ficant generation of pressure waves and subsequent radiation of acoustic waves in all directions (0.4MFigure 12 ). For example, some of the waves radiating in to the external flow (e.g., wave A) were argued to have minor effects on th e oscillations inside the cavity. However, the effect of the waves propagating inside of the cavity was deemed more significant. The resulting acoustic waves included the upstream propagating waves (e .g., wave C) and the reflected waves from the floor (e.g., wave F) and the upstream wall (e.g., wave E). Subsequent reflections of the acoustic waves by the walls, the cavity, or the shear layer we re deemed negligible. They concluded that the directly radiated wave and the first reflecte d waves by the floor and upstream end wall of the cavity provided the energy to excite the instability waves of the shear layer. These disturbances within the shear layer were then amplified as the instability waves propa gate downstream. When the disturbances amplitudes became large, nonlinear effects were important and ultimately established the amplitude of the discrete tones. A mathematical model of the cavity oscillation 24 PAGE 25 and acoustic field were developed. In order to calculate the phas es and waves generated at the trailing edge, a periodic line sour ce was simulated at the trailing edge of the cavity. In addition, the reflections of the acoustic waves by the cavity walls were m odeled by periodic line image sources about the cavity walls. Their model account s for the finite shear layer thickness effects and produces a more accurate estimation of the re sonance frequencies than Rossiters model. However, their resulting model is complicated and difficult to employ for control law design. Rowley et al. (2002 b, 2003, 2006) provided an alternative viewpoint for understanding flowinduced cavity oscillations. They showed that selfsustained oscillations existed only under certain conditions. The resonant frequencies were due to the instabiliti es in the shear layer interacting with the flow and acoustic fields. The amplitude of the oscillations was determined by nonlinear saturation. However, at other c onditions, the cavity oscillations could be represented as a lightly damped but stable linear system. The oscillat ions were caused by the amplification of external disturbances via the cl osedloop dynamics of the cavity. The amplitude of each mode was determined by the amplitude of the external forcing disturbances and some frequencydependent gain of the system. They modeled the dynamics of the shear layer as a secondorder system and the acoustic propagation process via a onedimensional, standingwave model. The impingement and receptivity procedures were simply modeled as a constant unity gain. Finally, the Rossiter formula was derive d under some specific conditions. The derivation of this model is provided in Appendix B. They also used Gaussian white noise as input and examined the probability density function (PDF) and the phase portrait of the output pressure signal at different Mach numbers. Their results showed that under some conditions, the selfsustained regime of Rossiter modes was valid. However, at other conditions, called the forced regime, open cavity oscillations may 25 PAGE 26 be represented as lightly damped stable linear sy stems. External random forces drove the finiteamplitude cavity oscillations, which implies they will disappear if the external forces were removed. This physical linear model was also proposed as a potential model for controller design. Kerschen and Tumin. (2003) and Alvarez et al. (2004) provided a pr omising global model to describe the flowinduced cavity oscillation problem for two different flow patterns ( Figure 11). Their model combined scattering analyses for the two ends of the cavity and a propagation analysis of the cavity shear layer, internal re gion of the cavity, and acoustic nearfield. They solved a matrix eigenvalue problem to identify th e resonant frequencies of the cavity oscillation. From their resulting characteristic functions, four and twelve closed loops could be identified for the supersonic flow and subsonic flow cases, re spectively. One more feedback loop makes the subsonic flow much more complex than the supers onic flow. For example, some of these closed loops were major loops, such as closed loop and (, SU UDFigure 11 ), while the other closed loops were considered minor loops. The combin ed effects of these loops caused the cavity resonances in the cavity flow. Besides these closed loops, the fo rward propagation paths, such as and also have critical effects on the amplit ude of the oscillations. This global model provides more insights for controller desi gn. A detailed derivation of this model is provided in Appendix B. dSDEEsfEClearly, the physicsbased models described a bove provide physical insight concerning flowinduced cavity oscillations. However, the original Rossiter mode l and the global model derived by Kerschen and Tumin. (2003) can only estimate the resonance frequencies of the cavity flow. The linear model derived by Rowley et al. (2002b, 2003, 2006) is transfer function based model but is not sufficiently accurate to de sign a control system. A transfer function based 26 PAGE 27 model, which is an extension of Kerschen et al. s model, of cavity acoustic resonances is derived and given in Appendix D. For this approach, a signal flow graph is firs t constructed from the block diagram of the Kerschen et als physical model, and then Masons rule (Nise 2004) is applied to obtain the transfer f unction from the disturbance input to the selected system output. This method can give predictions for both the re sonant frequencies of the flowinduced cavity oscillations and the amplitude of the cavity tones. In addition, this method also provides a linear estimate for the system transfer function from th e disturbance input at th e leading edge and the pressure sensor output within the cavity walls. Th erefore, this model is a potential global model for controller design in this research. Numerical Simulations Some computational fluid dynamic (CFD) method s, such as Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) and Reynolds Averaged Navierstokes (RANS), provide useful information for understand ing the issues of phys ical modeling of cavity oscillations. A review paper by Colonius (2001) gives a summary of issues related to each of these topics. More recent research on these topics can be found by Rizzetta et al. (2002, 2003) and Gloerfelt (2004). The Detached Eddy Simulation (DES) method, whic h is a involves a hybri d turbulence modeling methodology, has also been used to calculate the flow and acoustic fields of the cavity (Allen and Mendonca 2004; Hamed et al. 2003, 2004). Another hybrid RANSLES turbulence modeling approach is presented by Arunajate san and Sinha (2001, 2003). They model the upstream boundary layer flow field and the sh ear layer region via RANS and LES models, respectively. All of thes e computational methods provide, at a minimum, good flow visualization and physical insight, and, at a maxi mum, quantitative information on the details of flow dynamics. 27 PAGE 28 PODType Models The previous analytical physical models are not accurate enough to design a control system. Furthermore, CFD methods are far too co mputationally intensive at the present time to provide a reasonable framework to design and test potential controllers. This translates into the need for new methods to develop more accurate reducedorder models. Therefore, simulation and experimental data based models were propos ed and later used for the controller design. Rowley et al. (2001) introduced a nonlinear dynamical model for flowinduced rectangular cavity oscillations, which was based on the method of vectorvalued proper orthogonal decomposition (POD) and Galerkin projection. The POD method obtains lowdimensional descriptions of a highorder system (Chatterj ee 2000). For the cavity flow problem, data resulting from the temporalspatial evolution of the numerical simu lations or experiments is used to construct a loworder subspace system that capt ures the main features (coherent structures) of the cavity flow. A more detailed explanati on of POD methods for cavity flow are given by Rowley et al. (2000, 2001, 2002c, 2003a). Some of the control methodologies discussed in next section can be constructed based on the result ant model obtained by POD (Caraballo et al. 2003, 2004, 2005; Samimy et al. 2003, 2004; Yuan et al. 2005). Instead, we turn our attention to an alternat ive experimentalbased modeling approach that employs system identification techniques. He re, the nonlinear infini tedimensional governing equations are modeled by a reduced set of differential (in continuous) or difference (in discrete time) equations. This method is the focus of this study and is discussed in the following section. OnLine System ID and Active ClosedLoop Control Methodologies Previous studies aimed at suppression of the flowinduced cavity tones have employed mainly passive or openloop active flow control methodologies. The standard classification of the flow control techniques is shown in Figure 13 The review paper by Cattafesta et al. (2003) 28 PAGE 29 provides a detailed overview of various passive and openloop control methodologies. However, passive and openloop approaches are only effective for a limite d range of flow conditions. Active feedback flow control has recently been applie d to the flowinduced cavity oscillation problems. The closedloop control approaches have advantages of reduced energy consumption (Cattafesta et al. 1997), no a dditional drag penalty, and robustn ess to parameter changes and modeling uncertainties. In genera l, closedloop flow control measures and feeds back pressure fluctuations at the surf ace of the cavity walls (or floor) to an actuator at the cavity leading edge to suppress the cavity oscillati ons in a closedloop fashion. In general, past active control strategies have taken one of two approaches for the purpose of reducing cavity resonance. First, they can thicken the boundary layer in order to reduce the growth of the instabilities in the shear layer. A lternatively, they can be us ed to break the internal feedback loop of the cavity dynamics. Most cl osedloop schemes exploit the latter approach. Early closedloop control applications used manual tuning of the gain and delay of simple feedback loops to suppress resonance (Gharib et al. 1987; Williams et al. 2000a,b). Mongeau et al. (1998) and Kook et al. (2002) used an active spoiler driven at the leading edge and a loopshaping algorithm to obtain significant attenuation with small actuation effort. Debiasi et al. (2003, 2004) and Samimy et al. (200 3) proposed a simple logicba sed controller for closedloop cavity flow control. Loworder modelbased co ntrollers with different bandwidths, gains and time delays have also been designed and implemented (Rowley et al. 2002, 2003, Williams et al. 2002, Micheau et al. 2004, Debiasi et al. 2004). Linear optimal c ontrollers (Cattafesta et al. 1997, Cabell et al. 2002, Debiasi et al. 2004, Samimy et al. 2004, Caraballo et al. 2005) have been successfully designed for operation at a sing le flow condition. These models are all based on reducedorder system models, and most of these controller design methods are based on 29 PAGE 30 model forms of the frequency response function, rational discrete/continu ous transfer function, or statespace form. However, the coefficients of these model forms are assumed to be constant, and this assumption requires that the system is time invariant or at least a quasistatic system with a fixed Mach number. Although the physical models of flowinduced cavity oscillations have been explored extensively, they are not convenien t for control realization. This is because these models are highly dependent on the accuracy of the estimated internal states of the cavity system. In addition, cavity flow is known to be quite sensitive to slight changes in flow parameters. So a small change in Mach number can deteriorat e the performance of a singlepoint designed controller (Rowley and Williams 2003). Therefore, adaptive control is certainly a reasonable approach to consider for reducing oscillations in the flow past a cavity. Adaptive control methodology combines a general control strategy and system identification (ID) algorithms. This method is thus potentially able to adapt to the changes of the cavity dimension and flow conditions. It updates the cont roller parameters for optimum performance automatically. The structure of this method is illustrated in Figure 14 Two distinct loops can be observed in the controller. The outer loop is a standard feedback control system comprised of the process block and the controlle r block. The controller operates at a sample rate that is suitable for the discrete process under control. The inner loop consists of a parameter estimator block and a controller design block. An ID algorithm and a specified cost function are then used to design a controller that will minimize the output. The steps for realtime flow control include: (i) Use a broadband system ID input from the actua tor(s) and the measured pressure fluctuation output(s) on the walls of the cavity to estimate the sy stem (plant and disturbance) parameters. (ii) 30 PAGE 31 Design a controller based on the estimated system parameters. (iii) Control the whole system to minimize the effects of the disturbance, measur ed noise, and the uncertainties in the plant. Based on this adaptive control methodol ogy, some adaptive algorithms adjust the controller design parameters to track dynamic changes in the system. However, only a few researchers have demonstrated the online adap tive closedloop control of flowinduced cavity oscillations. Cattafesta et al. (1999 a, b) applied an adaptive di sturbance rejection algorithm, which was based upon the ARMARKOV/Toeplitz models (Akers and Bernstein 1997; Venugopal and Bernstein 2000, 2001), to identify and control a cavity flow at Mach 0.74 and achieved 10 dB suppression of a single Rossiter model. Other modes in the cavity spectrum were unaffected. Insufficient act uator bandwidth and authority limited the control performance to a single mode. Williams and Morrow (2001) applied an adaptive filteredX LMS algorithm to the cavity problem and demonstrated multiple ca vity tone suppression at Mach number up to 0.48. However, this was accompanied by simultaneous amplification of other cavity tones. Numerical simulations using the least mean squares (LMS) algorithm were shown by Kestens and Nicoud (1998) to minimize the output of a singl e error sensor. The reduction was associated with a single Rossiter mode, but only within a small spatial region around the error sensor. Kegerise et al. (2002) implemen ted adaptive system ID algorithms in an experimental cavity flow at a single Mach number of 0.275. They also summarized the typical finiteimpulse response (FIR) and infiniteimpulse response (IIR) based system ID algorithms. They concluded that the FIR filters used to represent the flow induced cavity process were unsuitable. On the other hand, IIR models were able to model the dynamics of the cavity system. LMS adaptive algorithm was more suitable for realtime control than the recursiveleast square (RLS) adaptive algorithm due to its reduced computational comple xity. Recently, more advanced controllers, 31 PAGE 32 such as direct and indirect synthesis of the ne ural architectures for bot h system ID and control (Efe et al. 2005) and the generalized predictiv e control (GPC) algorithm (Kegerise et al. 2004), have been implemented on the cavity problems. From a physical point of view, the closedloop controllers have no effect on the mean velocity profile (Cattafesta et al. 1997). Howeve r, they significantly affect streamwise velocity fluctuation profiles. This control effect eliminates the strength of the pressure fluctuations related to flow impingement on the trailing edge of the cavity. Although closedloop control has provided promising results, the peaking (i.e., gene ration of new oscillation frequencies), peak splitting (i.e., a controlled peak splits into two sidebands) and mode switching phenomena (i.e., nonlinear interaction between two different Rossiter frequencies) often appear in active closedloop control experiments (Cattafesta et al. 1997, 1999 b; Williams et al. 2000; Rowley et al. 2002 b, 2003; Cabell et al. 2002; Kegerise et al. 2002, 2004a). Explanations of these phenomena are provi ded by Rowley et al. (2002b, 2006), Banaszuk et al. (1999), Hong and Bernstei n (1998), and Kegerise et al (2004). Rowley et al. (2002b, 2003) concluded that if the viewpoint of a lin ear model was correct, a closedloop controller could not reduce the amplitude of oscillations at all frequencies as a consequence of the Bode integral constraint. Banaszuk et al. (1999) gave explanations of the peaksplitting phenomenon. They claimed that the peak splitting effect was caused by a large delay and a relatively low damping coefficient of the openloop plant. Cabell et al. (2002) expl ained these phenomena by the combination of inaccuracies in the identified plant model, hi gh gain controllers, large time delays and uncertainty in system dynamics. In addition, narrowbandwidth actuators and controllers may also lead to a peaksplitting pheno menon (Rowley et al. 2006). 32 PAGE 33 Hong and Bernstein defined the closedloop sy stem disturbance amplification (peaking) phenomenon as spillover. They illustrated that the spillover problem was caused by the collocation of disturbance source and control signal or the co llocation of the performance and measurement sensors. For this reason, the redu ction of broadband pressure oscillations was not possible if the control input was collocated with the disturbance signal at the leading edge of cavity. Therefore, Kegerise et al. (2004) s uggested a zero spillover controller which utilized actuators at both the leading a nd trailing edges of the cavity fo r closedloop flow control. Unresolved Technical Issues Although the flowinduced cavity oscillation prob lem has been explored extensively, there are still some unresolved issues th at need to be studied further. A suitable theoretical model does not exist that estimates both the disc rete frequencies as well as the amplitude of the peaks. A feedback controller that re duces both broadband and tonal noise over a wide range of Mach numbers has not been achieved. An ad aptive zero spillover control algorithm may reduce both the tones and broa dband acoustic noise associated with cavity oscillations. The necessity for a highorder system model is a critical problem for controller design and implementation, because this highorder syst em results in significant computational complexity for application in digital signal processing (DSP) hardware. As such, the convective delays between the control inputs and the pressure sensor outputs must be specifically addressed in the control architecture. Closedloop control of high subsonic and s upersonic flows is an unexplored area. Technical Objectives According to those unresolved t echnical issues, the ultimate goa ls of this dissertation are summarized as follows. A feedback control methodology will be developed for reducing flowinduced cavity oscillation and broadband pressure fluctuations. Adaptive system ID and control algorithms will be combined and implemented in realtime. 33 PAGE 34 The relevant flow physics and the design of appropriate active flow control actuators will be examined in this research. The performance, adaptabilit y, costs (computational and ener gy), and limitations of the algorithms (spillover, etc.) will be investigated. Approach and Outline In order to achieve these objectives, some design and application approaches warrant additional consideration. First, a potential theoretical model of cavity acoustic resonances is derived based on the global model of Kerschen and Tumin. (2003). This model (derived in Appendix D) provides the framework to estimate the amplitudes and frequencies of the cavity tones. This model has a low system order and also accounts for the c onvective delay between the disturbance input and the output pressure measurement. Second, during the controller design, the controlled system is a continuous syst em; therefore, all the sensors measurements and the actuators inputs are analog signals. However, for the present realtime application, the control algorithms are implemented using a DSP. For this reason, additional hardware, such as analogtodigital converters (ADC), digitaltoana log converters (DAC), antialiasing filters, and power amplifier, must be included in the whol e control design procedure. Finally, multiple actuators and multiple sensors are employed in this study in order to design an adaptive zero spillover control algorithm to explore the possibility of achieving broadband acoustic noise reduction in addition to suppression of the cavity tones themselves. This active control method development proc edure can be summarized as the following stages according to Elliott (2001). Study the simplified analytical system model and understand the fundamental physical limitations of the proposed control strategy. Obtain the sensor output and derive the states or coefficients from the system ID algorithms using offline or online methods. 