CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS

Transcription

1 CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS By QI SONG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008

2 2008 Qi Song 2

3 To my wife, Jingyan Wang; and my lovely son, Lawrence W. Song 3

4 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. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS...4 LIST OF TABLES...8 LIST OF FIGURES...9 LIST OF ABBREVIATIONS...4 ABSTRACT...6 CHAPTER INTRODUCTON...8 page Literature Review...20 Physical Models...2 Physics-Based Models...2 Numerical Simulations...27 POD-Type Models...28 On-Line System ID and Active Closed-Loop Control Methodologies...28 Unresolved Technical Issues...33 Technical Objectives...33 Approach and Outline SYSTEM IDENTIFICATION ALGORITHMS...38 Overview...38 SISO IIR Filter Algorithms...39 IIR OE Algorithm...40 IIR EE Algorithm...4 IIR SM Algorithm...4 IIR CE Algorithm...4 Recursive IIR Filters Simulation Results and Analyses...4 Accuracy comparison for sufficient system...43 Accuracy comparison for insufficient system...43 Convergence rate...44 Computational complexity...44 Conclusions...45 MIMO IIR Filter Algorithm GENERALIZED PREDICTIVE CONTROL ALGORIHTM...62 Introduction...62 MIMO Adaptive GPC Model

6 MIMO Adaptive GPC Cost Function...66 MIMO Adaptive GPC Law...66 MIMO Adaptive GPC Optimum Solution...67 MIMO Adaptive GPC Recursive Solution TESTBED EXPERIMENTAL SETUP AND TECHNIQUES...7 Schematic of the Vibration Beam Test Bed...7 System Identification Experimental Results...72 Computational Complexity...72 System Identification...73 Disturbance Effect...74 Closed-Loop Control Experiment Results...74 Computational Complexity...74 Closed-Loop Results...74 Estimated Order Effect...75 Predict Horizon Effect...76 Input Weight Effect...76 Disturbance Effect for Different SNR Levels During System ID...76 Summary WIND TUNNEL EXPERIMENTAL SETUP...90 Wind Tunnel Facility...90 Test Section and Cavity Model...9 Pressure/Temperature Measurement Systems...93 Facility Data Acquisition and Control Systems...94 Actuator System WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION...3 Background...3 Data Analysis Methods...5 Noise Floor of Unsteady Pressure Transducers...6 Effects of Structural Vibrations on Unsteady Pressure Transducers...6 Baseline Experimental Results and Analysis...7 Open-Loop Experimental Results and Analysis...8 Closed-Loop Experimental Results and Analysis SUMMARY AND FUTURE WORK...40 Summary of Contributions...40 Future Work...4 APPENDIX A MATRIX OPRATIONS

7 Vector Derivatives...43 Definition of Vectors...43 Derivative of Scalar with Respect to Vector...43 Derivative of Vector with Respect to Vector...43 Second Derivative of Scalar With Respect to Vector (Hessian Matrix)...44 Table of Several Useful Vector Derivative Formulas...44 Proof of the Formulas...45 Proof (a)...45 Proof (b)...45 Proof (c)...46 Proof (d)...46 The Chain Rule of the Vector Functions...48 The Derivative of Scalar Functions Respect to a Matrix...50 B CAVITY OSCILLATION MODELS...55 Rossiter Model...55 Linear Models of Cavity Flow Oscillations...56 Global Model for the Cavity Oscillations in Supersonic Flow...59 Global Model for the Cavity Oscillations in Subsonic Flow...63 C DERIVATION OF SYSTEM ID AND GPC ALGORITHMS...69 MIMO System Identification...69 Generalized Predictive Control Model...7 D A POTENTIAL THEORETICAL MODEL OF OPEN CAVITY ACOUSTIC RESONANCES...79 Mason s Rule...79 Global Model for a Cavity Oscillation in Supersonic Flow...80 Global Model for a Cavity Oscillation in Subsonic Flow...83 E CENTER VELOCITY OF ACTUATOR ARRAY...96 F PARAMETRIC STUDY FOR OPEN-LOOP CONTROL LIST OF REFERENCES...29 BIOGRAPHICAL SKETCH

8 LIST OF TABLES Table page 2- Summary of the IIR OE algorithm Summary of the IIR EE algorithm Summary of the IIR SM algorithm Summary of the IIR CE algorithm Simulation results of IIR algorithms for sufficient case Simulation results of IIR algorithms for insufficient case Simulation conditions of IIR algorithms for sufficient case Summary of the IIR/LMS algorithms Parameters selection of the vibration beam experiment Summary of the results of the adaptive GPC algorithm Physical and piezoelectric properties of APC 850 device Geometric properties and parameters for the actuator Resonant frequencies with respective centerline velocities for each input voltage A- Vector derivative formulas D- Components of the Mason s formula for supersonic case...89 D-2 Components of the Mason s formula for subsonic case...90 D-3 Components of the Mason s formula for subsonic case

9 LIST OF FIGURES Figure page - Schematic illustrating flow-induced cavity resonance for an upstream turbulent boundary layer Tam and Block (978) model of acoustic wave field inside and outside the rectangular cavity Classification of flow control Block diagram of system ID and on-line control Linear time-invariant (LTI) IIR Filter Structure Simulation structure of the adaptive IIR filter z-plane of the test model D plot of the MSOE performance surface of the insufficient order test system Contour plot of the MSOE performance surface Simulation results of weight track of the IIR algorithms for sufficient case Simulation results of weight track of the IIR algorithms for insufficient case Learning curve of IIR algorithms for sufficient case Computational complexity results from the experiment Model predictive control strategy Schematic diagram of the vibration beam test bed Schematic of the wind tunnel facility Schematic of the test section and the cavity model Schematic of the control hardware setup Bimorph bender disc actuator in parallel operation Designed ZNMF actuator array Dimensions of the slot for designed actuator array ZNMF actuator array mounted in wind tunnel

10 5-8 Bimorph 3 centerline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal The comparison plot of the experiment and simulation result of the actuator design code for bimorph Current saturation effects of the amplifier Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with α=0.25, κ=0.7) Schematic of a single periodic cell of the actuator jets and the proposed interaction with the incoming boundary layer Schematic of simplified wind tunnel and cavity regions acoustic resonances for subsonic flow Noise floor level comparison at different discrete Mach numbers with acoustic treatment at trailing edge floor of the cavity with L/D= x -acceleration unsteady power spectrum (db ref. g) for case with acoustic treatment and no cavity y -acceleration unsteady power spectrum (db ref. g) for case with acoustic treatment and no cavity z -acceleration unsteady power spectrum (db ref. g) for case with acoustic treatment and no cavity Spectrogram of pressure measurement (db ref. 20e-6 Pa) on the trailing edge floor of the cavity for the case with acoustic treatment and no cavity Spectrogram of pressure measurement (ref 20e-6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D= Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with α=0.25, κ=0.7) Noise floor of the unsteady pressure level at the surface of the trailing edge of the cavity with and without the actuator turned on Open-loop sinusoidal control results for flow-induced cavity oscillations at trailing edge floor of the cavity Running error variance plot for the system identification algorithm Closed-Loop active control result for flow-induced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity

11 6-3 Input signal of the Closed-Loop active control result for flow-induced cavity oscillations at Mach 0.27 at the trailing edge floor of the L/D =6 cavity Sensitivity function (Equation 4-) of the closed-loop control for M=0.27 upstream flow condition Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream flow condition with varying estimated order Unsteady pressure level of the closed-loop control for M=0.27, L/D=6 upstream flow condition with varying predictive horizon s Unsteady pressure level comparison between the open-loop control and closed-loop control for M=0.27 upstream flow condition...39 B- Schematic of Rossiter model B-2 Block diagram of the linear model of the flow-induced cavity oscillations...66 B-3 Block diagram of the reflection model B-4 Global model for the cavity oscillations in supersonic flow...67 B-5 Block diagram of the global model for a cavity oscillation in supersonic flow B-6 Global model for a cavity oscillation in subsonic flow B-7 Block diagram of the global model for a cavity oscillation in subsonic flow D- Global model for a cavity oscillation in supersonic flow...93 D-2 Block diagram of the global model for a cavity oscillation in supersonic flow...93 D-3 Signal flow graph of the global model for a cavity oscillation in supersonic flow...94 D-4 Global model for a cavity oscillation in subsonic flow D-5 Block diagram of the global model for a cavity oscillation in subsonic flow D-6 Signal flow graph of the global model for a cavity oscillation in subsonic flow E- Hot-wire measurement for actuator array slot a...97 E-2 Hot-wire measurement for actuator array slot b E-3 Hot-wire measurement for actuator array slot 2a...99 E-4 Hot-wire measurement for actuator array slot 2b

12 E-5 Hot-wire measurement for actuator array slot 3a...20 E-6 Hot-wire measurement for actuator array slot 3b E-7 Hot-wire measurement for actuator array slot 4a E-8 Hot-wire measurement for actuator array slot 4b E-9 Hot-wire measurement for actuator array slot 5a E-0 Hot-wire measurement for actuator array slot 5b F- Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 500 Hz and 00 Vpp voltage F-2 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 500 Hz and 50 Vpp voltage F-3 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 600 Hz and 00 Vpp voltage F-4 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 600 Hz and 50 Vpp voltage F-5 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 700 Hz and 00 Vpp voltage F-6 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 700 Hz and 50 Vpp voltage F-7 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 800 Hz and 00 Vpp voltage....2 F-8 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 800 Hz and 50 Vpp voltage....2 F-9 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 900 Hz and 00 Vpp voltage F-0 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 900 Hz and 50 Vpp voltage F- Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 000 Hz and 00 Vpp voltage F-2 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 000 Hz and 50 Vpp voltage

13 F-3 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 00 Hz and 00 Vpp voltage F-4 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 00 Hz and 50 Vpp voltage F-5 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 200 Hz and 00 Vpp voltage F-6 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 200 Hz and 50 Vpp voltage F-7 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 300 Hz and 00 Vpp voltage F-8 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 300 Hz and 50 Vpp voltage F-9 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 400 Hz and 00 Vpp voltage F-20 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 400 Hz and 50 Vpp voltage F-2 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 500 Hz and 00 Vpp voltage F-22 Open-Loop control result for M=0.3 and excitation sinusoidal input with frequency 500 Hz and 50 Vpp voltage

14 LIST OF ABBREVIATIONS D L M Cavity depth Cavity length Freestream flow Mach number U a m v w 0 r ζ α κ γ λ v τ a τ s ADC ARMA CARIMA CE DAC DNS DSP EE Freestream flow velocity Mean sound speed inside the cavity Mode number (integer number,2 ) Natural frequency of second order system Reflection coefficient Damping ratio Phase lag factor Proportion of the vortices speed to the freestream speed Ratio of the specific heats Spacing of the vortices Time delay inside the cavity Time delay inside the shear layer Analog to digital converter Autoregressive and moving-average Auto-regressive and integrated moving average Composite error Digital to analog converter Direct Numerical Simulations Digital signal processing Equation error 4

15 FFT FIR FRF GPC ID IIR JTFA LES LMS LQG LTI MIMO MPC MSOE OE PDF POD RANS RLS SISO SM SNR SPL STR TITO Fast Fourier transform Finite impulse response Frequency response function Generalized predictive control Identification Infinite impulse response Joint-time frequency analysis Large Eddy Simulations Least mean square Linear quadratic Gaussian Linear time-invariant Multiple-input multiple-output Model predictive control Mean square output error Output error Probability density function Proper orthogonal decomposition Reynolds Averaged Navier-stokes Recursive least square Single-input single-output Steiglitz and McBride Signal to noise ratio Sound pressure level Self-tuning regulator Two-input Two-output 5

16 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CLOSED-LOOP CONTROL OF FLOW-INDUCED CAVITY OSCILLATIONS Chair: Louis Cattafesta Major: Aerospace Engineering By Qi Song May 2008 Flow-induced cavity oscillations are a coupled flow-acoustic problem in which the inherent closed-loop system dynamics can lead to large unsteady pressure levels in and around the cavity, resulting in both broadband noise and discrete tones. This problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, and automobile sunroofs and windows. Researchers in both fluid dynamics and controls have been working on this problem for more than fifty years. This is because not only is the physical nature of this problem rich and complex, but also it has become a standard test bed for controller deign and implementation in flow control. The ultimate goal of this research is to minimize the cavity acoustic tones and the broadband noise level over a range of freestream Mach numbers. Although open-loop and closed-loop control methodologies have been explored extensively in recent years, there are still some issues that need to be studied further. For example, a low-order theoretical model suitable for controller design does not exist. Most recent flow-induced cavity models are based either on Rossiter s semi-expirical formula or a proper orthogonal decomposition (POD) based models. These models cannot be implemented in adaptive controller design directly. In addition, closedloop control of high subsonic and supersonic flows remains an unexplored area. 6

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 identification algorithm is derived and combined with the generalized prediction control (GPC) algorithm. The resultant integration of adaptive system ID and GPC algorithms can potentially track nonstationary cavity dynamics and reduce the flow-induced oscillations. A novel piezoelectric-driven synthetic jet actuator array is designed for this research. The resulting actuator produces high velocities (above 70 m/s) at the center of the orifice as well as a large bandwidth (from 500 Hz to 500 Hz) which is sufficient to control the Rossiter modes of interest at low subsonic Mach numbers. A validation vibration beam problem is used to demonstrate the combination of system ID and GPC algorithms. The result shows a ~20 db reduction at the single resonance peak and a ~9 db reduction of the integrated vibration levels. Both open-loop control and closed-loop control are applied to the flow-induced cavity oscillation problem. Multiple Rossiter modes and the broadband level at the surface of the trailing edge floor are reduced for both cases. The GPC controller can generate a series of control signals to drive the actuator array resulting in db reduction for the second, third, and fouth Rossiter modes by 2 db, 4 db, and 5 db, respectively. In addition, the broadband background noise is also reduced by this closed-loop controller (i.e., the OASPL reduction is 3 db). The relevant flow physics and active flow control actuators are examined and explained in this research. The limitations of the present setup are discussed. 7

