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PAGE 1 ADAPTIVE CONTROL OF SEPARATED FLOW By YE TIAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 1 PAGE 2 2007 Ye Tian 2 PAGE 3 ACKNOWLEDGMENTS First, I thank my advisor, Dr. Louis Catta festa, for his contin uous support throughout my research; his insightful advice a nd comments made this dissertati on possible. Second, I thank my Ph.D. committee members: Warren Dixon, Bruce Carroll, Toshikaza Nishida for their advice and recommendations. The funding for this rese arch is from the NASA Langley Research Center and the Air Force Office of Scientific Re search. Additionally, I thank the student body in the Interdisciplinary Microsystems Group in the University of Flor ida for their great help to my research. Finally and most importantly, I thank my wife, Bei Wang, for her unconditional support. 3 PAGE 4 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ............................................................................................................... 3LIST OF TABLES ...........................................................................................................................7LIST OF FIGURES .........................................................................................................................8ABSTRACT ...................................................................................................................... .............16 CHAPTER 1 INTRODUCTION ................................................................................................................ ..18Overview ...................................................................................................................... ...........18Motivation ...............................................................................................................................21Background .................................................................................................................... .........21TwoDimensional Separation Flow Physics ...................................................................21Effects of Flow Separation ..............................................................................................22Control of Flow Separation .............................................................................................23Openloop separation control ...................................................................................24Closedloop separation control .................................................................................31ClosedLoop Control Algorithms ....................................................................................34Optimization algorithms ...........................................................................................34System identification and disturbance rejection algorithms ....................................36Objectives .................................................................................................................... ...........39Approach ...................................................................................................................... ...........39Outline of This Dissertation ....................................................................................................402 THEORETICAL BACKGROUND........................................................................................44Optimization Algorithms ....................................................................................................... .44Downhill Simplex Algorithm ..........................................................................................44Extremum Seeking Algorithm .........................................................................................45System Identification Algorithms ...........................................................................................48ARMARKOV/LS Algorithm ..........................................................................................49ARMARKOV/LS/ERA Algorithm .................................................................................52Recursive ARMARKOV/Toeplitz Algorithm .................................................................53Adaptive Disturbance Rejection Algorithms ..........................................................................55ARMARKOV Disturbance Rejection Algorithm ...........................................................563 SIMULATION AND VALI DATION EXPERIMENTS .......................................................63Optimization Simulations ...................................................................................................... .63Downhill Simplex Simulation Results ............................................................................63 4 PAGE 5 Extremum Seeking Si mulation Results ...........................................................................64Vibration Control Testbed Setup ............................................................................................65Results of the Vibration Control Tests ...................................................................................67Computational Tests ........................................................................................................67System Identification .......................................................................................................68Adaptive Disturbance Rejection ......................................................................................704 EXPERIMENTAL SETUP AND DATA ANALYSIS METHOD ........................................92NACA 0025 Airfoil Model .....................................................................................................92Synthetic Jet Actuators ...........................................................................................................93Experimental Methods .......................................................................................................... ..94Flow Visualization ...........................................................................................................9 4Lift/Drag Balance ............................................................................................................94Dynamic Pressure Transducers .......................................................................................96Hot Wire Anemometry ....................................................................................................97Control System Hardware and Software ................................................................................98Higher Order Statistical Analysis (HOSA) .............................................................................985 RESULTS AND DISCUSSION ...........................................................................................109Dynamic Feedback Control ..................................................................................................109Experimental Configuration ..........................................................................................109System Identification .....................................................................................................109Coherent flow structures ........................................................................................109Linear prediction ....................................................................................................110Frequency response and the performance of system ID.........................................111Acoustic contamination ..........................................................................................112Disturbance Rejection ...................................................................................................113Closedloop control ................................................................................................113Effect of control on su rface pressure signals .........................................................114Quantitative flow visualization ..............................................................................115Control input ..........................................................................................................116Discussion ..............................................................................................................117Nonlinear Control .................................................................................................................118Experimental Configuration ..........................................................................................118Flow Instabilities ...........................................................................................................119Actuator Calibration ......................................................................................................120Frequency response ................................................................................................120Types of actuation waveforms ...............................................................................121C and electrical power calibration .......................................................................123Adaptive Control Results ..............................................................................................1256 SUMMARY AND FUTURE WORK ..................................................................................140APPENDIX 5 PAGE 6 HIGHER ORDER SPECTRUM EXAMPLES ............................................................................144Example 1: Quadratic System .............................................................................................144Case 1 ............................................................................................................................144Case 2 ............................................................................................................................145Example 2: Cubic System ....................................................................................................14 