34 PAGE 35 Calculate the optimum performance using differe nt control strategies and find the control law for realization. Simulate the different control strategies a nd tune the candidate c ontroller for different operating conditions. Implement the candidate controlle r in realtime experiments. The thesis is organized as follows. Severa l SISO IIR system ID algorithms and a more general MIMO system ID algorithm are derived and discussed in the next Chapter. Then the MIMO adaptive GPC algorithm is described in Chapter 3. This is followed by a description of the sample experimental setup a nd the discussion of preliminary e xperimental results. Chapter 5 describes the wind tunnel facilities and the da ta processing methods. Wind tunnel experimental results for both openloop (baseline) and closed loop are then presented and discussed in Chapter 6. Finally, the conclusions and future work are presented in Chapter 7. 35 PAGE 36 L Turbulent Boundary Layer D 1 M s E f E S D U x y A 1 L Turbulent Boundary LayerD M dEuESDU x y B Figure 11. Schematic illustra ting flowinduced cavity resona nce for an upstream turbulent boundary layer. A) In supersonic flow.B) In subsonic flow. L D U Simulated Line Source A C E F x y Figure 12. Tam and Block (1978) model of acoustic wave fi eld inside and outside the rectangular cavity. 36 PAGE 37 37 Flow Control Approaches Passive Control Active Control OpenLoop ClosedLoop QuasiStatic Dynamic Figure 13. Classification of flow control. (after Cattafesta et al. 2003) input u(k) output y(k) disturbance reference Controller Plant Controller Design Parameter Estimator process part ID part control part ID input u(k) System ParametersControl Parameters Measured Noise Uncertainties Figure 14. Block diagram of sy stem ID and online control. PAGE 38 CHAPTER 2 SYSTEM IDENTIFICATION ALGORITHMS Equation Section 2 This chapter provides a detail discussion of the system identification algorithms. Several typical adaptive SISO IIR structure fi lters are chosen as the candidate digital filters. These algorithms are applied to an example from Johnson and Larimore (1977) for simulation analysis. Then, four interested aspects of these filters, accuracy, convergence, computational complexity, and robustness, are examined and summarized. Finally, a more general MIMO sy stem ID algorithm is derive d from one of the promising SISO system ID algorithms. The resulting model is used to combine with the MIMO adaptive GPC model which is discussed in next chapter. Overview As discussed in the first chapter, IIR struct ure filter is an applicable mathematical model to capture the cavity dynamics. Furt hermore, this kind of structure can be a starting point and easily combined with many controller design strategi es. Therefore, in this Chapter, several system ID algorithms ba sed on the IIR filter structure are examined. The ideal of the system ID is to construct a predefined IIR structure filter, which has the similar frequency response of the actual dynamic system, using the information from the previous and present input and output time series data of the dynamic system. In general, the system ID algorithms fall into two big cat egories, the batch me thod and the recursive method. The batch method directly identifies the final system parameters in onetime calculation using a block data from the i nput and a block data from the output. Nevertheless, the recursive method updates the estimated system para meters within each sampling period using the latest input and output data in time domain. At each iteration of calculation, the system parameters may not be the optimal values. However, these 38 PAGE 39 estimated parameters will finally converge to th e true values of the system internal states. Successful identifying the system internal states depends on two major assumptions. First, the input signal and the output signa l must have a good correlation. Then, the system ID model has the same structure of that of the estimated system model. The recursive method is more attractive for pres ent experiment, because this updating method is more suitable for online implementation and it can also track the change of the system dynamics. Furthermore, the computational complexity of recursive method is much lower than the batch method. SISO IIR Filter Algorithms Netto and Diniz (1995) give a summary of some popular ad aptive IIR filter algorithms. In this section, the Output E rror (OE), Equation Erro r (EE), Steiglitz and McBride (SM), and Composite Error (CE) algo rithms are selected and illustrated. The general structure of an IIR filter is shown in Figure 21 The filter output may be expressed as 10 ()()() ()()abnn ij ij Tykaykibxkj kk (21) where represents the estimation values. and are the adjustable coefficients of the model, while and is the estimated order of the feedback loop and forward path, respectively. ia jb an bn kyi)x(kj) () (k ()T ijb ka and This IIR filter structure, Equation abi=1,...,n;j=0,1,...,n 21 is the same as the au toregressive and movingaverage (ARMA) model (Haykin 2002). 39 PAGE 40 Based on different error, the value diffe rence between the filter output and the system output, definitions, quite a few IIR filter algorithms have been presented by Netto and Diniz (1995). In their simulations, they us e an insufficient model, which models a secondorder system using a firstorder system to test each algorithm. The results from their paper show that the Modified Out put Error (MOE) algorithm may converge to a meaningless stationary point. The same result is also shown by Johnson and Larimore (1977). The Simple Hyperstable Algorithm for Recursive Filters (SHARF) algorithm, the modified SHARF algorithm, and the Bias Remedy LeastMeanSquare Equation Error algorithm (BRLE) also show poor convergence rates. The Composite Regressor (CR) algorithm has similar problems as the MOE algorithm, since this algorithm combines the EE and MOE methods. Therefore, in this section, tests of these poor performing algorithms are not discussed. Fundamentally, there are two approaches for an adaptive IIR filter, the OE algorithm and the EE algorithm, which ha ve been derived by Haykin (2002) and Larimore et al. (2001), respectiv ely. Many other adaptive IIR filter algorithms are mainly derived from these algorithms, or combine some good features from the OE and the EE filters. Therefore, a summary of each of these two algorithms is provided in the following section. Two other algorithms, th e SteiglitzMcBride algorithm (SM) and the Composite Error algorithm (CE), are also in troduced, because both these algorithms also show good performance in our Simulink simulations. IIR OE Algorithm The IIR OE algorithm is summarized in Table 21 To ensure the stability of the algorithm, generally, the upper bound of step size is set to max2 where max is the 40 PAGE 41 maximum eigenvalue of the autocorrelat ion matrix of the regress vector The step sizes of the following algorithms are also satisfying this criterion. Furthermore, in order to guarantee the co nvergent approximation of ()OEkj andi this algorithm requires slow adaptation rates for small values of and (Haykin 2002). an bnIIR EE Algorithm The IIR EE algorithm is summarized in Table 22 Since the desired response is the supervisory signal supplied by the actual out put of plant during the training period, the EE algorithm may lead to faster converg ence rate of the adaptive filter (Haykin 2002). IIR SM Algorithm The IIR SM algorithm is summarized in Table 23 Since the EE algorithm and the OE algorithm possess their own advantages as well as drawbacks (discuss later), the motivation of the SM algorithm is to combine the desirable characteristics of the OE and the EE methods. IIR CE Algorithm This algorithm tries to combine both th e EE algorithm and the OE algorithm in another way. As shown in Table 24 a parameter is used to switch this algorithm between the EE algorithm and the OE algorithm. Recursive IIR Filters Simulation Results and Analyses In adaptive control experiments, the accuracy, the convergent rate, the computational complexity and the robustness are the main issues of the system ID algorithms. Here, computer simulations are ex amined in order to compare these aspects of the four system ID algorithms. 41 PAGE 42 The setup for the following Simulink simulation is shown in Figure 22 A Gaussian broad band white noise with zero mean and unity variance is chosen as the reference input signal. The prototype test model is a s econdorder dynamical system (Johnson and Larimore 1977) with the transfer function 1 1 10.050.4 () 11.13140.25 z Hz zz 2 (22) From the zplane plot ( Figure 23 ), it clearly shows that this test model is a stable and nonminimum phase system, which has two real poles at 0.3011 z and and two zeros at and 0.8303 z 0 z 8 z In the following simulations, a sufficient order identification problem is firstly examined, which means a secondorder syst em model with the transfer function 1 1 01 1 12 ()() (,) 1()() bkbkz Hzk akzakz 2 is used to estimate the test model. Then, an insufficient order identification problem is investigated. This approach uses a firstorder system model with the transfer function 1 0 1 1 () (,) 1() bk Hzk akz to estimate the test model. The mean square output error (MSOE) surface of this insufficient order dynamical system is obtained by Shynk (1989) 2 0 01 2 1 2() 1yb MSOEbHa a (23) A 3D surface plot and a contour plot of the MSOE performance surface are shown in Figure 24 and Figure 25 respectively. The plots show that the MSOE surface of the test model is bimodal with a global minimum (denoted by *) 42 PAGE 43 at which yields and a local minimum (marked by +) at( which corresponds to **(,)(0.311,0.906) ba ,)(0.114, ba*0.277 MSOE 0.519) 0.976 MSOE The input () x k and the test model output () y k (with or without disturbance ) are introduced to the ad aptive IIR filter algori thms at the same time. The adaptive IIR filter algorithms calculate th e error signal and update the weights at each iteration. () vkAccuracy comparison for sufficient system Table 25 and Figure 26 show the simulation results a nd weight tracks of the four IIR algorithms for the sufficient case, respectiv ely. For the sufficient case, the algorithms minimize the mean square error between the system output and the filter output, and the estimated weights converge to the origin al coefficients of the test model. Accuracy comparison for insufficient system The simulation results and weight tracks of the IIR algorithms for the insufficient case are shown in Table 26 and Figure 27 respectively. The OE algorithm starts from two different initial conditions. One point is closer to the global minimum, and the other one is closer to the local minimum. This method adjusts its weights via stochastic gradient estimation to the closest stat ionary point of the initial condition. Similarly, two initial conditions are sele cted for EE algorithm. One of them is close to the global minimum, and the other on e is much closer to the local minimum. This algorithm can avoid the local minimum an d adjust its weights to let the final mean square error value arrive at the area near the global minimum. However, for this insufficient order situation, the final soluti on exhibits bias compared to the optimum solution. 43 PAGE 44 The SM algorithm combines the advantages of the OE algorithm and the EE algorithm. This algorithm avoids the local minima and converges to the global minimum with different initial points, which is like th e EE algorithm. At the same time, the final solution for this algorithm is very close to the optimum solution. As addressed above, the CE algorithm is a combination of the OE algorithm and the EE algorithm. It uses a weighting parameter to switch and weight between the OE algorithm and the EE algorithm. For this in sufficient identification problem, this method performs well. If the weighting parameter is close to 0, this algorithm is more like the OE algorithm, and the interesting feature of th is algorithm shows that it converges to the global minimum in the MSOE surface. However, when is close to 1, the biased characteristic of the EE algorithm is apparent in the results. Convergence rate For convergence rate comparis on, the same step size and number of iterations are chosen for simulations. The simulation c onditions and the learning cures of the IIR algorithms for the sufficient case are shown in Table 27 and Figure 28 respectively. Obviously, the EE and SM algorithms converge faster than the OE and CE algorithms. Computational complexity In order to apply the ID al gorithm on an adaptive control algorithm for realtime implementation, the computational complexity for one iteration of the ID algorithm have to be less than the sampling time of the DPS processor used for realtime experiment. Four algorithms are compared for computa tional complexity by th e turnaround time with the increase of the number of unknown for each algorithm ( Figure 29 ). The hardware used for experiment is PowerPC 750 (480M Hz) microprocessor (12.6 SPECfp95). The 44 PAGE 45 experimental results are shown in Figure 29 The computational complexity of all of the IIR algorithms is approximately linear. A nd the CE algorithm needs more computational time for each iteration than time requirements for the other three algorithms. Conclusions Varies of IIR adaptive filters are examined in this Chapter, the objective of these digital filters is to identify the system coe fficients (internal stat es) from the input and output signals. The OE algorithm and the EE algorithm are two basic structures of an adaptive IIR filter. Beyond that, two other algorithms, the SM algorithm and the CE algorithm, are also examined. Simulation resu lts show that the mean square error value calculated by the OE algorithm converges to the optimum solution for both the sufficient case and the insufficient case if the proper initial condition is chosen. This means that the OE algorithm may converge to local minima in the MSOE surface. Furthermore, this algorithm does not guarantee that the poles of the ARMA model always lie inside the unit circle in the zplane. T hus, the OE method may become unstable (Haykin 2002) during the experiment. Therefore, a small enough st ep size and stability monitoring are required to ensure the convergence of the algorithm. However, the optimum step size is unknown, and the stability monitoring highly increases the computational complexity. These are the main drawbacks that should be considered in applications. The mean square error value calculat ed by the EE algorithm avoids the local minima and converges to the global minimum in the MSOE surface. The convergent rate and the computational complexity are good for realtime implementation. Unfortunately, the final solution is biased when the test model uses a lowerorder system to model a higherorder system (Shynk 1989, Netto and Diniz 1995). 45 PAGE 46 Both the SM algorithm and the CE algorith m can find the global minimum in the MSOE surface. However, the good performan ce of the SM algorithm does not occur in general and, in fact, cannot be assured in practice (Netto and Diniz 1992). Moreover, the CE algorithm produces good results when 0.041 Within these limits, the algorithm is unimodal, and the bias is neglig ible (Netto and Diniz 1992). However, the stability of the CE algorithm mu st still be monitored, and the computational complexity is also high for this algorithm. A summary of the four algorithms is given in Table 28 The robustness results of each ID algorithms come from the experiment discussed in Chapter 4. As the results, the EE algorithm is the best algorithm compari ng to the other three ID algorithms. Therefore, in next step, a MIMO IIR filter is going to be derived based on this algorithm. MIMO IIR Filter Algorithm In this section, a MIMO system ID al gorithm is developed based on the SISO IIR EE algorithm. First, a linear system model is summed with the r inputs 1 ru and the m outputs 1 my. For simplification, the order p of the feedback loop is assumed the same as the order of the forward path. At specific time index the system can be expressed as k 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuu p (24) where 46 PAGE 47 1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (25) Define the observer Markov parameters 10 **(1)()pp mmprpk (26) and the regression vector **(1)(1) () () () ()mprpk kp k k kp1 y y u u (27) substituting Equation 26 and Equation 27 into Equation 24 yields a matrix equation for the filter outputs 1 (**(1)) ()() ()m mmprpkkk y( ( 1 ) ) 1 m p r p (28) Furthermore, the errors are defined as 11 ()()()mmkkk1 m yy) k (29) Finally, the observer Markov parameters 26 can be identified recursively by (210) (1)()()(TkkkIn order to automatically upda te the step size, choose 21 (211) 47 PAGE 48 where is a small number to avoid the singularity when 2 is zero. The main steps of the MIMO identification for one iteration are summarized as follows Step 1: Initialize (**(1)) ()mmprpk 0. Step 2: Construct regression vector **(1)1()mprpk according to Equation 27 Step 3: Calculate the output error 1()mk according to Equation 29 Step 4: Calculate the step size according to Equation 211. Step 5: Update the observe r Markov parameters matrix *(1)()mmprpk according to Equation 210. Then, the calculation for the ne xt iteration goes back to step 2. The detail derivation for this MIMO ID al gorithm is given in Appendix B. And the experimental results of the algorithm, the co mputational complexity, and the disturbance effects will be discussed in later Chapters. The calculation result of this MIMO ID gives an estimated model of the system with the form of Equation 24 In the following Chapter, a MIMO control algorithm is de veloped based on this MIMO ID model. 