18 CHAPTER INTRODUCTON Equation Section Flow-induced cavity oscillations have been studied for more than fifty years, and the problem has attracted researchers in both fluid dynamics and controls. First, this problem exists in many practical environments, such as landing gear bays and weapon delivery systems on aircraft, sunroofs and windows buffeting in automobiles, and junctions between structural and aerodynamic components in both (Kook et al. 997). The flow-acoustic coupling inherent in cavity resonance can lead to high unsteady pressure levels (both broadband noise and discrete tones), and can cause fatigue failure of the cavity and its contents. For example, the measured sound pressure levels in and around a weapons bay can exceed 70 db ref 20 μ Pa. For this reason, researchers are usually interested in suppression of flow-induced open cavity oscillations. Furthermore, this problem has become a standard test problem for designing, testing, and analyzing real-time feedback control systems. Although the standard rectangular cavity geometry is relatively simple, the physical nature of this problem is both rich and complex. Several good review articles on the flow-induced cavity oscillation problem are available in the literature (Rockwell, 978, Komerath 987, Colonius 200, Cattafesta 2003). Figure - 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, δ, and momentum thickness, θ, 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 8

19 thicker shear layer with broadband disturbances, which leads to overall lower levels of oscillations (Tam and Block 978; Colonius 200). 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 f, and the low speed or slow acoustic wave, downstream in the supersonic flow case. In the subsonic flow case, only the so-called E s, propagate disturbance wave,, propagates downstream. In both cases, the shear layer instability, S, E d develops based upon its initial conditions coupled by the upstream traveling acoustic feedback wave, U, inside the cavity (and E u outside the cavity for subsonic flow). Kelvin-Helmholtz-type (Tam and Block 978) convective instability waves develop and amplify in the shear layer as they propagate downstream and finally saturate due to nonlinearity. In particular, the instability waves grow and form large-scale vortical structures that convect downstream at a fraction of the freestream velocity. These structures then impinge near the trailing edge of open cavities ( L/ D< 0). The reattachment region acts as the primary acoustic source, which has been modeled as a monopole (Tam and Block 978) or a dipole (Bilanin and Covert 973) source. As a result, an upstream traveling acoustic wave, U, is generated inside the cavity. In subsonic flow, an additional acoustic wave, E u, 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 9

20 waves and the acoustic disturbances differ, only those waves that are in-phase ensure reinforcement of disturbances at that frequency. Therefore, this process is normally considered an introduction of a disturbance into the system, ultimately resulting in large-amplitude discrete tones inside and around the cavity. The measured broadband noise component is mainly due to the turbulent shear layer. * The relevant dimensionless parameters are: LD, LW, L θ, and shape factor H = δ θ with the freestream flow parameters, Reynolds number Re θ and Mach number M, all of which lead to tones (with Strouhal number St = fl U ) characterized by their strength as unsteady pressure normalized by the freestream dynamic pressure, prms q. In this study, three dimensional effects are not considered, since the cavity tones are generated by the interaction of the freestream flow and the longitudinal modes (coupled with vertical depth modes). Width modes are not relevant in this feedback loop if the width is small enough to prevent higher-order spanwise modes but large enough so that the mean flow over the cavity length is approximately two dimensional. Note that the width of the cavity does affect the amplitude of the cavity oscillations (Rossiter 966) but is of secondary importance (Cain 999). Therefore, a two-dimensional model is reasonable from a physical perspective even though the unsteady turbulent motion is inevitably three-dimensional (Bilanin and Covert 973). Literature Review In this section, some published results related to the physics of flow-induced cavity oscillations are discussed. Since the ultimate goal of this research is to minimize the cavity acoustic tones and perhaps the broadband noise level, potential control methodologies and algorithms are also reviewed. A recent review paper by Cattafesta et al. (2003) gives a summary of the various passive and open-loop cavity suppression studies. 20

21 Physical Models In order to suppress the discrete tones and the broadband acoustic level of flow-induced open cavity resonance, an understanding of the physics is essential. From a control engineer s point of view, a simplified and low-order model is desirable in order to predict the resonant frequencies and amplitudes over a broad range of the governing dimensionless parameters. Physics-Based Models Rossiter (964) performed an extensive experimental study on the measurement of the unsteady pressure in and around a rectangular open cavity ( 2 ft.5 ft) in a subsonic and transonic freestream air flow ( 0. 5 M.2 ). He observed broadband noise and a series of unsteady acoustic tones generated in the cavity. For the deeper cavities ( ), there was usually a single dominant tone, and the dominant frequency was observed to jump between different cavity tones. For the shallower cavities ( L/ D > 4), two or more peaks were often observed and were approximately equal in magnitude. He proposed that the flow entering the cavity caused the external stream to accelerate, and then the flow decelerated near the reattachment region. As a result, pressure was lower near the separation region (leading edge) and higher near the reattachment region (trailing edge). As a result, he suggested that large eddies developed within the cavity due to this pressure gradient. He also used shadowgraphs to illustrate that the shear layer separates from the cavity leading edge, and the instability waves develop into discrete vortices that are shed at regular time internals from the front lip of the cavity (at Mach number 0.6 and with two-dimensional cavity L/ D< 4 L/ D = 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. 2

22 When the frequency of one of these components is close to the natural frequency of the cavity, resonance occurs. In his study, Rossiter gave a semi-empirical formula for predicting the resonant frequencies of these peaks at a specific Mach number. The derivation of the Rossiter equation is given in Appendix B, and the resulting formula for the dimensionless Strouhal number is St ( α ) f L = m m = v U + M κ (-) where f m is the resonance frequency for integer mode m, L is the length of the cavity, U is the freestream velocity, α is the phase lag factor (in fractions of a wavelength), κ is the ratio of the vortex propagation speed to the freestream velocity, and M is the freestream Mach number. Empirical constant values of κ = 0.57 and α = 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 κu of the vortices is a weak function of M, L θ v and D θ (Colonius 200). Different integer values m v give different frequencies, commonly referred to as shear layer or Rossiter modes. In conclusion, Rossiter s formula is based on an integer number of 2π phase shifts, 2kπ, around a resonant feedback loop consisting of a downstream unstable shear layer disturbance and an upstream feedback acoustic wave inside the cavity. This phase shift is a necessary condition for self-sustaining oscillations (Cattafesta et al. 999a). However, Rossiter s expression does not account for the depth or width of the cavity and only successfully predicts the longitudinal cavity resonant frequencies at moderate-to-high Mach numbers. It also does not predict the amplitude of the oscillations. 22

23 Heller and Bliss (975) 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 St ( α ) fml m = = v U M + κ γ 2 + M 2 (-2) where γ 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 (973) 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. 23

24 Tam and Block (978) carried out extensive experimental investigations at low subsonic Mach numbers ( M < 0.4 ) and postulated that vortex shedding was probably not the main factor for cavity resonance over the entire Mach number range. They made two key assumptions, namely that the rectangular cavity flow was two-dimensional, and the mean flow velocity inside the cavity was zero. These two assumptions were based on experimental evidence of little correlation between the mean flow and the acoustic feedback inside the cavity. Tam and Block proposed a process of flow-induced cavity oscillations as follows. The shear layer oscillated up and down at the trailing edge of the cavity. The upward movement was uncorrelated with the generation of the acoustic waves, because if the shear layer covered the trailing edge, then the external flow passes over the trailing edge without impingement. They argued that only the downward motion of the shear layer into the cavity caused significant generation of pressure waves and subsequent radiation of acoustic waves in all directions ( Figure -2). For example, some of the waves radiating into the external flow (e.g., wave A) were argued to have minor effects on the oscillations inside the cavity. However, the effect of the waves propagating inside of the cavity was deemed more significant. The resulting acoustic waves included the upstream propagating waves (e.g., wave C) and the reflected waves from the floor (e.g., wave F) and the upstream wall (e.g., wave E). Subsequent reflections of the acoustic waves by the walls, the cavity, or the shear layer were deemed negligible. They concluded that the directly radiated wave and the first reflected waves by the floor and upstream end wall of the cavity provided the energy to excite the instability waves of the shear layer. These disturbances within the shear layer were then amplified as the instability waves propagate downstream. When the disturbances amplitudes became large, nonlinear effects were important and ultimately established the amplitude of the discrete tones. A mathematical model of the cavity oscillation 24

25 and acoustic field were developed. In order to calculate the phases and waves generated at the trailing edge, a periodic line source was simulated at the trailing edge of the cavity. In addition, the reflections of the acoustic waves by the cavity walls were modeled by periodic line image sources about the cavity walls. Their model accounts for the finite shear layer thickness effects and produces a more accurate estimation of the resonance frequencies than Rossiter s model. However, their resulting model is complicated and difficult to employ for control law design. Rowley et al. (2002 b, 2003, 2006) provided an alternative viewpoint for understanding flow-induced cavity oscillations. They showed that self-sustained oscillations existed only under certain conditions. The resonant frequencies were due to the instabilities in the shear layer interacting with the flow and acoustic fields. The amplitude of the oscillations was determined by nonlinear saturation. However, at other conditions, the cavity oscillations could be represented as a lightly damped but stable linear system. The oscillations were caused by the amplification of external disturbances via the closed-loop dynamics of the cavity. The amplitude of each mode was determined by the amplitude of the external forcing disturbances and some frequency-dependent gain of the system. They modeled the dynamics of the shear layer as a second-order system and the acoustic propagation process via a one-dimensional, standing-wave model. The impingement and receptivity procedures were simply modeled as a constant unity gain. Finally, the Rossiter formula was derived under some specific conditions. The derivation of this model is provided in Appendix B. They also used Gaussian white noise as input and examined the probability density function (PDF) and the phase portrait of the output pressure signal at different Mach numbers. Their results showed that under some conditions, the self-sustained regime of Rossiter modes was valid. However, at other conditions, called the forced regime, open cavity oscillations may 25

26 be represented as lightly damped stable linear systems. 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 promising global model to describe the flow-induced cavity oscillation problem for two different flow patterns (Figure - ). Their model combined scattering analyses for the two ends of the cavity and a propagation analysis of the cavity shear layer, internal region of the cavity, and acoustic near-field. They solved a matrix eigenvalue problem to identify the resonant frequencies of the cavity oscillation. From their resulting characteristic functions, four and twelve closed loops could be identified for the supersonic flow and subsonic flow cases, respectively. One more feedback loop makes the subsonic flow much more complex than the supersonic flow. For example, some of these closed loops were major loops, such as closed loop S, U and U, D ( Figure -), 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 s, and E f, also have critical effects on the amplitude of the oscillations. This global model provides more insights for controller design. A detailed derivation of this model is provided in Appendix B. Clearly, the physics-based models described above provide physical insight concerning flow-induced cavity oscillations. However, the original Rossiter model and the global model derived by Kerschen and Tumin. (2003) can only estimate the resonance frequencies of the cavity flow. The linear model derived by Rowley et al. (2002b, 2003, 2006) is transfer function based model but is not sufficiently accurate to design a control system. A transfer function based 26

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 first constructed from the block diagram of the Kerschen et al s physical model, and then Mason s rule (Nise 2004) is applied to obtain the transfer function from the disturbance input to the selected system output. This method can give predictions for both the resonant frequencies of the flow-induced cavity oscillations and the amplitude of the cavity tones. In addition, this method also provides a linear estimate for the system transfer function from the disturbance input at the leading edge and the pressure sensor output within the cavity walls. Therefore, this model is a potential global model for controller design in this research. Numerical Simulations Some computational fluid dynamic (CFD) methods, such as Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) and Reynolds Averaged Navier-stokes (RANS), provide useful information for understanding the issues of physical modeling of cavity oscillations. A review paper by Colonius (200) gives a summary of issues related to each of these topics. More recent research on these topics can be found by Rizzetta et al. (2002, 2003) and Gloerfelt (2004). The Detached Eddy Simulation (DES) method, which is a involves a hybrid turbulence modeling methodology, has also been used to calculate the flow and acoustic fields of the cavity (Allen and Mendonca 2004; Hamed et al. 2003, 2004). Another hybrid RANS-LES turbulence modeling approach is presented by Arunajatesan and Sinha (200, 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. 27

28 POD-Type Models The previous analytical physical models are not accurate enough to design a control system. Furthermore, CFD methods are far too computationally intensive at the present time to provide a reasonable framework to design and test potential controllers. This translates into the need for new methods to develop more accurate reduced-order models. Therefore, simulation and experimental data based models were proposed and later used for the controller design. Rowley et al. (200) introduced a nonlinear dynamical model for flow-induced rectangular cavity oscillations, which was based on the method of vector-valued proper orthogonal decomposition (POD) and Galerkin projection. The POD method obtains low-dimensional descriptions of a high-order system (Chatterjee 2000). For the cavity flow problem, data resulting from the temporal-spatial evolution of the numerical simulations or experiments is used to construct a low-order subspace system that captures the main features (coherent structures) of the cavity flow. A more detailed explanation of POD methods for cavity flow are given by Rowley et al. (2000, 200, 2002c, 2003a). Some of the control methodologies discussed in next section can be constructed based on the resultant model obtained by POD (Caraballo et al. 2003, 2004, 2005; Samimy et al. 2003, 2004; Yuan et al. 2005). Instead, we turn our attention to an alternative experimental-based modeling approach that employs system identification techniques. Here, the nonlinear infinite-dimensional governing equations are modeled by a reduced set of differential (in continuous) or difference (in discrete time) equations. This method is the focus of this study and is discussed in the following section. On-Line System ID and Active Closed-Loop Control Methodologies Previous studies aimed at suppression of the flow-induced cavity tones have employed mainly passive or open-loop active flow control methodologies. The standard classification of the flow control techniques is shown in Figure -3. The review paper by Cattafesta et al. (2003) 28