5Example 3: x x ..................................................................................................................146LIST OF REFERENCES .............................................................................................................155BIOGRAPHICAL SKETCH .......................................................................................................162 6 PAGE 7 LIST OF TABLES Table page 31 Parameters for the simulations. ..........................................................................................7 332 Suppression performance of the di sturbance rejecti on algorithm with 20c and varying .....................................................................................................730.1 cn 33 Suppression performance of the di sturbance rejecti on algorithm with and varying 2cn 0.1 c ...................................................................................................................7334 Suppression performance of the di sturbance rejecti on algorithm with 2cn 20c and varying ....................................................................................................7335 Suppression performance of the disturba nce rejection algorithm at different modes with 2cn 20c and ....................................................................................740.1 51 Case descriptions and performance for di sturbance rejection e xperiments. (AoA=12 and Re=120,000) ..............................................................................................................12752 Summary of parameters in di sturbance rejection algorithm. ...........................................12753 Constrained optimization results using the AM, BM and PM signals. (Baseline L/D=1.01 at AoA=12 and Re=120,000) .........................................................................12754 Constrained optimization results using the AM, BM and PM signals. (Baseline L/D=1.1 at AoA=20 and Re=120,000) ...........................................................................12855 Results using sinusoidal excitations. ................................................................................128 7 PAGE 8 LIST OF FIGURES Figure page 11 Separation of flow over an airfoil. .....................................................................................4112 Types of velocity profiles as a func tion of pressure grad ient (White 1991). .....................4113 Lift and drag coefficien ts of NACA 0025 airfoil at Re100,000 ..................................4214 Classification of flow contro l. (Cattafesta et al. 2003) ......................................................4215 Characterization of possible frequency scales in separated flow (Mittal et al. 2005). ......4316 Comparison between the lumped elemen t model () and experimental frequency response () measured using phaselocked LDV for a prototypical synthetic jet (Gallas et al. 2003). ......................................................................................................... ...4321 Flow chart of downhill simplex algorithm. ........................................................................6122 Block diagram for the extremum seeking control..............................................................6223 Block diagram of disturbance rejection control. ................................................................6231 Onedimensional example of the downhill simplex algorithm. .........................................7432 Twodimensional example of the downhill simplex algorithm. ........................................7533 Simulation block diagram for extremum seeking control. .................................................7534 Single global maximum test model: 2 *ff* where *10f and .......76*5 35 Double hump model with one local ma ximum and one global maximum. The function is fitted by a polynomial: 1281077655 3423121.2e1.6e+1.4e2.3e +1.5e4.4e+3.8e2.35.4f ..........7636 converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while ..............77*f10 a50 w 37 converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while .........77*f10 w 0.001 a 38 converges to the local optim al input (see Figure 35) *14 ( ). ...780.001,50 aw 39 f converges to the local maximum (see Figure 35) 40 f (0.001,50 aw ). ........78 8 PAGE 9 310 Vibration Control Testbed. ............................................................................................... .79311 Block diagram of vibration cont rol with ID, then control. .............................................79312 Block diagram of vibration c ontrol with ID and control. ...............................................80313 Effects of varying or n p on the computational intensity of the ARMARKOV/Toeplitz system ID. ...................................................................................80314 Effects of varying p on the computational intensity of the ARMARKOV/Toeplitz system ID. .................................................................................................................... ......81315 Effects of varying p on the growth rate of the computational intensity of the ARMARKOV/Toeplitz system ID with respect to n ................................................81316 Effects of varying or cnc on the computational intensity of the ARMARKOV disturbance rejection. ........................................................................................................ .82317 Zeropole map of the nonparametric fit of the frequency response. ................................82318 Measured output and fitted output by the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. ..................................................................................83319 Weight tracks of the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. .......................................................................................................................83320 Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 1 and SNR=20 dB. ...................84321 Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. .................84322 Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 20 and SNR=20 dB. .................85323 Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2 pn 30 and SNR=20 dB. .................85324 Comparison of MSE of the ARMARKOV/Toeplitz system ID with SNR=20 dB with different 2 pn ...........................................................................................86325 Comparison of MSE of the ARMARKOV/Toeplitz system ID with 2 pn 20 with different SNR. ................................................................................................86 9 PAGE 10 326 Performance signal of the ARMARKOV disturbance rejection to bandlimited white noise (0150 Hz) with 2cn 20c and 1 ............................................................87327 Control signal of the ARMA RKOV disturbance rejection with 2cn 20c and ...................................................................................................................................871 328 Power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 20c and ........................................................................................881329 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection with 0.1 20c and different .........................................88cn330 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection with 2cn 0.1 and different c ...........................................89331 Comparison of convergence of the ARMARKOV disturba nce rejection with 2cn 20c and different ...................................................................................................89332 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection with 2cn 20c and different ...........................................90333 Control signal of the AR MARKOV disturbance rejection at ID and control mode with 2cn 20c and ....................................................................................900.1 334 Comparison of convergence of the ARMARKOV disturbance reje ction at different modes with 2cn 20c and 0.1 .........................................................................91335 Comparison of power spectra of th e performance signals of the ARMARKOV disturbance rejection at different modes with 2cn 20c and ....................910.1 41 NACA 0025 airfoil model with actuators and pressure transd ucers installed. (Adapted from Holman et al. 2003) .................................................................................10042 Schematic of a synthetic jet actuator. ..............................................................................10043 Synthetic jet array. (Adapt ed from Holman et al. 2003).................................................10144 Forces on NACA0025 airfoil. ..........................................................................................10145 Closer view of th e strain gauges. .....................................................................................10246 Wheatstone bridge configur ation of the balance. ............................................................10247 Normal force vs. balance output. .....................................................................................103 10 PAGE 11 48 Axial force vs. balance output. ........................................................................................10349 Static pressure dist ributions on the airfoil su rface at different AOA at ..104Re150,000 410 Comparison of lift and drag coefficients measured by the static pressure and the balance at .................................................................................................105Re100,000 411 Comparison of lift and drag coefficients measured by the static pressure and the balance at .................................................................................................106Re150,000 412 Picture of a Kulite transducer. .........................................................................................1 07413 Linear response (at 500Hz) of a typical Kulite transducer. .............................................107414 Frequency response of a typical Kulite transducer. .........................................................108415 Hot wire calibration curve. .............................................................................................. 