48 PAGE 49 Table 21. Summary of the IIR OE algorithm. Initialization: where ()[(0)(0)]T ijkab 0 1,...,,0,1,...,abinj n Computation: For 1,2,... k 10 ()()()abnn OE iOE j ij y kaykibxkj a b) k ()()()OE OEekykyk Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an iO Eki l n jk j lkykiaklforin kxkjaklforjn ()[()()]T OE ijkkk (1)()()(OEOEkkek where is the step size. In practice: Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an ff OE OE kOE a l n ff OE kOE b l y kiykiaykilforin x kjxkjaxkjlforjn ()[()()] f ff OE OE OEkykixkj T) k (1)()()(f OEOEkkek 49 PAGE 50 Table 22. Summary of the IIR EE algorithm. Initialization: T ij(k)=a(0)b(0)= 0 where 1,...,,0,1,...,abinj n Computation: For 1,2,... k 10 ()()()abnn EE i j ij y kaykibxkj )k ()()()EE EEekykyk ()[()()]T EEkykixkj (1)()()(EEkkekEE where is the step size. 50 PAGE 51 Table 23. Summary of the IIR SM algorithm. Initialization: T ij(k)=a(0)b(0)= 0 where 1,...,,0,1,...,abinj n Computation: For 1,2,... k 10 ()()()abnn EE i j ij y kaykibxkj ()()()EE EEekykyk 1 11 () () aSM EE n i i iekek az Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an ff SM kSM a l n ff SM kSM b l y kiykiaykilforin x kjxkjaxkjlforjn ()[()()] f fT SM SM SMkykixkj (1)()()(SMkkekSM )k 51 PAGE 52 Table 24. Summary of the IIR CE algorithm. Initialization: T ij(k)=a(0)b(0)= 0where 1,...,,0,1,...,abinj n Computation: For 1,2,... k Step 1: 10 ()()()abnn OE iOE j ij y kaykibxkj ()()()OE OEekykyk Define: 1 1 ()()(),1,..., ()()(),0,1,...,a an ff OE OE kOE a l n ff OE kOE b l y kiykiaykilforin x kjxkjaxkjlforjn ()[()()] f ff OE OE OEkykixkj T Step 2: 10 ()()()abnn EE i j ij y kaykibxkj ()()()EE EEekykyk ()[()()]T EEkykixkj Step 3: ()()(1)()CE EE OEekekek ()()(1)() kkCE EE OE fk)k (1)()()(CEkkekCE where 0 1 52 PAGE 53 Table 25. Simulation results of IIR algorithms for sufficient case. Adaptive Structure Algorithm Parameters Initial Point 01 12 (0)(0) (0)(0) bb aa Number of iterations Final Point 01 12 ()() ()() bnbn anan OE 0.005 00 00 12000 [0.05,.4007] [1.13,0.2484] EE 0.01 00 00 8000 [0.05 .4] [1.131 0.2493] SM 0.005 00 00 11000 [0.05022,0.4006] [1.13,0.249] 0.005,0.04 00 00 12000 [0.050,0.4006] [1.13,0.2485] CE 0.01,0.60 00 00 8000 [0.04997,0.400] [1.131,0.2496] 53 PAGE 54 Table 26. Simulation results of IIR algorithms for insufficient case. Adaptive Structure Algorithm Parameters Initial Point 01 (0),(0) ba Number of iterations Global Min. 0.311,0.906 Final Point A 0.001 [0.5,0.1] 5500 [0.3098,0.8998] OE B 0.003 [0.5,0.2] 12000 [0.0928,0.4896] A 0.001 [0.5,0.1] 7000 [0.04577,0.8755] EE B 0.001 [0.11,0.52] 7000 [0.05003,0.8719] A 0.0005 [0.5,0.1] 3000 [0.3132,0.9039] B 0.0005 [0.5,0.2] 4500 [0.3062,0.8992] SM C 0.0005 [0.11,0.52] 5500 [0.2967,0.9031] A 0.001,0.04 [0.5,0.1] 5000 [0.3037,0.9112] CE B 0.003,0.04 [0.5,0.2] 19000 [0.315,0.9181] 54 PAGE 55 Table 27. Simulation conditions of IIR algorithms for sufficient case. Adaptive Structure Algorithm Parameters Initial Point 01 12 (0)(0) (0)(0) bb aa Number of iterations OE 0.005 00 00 12000 EE 0.005 00 00 12000 SM 0.005 00 00 12000 CE 0.005,0.04 00 00 12000 55 PAGE 56 Table 28. Summary of the IIR /LMS algorithms. Rank Order: (High or Good) (Low or Bad) AB C D Accuracy Convergent Rate Computational Complexity Robustness OE C D B C EE B A A A SM A B B C CE A C D C 56 PAGE 57 1Z 1Z 1Z Output y(k)Input x(k) c(k) ana2 a 1ana1 a bnb 1bnb 2 b 1 b 0 b Figure 21. Linear timeinvari ant (LTI) IIR Filter Structure. Input x(k) Test Model Additive White Noise v(k) Test Model Output y(k) Adaptive IIR Filter Optimization MethodError e(k)LTI Discrete FilterFilter Output y(k) Figure 22. Simulation structure of the adaptive IIR filter. 57 PAGE 58 Figure 23. zplane of the test model. Figure 24. 3D plot of the MSOE performance surface of the insufficient order test system. 58 PAGE 59 Figure 25. Contour plot of the MSOE performance surface. Figure 26. Simulation results of weight track of the IIR algorithms for sufficient case. 59 PAGE 60 Figure 27. Simulation results of weight track of the IIR algorithms for insufficient case. Figure 28. Learning curve of IIR algorithms for sufficient case. 60 PAGE 61 61 Figure 29. Computational comple xity results from the experiment. PAGE 62 CHAPTER 3 GENERALIZED PREDICTIVE CONTROL ALGORIHTM Equation Section 3 This Chapter describes the background of the generalized predictive control (GPC) algorithm. Then, the GPC algorithm is deve loped based on the MIMO system ID model (discussed in Chapter 2). Both batch method and recursive version are given and discussed. Introduction The generalized predictive cont rol (GPC) algorithm belongs to a family of the most popular model predictive control (MPC). The MPC algorithm is a feedback control method, different choices of dynamic models, cost functi ons and constraints can generate different MPC algorithms. It was conceived near the end of th e 1970s and has been widely used in industrial process control. The methodology of MPC is represented in Figure 31 where is the time index number, are the input sequences, and are the actual output sequences. The and are estimated output and reference signals, respectively. k()uk()yk ()yk()rykTwo comments are made here to describe all MPC algorithms. First, at each time step, a specific cost function is cons tructed by a series of futu re control signals up to and a series of future error signals, which are the differences between the estimated output signals and the reference signals uks ) ) (ykj (rykj Second, a series of future inputs are calculated by minimizing this cost function, and only the first input signal is provided to the system. At the next sampling interval, new valu es of the output signals are obtained, and the future control inputs are calculated again accord ing to the new cost function. The same computations are repeated. ()ukjSome important MPC algorithms, such as model algorithmic control (MAC), dynamic matrix control (DMC) and GPC, have become p opular in industry. MAC explicitly uses an 62 PAGE 63 impulse response model and DMC applies the step response process model in order to predict the future control signals (Camacho 1995). The GPC method, which is inherited from generalized minimum variance (GMV) (Clarke 1979), was propos ed and explained by Clarke (1987 a, b). The GPC algorithm is an effective selftuning pred ictive control method (Clarke 1988). It uses controlled autoregressive and integrated moving average (CARIMA) model to derive a control law and can be used in real time applications. Juang et al. (1997, 2001) gi ve the derivation of the adaptive MIMO GPC algorithm. Th is algorithm is an effective control method for systems with problems of nonminimum phase, open loop unstable plants or lightly damped systems. It is also characterized by good control performance a nd high robustness. Furthermore, the GPC algorithm can deal with the multidimension case and can easily be combined with adaptive algorithms for selftuning realtime applicati ons. The problem of flowinduced open cavity oscillations exhibit several theses issues, ther efore, the GPC is considered as a potential candidate controller. Two modifications are made for this algorithm. First, a input weight matrix is integrated into the cost function, this control matrix can put the penalty for each control input signal and further to tune the performance for each input channels. Second, a recursive version of GPC is developed for realtime control application. MIMO Adaptive GPC Model In this section, a MIMO model, which has th e same form of the MIMO ID algorithm, is considered. A linear and time invariant system with inputs r 1ru and outputs m 1my at the time index k can be expressed as 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuup (31) 63 PAGE 64 where 1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (32) Shifting j step ahead from the Equation 31 the output vector () kj y can be derived as (33) () () 11 () (1) 00 () () () 01()(1)(1) ()()(1) ()(1)()jj p j p jj j pkjk kp kpkjkj kkkp yyy yuu uuu where () (1)(1) () (1)(1) 0101 11 1 2 () (1)(1) () (1)(1) 21 2 311 1 2 () (1) (1)() 11 11 () (1) 1,jjj jjj mr mm jjjjj mm mr jjjj pp p p mm jj pp mm + + ++ + (1) (1) 11 () (1) 1 jj pp mr jj pp mr + j 0 1 1 p p (34) and with initial (0) (0) 110 (0) (0) 221 (0) (0) 11 1 (0) (0),mm mr mm mr pp p mm mr ppp mm mr (35) The quantities () 0k ( ) are the impulse response sequence of the system. Defining the following the vector form 0,1, k 64 PAGE 65 1 1 1() () (1)(1) () ,() (1) () () (1) () (1)pj rp rj p mpukp uk ukp uk kp k uk ukj ykp ykp kp yk ( 1 ) 1 uu y s (36) the predictive index and 0,1,2,,1,,1 jqq 11() () (1) (1) () () (1)(1)ss rs msuk yk uk yk k, k uks yks uy (37) Finally, the predictive model for future outputs, s y is obtained, this future outputs consists of a weighted summation of future inputs, s u, previous inputs, and previous outputs, pupy (38) ()()()()ssppkkkpkpyTuBuAywhere 0 (1) 00 (1)(2) 00000 0ss msrs T (39) 11 (1) (1) (1) 11 (1)(1) (1) 11 pp pp sss pp msrp B (310) 65 PAGE 66 11 (1) (1) (1) 11 (1)(1) (1) 11 pp pp sss pp msmp A (311) The detail derivation of the GPC model is given in Appendix B. MIMO Adaptive GPC Cost Function Assume the control inputs (present input and future inputs) depend on the previous inputs and output and can be expressed as (*)1(*)[*()] [*()]1() () ()p s sr srpmr p pmrkp k kp u uH y (312) Two potential cost functions are li st below. The first one consis ts terms of future outputs and a trace of the feedb ack gain matrix ()()()T ssJkkktrT y Q y HH (313) and the second definition of cost function based on the total energy of future outputs as well as the inputs 1 ()()()()() 2TT ssssJkkkkk yQyuRu (314) The output weight matrix Q, input weight matrix and the control horizon are important parameters for tuning the controller. The horizon is usually chosen to be several times longer than the rise time of the plant in order to ensure a stable fee dback controller (Gibbs et al. 2004). Also, if the predict horizon range is from zero to infinity, the resu lting controller approaches the steadystate linear quadratic regulator (Phan et al. 1998). Rs sMIMO Adaptive GPC Law In order to minimize the cost function, th ree approaches are considered as follows. 66 PAGE 67 Based on Equation 313 the control coefficients can be update using adaptive gradient algorithm. Based on Equation 314 the optimum solution can be derived. However, this method requires the calculation of a ma trix inverse, so the comput ational complex is higher. Based on Equation 314 the control coefficients can be updated using an adaptive gradient algorithm. The first approach is examined by Kegerise et al. (2004). In the next section, the latter two approaches are derived. MIMO Adaptive GPC Optimum Solution Based on the cost function 314, the goal is to find (*)[*()] s rpmrH or (*)1()s srku to minimize the cost function. We will show that both minimizing the cost function 314 respect to control matrix (*)[*()] s rpmrH and input vector (*()sk)1sr u will provide the same result. To simplify the expression, lets define [*()]1 [*()]1() ()p p pmr p pmrkp kp u v y (315) Substituting the predictive model 38 and control law 312 into the cost function 314 gives 1 ()()()()() 2 1 2 1 2TT ssss T sps ppJkkkkk p TyQyuRu TuBAvQTuBAv HvRHv (316) with some algebraic manipulation, the gradient of cost function respect to the control matrix (*)[*()] s rpmrH can be obtained. The optimum solution is obtained when the gradient equal to zero. 67 PAGE 68 () 0TT sp pp TT sp ps p TT sp ppJk T T TTTQyvRHvv H TQTuBAvvRuv TQT+RuvTQBAvv (317) thus, 1 s p opt TTuTQT+RTQBA v (318) Alternatively, from Equation 316 setting the gradient of the cost function with respect to the input vector (*)1()s srku, to zero gives ()T sp sJk TTuBAvQTuR u =0s (319) thus, 1 s p opt TTuTQT+RTQBA v (320) A comparison of Equation 320 to Equation 318 shows that these two approaches yield the same result. It is easy to apply the optim al solution of the Equation 320 on the cavity problem. However, the matrix inversion calculation has high computational complexity. Only if the model order is low enough, the optimal inpu t can be used in realtime application. MIMO Adaptive GPC Recursive Solution To avoid calculating the invers e of the matrix in Equation 320, the stochastic gradient descend method can be used to update the control matrix using the following algorithm H () Jk (k+1)(k) (k) HH H (321) 68 PAGE 69 Substituting Equation 312 into Equation 317 gives ()T pp T pp T ppJk (k) T T TTTQT+RHvv H TQBAvv TQT+RHTQBAvv (322) therefore, the recursiv e solution is given by T pp(k+1)(k) (k) TT H HTQT+RHTQBAvv (323) Since only present controls r 1()ruk are applied to the system, only the first rows in Equation r 323 are used first r rows T pph(k+1)h(k) (k) TTTQT+RHTQBAvv (324) In next chapter, this adaptive feedback controller, which is the combination of the MIMO system ID (discussed in Chapter 2) and the GPC algorithm, is implementation on a vibration beam test bed. The output weight matrix Q, input weight matrix R and the control horizon are tuning for testing their effects to the control performance. s 69 PAGE 70 70 k1 k 3 k 2 k ks 1 k 2 k () yk () yk ()ryk () uk Prediction Horizon PastFuturekj Figure 31. Model pred ictive control strategy. PAGE 71 CHAPTER 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES Equation Section 4 In this Chapter, the MIMO system ID (dis cussed in Chapter 2) algorithm and the GPC algorithm (discussed in Chapter 3) are implemente d on a vibration beam test bed. Since the objective idea of this sample experiment is similar to the flowinduced cavity oscillation, which is the disturbance rejection problem, the results of this vibration beam experiment will give us some insights to guide the later flow control applications of using this realtime adaptive control mythology. First, computational complexity of ID algorithm, ID results in time domain and frequency domain, and the distur bance effect for ID algorithm are examined. Then, the output and input weight matrices as well as the c ontrol horizon are tuned for testing the control performance with varies of these parameters. Schematic of the Vibr ation Beam Test Bed Figure 41 shows a detailed sketch of the whole vi bration control testbed setup. A thin aluminum cantilever beam with one piezoce ramic (PZT5H) plate bonded to each side is mounted on a block base and connected to an elec trical ground. The two piezoceramic plates are used to excite the beam by applying an electrical field across their thickness. The piezoceramic plate bonded to the right side of the beam is called the disturb ance piezoceramic because it is used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the left side of the beam is called the control piezocer amic because it is supplied with the controller output signal to counteract the unknown disturba nce actuator. The beam system has a natural frequency of about 97 Hz. The goal of the disturbance rejection controller is to mitigate the tip deflection of the aluminum beam generated by an external unknown disturbance signal. Th e controller tries to generate a signal to counteract the vibration of the aluminum beam generated by the disturbance 71 PAGE 72 piezoceramic. The performance signal and the fee dback signal of the controller are collocated, which is measured at the center of the tip of the beam by a laseroptical displacement sensor (Model MicroEpsilon OptoNCDT 2000). This device gives an output sensitivity of 1 / with a resolution of Vmm0.5 m and a sample rate of 10. The performance signal is filtered by a high pass filter (Model Kemo VBF 35) with kHz1cz f H to filter out the offset of the displacement sensor and then amp lified by a highvoltage amplifie r (Model Trek 50/750) with a gain of 10. dcThe disturbance and control signals ar e generated by dSPACE (Model DS1005) DSP system with 466MHz Motorola PowerPC micr oprocessor and amplif ied by two separate channels of the power amplifier by the same gain of 50. The types and conditions of the signals are discussed in detail in the next section. The dSPACE sy stem has a 5channel 16bit ADC (DS2001) and a 6channel 16bit DAC (DS2102) board. The signals are acquired using Mlib/Mtrace programs in MATLAB through the d SPACE system. The block diagram of the vibration beam test bed is shown in Figure 42 System Identification Experimental Results Computational Complexity During the realtime adaptive control of flow induced cavity oscillations, computational complexity is an important issue. Kegerise et al. (2004) use 80 order estimated model for the system ID and 240 prediction horizon for the re cursive GPC algorithm to capture the dynamics of the cavity system. Therefore, the computati onal complexity of the online adaptive controller have to reach or beyond these lengths of parameters. Figure 43 shows the changes of turn around time with the increasing estimated system order of the MIMO system ID algorithm. It is clear that the computational complexity of this algorithm is approximately linear, and the time 72 PAGE 73 requirement to estimate the same system order for the two inputs and two outputs (TITO) system is approximately three times longer than the requ irement of the SISO system. And both cases have enough turn around time fo r subsonic cavity experiment. System Identification Before identifying the parameters of the system, the system order has to be estimated. Using the ARMARKOV/LS/ERA algorithm (Akers et al. 1997; Ljing 1998) the eigenvalues of the triangle matrix calculated from singular va lue decomposition (SVD) of the vibration beam system is shown in Figure 44 This plot shows that the minimum reasonable estimated order of the system is 2. For different ID algorithms, the estimated system order may be different in practice. Therefore, the resulti ng identified system transfer func tion should be checked in order to match the experimental data shape in both time domain and frequency domain. A periodic swept sine signal ( Figure 45 ) is chosen as the ID input signal (without external disturbance). The sampling frequency is 1024, the sweep frequency produced by dSPACE system is from to 150, and the amplitude of the sweep sine signal is The performance of the system ID algorithm improves with increasing estimated system order. For this case, the estimated order of system ID block is set to 10. Hz 0 Hz Hz 0.25 voltFigure 45 shows that the output of the system ID algorithm matches the system output very well in the time domain. The coherence function ( Figure 46 ) shows good correlation between th e input and output signals. The zeropole location and the transfer func tion between the input and sensor output are shown in Figure 47 and Figure 48 respectively. Three system ID methods are used for comparison. Two batch methods calculate tran sfer function in frequency domain using the experimental data and the FRF method to fit th e frequency domain data One recursive method updates the system coefficients in realtime. Notice that all of these three methods give the 73 PAGE 74 similar shape in frequency domain and capture the two dominant poles of the system ( Figure 47). However, the FRF fit batch method gives a lower order model than the recursive method. Disturbance Effect External disturbance degrades the perfo rmance of the system ID algorithm. Figure 49 shows the vibration shim experiment system ID re sult with different extern al disturbance levels. A larger SNR (lower external disturbance level) in the input signal generally give more accurate identified system models. However, although the lower SNR input signal may result in a suboptimal system model, the closedloop cont rol implementation base d on this model still works well. The results are shown later. ClosedLoop Control Experiment Results Computational Complexity The controller design block is the most tim e consuming blocks in the entire adaptive control implementation. Estimated model order and the length of the predict horizon are two main parameters effecting the computational complexity. Figure 410 illustrates the computational complexity of the main C code Sfunction block which maps the observer Markov parameters (discussed in Chapter 2) to predict mo del coefficients (discussed in Chapter 3). The result shows that the turnaround time increases more quickly with increasing estimated system order than increasing of the prediction horizon. ClosedLoop Results The experimental parameters for th e closedloop control are list in Table 41 and some result plots are presented here. Figure 411 shows the sensor output signal in time domain, in which the control signal is initiated at time 0. 74 PAGE 75 Power spectra of open loop (base line) vs. cl osedloop sensor output and the closedloop sensitivity are shown in Figure 412 and Figure 413 respectively. The sensitivity function is define as 2 2() ()cl op y f s y f (41) Equation 41 provides a scalar measurement of distur bance rejection. A value less than one (negative log magnitude) indicates disturbance attenuation, while a value greater than one (positive log magnitude) indicates disturbance amplification. Although the res onance of the open loop system can be mitigated by the closedloop controller, a spillover phenomenon is also observed in Figure 413 As discussed in Chapter 1, the spillover problem is generated because, for this special case, the performance sens or output and the measurement sensor output (feedback signal) are collocated. Next, the effects of the adaptive GPC parameters are examined. Figure 414 shows the effect of the changes of the estimated model order. Figure 415 shows the effect of the changes of the predict horizon. Figure 416 shows the effect of the ch anges of the input weight. Figure 418 shows the effect of the different level of disturbance (SNR) during the system ID. The results for each case are discussed below. Estimated Order Effect In general, increasing the estimated order of the GPC, up to a certain point, can improve the performance of the closedloop control ( Figure 414 ). The experimental result shows that when the estimated model order is greater than 4, the closedloop controll er can not improve the performance any more. 75 PAGE 76 Predict Horizon Effect It is clearly see that increasing the predic t horizon can improve th e performance of the closedloop controller ( Figure 415 ). Input Weight Effect The input weight penalizes the magnitude of the input signal. For this experiment, saturation is given to the input signal to avoid the damage of the actuator. In order to restrict the input signal within the limits of th e saturation, the input we ight should be carefully tuned to obtain a realizable GPC. Although a sm aller input weight improve performance of the closedloop controller (0.75 volt Figure 416 ), it also generates a larger control signal ( Figure 417 ). Therefore, the tuning idea is to decrease the input weight as low as possible under the input saturation constraints. Disturbance Effect for Different SNR Levels During System ID As mentioned above, the level of the external disturbance sign al (different SNR) is an important issue for the accuracy of the system ID ( Figure 49 ). However, the adaptive closedloop controller gives the surprising results ( Figure 418 ). Three cases are examined and compared in this section. First, the open l oop (base line) case is th e power spectrum of the output measurement of laser sensor without any control input. Second, th e external disturbance is turned off during the system ID. Finally, the external disturbance is turned on with some level during the system ID. The result shows that th e higher disturbance leve l (low SNR) does not have a detrimental effect on the performance of th e closedloop system. In fact, the performance of the closedloop cont roller with low SNR is improved slightly. 