29 provides a detailed overview of various passive and open-loop control methodologies. However, passive and open-loop approaches are only effective for a limited range of flow conditions. Active feedback flow control has recently been applied to the flow-induced cavity oscillation problems. The closed-loop control approaches have advantages of reduced energy consumption (Cattafesta et al. 997), no additional drag penalty, and robustness to parameter changes and modeling uncertainties. In general, closed-loop flow control measures and feeds back pressure fluctuations at the surface of the cavity walls (or floor) to an actuator at the cavity leading edge to suppress the cavity oscillations in a closed-loop fashion. In general, past active control strategies have taken one of two approaches for the purpose of reducing cavity resonance. First, they can thicken the boundary layer in order to reduce the growth of the instabilities in the shear layer. Alternatively, they can be used to break the internal feedback loop of the cavity dynamics. Most closed-loop schemes exploit the latter approach. Early closed-loop control applications used manual tuning of the gain and delay of simple feedback loops to suppress resonance (Gharib et al. 987; Williams et al. 2000a,b). Mongeau et al. (998) 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. (2003) proposed a simple logic-based controller for closed-loop cavity flow control. Low-order model-based controllers with different bandwidths, gains and time delays have also been designed and implemented (Rowley et al. 2002, 2003, Williams et al. 2002, Micheau et al. 2004, Debiasi et al. 2004). Linear optimal controllers (Cattafesta et al. 997, Cabell et al. 2002, Debiasi et al. 2004, Samimy et al. 2004, Caraballo et al. 2005) have been successfully designed for operation at a single flow condition. These models are all based on reduced-order system models, and most of these controller design methods are based on 29

30 model forms of the frequency response function, rational discrete/continuous transfer function, or state-space form. However, the coefficients of these model forms are assumed to be constant, and this assumption requires that the system is time invariant or at least a quasi-static system with a fixed Mach number. Although the physical models of flow-induced cavity oscillations have been explored extensively, they are not convenient for control realization. This is because these models are highly dependent on the accuracy of the estimated internal states of the cavity system. In addition, cavity flow is known to be quite sensitive to slight changes in flow parameters. So a small change in Mach number can deteriorate the performance of a single-point designed controller (Rowley and Williams 2003). Therefore, adaptive control is certainly a reasonable approach to consider for reducing oscillations in the flow past a cavity. Adaptive control methodology combines a general control strategy and system identification (ID) algorithms. This method is thus potentially able to adapt to the changes of the cavity dimension and flow conditions. It updates the controller parameters for optimum performance automatically. The structure of this method is illustrated in Figure -4. Two distinct loops can be observed in the controller. The outer loop is a standard feedback control system comprised of the process block and the controller block. The controller operates at a sample rate that is suitable for the discrete process under control. The inner loop consists of a parameter estimator block and a controller design block. An ID algorithm and a specified cost function are then used to design a controller that will minimize the output. The steps for real-time flow control include: (i) Use a broadband system ID input from the actuator(s) and the measured pressure fluctuation output(s) on the walls of the cavity to estimate the system (plant and disturbance) parameters. (ii) 30

31 Design a controller based on the estimated system parameters. (iii) Control the whole system to minimize the effects of the disturbance, measured noise, and the uncertainties in the plant. Based on this adaptive control methodology, some adaptive algorithms adjust the controller design parameters to track dynamic changes in the system. However, only a few researchers have demonstrated the on-line adaptive closed-loop control of flow-induced cavity oscillations. Cattafesta et al. (999 a, b) applied an adaptive disturbance rejection algorithm, which was based upon the ARMARKOV/Toeplitz models (Akers and Bernstein 997; Venugopal and Bernstein 2000, 200), to identify and control a cavity flow at Mach 0.74 and achieved 0 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 (200) applied an adaptive filtered-x LMS algorithm to the cavity problem and demonstrated multiple cavity tone suppression at Mach number up to 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 (998) 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 They also summarized the typical finite-impulse response (FIR) and infinite-impulse response (IIR) based system ID algorithms. They concluded that the FIR filters used to represent the flow-induced cavity process were unsuitable. On the other hand, IIR models were able to model the dynamics of the cavity system. LMS adaptive algorithm was more suitable for real-time control than the recursive-least square (RLS) adaptive algorithm due to its reduced computational complexity. Recently, more advanced controllers, 3

32 such as direct and indirect synthesis of the neural architectures for both system ID and control (Efe et al. 2005) and the generalized predictive control (GPC) algorithm (Kegerise et al. 2004), have been implemented on the cavity problems. From a physical point of view, the closed-loop controllers have no effect on the mean velocity profile (Cattafesta et al. 997). However, they significantly affect streamwise velocity fluctuation profiles. This control effect eliminates the strength of the pressure fluctuations related to flow impingement on the trailing edge of the cavity. Although closed-loop control has provided promising results, the peaking (i.e., generation of new oscillation frequencies), peak splitting (i.e., a controlled peak splits into two sidebands) and mode switching phenomena (i.e., non-linear interaction between two different Rossiter frequencies) often appear in active closedloop control experiments (Cattafesta et al. 997, 999 b; Williams et al. 2000; Rowley et al 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. (999), Hong and Bernstein (998), and Kegerise et al. (2004). Rowley et al. (2002b, 2003) concluded that if the viewpoint of a linear model was correct, a closed-loop controller could not reduce the amplitude of oscillations at all frequencies as a consequence of the Bode integral constraint. Banaszuk et al. (999) gave explanations of the peak-splitting phenomenon. They claimed that the peak splitting effect was caused by a large delay and a relatively low damping coefficient of the open-loop plant. Cabell et al. (2002) explained these phenomena by the combination of inaccuracies in the identified plant model, high gain controllers, large time delays and uncertainty in system dynamics. In addition, narrow-bandwidth actuators and controllers may also lead to a peak-splitting phenomenon (Rowley et al. 2006). 32

33 Hong and Bernstein defined the closed-loop system disturbance amplification (peaking) phenomenon as spillover. They illustrated that the spillover problem was caused by the collocation of disturbance source and control signal or the collocation of the performance and measurement sensors. For this reason, the reduction of broadband pressure oscillations was not possible if the control input was collocated with the disturbance signal at the leading edge of cavity. Therefore, Kegerise et al. (2004) suggested a zero spillover controller which utilized actuators at both the leading and trailing edges of the cavity for closed-loop flow control. Unresolved Technical Issues Although the flow-induced cavity oscillation problem has been explored extensively, there are still some unresolved issues that need to be studied further. A suitable theoretical model does not exist that estimates both the discrete frequencies as well as the amplitude of the peaks. A feedback controller that reduces both broadband and tonal noise over a wide range of Mach numbers has not been achieved. An adaptive zero spillover control algorithm may reduce both the tones and broadband acoustic noise associated with cavity oscillations. The necessity for a high-order system model is a critical problem for controller design and implementation, because this high-order system results in significant computational complexity for application in digital signal processing (DSP) hardware. As such, the convective delays between the control inputs and the pressure sensor outputs must be specifically addressed in the control architecture. Closed-loop control of high subsonic and supersonic flows is an unexplored area. Technical Objectives According to those unresolved technical issues, the ultimate goals of this dissertation are summarized as follows. A feedback control methodology will be developed for reducing flow-induced cavity oscillation and broadband pressure fluctuations. Adaptive system ID and control algorithms will be combined and implemented in realtime. 33

34 The relevant flow physics and the design of appropriate active flow control actuators will be examined in this research. The performance, adaptability, costs (computational and energy), and limitations of the algorithms (spillover, etc.) will be investigated. Approach and Outline In order to achieve these objectives, some design and application approaches warrant additional consideration. First, a potential theoretical model of cavity acoustic resonances is derived based on the global model of Kerschen and Tumin. (2003). This model (derived in Appendix D) provides the framework to estimate the amplitudes and frequencies of the cavity tones. This model has a low system order and also accounts for the convective delay between the disturbance input and the output pressure measurement. Second, during the controller design, the controlled system is a continuous system; therefore, all the sensors measurements and the actuators inputs are analog signals. However, for the present real-time application, the control algorithms are implemented using a DSP. For this reason, additional hardware, such as analog-to-digital converters (ADC), digital-to-analog converters (DAC), anti-aliasing filters, and power amplifier, must be included in the whole control design procedure. Finally, multiple actuators and multiple sensors are employed in this study in order to design an adaptive zero spillover control algorithm to explore the possibility of achieving broadband acoustic noise reduction in addition to suppression of the cavity tones themselves. This active control method development procedure can be summarized as the following stages according to Elliott (200). Study the simplified analytical system model and understand the fundamental physical limitations of the proposed control strategy. Obtain the sensor output and derive the states or coefficients from the system ID algorithms using off-line or on-line methods. 34

35 Calculate the optimum performance using different control strategies and find the control law for realization. Simulate the different control strategies and tune the candidate controller for different operating conditions. Implement the candidate controller in real-time experiments. The thesis is organized as follows. Several SISO IIR system ID algorithms and a more general MIMO system ID algorithm are derived and discussed in the next Chapter. Then the MIMO adaptive GPC algorithm is described in Chapter 3. This is followed by a description of the sample experimental setup and the discussion of preliminary experimental results. Chapter 5 describes the wind tunnel facilities and the data processing methods. Wind tunnel experimental results for both open-loop (baseline) and closed-loop are then presented and discussed in Chapter 6. Finally, the conclusions and future work are presented in Chapter 7. 35

36 M > Turbulent Boundary Layer δ y Es x E f S D U D L M < Turbulent Boundary Layer δ y Ed x A S Eu D U D B Figure -. Schematic illustrating flow-induced cavity resonance for an upstream turbulent boundary layer. A) In supersonic flow.b) In subsonic flow. L y A U x Simulated Line Source E F C D Figure -2. Tam and Block (978) model of acoustic wave field inside and outside the rectangular cavity. L 36

37 Flow Control Approaches Passive Control Active Control Open-Loop Closed-Loop Quasi-Static Dynamic Figure -3. Classification of flow control. (after Cattafesta et al. 2003) Uncertainties disturbance Measured Noise reference control part Controller input u(k) Plant output y(k) process part Control Parameters ID part Controller Design ID input u(k) Parameter Estimator System Parameters Figure -4. Block diagram of system ID and on-line control. 37

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 filters are chosen as the candidate digital filters. These algorithms are applied to an example from Johnson and Larimore (977) for simulation analysis. Then, four interested aspects of these filters, accuracy, convergence, computational complexity, and robustness, are examined and summarized. Finally, a more general MIMO system ID algorithm is derived from one of the promising SISO system ID algorithms. The resulting model is used to combine with the MIMO adaptive GPC model which is discussed in next chapter. Overview As discussed in the first chapter, IIR structure filter is an applicable mathematical model to capture the cavity dynamics. Furthermore, this kind of structure can be a starting point and easily combined with many controller design strategies. Therefore, in this Chapter, several system ID algorithms based on the IIR filter structure are examined. The ideal of the system ID is to construct a predefined IIR structure filter, which has the similar frequency response of the actual dynamic system, using the information from the previous and present input and output time series data of the dynamic system. In general, the system ID algorithms fall into two big categories, the batch method and the recursive method. The batch method directly identifies the final system parameters in one-time calculation using a block data from the input and a block data from the output. Nevertheless, the recursive method updates the estimated system parameters within each sampling period using the latest input and output data in time domain. At each iteration of calculation, the system parameters may not be the optimal values. However, these 38

39 estimated parameters will finally converge to the true values of the system internal states. Successful identifying the system internal states depends on two major assumptions. First, the input signal and the output signal must have a good correlation. Then, the system ID model has the same structure of that of the estimated system model. The recursive method is more attractive for present experiment, because this updating method is more suitable for on-line implementation and it can also track the change of the system dynamics. Furthermore, the computational complexity of recursive method is much lower than the batch method. SISO IIR Filter Algorithms Netto and Diniz (995) give a summary of some popular adaptive IIR filter algorithms. In this section, the Output Error (OE), Equation Error (EE), Steiglitz and McBride (SM), and Composite Error (CE) algorithms are selected and illustrated. The general structure of an IIR filter is shown in Figure 2-. The filter output may be expressed as nˆa nˆb yk ˆ( ) = ayk ˆ ˆ( i) + bxk ˆ ( j) i i= j= 0 T = φˆ ( k) θˆ ( k) j (2-) where ˆ represents the estimation values. a and b are the adjustable coefficients of the model, while n and n is the estimated order of the feedback loop and forward path, ˆa ˆb ˆi ˆj respectively. [ ˆ ] Τ φ( ˆ k)= y(k -i) x(k- j), θ ˆ( k) = aˆ ˆ i b j, and i =,...,n ˆ ˆ a; j = 0,,...,nb. T This IIR filter structure, Equation 2-, is the same as the autoregressive and movingaverage (ARMA) model (Haykin 2002). 39

40 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 (995). In their simulations, they use an insufficient model, which models a second-order system using a first-order system to test each algorithm. The results from their paper show that the Modified Output Error (MOE) algorithm may converge to a meaningless stationary point. The same result is also shown by Johnson and Larimore (977). The Simple Hyper-stable Algorithm for Recursive Filters (SHARF) algorithm, the modified SHARF algorithm, and the Bias Remedy Least-Mean-Square Equation Error algorithm (BRLE) also show poor convergence rates. The Composite Regressor (CR) algorithm has similar problems as the MOE algorithm, since this algorithm combines the EE and MOE methods. Therefore, in this section, tests of these poor performing algorithms are not discussed. Fundamentally, there are two approaches for an adaptive IIR filter, the OE algorithm and the EE algorithm, which have been derived by Haykin (2002) and Larimore et al. (200), respectively. Many other adaptive IIR filter algorithms are mainly derived from these algorithms, or combine some good features from the OE and the EE filters. Therefore, a summary of each of these two algorithms is provided in the following section. Two other algorithms, the Steiglitz-McBride algorithm (SM) and the Composite Error algorithm (CE), are also introduced, because both these algorithms also show good performance in our Simulink simulations. IIR OE Algorithm The IIR OE algorithm is summarized in Table 2-. To ensure the stability of the algorithm, generally, the upper bound of step size μ is set to 2, where λ λ max is the max 40