10851 NACA 0025 airfoil model with actuators, sensors and closedloop control system. ......12952 Phase averaged pulse response measur ed by six pressure sensors. The slow propagation velocity of the coherent flow structures is clearly visible. ..........................12953 Comparison between measured signal from the pressure sensor (#6 in Figure 51) and the fitted output by ARMARKOV system ID algorithm for long and short time intervals. Results show a reasonable match at low frequencies between measured and fitted outputs. For ARMARKOV ID: p=1, n=2, =10. .................................................13054 Mean Squared Error (Running MSE) between measured and fitted outputs. Results show that the ARMARKOV ID algorithm converges, i.e. error being minimized. .........13055 Comparison between frequency respons e (FR) and fitted response by ARMARKOV ID algorithm. Parameters for FR: s f =4096 Hz, NFFT=1024, 75% overlap and Hanning window. For ARMARKOV system ID: p=1, n=2, =10. ...............................13156 Dual signal paths from the actuator to the pressure sensor (acoustic and hydrodynamic). A digital filter is introduced to remove the acoustic component by turning off the flow to isolate the acoustic path. ..............................................................13157 Actual measured and predicted acoustic noise using a bandlimited random signal to the actuator. ......................................................................................................................13158 Actual measured and predicted acoustic noise using the same filter as in Figure 57 but with one half of the input amplitude. .........................................................................13159 Power spectra of the sensor signals (w ith wind tunnel running) before and after applying acoustic filter. ....................................................................................................1 32 11 PAGE 12 510 Performance surface pressure (S1) and c ontrol input signals (in Volt) before and after the ID and control is initiated for case #2. Control is established within 1 second or <100 convective time scales. ...........................................................................132511 Power spectra of the pressure transdu cer output for the baseline and the closedloop control cases measured by S6 (performance). .................................................................132512 Contours of streamwise velocity uU for (a) baseline and (b) closedloop control case #2 at AoA = 12 and Rec=120,000. .........................................................................133513 Contours of vorticity for baseline and closedloop control case #2 at AoA = 12 and Rec=120,000. ....................................................................................................................133514 Voltage and electrical po wer spectra of the actuator A1 input signal for the closedloop control case. ............................................................................................................ .133515 Performance comparison at different AoA. .....................................................................134516 Power spectra of the pressure transdu cer output for the baseline and the closedloop control cases measured by S6 (performance) at AoA = 20 and Rec=120,000. ..............134517 NACA 0025 airfoil model with actuators, sensors and closedloop control system. ......134518 Flow structures in separated flow. ...................................................................................135519 Wake (1 chord aft of TE) and shear la yer (near separation) power spectral density functions at peak rms location. ........................................................................................135520 Autobicoherence of the same velocity si gnal analyzed in Figure 519 using the same parameter settings. The autobicoheren ce is zero except where nonlinear phase quadratic phase coupling occurs due to in teractions between the shear layer and wake instabilities. ........................................................................................................... ..136521 Frequency response of ZNMF actuator A1. ....................................................................136522 Various waveforms of unit amplitude 1 A that can be used to excite multiple instabilities or modes in a separated flow. .......................................................................137523 Velocity response (a) and its power spectra l density (b) subject to an AM excitation for the ZNMF actuator A1. A 50 Vpp 50mf Hz and 1180cf Hz. Measurements were made outside the region of reverse flow. ........................................137524 C (a) and electrical power (b) profile subject to AM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. ...............................137525 C (a) and electrical power (b) profile subject to BM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. ...............................138 12 PAGE 13 526 C (a) and electrical power (b) profile subject to PM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. ...............................138527 Constrained search using AM: Cases (a ) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. ...................................................................................139528 Constrained search using BM: Cases (a ) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. ...................................................................................139529 Constrained search using PM: Cases (a) a nd (b). AOA = 20 deg. and Re = 120,000. ...139A1 Test model #1 for higher order spectrum analysis. ..........................................................147A2 Power spectrum of y for test model #1. ...........................................................................147A3 Autobicoherence of y for test model #1. ........................................................................148A4 Crossbicoherence between x and y for test model #1. ...................................................148A5 Test model #2 for higher order spectrum analysis. ..........................................................149A6 Power spectrum of y for test model #2. ...........................................................................149A7 Autobicoherence of y for test model #2. ........................................................................150A8 Crossbicoherence between x and y for test model #2. ...................................................150A9 Test model #3 for higher order spectrum analysis. ..........................................................151A10 Power spectrum of y for test model #3. ...........................................................................151A11 Autobicoherence of y for test model #3. ........................................................................152A12 Crossbicoherence between x and y for test model #3. ...................................................152A13 Test model #4 for higher order spectrum analysis. ..........................................................153A14 Power spectrum of y for test model #4. ...........................................................................153A15 Autobicoherence of y for test model #4. ........................................................................154A16 Crossbicoherence between x and y for test model #4. ...................................................154 13 PAGE 14 NOMENCLATURE c Airfoil chord length D C Drag coefficient ( Dqc) LC Lift coefficient ( Lqc) pC Static pressure coefficient ( p p q ) C Steady momentum coefficient ( Jqc) C Oscillatory momentum coefficient ( Jqc) D Drag (pdC dxc) Pressure recovery coefficient c f Filter cutoff frequency e f Excitation frequency m f Modulation frequency s ep f Shedding frequency of separated flow ( TEUX) wake f Wake shedding frequency ( wakeUW) F Reduced excitation frequency ( TE f XU ) h Slot width J Steady jet momentum ( ) 2 jUh J Oscillatory jet momentum ( ) 2 juhL Lift p Static local pressure p Free stream pressure q Free stream dynamic pressure ( 22U) ju Oscillatory jet velocity jU Mean jet velocity U Free stream velocity SEP X Distance from separation points to trailing edge TE X Distance from excitation slot to trailing edge Air density Boundary layer momentum thickness 2 s Variance of signal 2n Variance of noise 14 PAGE 15 ABBREVIATIONS AM Amplitude Modulation AOA Angle Of Attack BM Burst Modulation DAQ Data AcQuisition DSP Digital Signal Processing ID IDentification LDV Laser Doppler Velocimetry MSE Mean Square Error PIV Particle Image Velocimetry PM Pulse Modulation PSD Power Spectral Density SJA Synthetic Jet Actuator SNR (dB) Signal to Noise Ratio ( 22 1010log() s n ) 15 PAGE 16 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy B ADAPTIVE CONTROL OF SEPARATED FLOW By Ye Tian August 2007 Chair: Louis Cattafesta Major: Aerospace Engineering Flow separation has severe adverse effects on performance of flowrelated devices (e.g., lift loss of aircrafts). Active control of separa ted flow has received extensive attention as it is able to mitigate or eliminate flow separation eff ectively. Most research has been openloop in nature (i.e., manually adjusting control inputs to achieve best results). Closedloop control of separated flow has many potential advantages over openloop c ontrol, namely optimization in multidimensional domain with constraints, adaptabi lity to changing flow conditions, etc. In this research, adaptive closedloop control is used to reattach the separa ted flow over a NACA 0025 airfoil using multiple zeronetmassflux (ZNMF) actuators that cover the central 33% of the airfoil span. In particular, tw o distinct approaches are used. Adaptive disturbance rejection algorithms are used to apply dynamic feedback control of separated flow. The closedloop control results show ~ 7 x