76 PAGE 77 Summary Table 42 gives a summary of the experimental results of adaptive GPC algorithm. It can be seen that the GPC algorithm gives the better control performance with the larger estimated system order, the higher prediction horizon and the lower input weight. In Chapter 6, the similar control approach combining the system ID algorithm and the GPC algorithm will be implemented on the flowinduced cavity oscillations problem. Since the control ideas for both the vibra tion beam problem and the cavity oscillations problem are disturbance rejection, the successful implem entation of the system ID algorithm and GPC algorithm to the vibration beam test bed ma y give the guidance to the flow control. 77 PAGE 78 Table 41. Parameters selection of the vibration beam experiment. Fs Disturbance GPC White Noise Low Pass Filter (IIR Butterworth, 4th order) Input Weight Estimated Order Prediction Horizon 1024 Hz 0.09 Var 150c f Hz 1 10 10 78 PAGE 79 Table 42. Summary of the results of the adaptive GPC algorithm. Estimated Order Input Weight Prediction Horizon Integrated Reduction (In dB) Reduction at Resonance (In dB) 4 1 4 7.0 11.2 4 1 10 9.2 20.1 10 1 10 8.4 13.4 10 10 10 2.8 4.2 79 PAGE 80 PC MLIB/MTRACE DS1005 Processor Board DS2102 D/A Board OptoNCDT 2000 LaserOptical Displacement Sensor Kemo VBF35 Reconstruction Low Pass Filer Trek 50/750 Power Amplifier Aluminum Shim Disturbance Piezoceramic Control Piezoceramic Base Kemo VBF35 Antialiasing Low Pass Filer DS2001 A/D Board Disturbance Signal ID Input or Control Input Trek 50/750 Power Amplifier Figure 41. Schematic diagram of the vibration beam test bed. 16bit ADC LPF LPF AMP AMP System Sensor 16bit ADC System ID Controller Design Controller W(k) ID Input U(k) Y(k) Y(t) U(t) W(t) Actuator Analog Part Digital Part AMP BPF Figure 42. Block diagram of the vibration beam test bed. 80 PAGE 81 Figure 43. Computational complexity of the MIMO system ID. Figure 44. Eigenvalues of the triangle matr ix obtained by the SVD method of the vibration beam system. (Calculated by ARMARKOV/LS/ERA algorithms with 50 Markov parameters and estimated order of the denominator is 10). 81 PAGE 82 Figure 45. Input time series (top), system output and system ID algorithm output time series (bottom). Figure 46. Coherence function of system input and system output. 82 PAGE 83 Figure 47. Zeropole location of FRF (top) and system ID algorithm. The estimated order of FRF fit function is 2 and the estimated order of system ID algorithm is 10. Figure 48. Identified transfer function usi ng the experiment data by frequency response function (experiment), frequency response functi on fit(FRF fit) and tim e domain system ID algorithm (ID). The estimated orde r of FRF fit function is 2 and the estimated order of system ID algorithm is 10. 83 PAGE 84 Figure 49. Learning curve of syst em ID with different input SNR. The estimated system order is 10, sampling frequency is 1024 Hz. A Figure 410. Computational complexity of the ma in controller design C Sfunction. A) For SISO case. B) For TITO case. 84 PAGE 85 B Figure 410 Continued Figure 411. System output time series data. Th e control signal is intr oduced at 0 second, the estimated order is 10, predict horizon is 10, and the input weight is 1. 85 PAGE 86 Figure 412. Power spectrum of output signal wi th control and without control signal. The estimated order is 10, predict horizon is 10, and the input weight is 1. 0 50 100 150 1 0.8 0.6 0.4 0.2 0 0.2 Sensitivity Frequency (Hz)RMS Gain (in Log Scale) Figure 413. Sensitivity function of the system. Th e estimated order is 10, predict horizon is 10, and the input weight is 1. 86 PAGE 87 Figure 414. Power spectrum of output signals for different estimated order. Predict horizon is 10, and the input weight is 1. Figure 415. Power spectrum of output signals for different predict horizon. The estimated order is 4, and the input weight is 1. 87 PAGE 88 Figure 416. Power spectrum of output signals for different input weight. The estimated order is 10, predict horizon is 10. Figure 417. Control signals fo r different input wei ght. The estimated order is 10, predict horizon is 10. 88 PAGE 89 89 Figure 418. Power spectrum of output signals fo r different system ID disturbance conditions. The estimated order is 10, predict horiz on is 10, and the input weight is 1. PAGE 90 CHAPTER 5 WIND TUNNEL EXPERIMENTAL SETUP Equation Section 5 The experimental facilities and instruments used in this study are described in detail in this Chapter. These devices consist of a blowdown wind tunnel with a te st section and cavity model, unsteady pressure transducers, data acquisition systems, and a DSP realtime control system. Finally, the actuator used in this study is described. Wind Tunnel Facility The compressible flow control experiments ar e conducted in the University of Florida Experimental Fluid Dynamics Laboratory. A schematic of the supply portion of the compressible flow facility is shown in Figure 51 This facility is a pressuredriven blowdown wind tunnel, which allows for control of th e upstream stagnation pressure but without temperature control. The compressed air is generated by a Qu incy screw compressor (250 psi maximum pressure, Model 5C447TTDN7039BB). A desiccan t dryer (ZEKS Model 730HPS90MG) is used to remove the moisture and residual oil in the compressed air. The flow conditioning is accomplished first by a settling chamber. Th e stagnation chamber consists of a 254 mm diameter cast iron pipe supplied with the clean, dry compressed air. A computer controlled control valve (Fischer Contro ls with body type ET and Acuator Type 667) is situated approximately 6 meters upstream of the stagnation chamber with a 76.2 mm diameter pipe connecting the two. A flexible rubber coupler is located at the entrance of to the stagnation chamber to minimize transmitted vibrations from the supply line. The stagnation chamber is mounted on rubber vibration isol ation mounts. A honeycomb and two flow screens are located at the exit of the settling chambe r and the start of the contracti on section, respectively. The honeycomb is 76.2 mm in width (t he cell is 76.2 mm long) with a cell size of 0.35 mm. Two 90 PAGE 91 antiturbulence screens spaced 25.4 mm apart are used; these screens have open area and use 0.1 mm diameter stainless steel wire. For the current experiment, the facility was fitted with a subsonic nozzle that transitions from the 254 mm diameter circular crosssection to a square crosssection linearly over a distance of 355.6 mm. The profile designed for this contraction found in previous work provides good flow quality downstream of the contraction (Carroll et al. 2004). The overall area contraction ratio fr om the settling chamber to the test section is 19.6:1. For the present subsonic setup, the freestream Mach number can be altered from approximately 0.1 to 0.7, and the fa cility run times are approximately 10 minutes at the maximum flow rate due to the limited size of the two storage tanks, each with volume of 3800 gallons. 62%50.8 mm 50.8 mm Test Section and Cavity Model A schematic of the test section with an integrated cavity model is shown in Figure 52 The origin of the Cartesian coordinate system is s ituated at the leading edge of the cavity in the midplane. The test section connects the subs onic nozzle exit and the exhaust pipe with 431.8 mm long duct with a 50.8 m square cross section. m 50.8 mm The cavity model is contained inside this duct and is a canonical rectangular cavity with a fixed length of and width of 152.4 mm L 50.8 mm W and is installed along the floor of the test section. The dept h of the cavity model, can be adjusted continuously from 0 to This mechanism provides a range of cavity lengthtodepth ratios, D50.8 mm / L D, from 3 to infinity. The cavity model spans the width of the test section W. However, a small cavity width is not desirable, because the side wall bounda ry layer growth introduces threedimensional effects in the aft region of the cavity. As a result, the grow th of the sidewall boundary layers in the test section may result in modest flow acceleration. The boundary layers have not been 91 PAGE 92 characterized in this study. Ne vertheless, the cavity geometry a pplied in this study is consistent with previous efforts in the li terature (Kegerise et al. 2007a,b) considered to be shallow and narrow, so twodimensional longitudinal mode s will be dominant (Heller and Bliss 1975). Removable, optical quality plexiglas windows w ith 25.4 mm thickness bound either side of the cavity model to provide a full view of the cavity and the flow above it. The floor of the cavity is also made of 14 mm thick plexiglas for optical access. Two different wind tunnel cavity ceiling configurations are available. The first one is an aluminum plate with 25.4 mm thickness that can be considered a rigidwall boundary condition. This boundary condition helps excite the cavity vertical modes a nd the cuton frequencies of the cavity/duct configuration (Rowley and Williams 2006). The performance of this ceiling is discussed in the next chapter. In order to simulate an unbounded cavity fl ow encountered in practical bombbay configurations, a flushmounted acoustic treatment is construc ted to replace the rigid ceiling plate. The new cavity ceiling modifies the boundary conditions of th e previous sound hard ceiling. This acoustic treatment consists of a porous metal laminate (MKI BWM series, Dynapore P/N 408020) backed by 50.8 mm th ick bulk pink fiberglass insulation ( Figure 52 ). This acoustic treatment covers the whole cav ity mouth and extends 1 inch upstream and downstream of the leading edge a nd trailing edge, respec tively. This kind of acoustic treatment reduces reflections of acoustic waves. The perfor mance of this treatment is assessed in the next chapter. The exhaust flow is dumped to atmosphere via a 5 angle diffuser attached to the rear of the cavity model for pressure recovery. A custom rectangulartoround trans ition piece is used to connect the rectangular diffuser to the 6 inch diameter exhaust pipe. 92 PAGE 93 Three structural supports are used to reduce tunnel vi brations (Carroll et al. 2004). Two of these structural supports attach to both sides of the test section inlet flange, and the additional structural support is installed to support the iron exhaust pipe ( Figure 52 ). Pressure/Temperature Measurement Systems Stagnation pressure and temperature are monitored during each wind tunnel run and converted to Mach number via the standard isentropic re lations with an esti mated uncertainty of The reference tube of the pressure transducer is connected to static pressure port (shown in 0.01 Figure 52 ) using 0.254 mm ID vinyl tubing to measure the upstream static pressure of the cavity. The stagnation and static pressures are measured separa tely with Druck Model DPI145 pressure transducers (with a quoted measurement precision of 0.05% of reading). The stagnation temperature is measured by an OMEGA thermocouple (Model DP80 Series, with nominal resolution). 0.1 CTwo pressure transducers are located in th e test section to measure the pressure fluctuations. The first transducer is a flushmounted unsteady Kulite dynamic pressure transducer (Model XT19050A) and is an absolute transducer with a measured sensitivity V/Pa with a nominal 500 kHz natural frequency, Pa (50 psia) max pressure, and is 5 mm in diameter. This pressure transducer is located on the cavity floor (72.640.061053.44710 y D inch) 0.6 upstream from the cavity real wall ( x L ), and 8.89 mm ( ) away from the midplane. This position allows optical access from the midplane of cavity floor for flow visualization and avoids the possibility of coinciding with a pressure node along the cavity floor (Rossiter 19 64). The second pressure transdu cer is also an Kulite absolute transducer (with measured sensitivity 8.89 mm z 5.130.03107 V/Pa and nominal 400 kHz natural frequency, Pa (25 psia) max pressure, 5 mm in di ameter), and it is flush mounted in 51.72410 93 PAGE 94 the tunnel side wall 63.5 mm downst ream of the cavity as shown in Figure 52 From a series of vibration impact tests performed in a previous st udy (Carroll et al. 2004), the results indicated that the pressure transducer outputs are not a ffected by the vibration of the structure. An experiment to validate this hypothesis is discussed in the next Chapter. Due to a modification of the experimental setup, the second pressure sensor is moved to the cavity floor ( Figure 52 ) for both openloop control and closedloop control. A PC monitors the upstream Mach numb er, stagnation pressure, and stagnation temperature, as well as the static pressure. This computer is also used for remote pressure valve control ( Figure 51 ) in order to control the freestream M ach number using a PID controller. In addition, an Agilent E1433A 8channel, 16bit dynamic data acquisition system with builtin antialiasing filters acquires the unsteady pre ssure signals and communicates with the wind tunnel control computer via TCP/IP for synchroni zation. The code for both data acquisition and remote pressure control output generation are pr ogrammed in LabVIEW. The pressure sensor timeseries data are also collected for both the ba seline and controlled cavity flows for posttest analysis. Facility Data Acquisition and Control Systems The schematic of the control hardware setup is shown in Figure 53 For the realtime digital control system, the voltage signals from the dynamic pressure transducers are first preamplified and lowpass filtered using Kemo Model VBF 35. This filter has a cutoff range 0.1 Hz to 102 kHz, and three filter shap es can be used. Option 41 with nearly constant group delay (linear phase) in the pass band a nd 40dB/octave rolloff rate is c hosen. The cutoff frequency is 4 kHz for a sampling frequency of 10.24 kHz. The si gnal is then sampled with a 5channel, 16bit, simultaneous sampling ADC (dSPACE Model DS2001). 94 PAGE 95 The control algorithms are coded in SIMULINK and C code Sfunctions and are compiled via Matlab/RealTime Workshop (RTW). These codes are uploaded and run on a floatingpoint DSP (dSPACE DS1006 card with AMD Opteron Pr ocessor 3.0GHz) digital control system. The DSP was also used to collect input and out put data from the DS2001 ADC boards as well as computing the control signal once per time step. At each iteration, the computed control effort is converted to an analog signal accomplished us ing a 6channel 16bit DAC (DS2102). This signal is passed to a reconstruction filter (Kem o Model VBF 35 with identical settings to the antialias filters) to smooth the zeroorder hold si gnal from the DAC. The output from this filter is then sent to a highvoltage amplifier (PCB Model 790A06) to produce the input signal for the actuator. The computer is also able to access th e data with the dSPACE system via the Matlab mlib software provided by dSPACE Inc. Actuator System In order to achieve effective closedloop flow control, high bandwidth and powerful (high output) actuators are required. The following issues should be considered for selecting the actuators (Schaeffler et al. 2002). The selected actuators must produce an output consisting of multiple frequencies at any one instant in time. The bandwidth of the actuators should enable control of all signifi cant Rossiter modes of interest. The control authority must be large enough to counteract the natural disturbances present in the shear layer. According to Cattafesta et al (2003), one kind of actuator called Type A has these desirable properties. Such actuators include piez oelectric flaps and have successfully been used for active control of flowinduced cavity oscillations by Cattafe sta (1997) and Kegerise et al. (2002). Their results show that the external flow has no significant in fluence of the actuator 95 PAGE 96 dynamic response over the range of flow condition s. Their later work (Kegerise et al. 2004; 2007a,b) also shows that one bimorph piezoelectri c flap actuator is capable of suppressing multiple discrete tones of the cavity flow if the modes lie within the bandwidth of the actuator. Therefore, the piezoelectric bimorph actuator is a potential candidate for the present cavity oscillation problem. Another candidate actuator is the synthetic or zeronet massflux jet (Williams et al. 2000; Cabell et al. 2002; Rowley et al. 2003, 2006; Caraballo et al 2003, 2004, 2005; Debiasi et al. 2003, 2004; Samimy et al. 2003, 2004; Yuan et al. 2005) This actuator can be used to force the flow via zeronetmass flux pertur bations through a slot in the upstream wall of the cavity. Although the actuator injects zeronetmass through the slot during one cycle, a nonzero net momentum flux is induced by vor tices generated via periodic bl owing and suction through the slot. In this research, a piezoelectricdriven synthetic jet actuator array is designed. This type of synthetic jet based actuators normally gives a larger bandwidth than the piezoelectric flap type of actuators. A typical commercial parallel operation bimorph piezoel ectric disc (APC Inc., PZT5J, Part Number: P412013TJB) is used for this design. The physical and piezoelectric properties of the actuator material are listed in Table 51 The composite plate is a bimorph piezoelectric actuator, which includes two piezoelectric patche s on upper and lower sides of a brass shim in parallel operation ( Figure 54 ). The final design of the actuator array consists of 5 single actuator units. Each actuator unit contains one co mposite plate and two re ctangle orifices shown in Figure 55 The designed slot geometries fo r the actuator array are shown in Figure 56 Another advantage of this design is that it avoids the pressure imbalance problem on the two sides of the diaphragm during the experiment Since the two cavities on either side of a 96 PAGE 97 single actuator unit are vented to the local sta tic pressure, the diaphragm is not statically deflected when the tunnel static pressure deviates from atmosphere. The challenge is whether these actuators can provide strong enough jets to alter the shear layer in stabilities in a broad Mach number range and also whether the actuator s produce a coherent signal that is sufficient for effective system identification and control. A lumped element actuator design code (Galla s et al. 2003) was used together with an experimental trialanderror method to design the single actuator unit. The final designed geometric properties and parameters of the single actuator unit are listed in Table 52 To calibrate this compact actuator array, the centerl ine jet velocities from each slot are measured using constanttemperature hotwire anemomet ry (Dantec CTA module 90C10 with straight general purpose 1D probe model 55p11 and stra ight short 1D probe support model 55h20). A Parker 3axis traverse system is used to positi on the probe at the center of actuator slots. The sinusoidal excitation signal from the Agilent 33120A function generato r is fed to the 790A06 PCB power amplifier with a consta nt gain of 50 V/V. The piezoceremic discs are driven at three input voltage levels: 50 Vpp, 100 Vpp, and 150 V pp, respectively, over a range of sinusoidal frequencies from 50 Hz to 2000 Hz in steps of 50 Hz. Each bimorph disc serves as a wall between two cavities labeled side A and side B. The notation used to identify each bimorph and its corresponding slots is shown in Figure 57 The rms velocities of the slots 3A and 3B located in the centerline of the cavity are shown in Figure 58 as an example. The maximum centerline velocities measured at the three excitation voltages for each slot are listed in Table 53 A summary of the measurements of the centerline velocities and cu rrents to the actuator array for each slot are provided in Appendix E. The pi ezoelectric plate is tested over a range of frequencies and amplitudes to determine the curre nt saturation associated with the amplifier. 