41 maximum eigenvalue of the autocorrelation matrix of the regress vector φ ˆ ( k OE ). The step sizes of the following algorithms are also satisfying this criterion. Furthermore, in order to guarantee the convergent approximation of α and β, this algorithm requires slow adaptation rates for small values of n and n (Haykin 2002). IIR EE Algorithm ˆa The IIR EE algorithm is summarized in Table 2-2. Since the desired response is the supervisory signal supplied by the actual output of plant during the training period, the EE algorithm may lead to faster convergence rate of the adaptive filter (Haykin 2002). IIR SM Algorithm The IIR SM algorithm is summarized in Table 2-3. Since the EE algorithm and the OE algorithm possess their own advantages as well as drawbacks (discuss later), the motivation of the SM algorithm is to combine the desirable characteristics of the OE and the EE methods. IIR CE Algorithm This algorithm tries to combine both the EE algorithm and the OE algorithm in another way. As shown in Table 2-4, a parameter β 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. ˆb j i 4

42 The setup for the following Simulink simulation is shown in Figure 2-2. A Gaussian broad band white noise with zero mean and unity variance is chosen as the reference input signal. The prototype test model is a second-order dynamical system (Johnson and Larimore 977) with the transfer function H z ( ) z = (2-2) z 0.25z From the z-plane plot (Figure 2-3), it clearly shows that this test model is a stable and non-minimum phase system, which has two real poles at z = 0.30 and z = , and two zeros at z = 0 and z = 8. In the following simulations, a sufficient order identification problem is firstly examined, which means a second-order system model with the transfer function ˆ ˆ ˆ b ( k) + b( k) z H ( z, k ) = aˆ ( k) z 2 aˆ ( k) z 0 2 is used to estimate the test model. Then, an insufficient order identification problem is investigated. This approach uses a first-order system model with the transfer function ˆ ˆ 0 ( ) ( b k H z, k ) = to estimate the test aˆ ( k) z model. The mean square output error (MSOE) surface of this insufficient order dynamical system is obtained by Shynk (989) ˆ ˆ b MSOE = σ 2 b H ( aˆ ) + (2-3) 2 0 y 0 2 aˆ A 3D surface plot and a contour plot of the MSOE performance surface are shown in Figure 2-4 and Figure 2-5, respectively. The plots show that the MSOE surface of the test model is bimodal with a global minimum (denoted by * ) 42

43 * * * at ( b, a ) = ( 0.3,0.906), which yields MSOE = 0.277, and a local minimum (marked + + by + ) at ( b, a ) = (0.4, 0.59), which corresponds to MSOE + = The input x( k ) and the test model output y( k ) (with or without disturbance vk ( ) ) are introduced to the adaptive IIR filter algorithms at the same time. The adaptive IIR filter algorithms calculate the error signal and update the weights at each iteration. Accuracy comparison for sufficient system Table 2-5 and Figure 2-6 show the simulation results and weight tracks of the four IIR algorithms for the sufficient case, respectively. For the sufficient case, the algorithms minimize the mean square error between the system output and the filter output, and the estimated weights converge to the original coefficients of the test model. Accuracy comparison for insufficient system The simulation results and weight tracks of the IIR algorithms for the insufficient case are shown in Table 2-6 and Figure 2-7, respectively. The OE algorithm starts from two different initial conditions. One point is closer to the global minimum, and the other one is closer to the local minimum. This method adjusts its weights via stochastic gradient estimation to the closest stationary point of the initial condition. Similarly, two initial conditions are selected for EE algorithm. One of them is close to the global minimum, and the other one is much closer to the local minimum. This algorithm can avoid the local minimum and adjust its weights to let the final mean square error value arrive at the area near the global minimum. However, for this insufficient order situation, the final solution exhibits bias compared to the optimum solution. 43

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 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 β 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 β 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 β is close to, the biased characteristic of the EE algorithm is apparent in the results. Convergence rate For convergence rate comparison, the same step size and number of iterations are chosen for simulations. The simulation conditions and the learning cures of the IIR algorithms for the sufficient case are shown in Table 2-7 and Figure 2-8, respectively. Obviously, the EE and SM algorithms converge faster than the OE and CE algorithms. Computational complexity In order to apply the ID algorithm on an adaptive control algorithm for real-time implementation, the computational complexity for one iteration of the ID algorithm have to be less than the sampling time of the DPS processor used for real-time experiment. Four algorithms are compared for computational complexity by the turnaround time with the increase of the number of unknown for each algorithm (Figure 2-9). The hardware used for experiment is PowerPC 750 (480MHz) microprocessor (2.6 SPECfp95). The 44

45 experimental results are shown in Figure 2-9. The computational complexity of all of the IIR algorithms is approximately linear. And the CE algorithm needs more computational time for each iteration than time requirements for the other three algorithms. Conclusions Varies of IIR adaptive filters are examined in this Chapter, the objective of these digital filters is to identify the system coefficients (internal states) from the input and output signals. The OE algorithm and the EE algorithm are two basic structures of an adaptive IIR filter. Beyond that, two other algorithms, the SM algorithm and the CE algorithm, are also examined. Simulation results show that the mean square error value calculated by the OE algorithm converges to the optimum solution for both the sufficient case and the insufficient case if the proper initial condition is chosen. This means that the OE algorithm may converge to local minima in the MSOE surface. Furthermore, this algorithm does not guarantee that the poles of the ARMA model always lie inside the unit circle in the z-plane. Thus, the OE method may become unstable (Haykin 2002) during the experiment. Therefore, a small enough step size and stability monitoring are required to ensure the convergence of the algorithm. However, the optimum step size is unknown, and the stability monitoring highly increases the computational complexity. These are the main drawbacks that should be considered in applications. The mean square error value calculated by the EE algorithm avoids the local minima and converges to the global minimum in the MSOE surface. The convergent rate and the computational complexity are good for real-time implementation. Unfortunately, the final solution is biased when the test model uses a lower-order system to model a higher-order system (Shynk 989, Netto and Diniz 995). 45

46 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 992). Moreover, the CE algorithm produces good results when 0.04 < β <. Within these limits, the algorithm is unimodal, and the bias is negligible (Netto and Diniz 992). However, the stability of the CE algorithm must still be monitored, and the computational complexity is also high for this algorithm. A summary of the four algorithms is given in Table 2-8. The robustness results of each ID algorithms come from the experiment discussed in Chapter 4. As the results, the EE algorithm is the best algorithm comparing to the other three ID algorithms. Therefore, in next step, a MIMO IIR filter is going to be derived based on this algorithm. MIMO IIR Filter Algorithm In this section, a MIMO system ID algorithm is developed based on the SISO IIR EE algorithm. First, a linear system model is summed with the r inputs [ u ] r and the m outputs [ y ] m. 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) = α y( k ) + α y( k 2) + + α y( k p) 2 + β u( k) + β u( k ) + β u( k 2) + + β u( k p) 0 2 p p (2-4) where 46

47 u( k) y ( k) u2( k) y2( k) u ( k) = [ u( k) ] =,y( k) = [ y( k) ] = r m u ( k) y ( k) [ ] [ ] r r m m α = α, α = α,, α = α β = [ β ], β = [ β ],, β = β m m 2 2 m m p p m m 0 0 m r m r p p m r (2-5) Define the observer Markov parameters and the regression vector θ = α α β β (2-6) ( k) p 0 p [ * *( ) ] [ m m* p+ r*( p+ ) y( k ) y( k p) ϕ ( k) = (2-7) u( k) u( k p) m p+ r p+ substituting Equation 2-6 and Equation 2-7 into Equation 2-4 yields a matrix equation for the filter outputs [ ] Furthermore, the errors are defined as yˆ( k) = ˆ( k) θ ϕ( k) m m ( m* p+ r*( p+ )) [ ] [ ] [ ( m* p r*( p )) ] ] + + (2-8) [ ( )] = [ yˆ ( )] [ y( )] ε k k k m m m (2-9) Finally, the observer Markov parameters 2-6 can be identified recursively by ˆ ˆ T θ( k+ ) = θ( k) με( k) ϕ ( k) (2-0) In order to automatically update the step size, choose μ = σ + ϕ 2 (2-) 47

48 where σ is a small number to avoid the singularity when ϕ is zero. 2 follows The main steps of the MIMO identification for one iteration are summarized as Step : Initialize [( * *( )) ] [ 0] ˆ( θ k ) =. m m p+ r p+ Step 2: Construct regression vector [ ( k) ] [ m* p + r*( p + ) Step 3: Calculate the output error [ ( k )] m ϕ according to Equation 2-7. ] ε according to Equation 2-9. Step 4: Calculate the step size according to Equation 2-. Step 5: Update the observer Markov parameters matrix [ ( k) ] m [ m * p + r ( p + ) θ according to Equation 2-0. Then, the calculation for the next iteration goes back to step 2. The detail derivation for this MIMO ID algorithm is given in Appendix B. And the experimental results of the algorithm, the computational complexity, and the disturbance effects will be discussed in later Chapters. The calculation result of this MIMO ID gives an estimated model of the system with the form of Equation 2-4. In the following Chapter, a MIMO control algorithm is developed based on this MIMO ID model. ] 48

49 Table 2-. Summary of the IIR OE algorithm. Initialization: θˆ ( k) = [ aˆ (0) bˆ (0)] T = 0, where i =,..., nˆ, j = 0,,..., nˆ Computation: For k =,2,... nˆa i nˆb j yˆ ( k) = aˆ yˆ ( k i) + bˆ x( k ) OE i OE j i= j= 0 e ( k) = y( k) yˆ ( k) OE OE j nˆ a α ( ) ˆ ( ) ˆ i k yoe k i + akαi( k l), for i =,..., nˆa l= Define: nˆ a β ( ) ( ) ˆ ( ), 0,,..., ˆ j k x k j + akβ j k l for j = n l= φ ˆ ( k) = [ α ( k) β ( k)] T OE i j θˆ( k+ ) θˆ( k) +μe ( ) ˆ OE k φ OE ( k) where μ is the step size. In practice: nˆ a f f yˆ ( ) ˆ ( ) ˆ ˆ OE k i = yoe k i + ak yoe ( ki l), for i =,..., nˆa l= Define: nˆ a f f xˆ ( k j) = x( k j) + aˆ xˆ ( k j l), for j = 0,,..., nˆ l= f f f T φˆ ( ) [ ˆ ( ) ˆ OE k = yoe ki xoe ( k j)] ˆ ˆ f θ( k+ ) θ( k) +μe ( k) φ ˆ ( k) OE k OE b OE OE a b b 49

50 Table 2-2. Summary of the IIR EE algorithm. Initialization: θ ˆ(k)= a ˆ ˆ i(0) b j(0) = 0 where i =,..., nˆ, j = 0,,..., nˆ Computation: For k =,2,... nˆa a nˆb j yˆ ( k) = aˆ y( k i) + bˆ x( k ) EE i j i= j= 0 e ( k) = y( k) yˆ ( k) EE EE φ ˆ ( k ) = [ y ( ki ) x ( k j )] T EE θˆ( k+ ) = θˆ( k) + μe ( ) ˆ EE k φ EE( k) where μ is the step size. b T 50

51 Table 2-3. Summary of the IIR SM algorithm. Initialization: θ T ˆ(k)= a ˆ ˆ i(0) b j(0) = 0 where i =,..., nˆ, j = 0,,..., nˆ Computation: For k =,2,... nˆa a nˆb j yˆ ( k) = aˆ y( k i) + bˆ x( k ) EE i j i= j= 0 e ( k) = y( k) yˆ ( k) EE EE e ( k) e ( k) SM Define: = nˆ a i ˆ az i i= EE b nˆ a f f yˆ ( ) ˆ( ) ˆ ˆ SM k i = y k i + ak ysm ( ki l), for i =,..., nˆa l= xˆ k j = x k j + a x k j l for j = n nˆ a f f ( ) ( ) ˆ ˆ SM k SM ( ), 0,,..., ˆb l= φ ˆ ( ) [ ˆ ( ) ˆ SM k = ysm ki xsm ( k j)] θˆ( k+ ) = θˆ( k) + μe ( k) φ ˆSM ( k ) f f T SM 5

52 Table 2-4. Summary of the IIR CE algorithm. Initialization: θ T ˆ(k)= a ˆ ˆ i(0) b j(0) = 0 where i =,..., nˆ, j = 0,,..., nˆ Computation: For k =,2,... Step : nˆa a nˆb j yˆ ( k) = aˆ yˆ ( k i) + bˆ x( k ) OE i OE j i= j= 0 b e ˆ OE ( k) = y( k) yoe ( k) nˆ a f f yˆ ( ) ˆ ( ) ˆ ˆ ( ),,..., ˆ OE k i = yoe k i + ak yoe ki l for i = na l= Define: nˆ a f f xˆ ( k j) = x( k j) + aˆ xˆ ( k j l), for j = 0,,..., nˆ l= f f f T φˆ ( k) = [ yˆ ( ki) xˆ ( k j)] OE k OE b OE OE OE Step 2: nˆa nˆb j yˆ ( k) = aˆ y( k i) + bˆ x( k ) EE i j i= j= 0 e ( ) ( ) ˆ EE k = y k yee ( k) φ ˆ ( k ) = [ y ( ki ) x ( k j )] T EE Step 3: ece ( k) = βeee ( k) + ( β ) eoe ( k) f φˆ ( k) = β ˆ ( k) + ( β) ˆ CE φee φoe( k) θˆ( k+ ) = θˆ( k) + μe ( ) ˆ CE k φ CE( k) where 0 β 52

53 Table 2-5. Simulation results of IIR algorithms for sufficient case. Initial Point Adaptive Algorithm bˆ 0(0) bˆ (0) Number of Structure Parameters iterations aˆ ˆ (0) a2(0) OE μ = EE μ = SM μ = CE μ = 0.005, β = μ = 0.0, β = Final Point bˆ 0( n) bˆ ( n) aˆ ˆ ( n) a2( n) [0.05, ] [-.3,0.2484] [ ] [ ] [ , ] [-.3,0.249] [0.050, ] [-.3,0.2485] [ ,-0.400] [-.3,0.2496] 53