improvements in th e lift/drag ratio, with a corresponding increase in lift and reduced drag and concomitant reductions in the fluctuating surface pressure spectra. On the other hand, a simplex optimization approach uses the lift and drag measured by a straingauge balance for feedback and searches for th e optimal actuation parameters in a closedloop fashion. The constrained optimization result s seeking to maximize lifttodrag ratio are promising and reveal the importance of forcing nonlinear interactions between the shear layer 16 PAGE 17 and wake instabilities. To the authors knowledg e, this is the first time that these types of closedloop control schemes are implemen ted to control separated flow. 17 PAGE 18 CHAPTER 1 INTRODUCTION The primary goal of this research is to implement a closedloop control system to control separated flow and to evaluate the performance of the controller. A control system that includes an array of actuators, sensors (pressure sensors or lift/drag balance) and a digital controller is proposed to control flow separati on in a closedloop fashion. This first chapter introduces the flow physics and active co ntrol approaches of flow separation. It is organi zed as follows. First, a brief overview of separation control is provided to orient the reader, followed by the motivation. Th en a technical background section is presented to review previous work reported in the lite rature. Finally, the objectives and technical approaches of this research are presented. Overview Flow separation is identified as one of the most important flow phenomena due to its severe adverse effects on flowrelated devices. Following the introduction of the concept of the boundary layer by Prandtl (1904), flow separation has received considerable attention in the fluid dynamics community. Flow separation is the breakaway or detachment of fluid from a solid surface (Greenblatt and Wygnanski 2000). Flow separation incurs a large amount of energy/lift loss and limits the performance of many flowrelated devices (e.g., airplanes, diffuse rs, etc.). Researchers have been trying to eliminate or at least mitigate flow separation for over a century because of its large potential payoff in many applications. As shown in Figure 14 control of separated flow is di vided into two main categories: active control and passive control. Active contro l provides external energy into the flow while passive control does not. Some passive separatio n control methods, such as geometrical shaping 18 PAGE 19 and turbulators (i.e., turbulence generators), ar e commonly used because of their simplicity and feasibility. On the other hand, tremendous progre ss has been made in act ive separation control over the past twenty years. Traditional ac tive separation control methods, such as steady blowing and suction, were initially used to control flow separation (GadelHak 2000). These methods were able to control of separation to some extent. However, they were far from optimal because the overall energy required input re quired to gain a meaningful lift increase or drag reduction was comparable to the energy saved via control of separation (Greenblatt and Wygnanski 2000). Schubauer and Skramstad (1948) first introduced a breakthrough in activ e flow control: periodic excitation. This technique requires much less energy than traditional steady active methods and accelerates and regulates the genera tion of large coherent structures that are primarily responsible for the transport of momentum across the flow (Greenblatt and Wygnanski 2000). The increased large coherent structures make the flow mo re resistant to separation. Periodic excitation has subsequent ly been shown to be superior to steady boundary layer control methods by many researchers (Seifert 1996; Greenblatt and Wygnanski 2000; Nishri and Wygnanski 1998). Because of thes e reasons, periodic exc itation is now widely used to control flow separation. Optimal excitation locations, waveforms shapes, and frequencies of periodic perturbations have been systematically studied by numerous research ers (Seifert and Pack 2003A, Amitay et al. 2001). Yet none of these st udies has used feedback control to optimize the excitation waveform. One of the most important aspects of separa tion control is the actuation mechanism that introduces periodic perturbations in to the flow structure. Intern al acoustic excitation (Hsiao et al. 1990; Huang et al. 1987), speakers (Naray anan and Banaszuk 2003), oscillatory blowing 19 PAGE 20 valves (Allen et al. 2000), and MEMSbased ac tuators (Rathnasingham a nd Breuer 2003), etc. have been investigated. Among th ese, synthetic jet or zeronet mass flux (ZNMF) actuators have been the focus of significant research for the past decade due to their utility in flow control applications (Glezer and Amita y 2002). ZNMF actuators utili ze the working fluid and do not require an external fluid source, which ma kes them very attractive from a systems implementation perspective. Significant progress has been made in the modeling and design of such devices (Gallas et al. 2003, 2005). More details of the synthetic jet ac tuators used in this research are described in Chapter 4. The drivi ng frequency, location, and momentum coefficient of the actuation are the primary parameters that characterize their perf ormance (Amitay et al. 2001). Although separation control has received extensiv e attention, to date most studies have focused on openloop separation control. In the au thors opinion, this openloop approach is due to a fluid mechanics bias to avoid using a more complex closedloop control approach. Closedloop separation control has the potential to save more energy than openloop methods (Cattafesta et al. 1997) and make separation control systems adaptable to different flow conditions. Few experimental studies have focused on closedloop separation control. For example, Allan et al. (2000) attempted to tune a PID controller for cl osedloop separation control and showed that the integral gain was the most effective as a result of the large time constant of their low bandwidth actuator system. However, the realized model and controller were simple. Their results merely scratched the surface of what can possibly be accomp lished. Therefore, it is believed that control of flow separation using an ar ray of high bandwidth actuators and surface sensors (pressure or shear stress) is an excellent candidate for clos edloop separation control. Hence, implementation 20 PAGE 21 of feedback controllers including more advanc ed modeling and control algorithms to flow separation control is proposed and is the focus of this research. Motivation Numerous applications of separa tion control, each with significant potential payoffs, have been identified (Greenblatt and Wygnanski 2000). Many separation control strategies have been applied on civil and military aircrafts and unde rwater vehicles. However, most of the applications are openloop in nature because of their simplicity. Although some closedloop separation control resear ch has been done (Allan et al. 2000; Banaszuk et al. 2003, etc), they are not sufficiently developed to be implemented on real vehicles. The goal of this research is to design and implement various closedloop control sy stems for control of se parated flows and to seek physical insights behind the control sche mes. The main advantages of closedloop separation control potentially include better perf ormance, energy efficiency and adaptability to changing of flow conditions. Background TwoDimensional Separation Flow Physics Under the circumstances of an adverse pressure gradient ( 0 dpdx ), fluid particles are retarded by both the increasing pressure as well as wall skin friction. If the adverse pressure gradient is of sufficient strength, fluid particles near the wall are likely to separate from the wall and move upstream. This is due to the fact th at these particles have finite kinetic energy and cannot penetrate far into the adve rse pressure gradient region. The flow separates from the boundary layer and forms large scale vortical structures in the separated region ( Figure 11 ). Assuming twodimensional, incompressible, steady flow with negligible gravity, the streamwise ( x ) component of the momentum equation at the wall reduces to 21 PAGE 22 22 221 udpu u u x dxxy (1.1) or 22 21 udpu u ydxx2u x (1.2) where is the kinematic viscosity, y is the wall normal coordinate, x is the streamwise coordinate with a corresponding velocity and streamwise pressure gradient u dpdx. From eqn. (1.2) we can see that only an a dverse pressure gradient ( 0 dpdx ) can cause a point of inflection in the velocity profile and the curvature changing sign to make the profile Sshape. In this case, separation will occur when the adverse pressure gradient is strong enough to make the right hand side of eqn. (1.2) positive (shown in Figure 11 ). Effects of Flow Separation In the separation region, the normal velocity co mponent significantly in creases as well as the thickness of boundary layer. Therefore, th e boundary layer approxim ations are no longer valid and the problem can no longer be solved using boundary layer theory. Flow separation significantly changes the pre ssure distribution around the surface. Such deviations are usually detrimental. As an example, Figure 13 shows the and LC D C100 of a NACA0025 airfoil versus angle of attack m easured by a lift/drag balance at When the angle of attack increases from zero degree, both and Re,000LC D C increase as expected. However, drops dramatically due to flow separation at about 13 degrees of angle of attack. At the same time, LC D C continues to increase be yond the inception of stall. Both of these effects generally have a negative impact on the airplane performance. However, some applications utilize flow separation. For ex ample, the use of spoilers on ai rplanes during landing reduces the lift and increases drag to allow the brakes to work more efficiently. 