97 PAGE 98 Figure 59 shows the simulation result calculated by the LEM actuator design code and is superposed on the experimental result, Figure 58 The results show that, the LEM actuator design code provides a pretty accurate rms veloci ty estimation of synthetic jet over a large frequency range between 50 Hz and 2000 Hz. Finally, the measure input current level to the actuator array after the amplifier is measured. The results are shown in Figure 510 and indicate that the input current will satu rate above 136mApp, which means if the input voltage is larger than 100 Vpp, the current to the actuator will ke ep a constant value. During the closedloop experiments, an upper limit of 150 Vpp is used since the current pr obe is unavailable. Figure 511 shows the spectrogram of the pressure measurement on the cavity floor with acoustic treatment. The Rossiter modes (Equation 12 ) with =0.25, =0.7 are superimposed on this figure. The experimental deta ils are explained in the next chap ter. For this dissertation, the lower portion of the Mach number range (from 0.2 to 0.35) is our control target as an extension to previous work by Kegerise et al. (2007a,b). The desired ba ndwidth of the designed actuator should cover the dominant peaks of Rossiter mode 2, 3 and 4, which is between 500 Hz and 1500 Hz. (Rossiter mode 1 is usually weaker compar ed to Rossiter modes 2, 3 and 4.) Over this frequency range, the designed actua tor can generate larg e disturbances. In addition, the array produces normal oscillating jets that seek to pene trate the boundary layer, resulting in streamwise vortical structures. In essence, it acts like a vi rtual vortex generator. A simple schematic of the actuator jets interacting with the fl ow vortical structures is shown in Figure 512 The approach boundary layer contains spanwise vorticity in the xy plane (the coordinate is shown in Figure 52 ). By interacting with the ZNMF actuator jets, the 2D shape of the vortical structures transform to a 3D shape with spanwise vortical structures. These streamwise vortical disturbances seek to destroy the spanwise coherence of the shear layer, and the corresponding Rossiter modes are 98 PAGE 99 disrupted (Arunajatesan et al. 2003 ). Alternatively, th e introduced disturbances may modify the stability characteristics of the mean flow, so th at the main resonance peaks may not be amplified (Ukeiley et al. 2003). Unfortunate ly, the flow interaction was not characterized in this dissertation and will be addressed in future work. Instead of using one specific amplitude and one frequency in openl oop control, a closedloop control algorithm is used in this study to ex am the effects of the di sturbance with multiple amplitudes and multiple frequencies. Thus, the present actuator represents a hybrid control approach, in which we seek to reduce both the Rossiter tones and the broadband spectral level. 99 PAGE 100 Table 51. Physical and piezoelectric proper ties of APC 850 device. Shim (Brass) Piezoceramic Bond Elastic Modulus (Pa) 108.96310 10510 83.9810 Poissons Ratio 0.324 0.31 0.3446 Density ( ) 3/ kgm 8700 7400 1060 Relative Dielectric Const. 2400 31d (m/V) 1220010 Maximum Voltage Loading 200 Vpp/mm for 0.15 mm thickness is 30 Vpp Resonant Resistance () 200 Electrostatic Capacitance (pF) 210,00030% Operating Temp. (C) 20~70 100 PAGE 101 Table 52. Geometric properties and parameters for the actuator. Geometric Properties of the Diaphragm APC PZT5J, P412013TJB Piezo. Configuration Bimorph Disc Bender Shim Diameter (mm) 41 Clamped Diameter (mm) 37 Shim Thickness (mm) 0.1 Piezoceramic Diameter (mm) 30 Piezoceramic Thickness (mm) 0.15 Ag Electrode Diameter (mm) 29 Total Bond Thickness (mm) 0.03 ( 0.015 on each side) Radius (mm) 0a 1/2 Length of the Orifice (mm) L 1 Width of the Orifice (mm) dw 3 Volume in side A ( ) 3mm 4064 Volume in side B ( ) 3mm 2989 Damping 0.04 101 PAGE 102 Table 53. Resonant frequencies with respec tive centerline velocities for each input voltage. Input Voltage 50 Vpp 100 Vpp 150 Vpp Slot Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) Fres (Hz) Vcenter (m/s) 1A 1150 35. 1100 58 1150 62 1B 1150 44 1100 69 1150 74 2A 1150 27 1100 47 1150 52 2B 1150 37 1100 60 1150 66 3A 1150 27 1100 45 1150 49 3B 1150 38 1100 60 1150 65 4A 1150 36 1150 58 1200 62 4B 1150 44 1150 68 1200 73 5A 1150 40 1150 60 1150 65 5B 1150 49 1100 72 1150 78 102 PAGE 103 Pressure Valve Settling Chamber Honeycomb Screens Manual Valve Subsonic Nozzle Connect to Test Section Figure 51. Schematic of the wind tunnel facility. Connect to Nozzle Cavity Model Perforated Metal Plate Fiberglass Dynamic Pressure Sensor P1 Static Pressure Port Dynamic Pressure Sensor P2 x y Exhaust Structural Support Inlet Structural Supports Unit: Inch Figure 52. Schematic of the test section and the cavity mode l. Dimensions are inches. 103 PAGE 104 16bit ADC Actuator Array Power Amplifier Fc Cavity FlowReconstruction Filter FcAntialiasing Filter 16bit ADC Controller dSPACE System Fs=10.240 kHz Fc = 4 kHz Fc = 4 kHz Gain = 50x Figure 53. Schematic of th e control hardware setup. pt s t Shim Piezoceramic acVp R s R ++++++ ++++++ Figure 54. Bimorph bender disc actuator in paralle l operation. The physical and geometric properties are shown in Table 51 and Table 52 104 PAGE 105 Side A Side B A B C D Figure 55. Designed ZNMF actua tor array. A) Operation plot. B) Assembly diagram of single unit. C) Singe unit of the actuator. D) Actuator array. 105 PAGE 106 Figure 56. Dimensions of the sl ot for designed actuator array. 106 PAGE 107 Cavity Floor Flow Direction Figure 57. ZNMF actuator array mounted in wind tunnel. 107 PAGE 108 0 500 1000 1500 2000 0 10 20 30 40 50 Frequency [Hz]Velocity [m/s] 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s] 50 Vpp 100 Vpp 150 Vpp B Figure 58. Bimorph 3 centerlin e rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid inpu t signal. A) For side A. B) For side B. 108 PAGE 109 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 10 20 30 40 50 60 70 80 Frequency (Hz)Jet Velocity Vout (m/s) Sim 50 Vpp Exp. 50 Vpp Sim 100 Vpp Exp. 100 Vpp Figure 59. The comparison plot of the experime nt and simulation result of the actuator design code for bimorph 3. The output is the cen terline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal for side B. 109 PAGE 110 200 400 600 800 1000 1200 1400 1600 1800 2000 0 20 40 60 80 100 120 140 160 Frequency (Hz)Current (mApp) 50 Vpp 100 Vpp 150 Vpp Figure 510. Current saturati on effects of the amplifier. 110 PAGE 111 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 70 80 90 100 110 120 130 140 150 Ri Figure 511. Spectrogram of pressure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7). 111 PAGE 112 112 x Y x z Y Leading Edge of the Cavity Figure 512. Schematic of a single periodic cell of the actuator jets and the proposed interaction with the incoming boundary layer. PAGE 113 CHAPTER 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION Equation Section 6 Experimental results for the baseline unc ontrolled and controlled cavity flows are presented in this chapter. Firs t, the effects of structural vibrations on the unsteady pressure transducers are illustra ted. Then, a joint timefrequency analysis of the unsteady pressure measurement for an uncontrolled cavity flow is sh own. Flowacoustic features are deduced from the results. An improved test environment is established by re placing the original hardwall ceiling of the wind tunnel with an acoustic liner. This new test section minimizes the effects of the vertical acoustic modes. Finally, the results of both openloop and adaptive closedloop control experiments using the ZN MF actuator array is presented and discussed in detail. The ability of the actuator to alter both broadband and tonal content of the uns teady pressure spectra is demonstrated at low Mach numbers. Background As discussed in the first Chapter, flowinduced cavity oscillations are often analyzed via unsteady pressure measurements in and around th e cavity. However, these measurements are often contaminated by other dynamics associated with the specific characteristics of the wind tunnel test section. As a re sult, the unsteady pressure spec trum may be due to the cavity oscillations or other phenomena. The experimental results of Cattafesta et al. (1999), De biasi and Samimy (2004), and Rowley et al. (2005) show that some of the re sonant frequencies measured within the cavity track or lock on to vertical acoustic duct modes at some test conditions. This effect can be reduced by adding acoustic treatment at the ceiling above the mouth of the cavity (Cattafesta et al. 1998; Williams et al. 2000; Ukeiley et al. 2003; Rowley et al. 2005). This acoustic treatment modifies the soundhard boundary condition and thus mitigates the contribution of the cavity 113 PAGE 114 vertical resonance modes to th e unsteady pressure measurements Consequently, the modified cavity model will ideally exhibi t the behavior of an unbounded cav ity flow and be dominated by Rossiter modes. Alvarez et al. (2005) developed a theoretical prediction method and showed that the wind tunnel walls lead to a significant increase in the growth rate of a resonant mode for frequencies near the cuton fre quency of a crossstream mode. In the present baseline (i.e., uncontrolled) expe rimental study, flowacoustic resonances in the test section region and in the cavity region are examined. A schematic of the simplified wind tunnel model and the cavity region of th e experimental setup are shown in Figure 61 Using the same nomenclature of Alvarez et al. (2005), the domain is divided into three regions: an upstream tunnel region (0 x ), a cavity region (0 x L ), and a downstream region ( x L ). The Rossiter modes ( ...iRi 0,1, ) are the combined result of a receptivity process at instability growth in the unstable shear layer, sound gene ration due to impingement of the shear layer at 0 x x L and upstream and downstream propaga ting acoustic waves within the cavity region. The resulting flow osc illations are in teresting targets for fluid dynamics and control researchers to analyze and mitigate. Additional vertical cavity acoustic modes ( 0,1,...iVi ) and cavity cuton modes ( ) can also be present, as discussed a bove. These acoustic modes are generated by the reflections from the ceiling and area changes of the cavity model. During the wind tunnel experiments, the vertical modes ( ) are undesirable and should be reduced in order to mimic the unbounded bomb bay problem more accurately. 0,1,...iCiiV As explained in Alvarez et al. (2005), the upstream region can support duct cuton modes ( ) and upstream propagating duct modes ( ) due to the acoustic iu uD0,1,...i,0,1,...u iTi 114 PAGE 115 scattering process. Similarly, the down stream region can support duct cuton modes ( ) and downstream propagating tunnel modes ( ) due to the scattering process. Here, we focus our attention on the propagating modes in the cavity and the downstream tunnel regions. id uD0,1,... i,0,1,...d iTiHzData Analysis Methods A schematic of unsteady pressure transducer locations for this study was presented in Chapter 5 (Figure 52). and measure the unsteady pressure fl uctuations in the cavity and downstream regions, respectively. 1P2P The cavity and wind tunnel acoustic modes can be obtained experimentally using two approaches. One way is to measure the output of each unsteady dynamic pressure sensor for different fixed freestream Mach numbers and then find the spectral peaks for each discrete Mach number. However, with this method it is difficu lt to track the gradual frequency changes with Mach number. The other choice is to record each unsteady pressure se nsor output continuously as the Mach number is increased gradually over th e desired range. Then, a jointtime frequency analysis (JTFA) (Qian and Chen 1996) is applied to these recorded pressure time series data. JTFA provides information on the measurement in both the time and frequency domains. Finally, the time axis is converted to Mach num ber via synchronized measurements of the Mach number versus time. Similar analysis methods can be found in Cattafesta et al. (1998), Kegerise et al. (2004), and Rowley et al. (2005). In this study, the samp ling frequency for experimental data collection is 10.24 and the frequency resolution is 5 The cutoff frequency of the antialiasing filter is 4 and 500 continuous blocks of time series data are used in the analysis. During the experiment, Mach number sweeps from 0.1 to 0.7 in about 100 seconds. kHzkHz 115 PAGE 116 Noise Floor of Unsteady Pressure Transducers The effective insitu noise floor of the two uns teady pressure transduc ers is presented in Figure 62 Each noise floor measurement is compar ed with the spectra obtained at different discrete Mach numbers for the acoustically treated L/D=6 cavity. Within the tested frequency range, the signaltonoise ratio ( ) is in excess of 30 which demonstrates adequate resolution of unsteady pressure tr ansducers for the present experiments despite their large fullscale pressure ranges. SNR dBEffects of Structural Vibrations on Unsteady Pressure Transducers A series of initial impulse impact tests are pe rformed before the baseline experiments. As discussed in Chapter 5, with the wind tunnel turn ed off, the pressure transducer outputs are not affected by hammeror shakerinduced structural vibr ations. A simple test is described here to investigate the effects of structural vibrations while the wind tunnel is running. To avoid confounding cavity oscillations, the cavity floor is mounted flush with the tunnel floor (0 D ). A piezoceramic accelerometer (P CB Piezotronics Model 356A16) is used to measure the structural vibrations. It is attached to the test section outer wall us ing wax at the location indicated in Figure 52 which is close to one of the pressure transducers ( ). This piezoceramic accelerometer is connected to a multichannel signal conditioner (PCB Piezotronics Model 481A01). Three channels of the piezoceramic accelerometer corresponding to the 2P x and directions are measured. Th e coordinate directions of y z x and are shown in yFigure 52 and is the corresponding late ral direction using the righthand rule. The accelerometer is calibrated with a reference shaker (PCB Piezotronics, Model: 394C06) that provides 1 (rms) at 1000 rad/s (159.2 ). zg Hz 116 PAGE 117 The JTFA results ( Figure 63 to Figure 65 ) for all components of the accelerometer measurements show that the power of the structural vibration spreads is broadband with a few spectral peaks. A modest peak at 1000 is present in the lateral ( ) and vertical ( Hz z y ) directions. In addi tion, some higher frequency peaks (i.e., 2450 in the Hz x or flow direction, 1880 in the lateral direction and 3200 in the vertical direction) can also be detected. However, the JTFA results of (Hz Hz1P Figure 66 ) and do not display any of these resonances. These results confirm that the unsteady pressure transducers are not affected by structural vibrations. 2PBaseline Experimental Results and Analysis The rigid ceiling plate (no acoustic treatment) ab ove the mouth of the cavity is considered first. JTFA results of the unsteady pressure tr ansducer measurement for this case are shown in Figure 67 Numerous flowacoustic res onances can be observed in the plots. For easy reference in the subsequent discussion, thes e features are numbered 1 and 2. The final goal of the baseline experiment is to simulate the unbounded weapon bay using the cavity model in the test section. Therefore, the active flow control scheme targets the Rossiter modes (feature 1 in Figure 67 ). The other unknown acoustic features 2 in Figure 67 are undesirable features that we wish to eliminate. These acoustic modes come from the bounded wind tunnel walls, the mismatched acoustic impedance due to area change, and th e leading and trailing edges of the cavity. In order to better mimic an unbounded cavity flow in a closed wind tunnel, the boundary condition of the cavity ceiling must be altered to eliminate the unexpected modes within the cavity region. A flushmounted ac oustic treatment (discussed in Chapter 5) is fabricated to replace the previous solid tunnel ceiling. Th e new cavity ceiling modifies the zero normal velocity boundary condition of the previous sound hard top plate. 117 PAGE 118 The unsteady pressure transducer JTFA meas urement for the trailing edge floor of the cavity is shown in Figure 68 The results illustrate a very clea n flow field below Mach 0.6. The acoustic features 24 in Figure 67 are eliminated within the ca vity region. Therefore, the experimental Rossiter modes i R shown in JTFA plot ( Figure 68 ) now follow the estimated Rossiter curves. At higher upstream Mach numbers ( ), the experimental Rossiter modes deviate slightly from the expected Rossiter curves This is partly because the estimated curves use the upstream static temperatur e to calculate the speed of sound. This estimation does not account for the expected significant static temp erature drop due to the large flow acceleration near the aft cavity region seen by Zhuang et al. (2003). Another pos sible reason for these deviations of the flowacoustic resonance comes from the structural vibr ation coupling with the Rossiter modes. At high Mach numbers above 0.6, the structural vibratio ns may cause a lockon phenomenon with the Rossiter modes. For this study, all experiments are thus performed below M = 0.6. 