54 Table 2-6. Simulation results of IIR algorithms for insufficient case. Adaptive Algorithm Initial Point Global Min. Number of Structure Parameters bˆ ˆ 0(0), a(0) [ 0.3,0.906] iterations Final Point OE A μ = 0.00 [-0.5,0.] 5500 [ ,0.8998] B μ = [-0.5,-0.2] 2000 [0.0928, ] EE A μ = 0.00 [-0.5,0.] 7000 [ ,0.8755] B μ = 0.00 [0.,-0.52] 7000 [ ,0.879] A μ = [-0.5,0.] 3000 [-0.332,0.9039] SM B μ = [0.5,-0.2] 4500 [ ,0.8992] C μ = [0.,-0.52] 5500 [ ,-0.903] CE A μ = 0.00, β = 0.04 [-0.5,0.] 5000 [ ,0.92] B μ = 0.003, β = 0.04 [0.5,-0.2] 9000 [-0.35,0.98] 54

55 Table 2-7. Simulation conditions of IIR algorithms for sufficient case. Initial Point Algorithm Adaptive Structure bˆ 0(0) bˆ (0) Number of Parameters iterations aˆ ˆ (0) a2(0) OE μ = EE μ = SM μ = CE μ = 0.005, β =

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

57 Input x(k) c(k) ˆb 0 Output y(k) ˆ Z â ˆb Z â 2 ˆb 2 aˆ nˆ a ˆ bn ˆ b Z aˆ n ˆa bˆ nb ˆ Figure 2-. Linear time-invariant (LTI) IIR Filter Structure. Additive White Noise v(k) Input x(k) Test Model Test Model Output y(k) Optimization Method Error e(k) + LTI Discrete Filter Filter Output ŷ(k) Adaptive IIR Filter Figure 2-2. Simulation structure of the adaptive IIR filter. 57

58 Figure 2-3. z-plane of the test model. Figure D plot of the MSOE performance surface of the insufficient order test system. 58

59 Figure 2-5. Contour plot of the MSOE performance surface. Figure 2-6. Simulation results of weight track of the IIR algorithms for sufficient case. 59

60 Figure 2-7. Simulation results of weight track of the IIR algorithms for insufficient case. Figure 2-8. Learning curve of IIR algorithms for sufficient case. 60

61 Figure 2-9. Computational complexity results from the experiment. 6

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 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 970s and has been widely used in industrial process control. The methodology of MPC is represented in Figure 3-, where k is the time index number, uk ( ) are the input sequences, and ( ) are the actual output sequences. The yk yk ˆ( ) and yr ( 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 and a series of future error signals, which are the differences between the estimated output signals and the reference signals yk ˆ( + j) y ( k j) r u( k + s) +. Second, a series of future inputs uk ( + 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 62

63 impulse response model and DMC applies the step response process model in order to predict the future control signals (Camacho 995). The GPC method, which is inherited from generalized minimum variance (GMV) (Clarke 979), was proposed and explained by Clarke (987 a, b). The GPC algorithm is an effective self-tuning predictive control method (Clarke 988). It uses controlled auto-regressive and integrated moving average (CARIMA) model to derive a control law and can be used in real time applications. Juang et al. (997, 200) give the derivation of the adaptive MIMO GPC algorithm. This algorithm is an effective control method for systems with problems of non-minimum phase, open loop unstable plants or lightly damped systems. It is also characterized by good control performance and high robustness. Furthermore, the GPC algorithm can deal with the multi-dimension case and can easily be combined with adaptive algorithms for self-tuning real-time applications. The problem of flow-induced open cavity oscillations exhibit several theses issues, therefore, the GPC is considered as a potential candidate controller. Two modifications are made for this algorithm. First, a input weight matrix is integrated into the cost function, this control matrix can put the penalty for each control input signal and further to tune the performance for each input channels. Second, a recursive version of GPC is developed for real-time control application. MIMO Adaptive GPC Model In this section, a MIMO model, which has the same form of the MIMO ID algorithm, is considered. A linear and time invariant system with r inputs [ u ] r and m outputs [ y ] m, at the time index k can be expressed as y( k) = α y( k ) + α y( k 2) + + α y( k p) 2 + β u( k) + β u( k ) + β u( k 2) + + β u( k p) 0 2 p p (3-) 63

64 where u( k) y ( k) u2( k) y2( k) u ( k) = [ u( k) ] =,y( k) = [ y( k) ] = r m u ( k) y ( k) [ ] [ ] r r m m α = α, α = α,, α = α β = [ β ], β = [ β ],, β = β m m 2 2 m m p p m m 0 0 m r m r p p m r (3-2) Shifting j step ahead from the Equation 3-, the output vector y ( k + j) can be derived as y( k+ j) = α y( k ) + + α y( k p+ ) ( j) ( j) p + α y( k p) + β u( k+ j) + β u( k+ j) + ( j) () p β u( k) + β u( k ) + + β u( k p) ( j) ( j) ( j) 0 p (3-3) where ( + ) j ( ) ( j) ( j) ( j) α = α α+ α m m 2 β = α β β α = α α + α β = α β + β, ( j) ( j) ( j) ( j) p = m m p+ p p j α α α α β = α β m r p+ β p ( j) ( j) ( j) ( j) α p = α m m α p p p β = α β m r ( j) ( j ) ( j ) 0 m r 0 ( j) ( j) ( j) ( j) ( j) ( ) 2 m m 2 3 m r 2 ( ) ( j) ( ) (3-4) and with initial (0) (0) α = α m m β 0 = β m r 0 (0) (0) α 2 = α m m 2 β = β m r, (0) (0) α p = α m m p β p = β m r (0) (0) α p = α m m p β p = β m r p p (3-5) The quantities β ( k = 0,, ) are the impulse response sequence of the system. Defining the ( k ) 0 following the vector form 64

65 uk ( p) uk ( ) uk ( p+ ) uk ( + ) up( k p) =, uj+ ( k) = uk ( ) uk ( + j) yk ( p) yk ( p+ ) y p ( k p) = yk ( ) ( r p) ( r j+ ) ( m p) ( ) (3-6) the predictive index j = 0,, 2, q, q+,, s, and uk ( ) yk ( ) uk ( + ) yk ( + ) us( k ) =, ys( k) = uk ( + s ) yk ( + s) ( r s) ( m s) (3-7) Finally, the predictive model for future outputs, y s, is obtained, this future outputs consists of a weighted summation of future inputs, u s, previous inputs, up, and previous outputs, y p y ( k) = Tu ( k) + Bu ( k p) + Ay ( k p) s s p p (3-8) where T β0 0 0 () β β = ( s) ( s2) β0 β0 β0 ( m s) ( r s) (3-9) β p β p β () () () β p β p β B = ( s) ( s) ( s) β p β p β ( m s) ( r p) (3-0) 65

66 αp αp α () () () αp αp α A = ( s) ( s) ( s) αp αp α ( m s) ( m p) (3-) 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 [ u ( k) ] [ H] up ( k p) = p ( k p) y s (*) s r (*)[ s r p*( m+ r)] [ p*( m+ r)] (3-2) Two potential cost functions are list below. The first one consists terms of future outputs and a trace of the feedback gain matrix T ( ) T Jk ( ) = y ( k) Qy ( k) + γ trh H (3-3) s s and the second definition of cost function based on the total energy of future outputs as well as the inputs T T Jk ( ) = ( ys ( k) Qys( k) + us ( k) Ru s( k) ) (3-4) 2 The output weight matrix Q, input weight matrix R and the control horizon s are important parameters for tuning the controller. The horizon s is usually chosen to be several times longer than the rise time of the plant in order to ensure a stable feedback controller (Gibbs et al. 2004). Also, if the predict horizon range is from zero to infinity, the resulting controller approaches the steady-state linear quadratic regulator (Phan et al. 998). MIMO Adaptive GPC Law In order to minimize the cost function, three approaches are considered as follows. 66

67 Based on Equation 3-3, the control coefficients can be update using adaptive gradient algorithm. Based on Equation 3-4, the optimum solution can be derived. However, this method requires the calculation of a matrix inverse, so the computational complex is higher. Based on Equation 3-4, the control coefficients can be updated using an adaptive gradient algorithm. The first approach is examined by Kegerise et al. (2004). In the next section, the latter two approaches are derived. MIMO Adaptive GPC Optimum Solution Based on the cost function 3-4, the goal is to find [ H ] (*)[ *( + )] or [ s ( k )] ( * ) s r p m r u to minimize the cost function. We will show that both minimizing the cost function 3-4 respect to control matrix [ H ] and input vector [ ] (*)[ s r p*( m+ r)] ( ) ( * simplify the expression, let s define u will provide the same result. To s k s r ) s r up ( k p) v p = [ p*( m+ r)] p ( k p) y [ p*( m+ r)] (3-5) Substituting the predictive model 3-8 and control law 3-2 into the cost function 3-4 gives T T Jk ( ) = ( ys ( k) Qys( k) + us ( k) Rus( k) ) 2 T = ( Tus + [ B A] p ) ( s [ ] ) 2 v Q Tu + B A v p (3-6) T + ([ H] p ) ([ ] p ) 2 v R H v with some algebraic manipulation, the gradient of cost function respect to the control matrix [ ] (*)[ s r p*( m+ r)] H can be obtained. The optimum solution is obtained when the gradient equal to zero. 67

68 T ([ ] ) ( [ ] ) J( k) T = T Qy s vp + R H vp vp H T = T Q Tu s + B A vp vp + Rus vp T T T ( ) [ ] = T QT+ R u s vp + T Q B A v p vp = 0 T T T T (3-7) thus, s opt T T ( ) [ ] u T QT+ R T Q B A v (3-8) = p Alternatively, from Equation 3-6, setting the gradient of the cost function with respect to the input vector [ u s ( k )] ( * ) s r, to zero gives ( [ ] s p ) J ( k ) = + T T Tu B A v QT + us R u s =0 (3-9) thus, s opt T T ( ) [ ] u T QT+ R T Q B A v (3-20) = p A comparison of Equation 3-20 to Equation 3-8 shows that these two approaches yield the same result. It is easy to apply the optimal solution of the Equation 3-20 on the cavity problem. However, the matrix inversion calculation has high computational complexity. Only if the model order is low enough, the optimal input can be used in real-time application. MIMO Adaptive GPC Recursive Solution To avoid calculating the inverse of the matrix in Equation 3-20, the stochastic gradient descend method can be used to update the control matrix H using the following algorithm Jk ( ) H(k +) = H(k) μ H(k) (3-2) 68

69 Substituting Equation 3-2 into Equation 3-7 gives Jk ( ) T T = ( T QT + R)[ H] p p (k) v v H T T [ ] p p T T ( )[ ] [ ] + T Q B A v v { } T = T QT + R H + T Q B A v p vp (3-22) therefore, the recursive solution is given by T T {( ) [ ]} T = μ + p p H(k +) H(k) T QT + R H(k) T Q B A v v (3-23) Since only present r controls [ uk ( )] r are applied to the system, only the first r rows in Equation 3-23 are used { } T T [ h(k +) ] [ h(k) ] ( )[ (k)] [ ] T = μ + p p T QT+ R H T Q B A v v (3-24) first r rows In next chapter, this adaptive feedback controller, which is the combination of the MIMO system ID (discussed in Chapter 2) and the GPC algorithm, is implementation on a vibration beam test bed. The output weight matrix Q, input weight matrix R and the control horizon s are tuning for testing their effects to the control performance. 69

70 Past Future Prediction Horizon { y ( k) } r { yˆ ( k) } { y( k) } { u( k) } k 2 k k k + k + 2 k + 3 k + j k+ s Figure 3-. Model predictive control strategy. 70

71 CHAPTER 4 TESTBED EXPERIMENTAL SETUP AND TECHNIQUES Equation Section 4 In this Chapter, the MIMO system ID (discussed in Chapter 2) algorithm and the GPC algorithm (discussed in Chapter 3) are implemented on a vibration beam test bed. Since the objective idea of this sample experiment is similar to the flow-induced cavity oscillation, which is the disturbance rejection problem, the results of this vibration beam experiment will give us some insights to guide the later flow control applications of using this real-time adaptive control mythology. First, computational complexity of ID algorithm, ID results in time domain and frequency domain, and the disturbance effect for ID algorithm are examined. Then, the output and input weight matrices as well as the control horizon are tuned for testing the control performance with varies of these parameters. Schematic of the Vibration Beam Test Bed Figure 4- shows a detailed sketch of the whole vibration control testbed setup. A thin aluminum cantilever beam with one piezoceramic (PZT-5H) plate bonded to each side is mounted on a block base and connected to an electrical ground. The two piezoceramic plates are used to excite the beam by applying an electrical field across their thickness. The piezoceramic plate bonded to the right side of the beam is called the disturbance piezoceramic because it is used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the left side of the beam is called the control piezoceramic because it is supplied with the controller output signal to counteract the unknown disturbance actuator. The beam system has a natural frequency of about 97 Hz. The goal of the disturbance rejection controller is to mitigate the tip deflection of the aluminum beam generated by an external unknown disturbance signal. The controller tries to generate a signal to counteract the vibration of the aluminum beam generated by the disturbance 7

72 piezoceramic. The performance signal and the feedback signal of the controller are collocated, which is measured at the center of the tip of the beam by a laser-optical displacement sensor (Model Micro-Epsilon OptoNCDT 2000). This device gives an output sensitivity of V / mm with a resolution of 0.5μ m and a sample rate of 0 khz. The performance signal is filtered by a high pass filter (Model Kemo VBF 35) with f c = H to filter out the dc offset of the z displacement sensor and then amplified by a high-voltage amplifier (Model Trek 50/750) with a gain of 0. The disturbance and control signals are generated by dspace (Model DS005) DSP system with 466MHz Motorola PowerPC micro-processor and amplified by two separate channels of the power amplifier by the same gain of 50. The types and conditions of the signals are discussed in detail in the next section. The dspace system has a 5-channel 6-bit ADC (DS200) and a 6-channel 6-bit DAC (DS202) board. The signals are acquired using Mlib/Mtrace programs in MATLAB through the dspace system. The block diagram of the vibration beam test bed is shown in Figure 4-2. System Identification Experimental Results Computational Complexity During the real-time adaptive control of flow-induced cavity oscillations, computational complexity is an important issue. Kegerise et al. (2004) use 80 order estimated model for the system ID and 240 prediction horizon for the recursive GPC algorithm to capture the dynamics of the cavity system. Therefore, the computational complexity of the on-line adaptive controller have to reach or beyond these lengths of parameters. Figure 4-3 shows the changes of turn around time with the increasing estimated system order of the MIMO system ID algorithm. It is clear that the computational complexity of this algorithm is approximately linear, and the time 72