22 PAGE 23 More commonly, we want to mitigate or eliminat e flow separation. Typical applications of flow separation control include: separation control of various airfoils to increase for larger payload (Greenblatt and Wygnanski 2000; Seifert and Pack 2002; etc); to reduce engine power and noise at takeoff (GadelHa k 2000); to increase efficiency of diffusers (i.e. pressure recovery) (Banaszuk et al. 2003); etc. maxLCControl of Flow Separation Because of the effects mentioned above and th e large potential payoff, researchers have been preoccupied with delaying flow separation or eliminating it entirely. As suggested by Cattafesta et al. (2003), the cl assification of flow control is chosen as shown in Figure 14 to be consistent with terminology used in active noise and vibration control. Active control is subdivided into openloop versus closedloop co ntrol. Closedloop control can be further classified into quasistatic versus dynamic, the distinction between the tw o being whether or not the feedback control is performed on a time scale with the dynamical scales of the flow. Since fluid flows are inherently nonlin ear (Wu et al. 1998), the standa rd frequency preservation of a linear system does not hold. Consequently, non linear feedback contro l on a very slow time compared to the characteristic times scales of th e flow is, in fact, possi ble and attractive. In essence, this socalled quasistatic control becomes a nonlinea r optimization problem. This research will investigate both classe s of closedloop control shown in Figure 14 Other fluid dynamic issues have been studied extensively, such as the effects of Reynolds number, frequency, actuator and sensor locations momentum coefficient, surface curvature, and compressibility, etc.. Although the topic of this research is closedloop separation control, the results and conclusions from the openloop contro l studies should serve as a sound physical basis for effective control and are reviewed below. 23 PAGE 24 Openloop separation control Periodic excitation has been shown to be much more effective than steady forcing because it enhances the momentum transport across the flow domain at a substantial reduction in energy expenditure. It accelerates and regulates the generation of large coherent structures that are primarily responsible for the momentum trans port across the flow (Greenblatt and Wygnanski 2000). The enhanced momentum transport forces the separated flow to reattach to the surface and form a thick turbulent boundary layer in a tim eaveraged sense. The reattachment of the boundary layer regains the pressure suction zone on the upper surface of the airfoil and thus enhances the lift performa nce. Furthermore, the superpositi on of weak suction on the periodic excitation enhances the receptivity of the separa ted shear layer to the fundamental excitation frequency and thus the effec tiveness of periodic excitation (Seifert and Pack 2002). Given the improved performance of periodic excitation to contro l flow separation, researchers have sought to optimi ze separation control via timecons uming parametric variations. Significant parameters or conditions that affect the performance of separation control have been identified. Although they are di scussed separately below, one should keep in mind that these factors are all coupled with each other. Actuation frequency First, consider the characteristic flow structur es associated with se parated flow. Based on previous studies, Mittal et al (2005) summarize the three situati ons with regards to separated flow, as shown in Figure 15 In poststall flow (case C in Figure 15 ), leadingedge shear layer rollup and vortex shedding in the wake are two char acteristic features (Wu et al. 1998). Huerre and Monkewitz (1990) suggest that this type of shear flow (with a pocket of absolu te instability of sufficient size) may display in trinsic dynamics of the same natu re as in a closedflow system, in which disturbances can grow upstream (i.e. globa l instability). Therefore, it is reasonable to 24 PAGE 25 postulate that separated flow over an airfo il acts as a nonlinear multifrequency closedflow system. In such a system, the shear layer instability (with characteristic frequency SL f ) and the global wake instability (with vortex shedding frequency wake f ) may interact with each other in a nonlinear fashion. In case B, a closed separa tion bubble is present at some distance downstream of the leading edge. In this case there are poten tially three characteristic flow frequencies in the separated flow: SL f wake f and s ep f where the new scale, s ep f corresponds to the characteristic frequency of the separation bubble. The scales of the three frequencies are ~SLSLfU ~SLsep f UL and ~wake wakefUW, where SL is the shear layer thickness, s epL is the length of the separation bubble and is the width of wake. Prasad and W illiamson (1996) also show that wakeWReB SL wake f Af where and Since there are different relevant leng th scales that are included in the three characteristic frequencies, one should expect a significant variation in the observed frequency scales and the correspon ding optimal frequency. 0.0235 A 0.67 B The present study is focused on how flow sy stems respond to modulated (e.g. AM, BM, PM) unsteady excitations by ZNMF de vices targeting the inherent flow instabilities that lead to the presence of these characteristic flow frequenc ies. The goal is to search for optimal forcing schemes that most effectively mitigate flow separation via nonlinear interaction of the instabilities. Much research has been conducted to determ ine what excitation frequencies are most effective for separation control. However, ex cept for the general agreement that periodic excitation is far more effective than steady blowi ng, the range of optimal actuation frequencies is a current subject of intense deba te. A dimensionless actuation fr equency is typically defined for this purpose. However, three slightly differen t definitions have been given for a socalled 25 PAGE 26 dimensionless frequency : 1) F eTEFfXU where e f is the excitation frequency, TE X is the distance from the excitation slot to the trailing edge and U is the free stream velocity; 2) esepFfLU where s epL is the distance from separation to reattachment; and 3) eFfcU where c is the chord length. These three ar e nearly identical for poststall flow (where the separation bubble leng th is approximately the airfoil chord), but they scale very differently if a closed separation bubble of finite extent is present. One should notice that none of these definitions is related to the shear layer frequency (SL f ). Most researchers implicitly ignore this important frequency wh en studying separation control. Herein, some results regarding actuation fre quency in previous studies are summarized. Among studies that define eTEFfXU Wygnanski and his colleag ues conclude that the optimal excitation frequency is of order unity (Seifert et al. 1996, Nishri and Wygnanski 1998, Greenblatt and Wygnanski 2000) and have found that socalled high frequency fording is ineffective for their airfoil (NACA 0015) and fl ow conditions. Conversely, using the same definition of (1)FO (10)FOF Amitay et al. (2001) found that when the excitation frequency the lifttopressure drag ratio was larger than that when the excitation frequency Honohan et al (2000) also suggest ed that higher reduced frequencies ( ) can be effective. They argued that it is because the high frequency excitation produces a virtual aerodynamic surface modificati on that thins the turbulent boundary layer and results in a local favorable pressure gradient. (10)FO 4F10 FBesides this argument, there may be two other possible reasons accounting for this interesting discrepancy. First, the lengthscale TE X may not be appropriate for their airfoil 26 PAGE 27 because of the formation of a closed separation bubble. Instead, if s epL were used, this discrepancy might not exist. Second, as mentioned earlier, the shear layer frequency SL f may also be important (Mittal et al 2005). Here, SL f U where is the boundary layer momentum thickness and not TE X or s epL. The different frequency scales are indicative of different flow instabilities that may exist in the flow and, if present, may compete with each other (Wu et al 1998). When periodic excitation is introduced, one or more of these instabilities may be energized. The controlled flow may then be regulated, and thus lift performance may be enhanced. This may explain the observed variations of the optimal excitation frequency. Along these lines, an innovative forcing appro ach that uses multiple harmonically related frequencies is presented by Narayanan and Bana szuk (2003). They demonstrated improvements of this new approach versus single frequency si nusoidal forcing in control of separation in a diffuser, although its effectiveness requires further investigation. To extend this idea further, one can use excitations with multiple frequency components corresponding to the characteristic frequencies mentioned above. This idea will be investigated in this research. Excitation amplitude Another key control parameter in a ZMNF device is jet velocity J V (some characteristic velocity measure, e.g. the peak or an average ve locity). In the literature, the jet frequency is usually nondimensionalized as /UsepFfL where s epL is, for example, the length of separation region and U is the free stream velocity. The jet velocity is usually nondimensionalized by U. Various researchers have show n that control authority varies monotonically with for a sinusoidal excitation up to so me maximum value (Seifert et al. 1993, 1996, 1999; Glezer and Amitay 2002; Mittal and Rampunggoon 2002). In practice, /JVU 27 PAGE 28 especially in high speed flows, c ontrol authority is often lacking. From an efficiency standpoint, it is desirable to control a flow with minimal actuator input. Modulation signals Piezoelectric actuators have fast dynamic res ponse and low power consumption. However, the use of piezoelectric actuators has been lim ited because of the diminution in their response outside a narrow frequency band around their resonance frequency and the need for testing over a wide frequency range due to the issues discussed in