0.6 M In conclusion, the observed flowacoustic behavi or of the acoustically treated cavity model behaves as expected below M = 0.6 and is ther efore suitable for application of openloop and closedloop flow control. OpenLoop Experimental Results and Analysis The openloop and closedloop experimental resu lts using the designed actuator array are shown in this section. Before the control experi ments, measurements of the pressure sensor at the surface of the trailing edge of the cavity with the without the actuator turned on are shown in Figure 69 Without the upcoming flow, the noise floor shows a significant peak at 660 Hz and a small peak at 2000 Hz. The pressure sensor can also sense the acoustic di sturbances associated with the excitation frequency a nd its harmonics, and the measured unsteady pressure level can 118 PAGE 119 reach 115120 dB. The extent to which the measured levels deviate from theses values with flow on (considered below) indicates the relative impact of the actuator on the unsteady flow. First, openloop active control is explored. The purpose of the openloop experiments is to verify if the synthetic jets generated from the designed actuator array can affect and control the cavity flow. A parametric study for the openloop c ontrol is explored first. A sinusoidal signal is chosen as the excitation i nput with the frequency swept from 500 Hz to 1500 Hz. The openloop experimental results are shown in Appendix F. The opencontrol performance is best over the frequency range 1000 Hz to 1500 Hz, which co rresponds to the resonan ce frequencies of the actuator array. Since at the res onance frequency, the actuator arra y can generate larger velocity jet, and the blow coefficient /()c cavityBmUA increases. As a result, the control effect increases. For these openloop tests, the upstream flow Mach number is varied from 0.1 to 0.4. For illustration purposes, results are examined here for two sinusoidal signals with 200 Vpp and excitation frequencies at either 1.05 kHz or 1.5 kHz to drive the actuato r array. The 1.05 kHz excitation frequency is close to the resonance frequency of the actuato r, while the 1.5 kHz frequency lies between the second and third Ross iter modes. The experime ntal results shown in Figure 610 illustrate that this actuator array can successfully reduce multiple Rossiter modes, particularly at Mach number 0.2 and 0.3. In addition, the pressure fluctuation is mitigated at the broadband level on the surface of the cavity floor for all the tested flow conditions. However, new peaks are generated by the excitation freque ncies and their harmonics, especially at low Mach number 0.1. With increasing upstream Mach number, the unsteady pressure level also increases and the effect of the control is reduced Note the synthetic jets introduce temporal and spatial disturbances to modify the mean flow in stabilities and destroy the coherence structure in 119 PAGE 120 spanwise, respectively. The eff ectiveness of the actuato r scales with the mo mentum coefficient, which is inversely proportional to the square of the freestream velocity. So, as the upstream Mach number increases, the synthetic jets are eventually not strong e nough to penetrate the boundary layer and the control e ffect is reduced. Future work should perform detailed measurements to validate this hypothesis. The results of the openloop cont rol suggest that this kind of actuator array can generate significant disturbances not only along the flow propagation di rection but also in the spanwise direction of the cavity. The combination of thes e effects disrupts the KelvinHelmholtz type of convective instability waves, which are the source of the Rossiter modes. As a result, multiple resonances are reduced via active control. The experimental results also show the limitation of the openloop control. ClosedLoop Experimental Results and Analysis The openloop control results sugg est that this compact actuator array may be effective for adaptive closedloop control. As discussed above, the synthetic jets add di sturbances to disrupt the spanwise coherence structure of the shear layer and result in a broadband reduction of the oscillations. However, at the same time, the coherence between the driv e signal and the unsteady pressure transducer will be redu ced. High coherence is considered essential for accurate system identification methods. To exam the accuracy of th e system ID algorithm w ith the change of the estimated order, an offline system ID analysis is first performed. The nominal flow condition is chosen at M = 0.275 (to match that of Kegerise et al. 2007a,b) with a L/D=6 cavity, and two system ID signals, one with white noise (bro adband frequency and amplitude 0.29 Vrms ) and the other with a chirp signal (amplitude 0.86 Vrms and fL = 25 Hz to fH = 2500 Hz in T = 0.05 sec), are used as a broadband excitation sour ce to identify the system. The running error variances the system ID are shown in Figure 611 It is clear that the larger the estimated order, 120 PAGE 121 p, the more accurate is the system ID algorithm However, due to the limitations of the DSP hardware, we cannot choose very large values of the estimated order for system ID algorithm online. One potential advantage of the closedloop adap tive control algorithm is that it does not rely exclusively on accurate system ID. Figure 612 shows the result of the closedloop realtime adaptive system ID together with the GPC contro l algorithm for an upstream Mach number 0.27. Based on the above system ID results, due to th e DSP hardware limitation, the estimated GPC order and the predictive horizon are chosen as 14 and 6, respectively. The breakdown voltage of the actuator array restricts the excitation voltage level; therefore, the diagonal element of the input weight penalty matrix R (Equation 314 ) is chosen as 0.1. This research represents an extension of Kegerise et al (2007b) where the system ID algorithm and the closedloop controller design algori thm are used simultaneously in a realtim e application. It is important to note that only the system ID white noise or chirp signal is used to identify the openloop dynamics, and the feedback signal is not used for th is purpose. Clearly, the results show that the GPC controller can generate a seri es of control signals to drive the actuator array resulting in significant reductions for the second, third, and fourth Rossiter modes by 2 dB, 4 dB, and 5 dB, respectively. In addition, th e broadband background noise is also reduced by this closedloop controller; the OASPL reduction is 3 dB. The input signal is shown in Figure 613 The sensitivity function discussed in Chapter 4 (Equation 41) is shown in Figure 614 A negative amplitude value indicates disturbance attenuation, while a positive value indicates disturbance amplification. The results show that all the points are negative, which indicates the closedloop controller reduces the pressure fluctu ation power at all frequencies. The spillover phenomenon (Rowley et al. 2006) is not observed in Figure 614 As discussed in Chapter 1, the 121 PAGE 122 spillover problem is generated because either th e disturbance source and control signal or the performance sensor output and the measurement sens or output (feedback signa l) are collocated. The Bodes integral formula is shown in 61 0log()Re()k kSidp (61) where k p are the unstable poles of the loop gain of the closedloop system. So, for a stable system, any negative area at the left hand side of the Equation 61 must be balanced by an equal positive area at the left hand side of the Equation 61 However, for present closedloop control study, the left hand side of the B odes integral formula is 38 ra d/sec, which shows that Bodes integral formula does not hold here. Since this formula is valid for a linear controller, the combination of the adaptive system ID and cont roller is apparently nonlinear. A more detailed study is required in the future to validate this hyothesis. A parametric study of the GPC is then studi ed by varying the es timated order and the predictive horizon. Figure 615 and Figure 616 show that the control effects improve with increasing order and predictive horizon. This trend matches the simulation results shown in Chapter 4. The comparison between the openloop a nd closedloop results is shown in Figure 617 for the same flow condition. Notice that the base line measurement for a same flow condition can vary a little from case to case. The openloop control uses a sinusoida l input signal at 1150 Hz forcing and 150 Vpp and the rms value of the in put is 53V. The closed loop control uses the estimated order 14, the predictive horizon 6, and the input weight 0.1, and the input rms value is 43 V. 122 PAGE 123 iu uDid uDi R iViCd iTu iT x y Cavity Region Downstream Region Upstream Region H L D Figure 61. Schematic of simplified wind tunne l and cavity regions acoustic resonances for subsonic flow. 0 1000 2000 3000 4000 5000 6000 70 80 90 100 110 120 130 140 150 frequency [Hz]Unsteady Pressure Level [dB] with Pref=20e6 Pa 0.4 0.5 0.55 0.58 0.6 0.65 0.69 Noise Floor Noise Floor Figure 62. Noise floor level comparison at different discrete Mach numbers with acoustic treatment at trailing edge floor of the cavity with L/D=6. 123 PAGE 124 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 50 60 70 80 90 100 Figure 63. x acceleration unsteady power spectrum (d B ref. 1g) for case with acoustic treatment and no cavity. 124 PAGE 125 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 30 40 50 60 70 80 90 100 Figure 64. y acceleration unsteady power spectrum (dB ref. 1g) for case with acoustic treatment and no cavity. 125 PAGE 126 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 40 50 60 70 80 90 100 Figure 65. acceleration unsteady power spectrum (dB ref. 1g) for case with acoustic treatment and no cavity. z 126 PAGE 127 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 90 100 110 120 130 140 Figure 66. Spectrogram of pre ssure measurement (dB ref. 20e6 Pa) on the trailing edge floor of the cavity for the case w ith acoustic treatment and no cavity. Noise spike near 600 Hz is electronic noise. 127 PAGE 128 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 100 110 120 130 140 150 2 2 1 Figure 67. Spectrogram of pre ssure measurement (ref 20e6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D=6. Unknow n acoustics features are denoted as while the Rossiter modes are denoted as 128 PAGE 129 Mach numberFrequency (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 70 80 90 100 110 120 130 140 150 Ri Figure 68. Spectrogram of pr essure measurement (ref 20e6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with =0.25, =0.7). 129 PAGE 130 0 1000 2000 3000 4000 5000 6000 80 85 90 95 100 105 110 115 120 Frequency [Hz]UPL [dB] TE noisefloor TE 1050Hz150Vpp A 0 1000 2000 3000 4000 5000 6000 80 85 90 95 100 105 110 115 120 Frequency [Hz]UPL [dB] TE noisefloor TE 1500Hz and 150Vpp B Figure 69. Noise floor of the unsteady pressure level at the surface of the trailing edge of the cavity with and without the actuator turned on. A) The exciting sinusoidal input has frequency 1050 Hz and amplitude 150 Vpp. B) The exciting sinusoidal input has frequency 1500 Hz and amplitude 150 Vpp. The peaks near 600 Hz and 2000 Hz are electronic noise. 130 PAGE 131 0 500 1000 1500 2000 2500 3000 3500 4000 75 80 85 90 95 100 105 110 115 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e6 Pa Mach 0.1, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp A 0 500 1000 1500 2000 2500 3000 3500 4000 85 90 95 100 105 110 115 120 125 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e6 Pa Mach 0.2, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp B Figure 610. Openloop sinusoidal control results for flowinduced cavity oscillati ons at trailing edge floor of the cavity. A) At Mach number 0.1. B) At Mach number 0.2. C) At Mach number 0.3. D) At Mach number 0.4. The cavity model with 6 inch long and L/D=6. 131 PAGE 132 0 500 1000 1500 2000 2500 3000 3500 4000 95 100 105 110 115 120 125 130 135 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e6 Pa Mach 0.3, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp C 0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 frequency [Hz]Unsteady Pressure Level [dB] with reference pressure 20e6 P a Mach 0.4, Baseline 1.05 kHz, 200 Vpp 1.50 kHz, 200 Vpp D Figure 610 Continued 132 PAGE 133 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 107 Time(s)Runing Error Var p = 2 p = 4 p = 6 p = 8 p = 14 p = 50 p = 100 A 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 107 Time(s)Runing Error Var p = 2 p = 4 p = 6 p = 8 p = 14 p = 50 p = 100 B Figure 611. Running error variance plot for the system identification algorithm. A) With chirp signal as input. B) With white noise si gnal as input. Upstream Mach number is 0.275, L/D=6. 133 PAGE 134 0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Baseline, M=0.27 TE ClosedLoop, p=14, s=6 Figure 612. ClosedLoop active control result for flowinduced cav ity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity. The control algorithm uses an estimated order of 14 for both the system ID and GPC algorithms, and the predictive horizon is chosen as 6. A chirp signal is used as the system ID excitation source. 134 PAGE 135 0 500 1000 1500 2000 2500 3000 3500 4000 40 30 20 10 0 10 20 Frequency [Hz]Input Power [dB] Figure 613. Input signal of the ClosedLoop active control result fo r flowinduced cavity oscillations at Mach 0.27 at the trailing edge floor of th e L/D =6 cavity. The control algorithm uses an estimated order of 14 fo r both the system ID and GPC algorithms, and the predictive horizon is chosen as 6. A chirp signal is used as the system ID excitation source. 135 PAGE 136 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.3 0.2 0.1 0 0.1 Frequency [Hz]Sensitivity in Log Scale Disturbance Amplification Disturbance Attenuation Figure 614. Sensitivity function (Equation 41) of the closedloop control for M=0.27 upstream flow condition. The estimated order is 14, prediction horizon is 6, and the input weight R is 0.1. This sensitivity is calculated based on Figure 612 136 PAGE 137 0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Baseline, M=0.27 TE ClosedLoop, p=2, s=6 TE ClosedLoop, p=14, s=6 Figure 615. Unsteady pressure level of the closedloop control for M=0.27, L/D=6 upstream flow condition with varying estimated orde r. The prediction horizon is 6, and the input weight is 0.1. The excitation source for the system ID is a swept sine signal. 137 PAGE 138 0 500 1000 1500 2000 2500 3000 3500 4000 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE Baseline, M=0.27 TE ClosedLoop, p=14, s=2 TE ClosedLoop, p=14, s=6 Figure 616. Unsteady pressure level of the closedloop control for M=0.27, L/D=6 upstream flow condition with varying predictive horiz on s. The estimated order of the system is 6, and the input weight is 0.1. The ex citation source for the sy stem ID is a swept sine signal. 138 PAGE 139 139 0 500 1000 1500 2000 2500 3000 3500 4000 100 105 110 115 120 125 130 135 Frequency [Hz]UPL [dB] TE OpenLoop TE Baseline, M=0.27 TE ClosedLoop Figure 617. Unsteady pressure level comparis on between the openloop control and closedloop control for M=0.27 upstream flow condition. The openloop control uses a sinusoidal input signal at 1150 Hz forci ng and 150 Vpp and the rms value is 53 V. The closedloop control uses an estimated order 14, pr edictive horizon 6, input weight 0.1, and the input rms value is 43 V. PAGE 140 CHAPTER 7 SUMMARY AND FUTURE WORK Equation Section 7 This chapter summarizes the previously disc ussed work and presents contributions from this study. Future work is summarized that a ddresses detailed measurements of the actuator system, a systematic experimental analysis of the flow using various flow diagnostics, and a more detailed parametric study of openloop and closedloop control. Summary of Contributions The contributions of this research are summarized here. First, a global model of flowinduced cavity oscillation is derive d that provides insight into the required structure for a plant model used for subsequent control. When simp lified, this model matches the Rossiter model. Second, a novel piezoelectricdriven synthetic jet actuator array is designed for this research. The resulting actuator produces hi gh velocities (above 70 m/s) at th e center of the orifice as well as a large bandwidth (from 500 Hz to 1500 Hz) whic h is sufficient to cont rol the Rossiter modes of interest at low subsonic Mach numbers. Th is actuator array produces normal zeronet massflux jets that seek to penetrate the boundary laye r, resulting in streamwi se vortical structures. These streamwise vortical disturbances destroy th e spanwise coherence of the shear layer, and the corresponding Rossiter modes are disrupted. Alternatively, the introduced disturbances modify the stability characteristics of the mean flow, so that the main resonance peaks may not be amplified. Next, a MIMO system ID IIRb ased algorithm is developed ba sed on the structure inferred from the global model. This system ID algorithm combined with a GPC algorithm is applied to a validation vibration beam problem to demonstrat e its capabilities. The control achieves ~20 dB reduction at the single resonance peak and ~9 dB reduction of the integrat ed vibration levels. Finally, this control methodology is extended and applied to subsonic cavit y oscillations for on140 PAGE 141 line adaptive identification and control. Openl oop active control uses a sinusoidal signal with 200 Vpp and an excitation frequency of either 1. 05 kHz or 1.5 kHz, which are detuned from the Rossiter frequencies, to drive the actuator a rray. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced. However, when the upstream Mach number increases (greater than Mach number 0.4), the effects of the synthetic jets from this actuator are gradually reduced. Adaptive closedloop control is then applied for an upstream Mach number of 0.27; the estimated GPC order is 14 and the predictive horizon is 6. To avoid saturation in the control signal, the input weight penalty is ch osen as 0.1. The GPC controller can generate a series of control signals to driv e the actuator array resulting in dB reduction for the second, third, and fouth Rossiter modes by 2 dB 4 dB, and 5 dB, respectively. In addition, the broadband background noise is also reduced by this closed loop controller (i.e., the OASPL reduction is 3 dB). However, unlike previously reported closedloop cavity results, a spillover phenomenon is not observed in the closedloop cont rol result. As discus sed in Chapter 1, the spillover problem is generated by a linear controller because the disturbance source and control signal or the performance sensor output and the measurement sensor output (feedback signal) are collocated. The nonlinear nature of the adaptive system may be responsible for this effect. Future Work Recommended future work consists of the following items. The phaselocked centerline velocity of the act uator array should be measured using, at least, by hotwire anemometry. The actuator system for the active flow control needs to be explored in detail. Since the size of the rectangular slots are small (1 mm by 3 mm), the present size of the hotwire (~1 mm) cannot provide spatiallyresolved measurements. The hot wire is also not suitable to decipher the 3D velocity field resulting from the interaction of the jets with the boundary layer. Laser Doppler Velocimetry (LDV) or stereo Particle Image Velocimetry (PIV) measurements can provide good spat ial resolution of the 3D, tu rbulent velocity field that results from the interaction between the ZNMF jets and the grazing boundary layer. 