73 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. 997; Ljing 998), the eigenvalues of the triangle matrix calculated from singular value decomposition (SVD) of the vibration beam system is shown in Figure 4-4. This plot shows that the minimum reasonable estimated order of the system is 2. For different ID algorithms, the estimated system order may be different in practice. Therefore, the resulting identified system transfer function should be checked in order to match the experimental data shape in both time domain and frequency domain. A periodic swept sine signal (Figure 4-5) is chosen as the ID input signal (without external disturbance). The sampling frequency is 024 Hz, the sweep frequency produced by dspace system is from 0 Hz to 50 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 0. Figure 4-5 shows that the output of the system ID algorithm matches the system output very well in the time domain. The coherence function (Figure 4-6) shows good correlation between the input and output signals. The zero-pole location and the transfer function between the input and sensor output are shown in Figure 4-7 and Figure 4-8 respectively. Three system ID methods are used for comparison. Two batch methods calculate transfer function in frequency domain using the experimental data and the FRF method to fit the frequency domain data. One recursive method updates the system coefficients in real-time. Notice that all of these three methods give the 73

74 similar shape in frequency domain and capture the two dominant poles of the system (Figure 4-7). However, the FRF fit batch method gives a lower order model than the recursive method. Disturbance Effect External disturbance degrades the performance of the system ID algorithm. Figure 4-9 shows the vibration shim experiment system ID result with different external disturbance levels. A larger SNR (lower external disturbance level) in the input signal generally give more accurate identified system models. However, although the lower SNR input signal may result in a suboptimal system model, the closed-loop control implementation based on this model still works well. The results are shown later. Closed-Loop Control Experiment Results Computational Complexity The controller design block is the most time consuming blocks in the entire adaptive control implementation. Estimated model order and the length of the predict horizon are two main parameters effecting the computational complexity. Figure 4-0 illustrates the computational complexity of the main C code S-function block which maps the observer Markov parameters (discussed in Chapter 2) to predict model coefficients (discussed in Chapter 3). The result shows that the turnaround time increases more quickly with increasing estimated system order than increasing of the prediction horizon. Closed-Loop Results The experimental parameters for the closed-loop control are list in Table 4-, and some result plots are presented here. Figure 4- shows the sensor output signal in time domain, in which the control signal is initiated at time 0. 74

75 Power spectra of open loop (base line) vs. closed-loop sensor output and the closed-loop sensitivity are shown in Figure 4-2 and Figure 4-3, respectively. The sensitivity function is define as s y ( f ) cl 2 = (4-) y op ( f ) 2 Equation 4- provides a scalar measurement of disturbance rejection. A value less than one (negative log magnitude) indicates disturbance attenuation, while a value greater than one (positive log magnitude) indicates disturbance amplification. Although the resonance of the open loop system can be mitigated by the closed-loop controller, a spillover phenomenon is also observed in Figure 4-3. As discussed in Chapter, the spillover problem is generated because, for this special case, the performance sensor output and the measurement sensor output (feedback signal) are collocated. Next, the effects of the adaptive GPC parameters are examined. Figure 4-4 shows the effect of the changes of the estimated model order. Figure 4-5 shows the effect of the changes of the predict horizon. Figure 4-6 shows the effect of the changes of the input weight. Figure 4-8 shows the effect of the different level of disturbance (SNR) during the system ID. The results for each case are discussed below. Estimated Order Effect In general, increasing the estimated order of the GPC, up to a certain point, can improve the performance of the closed-loop control (Figure 4-4). The experimental result shows that when the estimated model order is greater than 4, the closed-loop controller can not improve the performance any more. 75

76 Predict Horizon Effect It is clearly see that increasing the predict horizon can improve the performance of the closed-loop controller (Figure 4-5). Input Weight Effect The input weight penalizes the magnitude of the input signal. For this experiment, ±0.75 volt saturation is given to the input signal to avoid the damage of the actuator. In order to restrict the input signal within the limits of the saturation, the input weight should be carefully tuned to obtain a realizable GPC. Although a smaller input weight improve performance of the closed-loop controller ( Figure 4-6), it also generates a larger control signal (Figure 4-7). Therefore, the tuning idea is to decrease the input weight as low as possible under the input saturation constraints. Disturbance Effect for Different SNR Levels During System ID As mentioned above, the level of the external disturbance signal (different SNR) is an important issue for the accuracy of the system ID (Figure 4-9). However, the adaptive closedloop controller gives the surprising results (Figure 4-8). Three cases are examined and compared in this section. First, the open loop (base line) case is the power spectrum of the output measurement of laser sensor without any control input. Second, the external disturbance is turned off during the system ID. Finally, the external disturbance is turned on with some level during the system ID. The result shows that the higher disturbance level (low SNR) does not have a detrimental effect on the performance of the closed-loop system. In fact, the performance of the closed-loop controller with low SNR is improved slightly. 76

77 Summary Table 4-2 gives a summary of the experimental results of adaptive GPC algorithm. It can be seen that the GPC algorithm gives the better control performance with the larger estimated system order, the higher prediction horizon and the lower input weight. In Chapter 6, the similar control approach combining the system ID algorithm and the GPC algorithm will be implemented on the flow-induced cavity oscillations problem. Since the control ideas for both the vibration beam problem and the cavity oscillations problem are disturbance rejection, the successful implementation of the system ID algorithm and GPC algorithm to the vibration beam test bed may give the guidance to the flow control. 77

78 Table 4-. Parameters selection of the vibration beam experiment. Fs Disturbance GPC White Noise Low Pass Filter Prediction Input Estimated (IIR Butterworth, Horizon 4 th Weight Order order) 024Hz Var = 0.09 f = 50Hz 0 0 c 78

79 Table 4-2. Summary of the results of the adaptive GPC algorithm. Integrated Estimated Prediction Input Weight Reduction Order Horizon (In db) Reduction at Resonance (In db) 79

80 Kemo VBF35 Anti-aliasing Low Pass Filer Trek 50/750 Power Amplifier PC MLIB/MTRACE Aluminum Shim OptoNCDT 2000 Laser-Optical Displacement Sensor Disturbance Piezoceramic DS005 Processor Board Control Piezoceramic Disturbance Signal DS200 A/D Board ID Input or Control Input Base DS202 D/A Board Trek 50/750 Power Amplifier Kemo VBF35 Reconstruction Low Pass Filer Figure 4-. Schematic diagram of the vibration beam test bed. W(t) LPF AMP LPF AMP Actuator System Sensor BPF AMP Y(t) 6-bit ADC U(t) 6-bit ADC Analog Part W(k) Controller Design System ID Digital Part ID Input U(k) Controller Y(k) Figure 4-2. Block diagram of the vibration beam test bed. 80

81 Figure 4-3. Computational complexity of the MIMO system ID. Figure 4-4. Eigenvalues of the triangle matrix obtained by the SVD method of the vibration beam system. (Calculated by ARMARKOV/LS/ERA algorithms with 50 Markov parameters and estimated order of the denominator is 0). 8

82 Figure 4-5. Input time series (top), system output and system ID algorithm output time series (bottom). Figure 4-6. Coherence function of system input and system output. 82

83 Figure 4-7. Zero-pole location of FRF (top) and system ID algorithm. The estimated order of FRF fit function is 2 and the estimated order of system ID algorithm is 0. Figure 4-8. Identified transfer function using the experiment data by frequency response function (experiment), frequency response function fit(frf fit) and time domain system ID algorithm (ID). The estimated order of FRF fit function is 2 and the estimated order of system ID algorithm is 0. 83

84 Figure 4-9. Learning curve of system ID with different input SNR. The estimated system order is 0, sampling frequency is 024 Hz. A Figure 4-0. Computational complexity of the main controller design C S-function. A) For SISO case. B) For TITO case. 84

85 Figure 4-0. Continued B Figure 4-. System output time series data. The control signal is introduced at 0 second, the estimated order is 0, predict horizon is 0, and the input weight is. 85

86 Figure 4-2. Power spectrum of output signal with control and without control signal. The estimated order is 0, predict horizon is 0, and the input weight is. Sensitivity 0.2 RMS Gain (in Log Scale) Frequency (Hz) Figure 4-3. Sensitivity function of the system. The estimated order is 0, predict horizon is 0, and the input weight is. 86

87 Figure 4-4. Power spectrum of output signals for different estimated order. Predict horizon is 0, and the input weight is. Figure 4-5. Power spectrum of output signals for different predict horizon. The estimated order is 4, and the input weight is. 87

88 Figure 4-6. Power spectrum of output signals for different input weight. The estimated order is 0, predict horizon is 0. Figure 4-7. Control signals for different input weight. The estimated order is 0, predict horizon is 0. 88

89 Figure 4-8. Power spectrum of output signals for different system ID disturbance conditions. The estimated order is 0, predict horizon is 0, and the input weight is. 89

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 test section and cavity model, unsteady pressure transducers, data acquisition systems, and a DSP real-time control system. Finally, the actuator used in this study is described. Wind Tunnel Facility The compressible flow control experiments are conducted in the University of Florida Experimental Fluid Dynamics Laboratory. A schematic of the supply portion of the compressible flow facility is shown in Figure 5-. This facility is a pressure-driven blowdown wind tunnel, which allows for control of the upstream stagnation pressure but without temperature control. The compressed air is generated by a Quincy screw compressor (250 psi maximum pressure, Model 5C447TTDN7039BB). A desiccant dryer (ZEKS Model 730HPS90MG) is used to remove the moisture and residual oil in the compressed air. The flow conditioning is accomplished first by a settling chamber. The stagnation chamber consists of a 254 mm diameter cast iron pipe supplied with the clean, dry compressed air. A computer controlled control valve (Fischer Controls with body type ET and Acuator Type 667) is situated approximately 6 meters upstream of the stagnation chamber with a 76.2 mm diameter pipe connecting the two. A flexible rubber coupler is located at the entrance of to the stagnation chamber to minimize transmitted vibrations from the supply line. The stagnation chamber is mounted on rubber vibration isolation mounts. A honeycomb and two flow screens are located at the exit of the settling chamber and the start of the contraction section, respectively. The honeycomb is 76.2 mm in width (the cell is 76.2 mm long) with a cell size of 0.35 mm. Two 90

91 anti-turbulence screens spaced 25.4 mm apart are used; these screens have 62% open area and use 0. mm diameter stainless steel wire. For the current experiment, the facility was fitted with a subsonic nozzle that transitions from the 254 mm diameter circular cross-section to a 50.8 mm 50.8 mm square cross-section linearly over a distance of 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 9.6:. For the present subsonic setup, the freestream Mach number can be altered from approximately 0. to 0.7, and the facility run times are approximately 0 minutes at the maximum flow rate due to the limited size of the two storage tanks, each with volume of 3800 gallons. Test Section and Cavity Model A schematic of the test section with an integrated cavity model is shown in Figure 5-2. The origin of the Cartesian coordinate system is situated at the leading edge of the cavity in the mid-plane. The test section connects the subsonic nozzle exit and the exhaust pipe with 43.8 mm long duct with a 50.8 m m 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 = 52.4 mm and width of W = 50.8 mm and is installed along the floor of the test section. The depth of the cavity model, D, can be adjusted continuously from 0 to 50.8 mm. This mechanism provides a range of cavity length-to-depth ratios, L / D, from 3 to infinity. The cavity model spans the width of the test section W. However, a small cavity width is not desirable, because the side wall boundary layer growth introduces three-dimensional effects in the aft region of the cavity. As a result, the growth of the sidewall boundary layers in the test section may result in modest flow acceleration. The boundary layers have not been 9

92 characterized in this study. Nevertheless, the cavity geometry applied in this study is consistent with previous efforts in the literature (Kegerise et al. 2007a,b) considered to be shallow and narrow, so two-dimensional longitudinal modes will be dominant (Heller and Bliss 975). 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 4 mm thick plexiglas for optical access. Two different wind tunnel cavity ceiling configurations are available. The first one is an aluminum plate with 25.4 mm thickness that can be considered a rigid-wall boundary condition. This boundary condition helps excite the cavity vertical modes and the cut-on frequencies of the cavity/duct configuration (Rowley and Williams 2006). The performance of this ceiling is discussed in the next chapter. In order to simulate an unbounded cavity flow encountered in practical bomb-bay configurations, a flush-mounted acoustic treatment is constructed to replace the rigid ceiling plate. The new cavity ceiling modifies the boundary conditions of the previous sound hard ceiling. This acoustic treatment consists of a porous metal laminate (MKI BWM series, Dynapore P/N ) backed by 50.8 mm thick bulk pink fiberglass insulation (Figure 5-2). This acoustic treatment covers the whole cavity mouth and extends inch upstream and downstream of the leading edge and trailing edge, respectively. This kind of acoustic treatment reduces reflections of acoustic waves. The performance of this treatment is assessed in the next chapter. The exhaust flow is dumped to atmosphere via a 5 angle diffuser attached to the rear of the cavity model for pressure recovery. A custom rectangular-to-round transition piece is used to connect the rectangular diffuser to the 6 inch diameter exhaust pipe. 92