the last section. Wiltse and Glezer (1993) introduced a clever amplitude modulation method to flow control problems to overcome this problem. The piezoelect ric actuator is resonantly driven with a carrier waveform, which is amplitude modulated with a timeharmonic wave train: () et ()[1sin()]sin()mmrcet tAt (1.3) where is the amplitude of the carrier signal, rA is the degree of modulation (01 ), c is the carrier frequency (or the resona nt frequency of the actuator) in /radsm is the modulation frequency (which is also the desired excitation fr equency or receptive frequency of the flow) in and /rad sm is the phase of the modula ting signal. By using trigono metric identities, one can show that contains frequency components at etc and cm However, when the excitation amplitude is high enough, et is demodulated by the nonlinear fluid dynamical system that is associated with the formation and coalescence of nominally spanwise vortices. This nonlinearity results in the presence of c and cm and also m in the flow. In practice, c is set at the resonance frequency of th e piezoelectric actuator (which is usually m ) and m is set at the desired low frequency corre sponding to the desired excitation frequency e f 28 PAGE 29 Along these lines, other modulation signals such as burst modulation and pulse modulation can also be used. This modula tion technique allows the actuato r operating at its resonant frequency to generate a significant flow distur bance while effectively manipulating flows at characteristic frequencies of the flow. It provid es a much more flexible approach than matching the resonant frequency of the actuator with the receptive frequencies of the flow. However, some features of the technique should al so be kept in mind. First, the actuator is driven continuously near its resonant frequency, so the probabili ty of mechanical failure is greater than when it is driven off resonance. Second, as mentioned above, demodulation of the waveform is due to nonlinearities of the flow and actuator. As a result, feedback controllers designed based on a linear assumption may not work as desired. This aspect will be studies in this research. Actuation location It is argued by many researchers that the optimal actuation location is at the vicinity of the point of separation (Amitay et al. 2001, Seifert et al. 1996, Seifert and Pack 2003). This is physically plausible since the disturbances intr oduced at this location can most effectively transport momentum between the free shear layer and the separated region. However, this has not been systematically studied because of some practical limitations, namely the difficulty of installing multiple actuators inside an airfoil. Amitay et al. (2001) used an unconventional airfoil that had an aft portion of a symmetric airfoil attached to a circular cylinder forebody with a synthetic jet slot that could be adjusted by rotating the cylinder. They state that the closer the control is located to the observed separation point, the less power is required to reattach the flow. They also made an interesting point that if ei ther the separation locat ion is unknown or practical 29 PAGE 30 limitations preclude control near the separa tion location, the momentum coefficient C may be manipulated to achieve optimal performance. Besides the effects of actuation location di scussed above, the inte raction of adjacent synthetic jet actuators has been investigated by Holman et al. (2003). They found that relative phasing between adjacent actuators does not appear to affect the effec tiveness of separation control significantly for their airf oil (NACA 0025) and flow conditions ( and ). 510eR12 AOAIn summary, based on the previous studies it is suggested that sli ghtly upstream of the separation location is the best place to introduce actuation. Furthermore, a combination of upstream leading edge and downstream trailing edge actuations may also be a good candidate and remains to be investigated (Mittal et al 2005). Wu et al (1998) discuss this idea in the context of the KuttaJoukowski lift formula (LU L ), which assumes the flow is incompressible and steady. In the formula, is the lift, U is free stream velocity and is the circulation (a counterclockwise circulation is assumed positive). Although the separation is an unsteady process, this formula still holds in a timeaveraged sense for the entire flow. Based on these arguments, if the combination of leading e dge and trailing edge actuation can be designed to alter the circulation of the airf oil, it should be able to control flow separation in some manner. Effects of Reynolds number and compressibility It is shown that control of flow separation is insensitive to the Reynolds number at high chord Reynolds numbers of 11~30 million (Sei fert and Pack 2003 A, B, Greenblatt and Wygnanski 2000). The Reynolds number has a very weak effect on pressure distributions around the surface, regardless of the Mach number. 30 PAGE 31 On the other hand, strong Reynolds number e ffects are identified in the airfoil baseline performance at moderately compressible flow conditions (Seifert and Pack 2001). Reynolds number effects weaken as the Mach number in creases and a stronger sh ock wave develops. Compressibility tends to elongate the separation bubble and reduce the capability of periodic excitation to shorten the separa tion bubble with similar excitation frequencies and momentum (Seifert and Pack 2001). It is also suggested by Seifert and Pack ( 2001) that in the presence of shock waves the excitation location should be sligh tly upstream of the shock wave. If the excitation is introduced well upstream of the shock wave, it has a detrimental effect on lift, drag a nd wake steadiness. Closedloop separation control Closedloop experimental separa tion control has not yet received significant attention. This section first reviews some development of the microelectromechanical systems (MEMS) based actuators because of th eir potential importance to high bandwidth closedloop control systems. Then the limited previous work on closedloop separation control is presented. For closedloop flow control systems, the desire d actuators should be fast, power efficient, and reliable. In previous separation control st udies, acoustic excitation (Hsiao et al. 1990 and Huang et al. 1987) seems facility dependent be cause the acoustic drivers stimulate the wind tunnel resonant modes to excite the separated flow ; oscillatory blowing valv es (Allen et al. 2000) appear to have slow dynamic response; active flexible wall transducers (Sinha 2001) have complicated structures despite its high actuation ef ficiency and ability to actuate and sense with the same hardware. These drawbacks have limited the use of these actuators. On the other hand, synthetic jet (ZNMF) act uators have been th e focus of significant research activities for the past decade due to thei r utility in flow control applications (Glezer and Amitay 2002). They utilize the working fluid and do not need external fluid injection. They can 31 PAGE 32 force the momentum transfer across the flow w ithout net mass flux (thus the name synthetic). The design of synthetic jets is also flexible and the working frequency range can be tuned according to different flow control applications. In addition, the recent paper by Gallas et al. (2003) presents a lumped element model of a piez oelectricdriven syntheti c jet actuator. They provide a novel method to design and model synthetic jets, which makes them very suitable for closedloop separation control. In lumped element modeling ( LEM), the individual components of a synthetic jet are modeled as elements of an equivalent electrical circuit using conjugate power variables (i.e., power = generalized flow x generalized effort). The frequency response function of the circuit is derive d to obtain an expression for outACQV, the volume flow rate per applied voltage. The compar ison between the LEM and experimental frequency response is shown in Figure 16 For a variety of reasons, closedloop contro l in a realtime experiment has been traditionally difficult to achieve. In reduceds cale laboratory experiments, the characteristic frequencies of separated turbulent flows are proportionally higher th an those on fullscale models, which requires high frequency sensing and actuating capabilities. Furthermore, realtime experiments require the digital control system to sample at a minimum of twice of the highest frequency of interest. Th e availability of hardware (inclu ding actuators, sensors and realtime control systems) therefore imposes significan t limitations on the complexity of the closedloop control system. Lower order system models are typically required to reduce the complexity of the system. Many modelbased approaches are being develo ped and have shown promising results. Proper Orthogonal Decomposition (POD) based low order models have been studied extensively (Holmes et al. 1998; Tadmor et al. 2007) owi ng to their relatively high resolution and low 32 PAGE 33 computational intensity. Other reducedbasis mode ls have also been studied (Coller et al. 2000; Wang et al. 2003). These models require that multiple measurements are simultaneously available in the flow field. However, this is impractical in feedback separation control and surface measurements are required in most applic ations. Ausseur et al. (2007) implemented a POD/mLSM proportional feedback c ontrol using the velocity field and surface pressure data to delay flow separation. Some nonmodel based control approaches ha ve gained favor because they bypass the complication of modeling separated flow while focusing on the primary control objectives. For example, Banaszuk et al. (2003) and Becker et al. (2006) used an extremumseeking closedloop control algorithm to optimize the pressure recovery and lift, resp ectively. The present author in Tian et al. (2006) used a multidimensional optim ization algorithm to optimize lifttodrag ratio over an airfoil. These approaches are capable of training the excitation signals to be most effective in terms of the objective functions (i.e., pressure recovery