141 PAGE 142 142 The turbulent boundary layer characteristics (e.g., incoming boundary layer thickness) at the leading edge of the cavity and the mean flowfield of th e baseline uncontrolled case should be measured. The impedance of the ceiling liner should be measured using an acoustic impedance tube. Another potential system order estimation algor ithm from information theory of empirical Bayesian linear regression by Stoica (1989, 1997) should be applied to this problem. Parametric studies are recommended to anal yze the performance, adaptability, cost function, and limitations (spillover, etc.) of the adaptive MIMO control algorithms over and above that of the present SISO experiments. The effectiveness of the adaptive closedloop co ntroller should be evaluated with changes in the upstream Mach number. PAGE 143 APPENDIX A MATRIX OPRATIONS Equation Section 1 Vector Derivatives In this appendix, finite length vectors derivatives are illustra ted. During most optimization method development, they are the fundamental t ools to find the optimum value. Only real numbers are considered in this appendix. Definition of Vectors Define the vectors and u y as following (A1) 111 1 222 1 1 11()(,,) ()(,,) ()(,,)n n nmmn nmuyyuu uyyuu uyyuu u u uy u1mFor special case, if or 1 n1 m the vector or is reduced to scalar, respectively. uyDerivative of Scalar with Respect to Vector 12 1n nyyyy uuu u (A2) Note the derivative of scalar is a row vector. Derivative of Vector with Respect to Vector 111 1 12 222 2 12 12 n n m mmm n mnyyy y uuu yyy y uuu y yyy uuu u y u u u (A3) 143 PAGE 144 Second Derivative of Scalar With Re spect to Vector (Hessian Matrix) 222 2 1121 1 222 2 2 2122 2 222 12 n T n n nnnn nnyyy y uuuuu u y yyy u uuuuuu y yyy u uuuuuu 2yy uuuu (A4) Example 1 Given: 1 2 1 12 2 2 2 32 21 3 31, 3 u y uu u y uu u uy (A5) Find: y u Solution: From the Equation A3, the derivative of matrix y u is computed as 111 1 123 2222 123 23 1 3 23210 032 yyy y uuu yyyy uuu u u y u u u (A6) Table of Several Useful Vector Derivative Formulas Table A1 lists the most common vector derivative results, these results are very useful for MIMO controller design. If nnA is symmetric, the last formulas in Table A1 can be expressed as 144 PAGE 145 12T nnAn y u u (A7) Proof of the Formulas Proof (a) 11 11 11 211 22 12 222 12 11 111122 1 211222 2 1122 1nnnn n n nnnn nn nn nn nn nnnnn nA yaaau yaaau yaaau auauau auauau auauau yun n (A8) According to Equation A3 1112 1 2122 2 12 n n nnnn nn nnaaa aaa y aaa A u (A9) Proof (b) 11 1112 1 2122 2 12 1 12 11121211122 1 T nnnn n n n n nnnn nn nnnn nnn nA aaa aaa uuu aaa auauauauauau yu (A10) According to Equation A3 145 PAGE 146 1121 1 1222 2 12 n n nnnn nn T nnaaa aaa y aaa A u (A11) Proof (c) 1111 1 2 12 1 1 222 12 11 T nn n n n n ny u u uuu u uuu uu (A12) According to Equation A2 12 1 12 1 1222 2 2n n n n T ny uuu uuu u u (A13) Proof (d) 111 1 1112 1 1 2122 2 2 12 1 12 1 1 2 11121211122 1 1 111212 11 1 T nnnn n n n n nnnnn nnn nnnn nnn n n n nn nyA aaau aaau uuu aaau u u auauauauauau u auauauuau uu122 11 nn n nauauu n (A14) According to Equation A2 146 PAGE 147 111212 1 111122 1 1122 1122 1 11121211122 1 111122 1 1122 1 1112 12 1 nn nn nnnnnnnnnn n nnnn nnn n nnnn nnn n n nauauauauauau y auauauauauau auauauauauau auauauauauau aaa uuu u 1 2122 2 12 1121 1 1222 2 12 1 12 11 n n nnnn nn n n n n nnnn nn TT T nnnnnnaaa aaa aaa aaa uuu aaa AA uu (A15) Example 2 Given: 1 2 3 31 11313331110 ,101 001Tu uA u yA u uu (A16) Find: y u Solution: 11313331 1 123 2 13 3 3331 1 121232 13 3 31 22 1233110 101 001TyA u uuu u u u uuuuuu u uuuu uu (A17) 147 PAGE 148 According to Equation A2, the derivative of y u is computed as 1323 1322 y uuuu u (A18) Now, calculate the gradient of y u using the Table A1 31333133 123 123 13 13 33 33 1212312133 13 13 1323 13110 110 101 100 001 011 22TT Ty AA uuu uuu uuuuuuuuuu uuuu uu u (A19) The result of Equation A19 is the same as that in Equation A18. The Chain Rule of the Vector Functions Define the vectors u y z and as following w 11 22 1 1 1 2 2 11() () () () () () () () ()nm n s r r1m s uy uy uy w z w z w z u u uy u z y z y zw z y (A20) From Equation A3 148 PAGE 149 111 12 222 12 12 n n rrr n rnzzz uuu zzz uuu zzz uuu z u (A21) Each element of Equation A21 may be expanded using chain rule as 12 12 1 iii i jjjm m ik k kjzzzzy yy uyuyuyu zy yu m j (A22) where 1,2,, 1,2,ir j n Substitute Equation A22 into Equation A21 111 111 12 222 111 12 111 12 mmm kkk kkk kkk n mmm kkk kkk kkk n rn mmm kkk rrr kkk kkk n rnyyy zzz yuyuyu yyy zzz yuyuyu yyy zzz yuyuyu z u 111 111 12 12 222 222 12 12 12 12 n m n m mmm rrr m n rm mn mn rmyyy zzz uuu yyy yyy zzz uuu yyy yyy zzz yyyuuu zy yu (A23) 149 PAGE 150 Similarly, for more vectors, the chain rule just builds the new derivatives to the right. s ns r rm m n wwzy uzyu (A24) The Derivative of Scalar Functions Respect to a Matrix Define a matrix 1112 1 2122 2 12 n n mmmn mnhhh hhh hhh H, (A25) and a scalar function ()Jf H (A26) The gradient of J with respect to H represented by 1112 1 2122 2 12 n n mmmn mnJJJ hhh JJJ J hhh JJJ hhh H. (A27) Example 3 Find the gradient matrix, if J is the trace of a square matrix H 1n ii iJ=tr h H (A28) According to Equation A27 the gradient of J with respect to H is 150 PAGE 151 100 010 001nn nn nnJ I H (A29) Example 4 Find the gradient matrix, if J is the trace of a square matrix T H H, where H is defined by Equation A25 and need not be square. The scalar function J can be expressed as 1121 11112 1 1222 2 2122 2 12 12 222222 1121 11222 2 222 12 2 11 mn mn nnmnmmmn nm mn nn m nnmn nm ij jiJtr hhhhhh hhhhhh tr hhhhhh hhhhhh hhh hm THH (A30) and the gradient of J is 1112 1 2122 2 12222 222 222 2n n nn mmmn mn mnhhh hhh J hhh H H (A31) Example 5 Define the vectors A and B H as following 151 PAGE 152 11 22 11 1112 1 2122 2 12,mn mn n n mmmn mnab ab ab hhh hhh hhh AB H (A32) Find the derivatives of a scalar function J respect to matrix H Express the scalar function as 1112 1 1 2122 2 2 12 1 12 1 1 2 111221 1 1122 1 1 11 11 1 T n n m m mmmnn mnn mm nnmmn n n n mm kk kknn kk m kkll kJAHB hhhb hhhb aaa hhhb b b ahahahahahah b ahbahb ahb H1 n l (A33) So, the derivative of J respect to the matrix H is 1112 1 2122 2 12 1 2 12 1 1 1 1 n n mn mmmn mn n n m m m nababab ababab J ababab a a bbb a H T=AB. (A34) 152 PAGE 153 Note that since J is a scalar function, and the following equation is also hold. TJJ TTBHA 1 1 mn mn m nJ HHTT TBHA =AB (A35) 153 PAGE 154 154 Table A1. Vector derivative formulas. y y u 11 11 1111 111 1() () () ()nnnn T nnnn T nn T nnnnaA bA cy dyA yu yu uu uu 1 112nn T nn T n TT nnnnnnA A AA u uuT PAGE 155 APPENDIX B CAVITY OSCILLATION MODELS Equation Section 2 Rossiter Model The derivation of the Rossiter Model depends on the following assumptions (Rossiter 1964) Frequencies of the acoustic radiation are th e same as the vortex shedding frequency. There are vm complete wave length of th e vertex motion at the time 0'ttt (e.g. 1,2,... ) vm The schematic of the Rossiter model is illustrated in Figure B1 and the symbols in the plot are listed as follows ,LD Cavity length and depth U Free stream velocity a Mean sound speed inside the cavity vm Mode number (integer number 1,2) Phase lag factor between impact of the large scale structure on the trailing edge and the generation of the acoustic wave. Proportion of the convective vortices speed to the free stream speed v Spacing of the vortices At specific initial time an acoustic wave forms at the trailing edge with the distance 0tt v ( Figure B1 ). Appropriately choose the time such that, the propaga ting wave front just reaches the cavity leading edge. By the assumption, the mode number is an integer number. Then, tvm 'Lat (B1) 155 PAGE 156 'vvvLmUt (B2) And the frequencies of the oscillations are re lated the phase speed and the wavelength of the vortical disturbance. vU f (B3) Substituting the Equation B1 in to Equation B2, 1vvv vvL LmU a ULm a (B4) Then, combining the Equation B3 and Equation B4 resulting 1 1v vU ULm af m fL U U a (B5) The Rossiter model is defined by 1 ,1,2,... 1v v vfL St U m St U a m St m M (B6) Linear Models of Cavity Flow Oscillations The block diagram of the linear model of th e flowinduced oscillations is show in Figure B2 (Rowley et al. 2002). And the closedloop cavity tr ansfer function can be expressed as 156 PAGE 157 ()()() () 1()()()()GsSsAs Ps GsSsAsRs (B7) Furthermore, the shear layer model is consid ered as a secondorder system with a time delay 2 0 0 2 00()() 22 s s s sw GsGse e sww (B8) and sL U (B9) where 0w Natural frequency of second order system Damping ratio s Time delay inside the shear layer The acoustics model can be represented as a reflection model ( ()AsFigure B3 ) where a Time delay inside the cavity r Reflection coefficient The closedloop transfer f unction of the reflection model can be written as 2() 1a as se As re (B10) and aL a (B11) To recover the Rossiter formula, additional assumptions are required 157 PAGE 158 Impingement model ()Ssand receptivity model () R s are unit gains. No reflections in acoustic model in B10, 0 r The shear layer model is only a constant phase delay 2 0()iGse (B12) Depending on these assumption, combine the Equation B7, Equation B10 and Equation B12, () 2 () 2()() () 1()() 1as ass i s iGsAs Ps GsAs ee ee (B13) In order to find the resonant of the system, substitute the poles locations into the characteristic function of the Equation siw B13, and then combine the Equation B9 and Equation B11 resulting () 2 2( 210 2()() 1 2as vaiw i im iw i v vee eee LL mw aU m wL U U a )s (B14) Define ,1,2,... 1v vfL St U m m M (B15) The linear model results Equation B15 matches the Rossiter formula Equation B6. 158 PAGE 159 Global Model for the Cavity Os cillations in Supersonic Flow The global model for cavity oscillations in supersonic flow is shown in Figure B4 To be consistent with the notation of Kerschen et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude upstream propagating cavity acoustic modes U and downstream propagating cavity acoustic modes and fast modes SD f E and slow modes s E nearfield acoustic waves in the supers onic stream. The local amplitudes of all quantities at the leading edge are denoted by the decoration while quantities at the trailing edge do not have the decoration. The scattering processing at th e leading edge is modeled by f sSU DU EU f EU sC S C D U C E C E (B16) SUC, D UC, and are the four scattering coefficients for the upstream end of the cavity. The first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at th e trailing edge is modeled by fEUCsEUC fsUSUDUEUE f s S D UCCCC E E (B17) USC, UDC f UEC, and s UEC are the four scattering coefficien ts for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by 159 PAGE 160 1233 22 ,diL iL iML iML ffssSSeDDe ELEeELEe (B18) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. 1 M and 2 M are the wavenumbers of the fast and slow downstream propagating nearfield acoustic waves, respectively. Finally, the upstream propagating acoustic cavity mode is uiLUUe (B19) where u is the complex wavenumber of this mode. The global model can be expressed in a block diagram ( Figure B5 ). Substituting Equation B19 into Equation B17 results (B20) 0u fs u fsiL USUDUEfUEs iL USUDUEUE f sUeUCSCDCECE U S eCCCC D E E Also, substituting Equation B18 into Equation B16, the following formula can be obtained 1 23 2 3 2 d f siL SU iL DU iML EU f f EU iML s sSe C S De C D U C LEe E C E LEe (B21) These four equations can be written in matrix form 160 PAGE 161 1 23 2 3 2000 000 0 000 000d f siL SU iL DU iML EU f iML s EUCe U Ce S D CL e E E CL e (B22) Now, combine Equation B20 and Equation B22 yields 1 23 2 3 2 000 000 0 000 000u fs d f siL USUDUE UE iL SU iL DU iML EU f s iML EUeCCCC U Ce S Ce D CL e E E CL e (B23) Define 1 23 2 3 2000 00 000 000 u fs d f siL USUDUE UE iL SU iL DU iML EU iML EU f seCCCC Ce Ce A CL e C U S X D E E 0 L e (B24) Therefore, Equation B23 can be written as 0 AX (B25) 161 PAGE 162 Notice that the quantities of X are the incident waves on the two ends of the cavity. The global mode has to satisfy Equation B25 which corresponds to the condition det( Calculating the determinant and simplifying, )0 A 123 2 30ff ss udiML iML EUUE EUUE iL iL iL SUUS DUUDLCCeCCe LeCCeCCe (B26) Assume a simple case where only 0,0USSUCC and all other scattering coefficients are zeros. Therefore, Equation B26 can be simplified to ()0 1 1u u uiL iL SUUS iL SUUS iL iL SU USeCCe CCe CeCe (B27) Enforce the phase criterion and notic e that the length is normalized by L L U and Equation B27 results ()1 Re()2uiL SUUS SU US uCCe ArgCArgC Lm (B28) or 2 Re /2 2R eSU US u SU US umArgCArgC L L U mArgCArgC L U (B29) Consider the normalized wave number /1 Re / / Re /uUU M cU (B30) 162 PAGE 163 and define 2 /2SU USL St U ArgCArgC (B31) Therefore, Equation B29 can be written as 2 Re ,1,2,... 1 2SU US umArgCArgC L L U Lm St m U M (B32) This matches the Rossiter formula Equation B6. Global Model for the Cavity Os cillations in Subsonic Flow The global model for cavity oscillations in subsonic flow is shown in Figure B6 To be consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude upstream modes U and downstream modes propagating acoustic modes in the cavity, and upstream modes and downstream modes propagating nearfields acoustic waves in the subsonic flow. The local amplitudes of all quantities at the leading e dge are denoted by the decoration SDdEuE while quantities at the trailing edge do not have the decoration. The scattering processing at th e leading edge is modeled by u u dd uSUSE DUDE u EUEE dS CC U DCC E CC E (B33) 163 PAGE 164 SUC D UC ,dEUCuSECu D EC and are scattering coefficients for the upstream end of the cavity. Again, the first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering proce ss at the trailing edge is modeled by duEEC d uuudUSUDUE u ESEDEE dS CCC U D E CCC E (B34) USC , and are scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by UDCdUECuESCuEDCudEEC 3 2 ,,diL iML iL ddSSeDDeELEe d (B35) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. d M is the complex wavenumber of the downstream propagating nearfield acoustic wave. Finally, the upstream propagating acoustic cavity modes are 3 2 ,uiL iML uuUUeELEeu (B36) where u and u M are the complex wavenumbers of the upstream traveling cavity mode and the upstream traveling acoustic ne arfield mode, respectively. The global model can be represented in a block diagram ( Figure B7 ). Again, substituting Equation B35 into Equation B34and Equation B36 into Equation B33, and combine the results, a matrix equation can be obtained, where 0 AX 164 PAGE 165 3 2 3 2 3 2 3 2 3 210 01 100 010 001dd d dd uuu d uu u uu u uu dd uiL iML iL US UD UE iL iML iL ES ED EE u iL iML SU SE iL iML DU DE d iL iML EU EECeCeLCe U CeCeLCe E S AX CeLCe D CeLCe E CeLCe (B37) The global mode has to satisfy 0 AX which corresponds to the condition det()0 A Calculating the determinant and simplifying, 3 2 33 3 22 3 2ud ud u dd ud u uu uu duud uu uu duu uu uuiLiL iML SUUS DUUD EUUE iML iML iMML SEES DEED EEEE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEESCCeCCeLCCe LCCeLCCeLCCe CCCCCCCC Le CCCCCCCC 3 31ududdudu duu duuddduu ududdudu dduu duuddduuSUESEEUEEUEESEUS iMML SUUSEEEEEUUESEES DUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEEDCCCCCCCC Le CCCCCCCC CCCCCCCC Le CCCCCCCC d u (B38) Assume a simple case where only 0,0USSUCC and all other scattering coefficients are zero. Therefore, Equation B38 can be simplified to ()1 1u uiL SUUS iL iL SU USCCe CeCe (B39) This matches the supers onic case results Equation B27 165 PAGE 166 L D v v U a0tt L D U a0' ttt 'vUt vvm Figure B1. Schematic of Rossiter model. G(s) S(s) A(s)(Leading Edge) Receptivity Shear Layer (Trailing Edge) Impingment Acoustics Feedback Sensor Output Actuator Input R(s) Figure B2. Block diagram of the linear mode l of the flowinduced cavity oscillations. 166 PAGE 167 a s e a s re Figure B3. Block diagram of the reflection model. L Turbulent Boundary Layer D 1 M s EfESD U x y Figure B4. Global model for the cavity oscillations in supersonic flow. f SUSUDUEUECCCC f SSU DU E U E UC C C C 23 2 iMLLe 13 2 iMLLe iLe N f s S D E E f s S D E EU uiLe diLe U Figure B5. Block diagram of the global model for a cavity oscillation in supersonic flow. 167 PAGE 168 168 1 L Turbulent Boundary LayerD M dEuESDU x y Figure B6. Global model for a cavit y oscillation in subsonic flow. N U uiLe 3 2uiML L e u u dd uSUSE DUDE E UEECC CC CC d uuudUSUDUE E SEDEECCC CCC 3 2diML L e diLe iLe uU E uEdS D E dS D E Figure B7. Block diagram of the global mode l for a cavity oscillation in subsonic flow. PAGE 169 APPENDIX C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS Equation Section 3 MIMO System Identification Assume a linear and time invariant system, with the inputs r 1ru and the m outputs 1my, at the time the system can be expressed as k 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuup (C1) where 1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (C2) Define 10 **(1)()pp mmprpk (C3) and **(1)(1) () () () ()mprpk kp k k kp1 y y u u (C4) and these yield the filter outputs 1 (**(1)) ()() ()m mmprpkkk y( ( 1 ) ) 1m p r p (C5) 169 PAGE 170 Therefore, the error between the two outputs is defined as 11 ()()() ()()()mmkkk kkk1m yy y (C6) and the scalar error cost function is defined by 1 ()()() 2TJkkk (C7) To identify the observer Markov parameters C3, the following equations based on the gradient descend method is developed () (1)() () Jk kk k (C8) where is the step size. Substituting Equation C6 into Equation C7 1 ()()() 2 1 ()()()()()() 2 1 2 1 2T T T TTJkkk kkkkkk T TTyy yy yyyy (C9) and the gradient of er ror cost function is 1 1**(1)()1 2 () 1 2 1 2 ()()m mprpJk k kk Tyy 2y 2y y (C10) Finally, substituting Equation C10 into Equation C8 and yielding 170 PAGE 171 (C11) (1)()()(Tkkk) k In order to automatically upda te the step size, choose 21 (C12) where is a small number to avoid the infinity number when 2 is zero. Here the main steps of the MIMO identification are given as follows Step 1: Initialize (**(1)) ()mmprpk 0. Step 2: Construct regression vector **(1)1()mprpk according to Equation C4. Step 3: Calculate the output error 1()mk according to Equation C6. Step 4: Calculate the step size according to Equation C12 Step 5: Update the observe r Markov parameters matrix *(1)()mmprpk according to Equation C11. And then go back to st ep 2 for next iteration. Generalized Predictive Control Model In this section, a MIMO model, which is the same as the model of the MIMO ID C1, is considered. Assume a linear a nd time invariant system, with the inputs r 1 ru and outputs m 1 m y at the time index the system can be expressed as k 12 012()(1)(2)() ()(1)(2)()p pkkkkp kkkk yyyy uuuu p (C13) where 171 PAGE 172 1 1 2 2 11 1 1 1122 0011() () () () ()() ()() () () ,,, ,,,rm m r r m pp mm mm mm pp mr mr mryk uk yk uk kuk kyk yk uk u, y (C14) Shifting Equation C13 one time step ahead and can be expressed as 12 012(1)()(1)(1) (1)()(1)(1p pkkkkp kkkkp yyyy uuuu ) (C15) Substituting Equation C13 into Equation C15 12 12 1 23 1011 1 012 122 123 11 1(1)(1)(2)() (1)(2)(1) ()(1)() (1)()(1)(1) (1)(2) (1)(p p p p pp pkkkkp kkkp kkkp kkkkp kk kp k yyyy yyy uuu uuuu +y+y +y y 0 101 112 11 1 (1) (2) 12 (1) (1) 1 (1) (1) 001)( () (1) (1)() (1)(2) (1)() (1)()(1)pp p mm mm pp mm mm mr mr mrpk kk kp kp kk kp kp kkk 1) u +u+u +u u yy yy uuu (1) (1) 1(1)()pp mr mrkp kp uu (C16) where 172 PAGE 173 (1) (1) 1 1120101 (2) (1) 21 2 311 1 (1) (1) 11 111 1 (1) (1) 11,mm mr mm mr pp p pp mm mr ppp mm mr ++ ++ ++ 2 p p (C17) The output vector is the linear combination of the past outputs, the past inputs and future inputs. By induction, the output vector (1) k y() kj y can be derived as (C18) () () () 121 () (1) 00 () () () 012 ()()(1)(2) ( ()()(1) ()(1)(2) ()jjj p mm mm mm j p mr mm mr jjj mr mr mr j p mrkj k k kp kpkjkj kkk kp yyyy yuu uuu u 1) where () (1)(1) () (1)(1) 0101 11 1 2 () (1)(1) () (1)(1) 21 2 311 1 2 () (1) (1)() 11 11 () (1) 1,jjj jjj mr mm jjjjj mm mr jjjj pp p p mm jj pp mm + + ++ + (1) (1) 11 () (1) 1 jj pp mr jj pp mr + j 0 1 1 p p (C19) with initial (0) (0) 110 (0) (0) 221 (0) (0) 11 1 (0) (0),mm mr mm mr pp p mm mr ppp mm mr (C20) The quantities () 0 k mr ( ) are the impulse response sequences of the system. 0,1,k 173 PAGE 174 Define the following the vector form 11 11 1 1 1 1 1 1 1() () (1) (1) () ,() (1) () () (1) () (1)rr rr pj r rp rj m m p m mpukp uk ukp uk kp k uk ukj ykp ykp kp yk 1 ( 1 ) 1 r uu y (C21) Substituting Equation C21 into Equation C18 and express it in matrix form as (C22) () (1) 0001 1 (1)1 (1) () () () 21 1 () () () 21 1() () () ()j j mm r rj mr mr mrj jjj pp rp mr mr mr mrp jjj pp mp mm mm mm mmpkj k kp kp yu u y Now, let the predictive index 0,1,2,,1,,1 j qqs and define 1 1 1 1 1 1 1 1() (1) () (1) () (1) () (1)r r s r rs m m s m msuk uk k uks yk yk k yks u y (C23) A predictive model can be expressed by 174 PAGE 175 1 1 1 1 1() (1) () (1)m m s ms m msyk yk k yks I IIIII y (C24) 1 1 0 1 1 1 (1) 1 00 1 1 1 (1) (2) 1 000 1() 0 00 0 () (1) 0 0 () (1) (1)r r mrmr mr mrs r rs r r mr mr mr mrs r rs r ss r mr mr mr mrs rk k k I k k ks u u u u u u 1 1 0 1 (1) 00 1 (1) (2) 1 00000 () 0 (1) (1)rs ms mr mr mr r mr mr mr r ss r rs mr mr mr msrsk k ks u u u 1 (C25) or for simplification 1()s msrs rsIk Tu (C26) where 175 PAGE 176 0 (1) 00 (1) (2) 00000 0mr mr mr mr mr mr msrs ss mr mr mr msrs T (C27) The matrix T is called Toeplitz matrix. A nd the second part of Equation C24 is 1 1 11 1 1 1 (1) (1) (1) 1 11 1 1 (1) (1) 1() (1) (1) () (1) (1)r r pp mr mr mr mrp r rp r r pp mr mr mr mrp r rp ss pp mrkp kp k kp kp II k u u u u u u 1 (1) 1 1 1 1 1 11 (1) (1) (1) 11 (() (1) (1)r s r mr mr mrp r rp ms pp mr mr mr pp mr mr mr s pkp kp k u u u 1 1 1) (1) (1) 1 1 11() (1) (1)r r ss r rp p mr mr mr msrpkp kp k u u u(C28) or for simplification 1()p msrp rpII kp Bu (C29) where 176 PAGE 177 11 (1) (1) (1) 11 (1) (1) (1) 11 pp mr mr mr pp mr mr mr msrp sss pp mr mr mr msrp B (C30) and the third part of Equation C24Is 1 1 11 1 1 1 (1) (1) (1) 1 11 1 1 (1) (1 1() (1) (1) () (1) (1)r r pp mm mm mm mmp r mp r r pp mm mm mm mmp r mp ss pp mmkp kp k kp kp III k y y y y y y 1 )( 1 ) 1 1 1 1 1 11 (1) (1) (1) 11 (() (1) (1)r s r mm mm mmp r mp ms pp mm mm mm pp mm mm mm s pkp kp k y y y 1 1 1) (1) (1) 1 1 11() (1) (1)r r ss r mp p mm mm mm msmpkp kp k y y y(C31) or for simplification 1()p msmp mpIII kp Ay (C32) where 177 PAGE 178 178 11 (1) (1) (1) 11 (1) (1) (1) 11 pp mm mm mm pp mm mm mm msmp sss pp mm mm mm msmp A (C33) Combine Equation C26, Equation C29 and Equation C32 in to Equation C24, 11 1 1() () () ()ssp ms msrs rs msrp rp p msmp mpkk kp yTuBu Ayk p (C34) PAGE 179 APPENDIX D A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES Equation Section 4 In this section, a potential theoretical mode l of cavity acoustic resonance is derived based on the model of Kerschen et al. (2003). The model combines scattering analyses for the two ends of the cavity and the propa gation analyses of the cavity shear layer, internal region of the cavity, and acoustic nearfield. Kerschen et al. so lve a matrix eigenvalue problem to identify the frequencies of the cavity oscillat ion. A different approach for characterizing the same model is illustrated in this section. A signal flow graph is first constructed from a block diagram of the physical model, and then Masons rule (Nise 2004) is applied to obtain the transfer function from the disturbance