93 Three structural supports are used to reduce tunnel vibrations (Carroll et al. 2004). Two of these structural supports attach to both sides of the test section inlet flange, and the additional structural support is installed to support the iron exhaust pipe (Figure 5-2). Pressure/Temperature Measurement Systems Stagnation pressure and temperature are monitored during each wind tunnel run and converted to Mach number via the standard isentropic relations with an estimated uncertainty of ±0.0. The reference tube of the pressure transducer is connected to static pressure port (shown in Figure 5-2) using mm ID vinyl tubing to measure the upstream static pressure of the cavity. The stagnation and static pressures are measured separately with Druck Model DPI45 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. C nominal resolution). Two pressure transducers are located in the test section to measure the pressure fluctuations. The first transducer is a flush-mounted unsteady Kulite dynamic pressure transducer (Model XT-90-50A) and is an absolute transducer with a measured sensitivity 7 ( 2.64 ± 0.06) 0 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 ( y = D ) 0.6 inch upstream from the cavity real wall ( x = L ), and 8.89 mm ( z = 8.89 mm ) away from the mid-plane. This position allows optical access from the mid-plane of cavity floor for flow visualization and avoids the possibility of coinciding with a pressure node along the cavity floor (Rossiter 964). The second pressure transducer is also an Kulite absolute transducer (with measured sensitivity ( ) 5.3 ± V/Pa and nominal 400 khz natural frequency, Pa (25 psia) max pressure, 5 mm in diameter), and it is flush mounted in 93

94 the tunnel side wall 63.5 mm downstream of the cavity as shown in Figure 5-2. From a series of vibration impact tests performed in a previous study (Carroll et al. 2004), the results indicated that the pressure transducer outputs are not affected by the vibration of the structure. An experiment to validate this hypothesis is discussed in the next Chapter. Due to a modification of the experimental setup, the second pressure sensor is moved to the cavity floor (Figure 5-2) for both open-loop control and closed-loop control. A PC monitors the upstream Mach number, stagnation pressure, and stagnation temperature, as well as the static pressure. This computer is also used for remote pressure valve control (Figure 5-) in order to control the freestream Mach number using a PID controller. In addition, an Agilent E433A 8-channel, 6-bit dynamic data acquisition system with built-in anti-aliasing filters acquires the unsteady pressure signals and communicates with the wind tunnel control computer via TCP/IP for synchronization. The code for both data acquisition and remote pressure control output generation are programmed in LabVIEW. The pressure sensor time-series data are also collected for both the baseline and controlled cavity flows for post-test analysis. Facility Data Acquisition and Control Systems The schematic of the control hardware setup is shown in Figure 5-3. For the real-time digital control system, the voltage signals from the dynamic pressure transducers are first preamplified and low-pass filtered using Kemo Model VBF 35. This filter has a cutoff range 0. Hz to 02 khz, and three filter shapes can be used. Option 4 with nearly constant group delay (linear phase) in the pass band and 40dB/octave roll-off rate is chosen. The cutoff frequency is 4 khz for a sampling frequency of 0.24 khz. The signal is then sampled with a 5-channel, 6-bit, simultaneous sampling ADC (dspace Model DS200). 94

95 The control algorithms are coded in SIMULINK and C code S-functions and are compiled via Matlab/Real-Time Workshop (RTW). These codes are uploaded and run on a floating-point DSP (dspace DS006 card with AMD Opteron Processor 3.0GHz) digital control system. The DSP was also used to collect input and output data from the DS200 ADC boards as well as computing the control signal once per time step. At each iteration, the computed control effort is converted to an analog signal accomplished using a 6-channel 6-bit DAC (DS202). This signal is passed to a reconstruction filter (Kemo Model VBF 35 with identical settings to the anti-alias filters) to smooth the zero-order hold signal from the DAC. The output from this filter is then sent to a high-voltage amplifier (PCB Model 790A06) to produce the input signal for the actuator. The computer is also able to access the data with the dspace system via the Matlab mlib software provided by dspace Inc. Actuator System In order to achieve effective closed-loop flow control, high bandwidth and powerful (high output) actuators are required. The following issues should be considered for selecting the actuators (Schaeffler et al. 2002). The selected actuators must produce an output consisting of multiple frequencies at any one instant in time. The bandwidth of the actuators should enable control of all significant Rossiter modes of interest. The control authority must be large enough to counteract the natural disturbances present in the shear layer. According to Cattafesta et al. (2003), one kind of actuator called Type A has these desirable properties. Such actuators include piezoelectric flaps and have successfully been used for active control of flow-induced cavity oscillations by Cattafesta (997) and Kegerise et al. (2002). Their results show that the external flow has no significant influence of the actuator 95

96 dynamic response over the range of flow conditions. Their later work (Kegerise et al. 2004; 2007a,b) also shows that one bimorph piezoelectric flap actuator is capable of suppressing multiple discrete tones of the cavity flow if the modes lie within the bandwidth of the actuator. Therefore, the piezoelectric bimorph actuator is a potential candidate for the present cavity oscillation problem. Another candidate actuator is the synthetic or zero-net mass-flux jet (Williams et al. 2000; Cabell et al. 2002; Rowley et al. 2003, 2006; Caraballo et al. 2003, 2004, 2005; Debiasi et al. 2003, 2004; Samimy et al. 2003, 2004; Yuan et al. 2005). This actuator can be used to force the flow via zero-net-mass flux perturbations through a slot in the upstream wall of the cavity. Although the actuator injects zero-net-mass through the slot during one cycle, a non-zero net momentum flux is induced by vortices generated via periodic blowing and suction through the slot. In this research, a piezoelectric-driven synthetic jet actuator array is designed. This type of synthetic jet based actuators normally gives a larger bandwidth than the piezoelectric flap type of actuators. A typical commercial parallel operation bimorph piezoelectric disc (APC Inc., PZT5J, Part Number: P4203T-JB) is used for this design. The physical and piezoelectric properties of the actuator material are listed in Table 5-. The composite plate is a bimorph piezoelectric actuator, which includes two piezoelectric patches on upper and lower sides of a brass shim in parallel operation (Figure 5-4). The final design of the actuator array consists of 5 single actuator units. Each actuator unit contains one composite plate and two rectangle orifices shown in Figure 5-5. The designed slot geometries for the actuator array are shown in Figure 5-6. Another advantage of this design is that it avoids the pressure imbalance problem on the two sides of the diaphragm during the experiment. Since the two cavities on either side of a 96

97 single actuator unit are vented to the local static pressure, the diaphragm is not statically deflected when the tunnel static pressure deviates from atmosphere. The challenge is whether these actuators can provide strong enough jets to alter the shear layer instabilities in a broad Mach number range and also whether the actuators produce a coherent signal that is sufficient for effective system identification and control. A lumped element actuator design code (Gallas et al. 2003) was used together with an experimental trial-and-error method to design the single actuator unit. The final designed geometric properties and parameters of the single actuator unit are listed in Table 5-2. To calibrate this compact actuator array, the centerline jet velocities from each slot are measured using constant-temperature hotwire anemometry (Dantec CTA module 90C0 with straight general purpose -D probe model 55p and straight short -D probe support model 55h20). A Parker 3-axis traverse system is used to position the probe at the center of actuator slots. The sinusoidal excitation signal from the Agilent 3320A 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, 00 Vpp, and 50 Vpp, respectively, over a range of sinusoidal frequencies from 50 Hz to 2000 Hz in steps of 50 Hz. Each bimorph disc serves as a wall between two cavities labeled side A and side B. The notation used to identify each bimorph and its corresponding slots is shown in Figure 5-7. The rms velocities of the slots 3A and 3B located in the centerline of the cavity are shown in Figure 5-8 as an example. The maximum centerline velocities measured at the three excitation voltages for each slot are listed in Table 5-3. A summary of the measurements of the centerline velocities and currents to the actuator array for each slot are provided in Appendix E. The piezoelectric plate is tested over a range of frequencies and amplitudes to determine the current saturation associated with the amplifier. 97

98 Figure 5-9 shows the simulation result calculated by the LEM actuator design code and is superposed on the experimental result, Figure 5-8. The results show that, the LEM actuator design code provides a pretty accurate rms velocity estimation of synthetic jet over a large frequency range between 50 Hz and 2000 Hz. Finally, the measure input current level to the actuator array after the amplifier is measured. The results are shown in Figure 5-0 and indicate that the input current will saturate above 36mApp, which means if the input voltage is larger than 00 Vpp, the current to the actuator will keep a constant value. During the closed-loop experiments, an upper limit of 50 Vpp is used since the current probe is unavailable. Figure 5- shows the spectrogram of the pressure measurement on the cavity floor with acoustic treatment. The Rossiter modes (Equation -2) with α=0.25, κ=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 500 Hz. (Rossiter mode is usually weaker compared to Rossiter modes 2, 3 and 4.) Over this frequency range, the designed actuator can generate large disturbances. In addition, the array produces normal oscillating jets that seek to penetrate the boundary layer, resulting in streamwise vortical structures. In essence, it acts like a virtual vortex generator. A simple schematic of the actuator jets interacting with the flow vortical structures is shown in Figure 5-2. The approach boundary layer contains spanwise vorticity in the x-y plane (the coordinate is shown in Figure 5-2). By interacting with the ZNMF actuator jets, the 2D shape of the vortical structures transform to a 3D shape with spanwise vortical structures. These streamwise vortical disturbances seek to destroy the spanwise coherence of the shear layer, and the corresponding Rossiter modes are 98

99 disrupted (Arunajatesan et al. 2003). Alternatively, the introduced disturbances may modify the stability characteristics of the mean flow, so that the main resonance peaks may not be amplified (Ukeiley et al. 2003). Unfortunately, the flow interaction was not characterized in this dissertation and will be addressed in future work. Instead of using one specific amplitude and one frequency in open-loop control, a closedloop 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. 99

100 Table 5-. Physical and piezoelectric properties of APC 850 device. Shim (Brass) Piezoceramic Bond Elastic Modulus (Pa) Poisson s Ratio Density ( kg / m ) Relative Dielectric Const d 3 (m/v) V pp /mm Maximum Voltage Loading - for 0.5 mm thickness - is 30 V pp Resonant Resistance ( Ω ) Electrostatic Capacitance (pf) - 20,000 ± 30% - Operating Temp. ( C) - -20~

101 Table 5-2. Geometric properties and parameters for the actuator. Geometric Properties of the Diaphragm APC PZT5J, P4203T-JB Piezo. Configuration Bimorph Disc Bender Shim Diameter (mm) 4 Clamped Diameter (mm) 37 Shim Thickness (mm) 0. Piezoceramic Diameter (mm) 30 Piezoceramic Thickness (mm) 0.5 Ag Electrode Diameter (mm) 29 Total Bond Thickness (mm) 0.03 ( 0.05 on each side) Radius a0 (mm) /2 Length of the Orifice L (mm) Width of the Orifice wd (mm) 3 3 Volume in side A ( mm ) Volume in side B ( mm ) 2989 Damping ζ

102 Table 5-3. Resonant frequencies with respective centerline velocities for each input voltage. Input Voltage 50 V pp 00 V pp 50 V pp Slot F res (Hz) V center (m/s) F res (Hz) V center (m/s) F res (Hz) V center (m/s) A B A B A B A B A B

103 Pressure Valve Manual Valve Honeycomb Settling Chamber Subsonic Nozzle Connect to Test Section Screens Figure 5-. Schematic of the wind tunnel facility. Inlet Structural Supports Perforated Metal Plate Fiberglass Exhaust Structural Support Connect to Nozzle y x Dynamic Pressure Sensor P2 Static Pressure Port Cavity Model Dynamic Pressure Sensor P Unit: Inch Figure 5-2. Schematic of the test section and the cavity model. Dimensions are inches. 03

104 Reconstruction Filter Anti-aliasing Filter Fc Power Amplifier Actuator Array Cavity Flow Fc Fc = 4 khz Gain = 50x Fc = 4 khz 6-bit ADC Controller 6-bit ADC Fs=0.240 khz dspace System Figure 5-3. Schematic of the control hardware setup. Piezoceramic V ac t p R p Shim t s R s Figure 5-4. Bimorph bender disc actuator in parallel operation. The physical and geometric properties are shown in Table 5- and Table

105 Side A Side B A B C D Figure 5-5. Designed ZNMF actuator array. A) Operation plot. B) Assembly diagram of single unit. C) Singe unit of the actuator. D) Actuator array. 05

106 Figure 5-6. Dimensions of the slot for designed actuator array. 06

107 Flow Direction Cavity Floor Figure 5-7. ZNMF actuator array mounted in wind tunnel. 07

108 50 50 V pp Velocity [m/s] V pp 50 V pp Frequency [Hz] A 70 Velocity [m/s] V pp 00 V pp 50 V pp Frequency [Hz] B Figure 5-8. Bimorph 3 centerline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal. A) For side A. B) For side B. 08

109 Sim 50 Vpp Exp. 50 Vpp Sim 00 Vpp Exp. 00 Vpp Jet Velocity V out (m/s) Frequency (Hz) Figure 5-9. The comparison plot of the experiment and simulation result of the actuator design code for bimorph 3. The output is the centerline rms velocities of the single unit piezoelectric based synthetic actuator with different excitation sinusoid input signal for side B. 09

110 Vpp 00 Vpp 50 Vpp 20 Current (mapp) Frequency (Hz) Figure 5-0. Current saturation effects of the amplifier. 0

111 Frequency (Hz) Ri Mach number Figure 5-. Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with α=0.25, κ=0.7)

112 Leading Edge of the Cavity Y x Y x z Figure 5-2. Schematic of a single periodic cell of the actuator jets and the proposed interaction with the incoming boundary layer. 2

113 CHAPTER 6 WIND TUNNEL EXPERIMENTAL RESULTS AND DISCUSSION Equation Section 6 Experimental results for the baseline uncontrolled and controlled cavity flows are presented in this chapter. First, the effects of structural vibrations on the unsteady pressure transducers are illustrated. Then, a joint time-frequency analysis of the unsteady pressure measurement for an uncontrolled cavity flow is shown. Flow-acoustic features are deduced from the results. An improved test environment is established by replacing the original hard-wall ceiling of the wind tunnel with an acoustic liner. This new test section minimizes the effects of the vertical acoustic modes. Finally, the results of both open-loop and adaptive closed-loop control experiments using the ZNMF actuator array is presented and discussed in detail. The ability of the actuator to alter both broadband and tonal content of the unsteady pressure spectra is demonstrated at low Mach numbers. Background As discussed in the first Chapter, flow-induced cavity oscillations are often analyzed via unsteady pressure measurements in and around the cavity. However, these measurements are often contaminated by other dynamics associated with the specific characteristics of the wind tunnel test section. As a result, the unsteady pressure spectrum may be due to the cavity oscillations or other phenomena. The experimental results of Cattafesta et al. (999), 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. 998; Williams et al. 2000; Ukeiley et al. 2003; Rowley et al. 2005). This acoustic treatment modifies the sound-hard boundary condition and thus mitigates the contribution of the cavity 3