lifttodrag ratio, etc.). The main drawback of the above a pproaches is that they operate on a time scale that is much larger than that of the flow dynamics. In other words, they work on timeaveraged objective functions by explicitly taking advantage of the nonlinear nature of the fl uid dynamics. This approach has the drawback of having to deal or cope with the nonlinear dynamics with no guarantee of success. This kind of approach is an exampl e of the quasistatic control scheme shown in Figure 14 On the other hand, the dynamic feedback control is used to model and control separated flow structures based on surface pressure data alone. The welldeveloped adaptive system identification (ID) algorithms in the controls co mmunity are utilized to model the flow system dynamics between the actuators and unsteady surface pressure sensors. The system ID 33 PAGE 34 algorithms generate known actuation signals and relate these signals with the surface pressure response measured by sensors. Linear dynamical equations are then used to model the relationship in a gradient descent sense (Hayki n 2002). The system therein includes the dynamics of the actuators, the flow structures excited by the actuation, and the dyna mics of the sensors. The system information is then used to pred ict the subsequent evolution of the pressure fluctuations. Control is applied using a span wise zeronetmassflux (ZNMF) actuator slot by attempting to reduce the power of the surface pressure fluctuations in a closedloop fashion, thus suppressing the unsteady flow fluc tuations based on predicted flow characteristics. A similar idea has been applied to control of flowinduced cavity oscillations (Catta festa et al. 1999) and turbulent boundary layer control (Rathnasingham and Breuer 2003). This kind of approach can be categorized as a dynamic control scheme shown in Figure 14 ClosedLoop Control Algorithms According to the classification in Figure 14 the control algorithms can be divided into two categories: quasistatic and dynamic. Optimiza tion algorithms are used in this research as quasistatic algorithms. They are used to optimize target functions (such as lift, pressure recovery, etc.) in a recursive but static or time averaged fashion. On the other hand, recursive system identification and disturba nce rejection algorithms are wide ly used in active noise control area as dynamic algorithms. No one has attempted to apply these algorithms to the closedloop separation control problem. This section gives a br ief review of the two types of the algorithms. Details will be given in chapter 2. Optimization algorithms Optimization algorithms are widely used by decisionmakers (e.g. economists, governments). They often need to choose an action to optimize target or cost functions, such as income, profit, etc. In a typically optimi zation problem, one is given a single function f that 34 PAGE 35 depends on one or more independent variables. The goal is to find the va lue of those variables where f is a maximum or a minimum value. In this research, various optimization algorithms are used to maximize/minimize different cost functi ons, such as lift, drag and pressure recovery. When using the optimization algorithms, some constraints are typi cally included in the algorithms. For example, one often seeks to limit the energy expenditure while optimizing the cost function. One should also notice that, unlike the applications used by the decisionmakers, the cost functions used in this research are meas ured by sensors instead of analytical functions. Some established minimization and maximiza tion algorithms are summarized by Press et al. (1992). Most optimization algorithms can be easily implemented in a multidimensional space. The downhill simplex algorithm and the Powells algorithm do not require derivative calculations. Between these two algorithms, the downhill simplex algorithm is more concise and selfcontained. Both of them require storage of order where is the number of dimensions or independent variables. Two other algorith ms, the conjugate gradient and quasiNewton methods, do require the calculation of derivatives. The conjugate gradient method requires only order storage, while the quasiNewton method requi res storage of order On the other hand, none of the algorithms mentioned above are guaranteed to find a global extremum. They can lead to local extrema. Finding a global extr emum is actually a very difficult problem. Two standard methods are typically used to improve the probability of finding a global extremum: 1) search for local extrema from various initial cond itions and pick the most extreme of these; 2) perturb a local extremum to see if the algorithm goes back to the same value or finds a better result. 2NN2NNThere are several global search algorithms that are currently ac tive in research(e.g. Genetic Algorithms (GA) (Holland 1975), Particle Sw arm Optimization (PSO) (Kennedy 1997) and 35 PAGE 36 Simulated Annealing Method (Haftka and Grdal 1992)). The genetic algorithms and the particle swarm optimization ar e both derived from biology. They are populationbased algorithms, namely they generate a populati on of points at each iteration and the population approaches an optimal solution. The GA and PS O take advantage of the large search population to increase probability of approaching a global op timum. The simulated annealing method is an analogy with thermodynamics, especially with the way metals cool and anneal, in which process nature finds the minimum energy state. The essen ce of the algorithm is to allow increase of cost function with some probability to improve the changes to find a global minimum. Another optimization algorithm that has been ap plied to flow control problems is called the extremumseeking algorithm. As a selfoptimizing control algorithm, the extremumseeking control was first introduced in the 1950s. After Krstic and Wa ng (1999) provided the stability studies, there has been a resurgen ce of interest of this control algorithm. Banaszuk et al. (2003) attempted to use this algorithm in the diffuser separation control problem. They were successful in maximizing the pressure recovery in the diffuse r. They also used this algorithm to control combustion instability (Banaszuk et al. 2000). System identification and di sturbance rejection algorithms System identification and disturbance rej ection technologies are well developed and various algorithms are available in the active noise control area. Cattafesta et al. (1999) have applied these algorithms to other flow control problems, such as cavity resonance control. No one has attempted to apply this kind of approach to the separation control problem. In this research, this approach is inve stigated. Some system identific ation and disturbance rejection algorithms are reviewed in this section. System identification algorithms 36 PAGE 37 In general, system identification (ID) uses m easured signals (i.e., inputs and outputs of the system) to identify (or estimate) the unknown syst em dynamics. It provides necessary system information for control algorithms. System iden tification algorithms can be divided into two categories: offline (or batch) and online (or recurs ive). Offline algorithms first acquire data and then try to estimate a loworder dynamical syst em model using these data offline. Online algorithms identify systems recursively while acquiring data in realtime. Online system identification is also known as adaptive filtering. Least square (LS) identification algorithm is a generally used offline algorithm. Akers and Bernstein (1997 A) applied this approach to the ARMARKOV/LS identifi cation algorithm with an ARMARKOV representation (see Chapter 2 for a detailed description of the algorithm). The ARMARKOV/LS identification algo rithm uses vectors comprised of inputoutput data with a leastsquares criterion to estimate a weight matrix containing a specified number of Markov (i.e., pulse response) parameters of the system. Th en the eigensystem rea lization algorithm (ERA) (Juang 1994) is used to construct a minimal state sp ace realization of the syst em. This is referred to as the ARMARKOV/LS/ERA identification algorithm. The ARMARKOV/LS/ERA identification algorith m has two clear advantages compared to the ARMA/LS identification algorit hm (Akers and Bernstein 1997 A). First, eigenvalues of the ARMARKOV representation are le ss sensitive to noise compared with eigenvalues of the ARMA representation. Second, the singular value decomposition of a block Hankel matrix constructed from the estimated Markov parameters provides an efficient model order indicator (Juang 1994, pp. 139). As far as online algorithms are concerned, the leastmeansquare (LMS) algorithm is the most commonly used algorithm. A more co mputationally intensive algorithm called the 37 PAGE 38 recursiveleastsquare (RLS) algorithm has faster convergence and smaller steadystate error than the LMS algorithm (Haykin 2002) but is mo re computationally intensive. Two different types of structure that can be applied to each of the algorithms are the finiteimpulseresponse (FIR) and the infiniteimpulseresponse (IIR) filter s. The FIR filter is widely used due to its simple architecture and inherent stability as an allzero model. However, its simple structure introduces difficulties for a system with low dampi ng. The IIR filter can solve this problem with significantly lowerorder and, therefore, lead to reduced computational expense. Unfortunately, the disadvantages of an IIR filter include more complicated adaptive algorithms compared with an FIR filter and the possible stability problems introduced by the pole(s) in the model (Haykin 2002; Shynk 1989; Netto and Diniz 1995). Applying the LMS algorithm to the ARMARKOV representation, Akers and Bernstein (1997 B) introduced the recursive ARMARKOV/Toepl itz algorithm that is based upon recursive identification of the Markov parameters of a system. It estimates the Markov parameters recursively using timedomain, i nputoutput data and then constr ucts the estimated model with the Markov parameters. Disturbance rejec tion algorithms As