input to the se lected system output. This me thod gives a prediction for the resonant frequencies of the flowinduced cavity os cillations. In addition, this method also provides a linear estimate for th e system transfer function. Masons Rule Mason's rule reduces a signal flow graph to a transfer f unction between of any two nodes in the network. A signal flow graph connects nodes, used to represent variables, by line segments, called branches. First, so me definitions are given as follows Input node: a node that has only outgoing branches. Output node: a node that has only incoming branches. Path: a string of connected branches and nodes. It contains the same branch and node only once. Forward path: a path that traverses from th e input node to the output node of the signal flow graph in the direction of signal flow. It touches the same node only once. Forward path gain: the product of gains found by traversing the path from the input node to the output node of the signal flow graph in the direction of signal flow. Loop: a path that starts at a node and ends at the same node without passing through any other node more than once and follow s the direction of the signal flow. 179 PAGE 180 Loop gain: the product of branch gains in a loop. Touching: two loops, a path and a loop, or tw o paths that have at least one common node. Nontouching loop: Loops that do not have any nodes in common. Nontouching loop gain: The product of loop gains from nontouching loops. The closed loop transfer function, of a linear dynamic system represented by a signal flow graph is (Nise 2004) () Ts 1()N kk kp Ts (D1) where N: number of forward paths k p : the forward path gain thk :1(loop gains)+(nontouching loop ga ins taken two at a time) (nontouching loop gains taken three at a time)+(nontouching loop gains taken four at a time)... th:(loop gain terms in that touch the forward path). In other words, is found by eliminating from those loop gains that touch the forward path.th k kk k Global Model for a Cavity Oscillation in Supersonic Flow The global model for cavity oscillations in supersonic flow is shown in Figure D1 To be consistent with the notation of Kerschen et al. (2003), the global model c onsists of a downstream traveling shear layer instability wave of amplitude upstream modes U and downstream modes propagating acoustic modes in the cavity, and fast modes SD f E and slow modes s E nearfield acoustic waves in the supersonic stream. The local amplitudes of all quantities at the 180 PAGE 181 leading edge are denoted by the decoration f s while quantities at the tr ailing edge do not have the decoration. The scattering processi ng at the leading edge is modeled by f sSU DU EU EUC S C D U C E C E (D2) SUC D UC and are the four scattering coefficients for the upstream end of the cavity. The first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering process at th e trailing edge is modeled by fEUCsEUC fsUS UEUE UD f s S D UCCC E E C (D3) USC UDC f UEC, and s UEC are the four scattering coefficien ts for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by 1233 22 iMD Ee ,diL iL L iML ffssSSeDe ELELEe (D4) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. 1 M and 2 M are the wavenumbers of the fast and slow downstr eam propagating nearfield acoustic waves, respectively. Finally, the upstream propagating acoustic cavity mode is uiLUUe (D5) 181 PAGE 182 where u is the complex wavenumber of this mode. The global model can be represented in a block diagram (Figure D2 ) or a signal flow graph ( Figure D3 ). From the signal flow graph ( Figure D3 ), the transfer function between the disturbance input and the upstream propagating cavity acoustic mode can be found. First, identify the components of Equation N'UD1 The results are listed in Table D1 The characteristic function of the system can be identified from Equation D1 124 1 1 3 21 1uu du ff ssk k iLiL iML iML SUUS DUUD EUUE EUUEl CCeCCeLCCeCCe u (D6) The numerator of the transfer function can be derived from D1 The terms are formed by eliminating from those loop gains that touch the forward path. kthk 13 4 2 1d ff ssiL iML iML iL kkSUUS DUUD EUUE EUUE kpCCeCCeLCCeCCe 2 (D7) Finally, the transfer function between the disturbance input and the upstream traveling wave in the cavity is N U 12 124 1 3 2 3 2() 1d ff ss uu duu ff sskk k UN iL iML iML iL SUUS DUUD EUUE EUUE iLiL iML iML SUUS DUUD EUUE EUUEp Ts CCeCCeLCCeCCe CCeCCeLCCeCCe (D8) The characteristic function in Equation D6 is the same as the eigenvalue relation derived by Kerschen et al (2003). And the transfer function in Equation D8 gives more 182 PAGE 183 information concerning the physical model. For example, this model can predict the resonant frequencies as well as the nodal re gions or zeros of the flow field. Global Model for a Cavity Os cillation in Subsonic Flow The global model for cavity oscillations in subsonic flow is shown in Figure D4 To be consistent with the notation of Alvarez et al. (2003), the global model consists of a downstream traveling shear layer instability wave of amplitude upstream modes U and downstream modes propagating acoustic modes in the cavity, and upstream modes and downstream modes propagating nearfields acoustic waves in the subsonic flow. The local amplitudes of all quantities at the leading e dge are denoted by the decoration SDdEuE while quantities at the trailing edge do not have the decoration. The scattering processing at th e leading edge is modeled by u u dd uSUSE DUDE u EUEE dS CC U DCC E CC E (D9) SUC D UC ,dEUCuSECu D EC and are scattering coefficients for the upstream end of the cavity. Again, the first subscript denotes the output, while the second subscript denotes the input. Similarly, the scattering proce ss at the trailing edge is modeled by duEEC d uuudUSUDUE u ESEDEE dS CCC U D E CCC E (D10) USC , and are scattering coefficients for the downstream end of the cavity. The downstream propagating components from the leading edge to the trailing edge are given by UDCdUECuESCuEDCudEEC 183 PAGE 184 3 2 ,,diL iML iL ddSSeDDeELEe d (D11) where and d are the complex wavenumbers of the shear layer instability wave and the downstream propagating acoustic cavity mode, respectively. d M is the complex wavenumber of the downstream propagating nearfield acoustic wave. Finally, the upstream propagating acoustic cavity modes are 3 2 ,uiL iML uuUUeELEeu (D12) where u and u M are the complex wavenumbers of the upstream traveling cavity mode and the upstream traveling acoustic nea rfield mode, respectively. The global model can be represented in a block diagram (Figure D5 ) or a signal flow graph ( Figure D6 ). From the signal flow graph ( Figure D6 ), the transfer functions between the disturbance input and the upstream propagating cavity acoustic modes, and can be found. The transfer function be tween the disturbance input and the upstream propagating cavity acoustic mode is calculated by first identi fying the components of Equation N U'uEN'UD1 The results are listed in Table D2 The characteristic function of the sy stem can be identified from Equation D1 184 PAGE 185 126 12 11 33 22 1 3 3 2 21 33 32 31 1ud u du dd uu udud uu udud dudu dudukk kk iLiL SUUS DUUD iML EUUE SUESDEUD SUESEEUE DUEDSEUS DUEDEEUE EUEESEUS EUEEDEll CCeCCe LCCeLCCCCI LCCCCILCCCCI LCCCCILCCCCI LCCCC 3 2 3 3 3 2 3 3 2 12 3 3 2 13 3u uu du du uu duud uu duud uu duud dduuiML UD SEES iML iMML DEED EEEE SUUSDEED SUUSEEEE DUUDSEES DUUDEEEE EUUESEEILCCe LCCeLCCe LCCCCILCCCCI LCCCCILCCCCI LCCCC 3 23 33 22 3 3 2 3 2 11dduu ud u du u dd uu du du uu duud uu uSE U U ED EE D iLiL SUUS DUUD iML iML EUUE SEES iML iMML DEED EEEE SUESDEUDDUEDILCCCCI CCeCCe LCCeLCCe LCCeLCCe CCCCCCC LI 3 2 3 3u uu uu ududdudu duuddduu ududdudu duuddduuSEUS SUUSDEEDDUUDSEES SUESEEUEEUEESEUS SUUSEEEEEUUESEES DUEDEEUEEUEEDEUD DUUDEEEEEUUEDEEDC CCCCCCCC CCCCCCCC LI CCCCCCCC CCCCCCCC LI CCCCCCCC (D13) where 123,,duu duu dduuiMLiMMLiMMLIeIeIe (D14) The numerator of the transfer function can be derived from D1 185 PAGE 186 9 1 3 3 2 3 3 2 333 2221 1 1du du uu duud ud u d uu duud ud d dd uu uukk k iML iMML iL SUUS DEED EEEE iML iMML iL DUUD SEES EEEE iML iML iML EUUE SEES DEEDp CCeLCCeLCCe CCeLCCeLCCe LCCeLCCeLCCe u 3 3 2 3 3 2 33ud ud uu udud du dud uu udud du dud dudu duduiML iMML SUESDEUD SUESEEUE iML iMML DUEDSEUS DUEDEEUE iMML iMML EUEESEUS EUEEDEUD SULCCCCeLCCCCe LCCCCeLCCCCe LCCCCe LCCCCe CC 3 2 3 2 3dd dd uu uu du uu uu ududdudu du duuddduuiL iML iL US DUUD EUUE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEEEEEUUESEESeCCeLCCe CCCCCCCC Le CCCCCCCC CCCCCCCC Le CCCCCCCC 3ududdudu ddu duuddduuDUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEEDCCCCCCCC Le CCCCCCCC (D15) Finally, the transfer function between the disturbance input and the upstream traveling wave in the cavity is N U 186 PAGE 187 ' 9 1 3 2 3 2 3()dd dd uu uu du uu uu ududdudu du dUN kk k iL iML iL SUUS DUUD EUUE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEETs p CCeCCeLCCe CCCCCCCC Le CCCCCCCC CCCCCCCC Le CCC 3 3 2 33 221uuddduu ududdudu ddu duuddduu ud ud u dd u uuEEEUUESEES DUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEED iLiL iML SUUS DUUD EUUE iML SEES DECCCCC CCCCCCCC Le CCCCCCCC CCeCCeLCCe LCCeLC 3 3 2 3du du uu duud uu uu duu uu uu ududdudu duu duudiML iMML ED EEEE SUESDEUDDUEDSEUS iML SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEEEEECeLCCe CCCCCCCC Le CCCCCCCC CCCCCCCC Le CCCCC 3dduu ududdudu dduu duuddduuUUESEES DUEDEEUEEUEEDEUD iMML DUUDEEEEEUUEDEEDCCC CCCCCCCC Le CCCCCCCC (D16) The characteristic function in Equation D13 is the same as the eigenvalue relation derived by Alvarez et al. (2003), but the transfer function in Equation D16 gives more information concerning the physical model. For example, this model can predict the resonant frequencies as well as the nodal regions or zeros of the flow field. Then, the transfer function be tween the disturbance input and the upstream propagating acoustic mode is calculated. Similarly, first, identify the components of Equation N'uE D1. The results are listed in Table D3 Because the loop gains and the nontouching loop gains are the same as before ( Table D2 ), only the forward path gains are listed. 187 PAGE 188 The characteristic function of the system is the same as Equation D13. And the numerator of the transfer function can be derived from D1 3 3 2 1d uud uiL iML iL kkSUESDUED EUEE kd d p CCeCCeLCCe (D17) Finally, the transfer function between the disturbance input and the upstream traveling wave in the cavity is N'uE 3 1 3 2 3 2 33 3 22 3 2() 1u dd uud u d ud ud u dd ud u uu uu duud dEN kk k iL iML iL SUESDUED EUEE iLiL iML SUUS DUUD EUUE iML iML iMML SEES DEED EEEE iMTs p CCeCCeLCCe CCeCCeLCCe LCCeLCCeLCCe Le 3 3uu uu uu uu uu ududdudu duu duuddduu udud dduuSUESDEUDDUEDSEUS L SUUSDEEDDUUDSEES SUESEEUEEUEESEUS iMML SUUSEEEEEUUESEES DUEDEEUE iMMLCCCCCCCC CCCCCCCC CCCCCCCC Le CCCCCCCC CCCCC Le d u dudu duuddduuEUEEDEUD DUUDEEEEEUUEDEEDCCC CCCCCCCC (D18) 188 PAGE 189 Table D1. Components of the Maso ns formula for supersonic case. Index Forward Paths Forward Path Gains 1 01261011 1iL SUUS p CCe 2 01371011 2diL DUUD p CCe 3 01481011 13 2 3ffiML EUUEpLCCe 4 01591011 23 2 4ssiML EUUEpLCCe Index Loops Loop Gains 1 12610 1 1 1uiL SUUSlCCe 2 13710 1 1 2udiL DUUDlCCe 3 14810 1 13 1 2 3u ffiML EUUElLCCe 4 159101 23 1 2 4u ssiML EUUElLCCe 189 PAGE 190 Table D2. Components of the Ma sons formula for subsonic case. Index Forward Paths Forward Path Gains 1 013691 1 1iL SUUS p CCe 2 014791 1 2diL DUUD p CCe 3 015891 1 3 2 3d ddiML EUUE p LCCe 4 01361024791 1 3 2 4ud uuiML SUESDEUDpLCCCCe 5 01361025891 1 3 5ud ududiMML SUESEEUEpLCCCCe 6 01471023691 1 3 2 6du uuiML DUEDSEUSpLCCCCe 7 014710258911 3 7dud ududiMM DUEDEEUEpLCCCCe L1 8 01581023691 3 8du duduiMML EUEESEUSpLCCCCe 9 015810247911 3 9dud duduiMML EUEEDEUDpLCCCCe Index Loops Loop Gains 1 1369 1 1 1uiL SUUSlCCe 2 1479 1 1 2duiL DUUDlCCe 3 1589 1 3 1 2 3du ddiML EUUElLCCe 4 136102479 1 3 1 2 4udu uuiML SUESDEUDlLCCCCe 5 1361025891 13 5udu ududiMML SUESEEUElLCCCCe 6 147102369 1 3 1 2 6duu uuiML DUEDSEUSlLCCCCe 7 147102589 1 13 7dudu ududiMML DUEDEEUElLCCCCe 8 1581023691 13 8duu duduiMML EUEESEUSlLCCCCe 9 158102479 1 13 9dudu duduiMML EUEEDEUDlLCCCCe 10 23610 2 3 1 2 10u uuiML SEESlLCCe 11 247102 3 1 2 11du uuiM DEEDlLCCe L 12 258102 13 12du duudiMML EEEElLCCe Index Nontouching Loops Nontouching Loop Gains 1 Loop 1 and Loop 11 3 2 2 1udu uuiM SUUSDEEDlLCCCCe L 2 Loop 1 and Loop 12 23 2udu duudiMM SUUSEEEElLCCCCe L 3 Loop 2 and Loop 10 3 2 2 3duu uuiM DUUDSEESlLCCCCe L 190 PAGE 191 4 Loop 2 and Loop 12 23 4dudu duudiMM DUUDEEEElLCCCCe L 5 Loop 3 and Loop 10 23 5duu dduuiMML EUUESEESlLCCCCe 6 Loop 3 and Loop 11 23 6dudu dduuiMML EUUEDEEDlLCCCCe 191 PAGE 192 Table D3. Components of the Ma sons formula for subsonic case. Index Forward Paths Forward Path gains 1 01361012 1uiL SUES p CCe 2 01471012 2d uiL DUED p CCe 3 01581012 3 2 3d dudiML EUEE p LCCe 192 PAGE 193 L Turbulent Boundary Layer D 1 M s EfESD U x y Figure D1. Global model for a cavity oscillation in supersonic flow. f SUSUDUEUECCCC f SSU DU E U E UC C C C 23 2 iMLLe 13 2 iMLLe iLe N f s S D E E f s S D E EU uiLe diLe U Figure D2. Block diagram of the global model for a cavity oscillation in supersonic flow. 193 PAGE 194 23 2 iMLLe 13 2 iMLLe iLe SSuiLe diLe NU U 0 1 2 3 4 5 6 7 8 9 10 USCSUCf E UCDUCS E UCUDC f UECSUEC 1 D f E s E D f E s E 1 11' U Figure D3. Signal flow graph of the global model for a cavity oscillation in supersonic flow. 1 L Turbulent Boundary LayerD M dEuESDU x y Figure D4. Global model for a cavit y oscillation in subsonic flow. 194 PAGE 195 195 N U uiLe 3 2uiML L e u u dd uSUSE DUDE E UEECC CC CC d uuudUSUDUE E SEDEECCC CCC 3 2diML L e diLe iLe uU E uEdS D E dS D E Figure D5. Block diagram of the global mode l for a cavity oscillation in subsonic flow. 3 2uiMLLeuiLe USCSUCd E UCDUCuSECUDCu E SC U 0 1 1 2uE 6 7 8 3 4 5 3 2diMLLe iLe SSdiLe D dE D dE 9U 10uE uDECdu E ECu E DCud E ECdUEC N 11 'uE U1 1 12 Figure D6. Signal flow graph of the global model for a cavity oscillation in subsonic flow. PAGE 196 APPENDIX E CENTER VELOCITY OF ACTUATOR ARRAY In this appendix, the center velocity of each slot (notation see Chapter 5) and corresponding current measurement of the actuator array are shown. 196 PAGE 197 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz] Velocity [m/s]slot 1a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 1a 50 Vpp 100 Vpp 150 Vpp B Figure E1. Hotwire measurement for actuator a rray slot 1a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 197 PAGE 198 0 500 1000 1500 2000 0 20 40 60 80 Frequency [Hz]Velocity [m/s]slot 1b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 1b 50 Vpp 100 Vpp 150 Vpp B Figure E2. Hotwire measurement for actuator a rray slot 1b. A) Center RMS velocity. B) Current measurement of the actuator array. 198 PAGE 199 0 500 1000 1500 2000 0 10 20 30 40 50 60 Frequency [Hz]Velocity [m/s]slot 2a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 2a 50 Vpp 100 Vpp 150 Vpp B Figure E3. Hotwire measurement for actuator a rray slot 2a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 199 PAGE 200 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 2b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 2b 50 Vpp 100 Vpp 150 Vpp B Figure E4. Hotwire measurement for actuator a rray slot 2b. A) Center RMS velocity. B) Current measurement of the actuator array. 200 PAGE 201 0 500 1000 1500 2000 0 10 20 30 40 50 Frequency [Hz]Velocity [m/s]slot 3a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 3a 50 Vpp 100 Vpp 150 Vpp B Figure E5. Hotwire measurement for actuator a rray slot 3a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 201 PAGE 202 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 3b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 3b 50 Vpp 100 Vpp 150 Vpp B Figure E6. Hotwire measurement for actuator a rray slot 3b. A) Center RMS velocity. B) Current measurement of the actuator array. 202 PAGE 203 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 4a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 4a 50 Vpp 100 Vpp 150 Vpp B Figure E7. Hotwire measurement for actuator a rray slot 4a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 203 PAGE 204 0 500 1000 1500 2000 0 20 40 60 80 Frequency [Hz]Velocity [m/s]slot 4b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 4b 50 Vpp 100 Vpp 150 Vpp B Figure E8. Hotwire measurement for actuator a rray slot 14b. A) Cent er RMS velocity. B) Current measurement of the actuator array. 204 PAGE 205 0 500 1000 1500 2000 0 10 20 30 40 50 60 70 Frequency [Hz]Velocity [m/s]slot 5a 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 5a 50 Vpp 100 Vpp 150 Vpp B Figure E9. Hotwire measurement for actuator a rray slot 5a. A) Cent er RMS velocity. B) Current measurement of the actuator array. 205 PAGE 206 206 0 500 1000 1500 2000 0 20 40 60 80 Frequency [Hz]Velocity [m/s]slot 5b 50 Vpp 100 Vpp 150 Vpp A 0 500 1000 1500 2000 0 0.01 0.02 0.03 0.04 0.05 0.06 Frequency [Hz]Current [Amps]slot 5b 50 Vpp 100 Vpp 150 Vpp B Figure E10. Hotwire measuremen t for actuator array slot 5b. A) Center RMS velocity. B) Current measurement of the actuator array. PAGE 207 APPENDIX F PARAMETRIC STUDY FOR OPENLOOP CONTROL In this appendix, a parametric study results for openloop control are shown. To illustrate the openloop control, a fixed fl ow condition (M=0.31) is chosen fo r all experimental cases. The frequencies of the excitation i nput signals to the actuator array are varied from 500 Hz to 1500 Hz, and for each frequency, two excitation volta ge levels, 100 Vpp and 150 Vpp, are chosen. 207 PAGE 208 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F1. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 500 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F2. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 500 Hz and 150 Vpp voltage. 208 PAGE 209 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F3. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 600 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F4. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 600 Hz and 150 Vpp voltage. 209 PAGE 210 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F5. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 700 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F6. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 700 Hz and 150 Vpp voltage. 210 PAGE 211 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F7. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 800 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F8. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 800 Hz and 150 Vpp voltage. 211 PAGE 212 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F9. OpenLoop control resu lt for M=0.31 and excitation si nusoidal input with frequency 900 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F10. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 900 Hz and 150 Vpp voltage. 212 PAGE 213 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F11. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1000 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F12. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1000 Hz and 150 Vpp voltage. 213 PAGE 214 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F13. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1100 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F14. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1100 Hz and 150 Vpp voltage. 214 PAGE 215 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F15. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1200 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F16. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1200 Hz and 150 Vpp voltage. 215 PAGE 216 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F17. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1300 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F18. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1300 Hz and 150 Vpp voltage. 216 PAGE 217 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F19. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1400 Hz and 100 Vpp voltage. 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F20. OpenLoop control result for M= 0.31 and excitation sinusoidal input with frequency 1400 Hz and 150 Vpp voltage. 217 PAGE 218 218 0 1000 2000 3000 4000 5000 6000 60 70 80 90 100 110 120 130 140 Frequency [Hz]UPL [dB] TE Baseline TE Open Loop Figure F21. 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