114 vertical resonance modes to the unsteady pressure measurements. Consequently, the modified cavity model will ideally exhibit the behavior of an unbounded cavity flow and be dominated by Rossiter modes. Alvarez et al. (2005) developed a theoretical prediction method and showed that the wind tunnel walls lead to a significant increase in the growth rate of a resonant mode for frequencies near the cut-on frequency of a cross-stream mode. In the present baseline (i.e., uncontrolled) experimental study, flow-acoustic resonances in the test section region and in the cavity region are examined. A schematic of the simplified wind tunnel model and the cavity region of the experimental setup are shown in Figure 6-. 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 L ), and a downstream region ( x > L ). The Rossiter modes ( R, i = 0,,...) are the combined result of a receptivity process at i 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 ( Vi, i = 0,,... ) and cavity cut-on modes ( C, i = 0,,... ) can also be present, as discussed above. These acoustic modes are generated by i the reflections from the ceiling and area changes of the cavity model. During the wind tunnel experiments, the vertical modes ( unbounded bomb bay problem more accurately. V i ) are undesirable and should be reduced in order to mimic the As explained in Alvarez et al. (2005), the upstream region can support duct cut-on modes u D ( i = 0,,... ) and upstream propagating duct modes ( T u, i = 0,,... ) due to the acoustic ui i 4

115 scattering process. Similarly, the downstream region can support duct cut-on modes d ( i = 0,,... ) and downstream propagating tunnel modes ( T, i = 0,,... ) due to the scattering process. Here, we focus our attention on the propagating modes in the cavity and the downstream tunnel regions. Data Analysis Methods A schematic of unsteady pressure transducer locations for this study was presented in Chapter 5 (Figure 5-2). and P measure the unsteady pressure fluctuations in the cavity and P 2 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, a joint-time frequency analysis (JTFA) (Qian and Chen 996) 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. (998), Kegerise et al. (2004), and Rowley et al. (2005). In this study, the sampling frequency for experimental data collection is 0.24 khz and the frequency resolution is 5 Hz. The cut-off frequency of the i d D u i anti-aliasing filter is 4 khz, and 500 continuous blocks of time series data are used in the analysis. During the experiment, Mach number sweeps from 0. to 0.7 in about 00 seconds. 5

116 Noise Floor of Unsteady Pressure Transducers The effective in-situ noise floor of the two unsteady pressure transducers is presented in Figure 6-2. Each noise floor measurement is compared with the spectra obtained at different discrete Mach numbers for the acoustically treated L/D=6 cavity. Within the tested frequency range, the signal-to-noise ratio ( SNR ) is in excess of 30 db, which demonstrates adequate resolution of unsteady pressure transducers for the present experiments despite their large fullscale pressure ranges. Effects of Structural Vibrations on Unsteady Pressure Transducers A series of initial impulse impact tests are performed before the baseline experiments. As discussed in Chapter 5, with the wind tunnel turned off, the pressure transducer outputs are not affected by hammer- or shaker-induced structural vibrations. A simple test is described here to investigate the effects of structural vibrations while the wind tunnel is running. To avoid confounding cavity oscillations, the cavity floor is mounted flush with the tunnel floor ( D = 0 ). A piezoceramic accelerometer (PCB Piezotronics Model 356A6) is used to measure the structural vibrations. It is attached to the test section outer wall using wax at the location indicated in Figure 5-2, which is close to one of the pressure transducers ( ). This piezoceramic accelerometer is connected to a multi-channel signal conditioner (PCB Piezotronics Model 48A0). Three channels of the piezoceramic accelerometer corresponding P 2 to the x, y, and z directions are measured. The coordinate directions of x and y are shown in Figure 5-2, and z is the corresponding lateral direction using the right-hand rule. The accelerometer is calibrated with a reference shaker (PCB Piezotronics, Model: 394C06) that provides g (rms) at 000 rad/s (59.2 Hz ). 6

117 The JTFA results (Figure 6-3 to Figure 6-5) for all components of the accelerometer measurements show that the power of the structural vibration spreads is broadband with a few spectral peaks. A modest peak at 000 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, 880 Hz in the lateral direction and 3200 Hz in the vertical direction) can also be detected. However, the JTFA results of P ( Figure 6-6) and do not display any of these resonances. These results confirm that the unsteady pressure transducers are not affected by structural vibrations. Baseline Experimental Results and Analysis The rigid ceiling plate (no acoustic treatment) above the mouth of the cavity is considered first. JTFA results of the unsteady pressure transducer measurement for this case are shown in Figure 6-7. Numerous flow-acoustic resonances can be observed in the plots. For easy reference in the subsequent discussion, these features are numbered 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 in Figure 6-7). The other unknown acoustic features 2 in Figure 6-7 are undesirable features that we wish to eliminate. These acoustic modes come from the bounded wind tunnel walls, the mismatched acoustic impedance due to area change, and the leading and trailing edges of the cavity. In order to better mimic an unbounded cavity flow in a closed wind tunnel, the boundary condition of the cavity ceiling must be altered to eliminate the unexpected modes within the cavity region. A flush-mounted acoustic treatment (discussed in Chapter 5) is fabricated to replace the previous solid tunnel ceiling. The new cavity ceiling modifies the zero normal velocity boundary condition of the previous sound hard top plate. P 2 7

118 The unsteady pressure transducer JTFA measurement for the trailing edge floor of the cavity is shown in Figure 6-8. The results illustrate a very clean flow field below Mach 0.6. The acoustic features 2-4 in Figure 6-7 are eliminated within the cavity region. Therefore, the experimental Rossiter modes R i shown in JTFA plot (Figure 6-8) now follow the estimated Rossiter curves. At higher upstream Mach numbers ( M > 0.6 ), the experimental Rossiter modes deviate slightly from the expected Rossiter curves. This is partly because the estimated curves use the upstream static temperature to calculate the speed of sound. This estimation does not account for the expected significant static temperature drop due to the large flow acceleration near the aft cavity region seen by Zhuang et al. (2003). Another possible reason for these deviations of the flow-acoustic resonance comes from the structural vibration coupling with the Rossiter modes. At high Mach numbers above 0.6, the structural vibrations may cause a lock-on phenomenon with the Rossiter modes. For this study, all experiments are thus performed below M = 0.6. In conclusion, the observed flow-acoustic behavior of the acoustically treated cavity model behaves as expected below M = 0.6 and is therefore suitable for application of open-loop and closed-loop flow control. Open-Loop Experimental Results and Analysis The open-loop and closed-loop experimental results using the designed actuator array are shown in this section. Before the control experiments, measurements of the pressure sensor at the surface of the trailing edge of the cavity with the without the actuator turned on are shown in Figure 6-9. Without the upcoming flow, the noise floor shows a significant peak at 660 Hz and a small peak at 2000 Hz. The pressure sensor can also sense the acoustic disturbances associated with the excitation frequency and its harmonics, and the measured unsteady pressure level can 8

119 reach 5-20 db. The extent to which the measured levels deviate from theses values with flow on (considered below) indicates the relative impact of the actuator on the unsteady flow. First, open-loop active control is explored. The purpose of the open-loop experiments is to verify if the synthetic jets generated from the designed actuator array can affect and control the cavity flow. A parametric study for the open-loop control is explored first. A sinusoidal signal is chosen as the excitation input with the frequency swept from 500 Hz to 500 Hz. The openloop experimental results are shown in Appendix F. The open-control performance is best over the frequency range 000 Hz to 500 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 B = m /( ρu A ) increases. As a result, the control effect c cavity increases. For these open-loop tests, the upstream flow Mach number is varied from 0. to 0.4. For illustration purposes, results are examined here for two sinusoidal signals with 200 Vpp and excitation frequencies at either.05 khz or.5 khz to drive the actuator array. The.05 khz excitation frequency is close to the resonance frequency of the actuator, while the.5 khz frequency lies between the second and third Rossiter modes. The experimental results shown in Figure 6-0 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.. 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 9

120 spanwise, respectively. The effectiveness of the actuator scales with the momentum coefficient, which is inversely proportional to the square of the freestream velocity. So, as the upstream Mach number increases, the synthetic jets are eventually not strong enough to penetrate the boundary layer and the control effect is reduced. Future work should perform detailed measurements to validate this hypothesis. The results of the open-loop control suggest that this kind of actuator array can generate significant disturbances not only along the flow propagation direction but also in the spanwise direction of the cavity. The combination of these effects disrupts the Kelvin-Helmholtz type of convective instability waves, which are the source of the Rossiter modes. As a result, multiple resonances are reduced via active control. The experimental results also show the limitation of the open-loop control. Closed-Loop Experimental Results and Analysis The open-loop control results suggest that this compact actuator array may be effective for adaptive closed-loop control. As discussed above, the synthetic jets add disturbances to disrupt the spanwise coherence structure of the shear layer and result in a broadband reduction of the oscillations. However, at the same time, the coherence between the drive signal and the unsteady pressure transducer will be reduced. High coherence is considered essential for accurate system identification methods. To exam the accuracy of the system ID algorithm with the change of the estimated order, an off-line system ID analysis is first performed. The nominal flow condition is chosen at M = (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 f L = 25 Hz to f H = 2500 Hz in T = 0.05 sec), are used as a broadband excitation source to identify the system. The running error variances the system ID are shown in Figure 6-. It is clear that the larger the estimated order, 20

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 closed-loop adaptive control algorithm is that it does not rely exclusively on accurate system ID. Figure 6-2 shows the result of the closed-loop real-time adaptive system ID together with the GPC control algorithm for an upstream Mach number Based on the above system ID results, due to the DSP hardware limitation, the estimated GPC order and the predictive horizon are chosen as 4 and 6, respectively. The breakdown voltage of the actuator array restricts the excitation voltage level; therefore, the diagonal element of the input weight penalty matrix R (Equation 3-4) is chosen as 0.. This research represents an extension of Kegerise et al. (2007b) where the system ID algorithm and the closed-loop controller design algorithm are used simultaneously in a real-time application. It is important to note that only the system ID white noise or chirp signal is used to identify the open-loop dynamics, and the feedback signal is not used for this purpose. Clearly, the results show that the GPC controller can generate a series of control signals to drive the actuator array resulting in significant reductions for the second, third, and fourth Rossiter modes by 2 db, 4 db, and 5 db, respectively. In addition, the broadband background noise is also reduced by this closed-loop controller; the OASPL reduction is 3 db. The input signal is shown in Figure 6-3. The sensitivity function discussed in Chapter 4 (Equation 4-) is shown in Figure 6-4. A negative amplitude value indicates disturbance attenuation, while a positive value indicates disturbance amplification. The results show that all the points are negative, which indicates the closed-loop controller reduces the pressure fluctuation power at all frequencies. The spillover phenomenon (Rowley et al. 2006) is not observed in Figure 6-4. As discussed in Chapter, the 2

122 spillover problem is generated because either the disturbance source and control signal or the performance sensor output and the measurement sensor output (feedback signal) are collocated. The Bode s integral formula is shown in 6-. log Si ( ω) dω π Re( p) 0 k (6-) k = where p k are the unstable poles of the loop gain of the closed-loop system. So, for a stable system, any negative area at the left hand side of the Equation 6- must be balanced by an equal positive area at the left hand side of the Equation 6-. However, for present closed-loop control study, the left hand side of the Bode s integral formula is -38 rad/sec, which shows that Bode s integral formula does not hold here. Since this formula is valid for a linear controller, the combination of the adaptive system ID and controller is apparently nonlinear. A more detailed study is required in the future to validate this hyothesis. A parametric study of the GPC is then studied by varying the estimated order and the predictive horizon. Figure 6-5 and Figure 6-6 show that the control effects improve with increasing order and predictive horizon. This trend matches the simulation results shown in Chapter 4. The comparison between the open-loop and closed-loop results is shown in Figure 6-7 for the same flow condition. Notice that the baseline measurement for a same flow condition can vary a little from case to case. The open-loop control uses a sinusoidal input signal at 50 Hz forcing and 50 Vpp and the rms value of the input is 53V. The closed-loop control uses the estimated order 4, the predictive horizon 6, and the input weight 0., and the input rms value is 43 V. 22

123 Upstream Region Cavity Region Downstream Region H u D u i u T i y C i V i R i d D u i d T i x D L Figure 6-. Schematic of simplified wind tunnel and cavity regions acoustic resonances for subsonic flow. Unsteady Pressure Level [db] with P ref =20e-6 Pa Noise Floor Noise Floor frequency [Hz] Figure 6-2. Noise floor level comparison at different discrete Mach numbers with acoustic treatment at trailing edge floor of the cavity with L/D=6. 23

124 Frequency (Hz) Mach number Figure 6-3. x -acceleration unsteady power spectrum (db ref. g) for case with acoustic treatment and no cavity. 24

125 Frequency (Hz) Mach number 30 Figure 6-4. y -acceleration unsteady power spectrum (db ref. g) for case with acoustic treatment and no cavity. 25

126 Frequency (Hz) Mach number Figure 6-5. z -acceleration unsteady power spectrum (db ref. g) for case with acoustic treatment and no cavity. 26

127 Frequency (Hz) Mach number Figure 6-6. Spectrogram of pressure measurement (db ref. 20e-6 Pa) on the trailing edge floor of the cavity for the case with acoustic treatment and no cavity. Noise spike near 600 Hz is electronic noise

128 Frequency (Hz) Mach number Figure 6-7. Spectrogram of pressure measurement (ref 20e-6 Pa) on the trailing edge cavity floor without acoustic treatment at L/D=6. Unknown acoustics features are denoted as 2, while the Rossiter modes are denoted as. 28

129 Frequency (Hz) Ri Mach number Figure 6-8. Spectrogram of pressure measurement (ref 20e-6 Pa) on the cavity floor with acoustic treatment superimposed with the Rossiter modes at L/D=6 (with α=0.25, κ=0.7)