mentioned earlier, one of the possible control schemes for closedloop separation control is to reduce velocity and pressure fluctu ations in the separated region. This control scheme is generally called disturbance rejection. Disturbance rejection controll ers have been widely used in active noise control applications (Kuo and Morgan 1996). Recently, researchers have started to apply adaptive controllers to flow control problems. For exam ple, Cattafesta et al. (1999) used an adaptive system to suppress the disturbance induced by the flow over a weaponsbay cavity. The advantages of using adaptive controllers are that they can adapt themselves according to different 38 PAGE 39 flow conditions and that they can potentially reduce the energy co st associated with the flow control problems. Cattafesta et al. (1997) showed that the control of cavit y flow with closedloop control requires one orderofmagnitude less power than that with open loop control. Commonly used disturbance rejection algor ithms include FilteredX LMS (FXLMS), FilteredU LMS (FULMS), FilteredX RLS (FXR LS) and FilteredU RLS (FXRLS) algorithms (Kuo and Morgan 1996). Besides these, the AR MARKOV adaptive control algorithm was first introduced by Venugopal and Berstein (1997) and fu rther developed by Sane et al. (2001). The underling model structure of the ARMARKOV adaptive control algor ithm is the ARMARKOV representation, which is an exte nsion of the ARMA representati on with explicit impulse response (Markov) parameters. The ARMARKOV adaptive c ontrol algorithm doesnt require a model of the controltoreference transfer function nor does it require a model of the transfer function from plant disturbances to sensors (S ane et al. 2001). The only transfer function needed is the control to performance transfer function, which can be identified simultaneously using the recursive ARMARKOV/Toeplitz system identi fication algorithm described in the previous section. Objectives To explore suitable linear and nonlinear contro l objectives and strate gies for closedloop control of separated flows. To implement optimization algorithms and sy stem identification/disturbance rejection algorithms for closedloop c ontrol of separated flow on a wind tunnel airfoil model (NACA 0025). To analyze performance, adaptability, cost s, and limitations of closedloop separation control algorithms. To investigate the relevant flow physics of successful feedback control strategies. Approach The proposed closedloop separa tion control includes two key parts: modeling and control strategies. As far as modeling is concerned, two types of appr oaches can be implemented to 39 PAGE 40 model the flow characteristics: 1) a reducedo rder flow model base d on the NavierStokes equations, 2) system identification techniques. The first approach is widely used in computational flow control simulations. This re search will concentrate on experimental studies by using system identification tec hniques that have not yet been a pplied to the separation control problem. The dynamical systems model will include the dynamics of actuators, sensors, and the flow system. Then the disturbanc e rejection algorithm is used to suppress flow fluctuations (e.g., measured by unsteady pressure transducers). One the other hand, for the nonmodel based optimization algorithms, no system identification is needed. The possible cost functions for the algorithm are summarized as follows. Since the suction pressure region of the upper surface of the airfoil is primarily responsible for lift generation and drag reduction, the static pressure recovery coefficient (pdCdxc) over the upper surface of the airfoil is a reasonable candidate as a cost function to maximize for feedback separation control. Other candidates for cost func tions are lift and drag or combinations of these (e.g., li ft/drag ratio). The benefit of us ing lift/drag is that L/D is a global or integrated quantity and is less sensitiv e to sensor location. The objectives for the controller are clear, i.e. to minimize drag and to maximize lif t or the ratio of lift/drag. The experimental setup uses a lift/dra g balance for this purpose. Outline of This Dissertation A theoretical background on system identificat ion, control, and op timization algorithms will be discussed in Chapter 2. Simulation results and validation experiments of the algorithms will then be presented in Chapter 3. Chapter 4 describes the experimental setup and techniques for this research. Chapter 5 presents experiment al results and discussion Summary and future work will be presented in the last chapter. 40 PAGE 41 Figure 11. Separation of flow over an airfoil. Figure 12. Types of velocity profiles as a function of pressure gradient (White 1991). 41 PAGE 42 Figure 13. Lift and dr ag coefficients of NACA 0025 airfoil at R e100,000 Figure 14. Classificatio n of flow control. (Cattafesta et al. 2003) 42 PAGE 43 43 Figure 15. Characteriza tion of possible frequency scales in separated flow (M ittal et al. 2005). 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 30 35 Frequency (Hz) Magnitude of maximum velocity (m/s) Figure 16. Comparison between the lumped element model () and experimental frequency response () measured using phaselocked LDV for a prototypical synthetic jet (Gallas et al. 2003). PAGE 44 CHAPTER 2 THEORETICAL BACKGROUND This chapter presents detailed descriptions a nd derivations of the al gorithms that are used in this research. The algorithms include optim ization, system identification, and disturbance rejection algorithms. Optimization Algorithms Some established minimization and maximiza tion algorithms are summarized by Press et al. (1992). The downhill simplex algorithm and th e Powells algorithm do not require derivative calculations, which makes them good candidates for this research since de rivative calculations are problematic for (usually no isy) experimental data. Be tween these two algorithms, the downhill simplex algorithm is more concise a nd selfcontained. The socalled extremumseeking algorithm has been applied to a flow c ontrol problem by Banaszuk et al. (2003). Thus, this algorithm is also summarized here. Downhill Simplex Algorithm The downhill simplex algorithm is implemente d to minimize an objective function (e.g., dragtolift ratio). The benefits of the algorithm are its simplicity, applicability to multidimensional optimization and robust performa nce. The algorithm searches downhill in a straightforward fashion that makes no prior assumptions about the function. The downhill simplex algorithm requires only function evaluations not derivatives. Since it does not make any assumptions about the function, it can be very slow sometimes. However, it can be very robust in the sense that it guarantees to find a mi nimum (at least a local minimum) (Press et al. 1992). A simplex is the geometrical object consisting, in dimensions, of points (or vertices) whereas the points span a dimensional vector space (Press et al. 1992). For N1N1 N N 44 PAGE 45 example, in two dimensions, a simplex is a tria ngle. In three dimensions, it is a tetrahedron, although not necessarily a regular tetrahedron. The downhill simplex algorithm makes use of the geometrical concept of a simplex and works its way in the local downhill direction until it encounters a (at least, local) minimum. The key steps of the downhill simplex algorithm are summarized as follows: 1) Evaluate the cost function at chosen initi al points. Note th at there should be 1 N initial points, defining an initial simplex. For two or higher dimensions, the initial points should not be li nearly dependent. 2) Take a series of steps to move in the downh ill direction. As an example, the steps for threedimensional search are illustrated in Figure 21 In the figure, Reflection means that the algorithm reflects the highest (i .e. worst) point about the center of the three lower (i.e. better) points with some coef ficients to the other side of the plane and then evaluates the cost function at the refl ected point. Expansion means to expand further along the reflection direction when the Reflection point does improve (i.e., lower the cost function). Contraction mean s to move the highest (i.e. worst) point towards the plane formed by the three lower (i.e. better) points, t hus contracting the original simplex. To summarize, all the necessary steps taken here are to move the worst point reference to the plane formed by the other better points to search for a better point. 3) Stop when some termination criteria are met. For example, the moving distance is smaller than some tolerance value. Extremum Seeking Algorithm Artiyur and Krstic (2003) pres ent the theoretic details and some applications of the extremumseeking algorithm. Simulations using the algorithm can be done following the block 45 PAGE 46 diagram in Figure 22 The simple proof that this algorithm will drive () f to its extremum is summarized below. First, assume that () f has a minimum f and can be approximated as the following form: 2 *() 2!f ff* (2.1) where is the optimal input and f is the local curvature of the cost function () f near Since it is assumed that () f has a minimum, f should be larger than zero for this case. Next, define the estimated error: (2.2) where is the estimated optimal input. From Figure 22 (2.3) sin()sin()awtawt Substituting equation (2.3) into equation (2.1) results in 2 *()sin() 2f yffawt (2.4) Expand equation (2.4) and apply 21cos(2) sin() 2 wt wt to obtain "2 "2 *2" "2"2 "2 *"sin()sin() 2 cos(2)sin() 44 2 fa f yf wtfawt fafa f fw t f a w 2 t (2.5) From Figure 22 this signal will pass through a high pass filter y hs sw Let 0hww then all the DC components in equation (2.5) will be removed while the oscillatory terms remain. 46 PAGE 47 "2 "cos(2)sin() 4fa wtfawt (2.6) Next, is multiplied by to give sin()awt "2 "cos(2)sin()sin() 4fa wtwtfawt 2 (2.7) Using the trigonometric identities 21cos(2) sin() 2 wt wt and sin(3)sin() cos(2)sin() 2 wtwt wtwt results in "2 ""2"2"sin(3)sin()1cos(2) 422 sin()sin(3)cos(2) 2882 fawtwt wt fa fafafafa wt wt wt (2.8) From Figure 22 this signal passes through a low pass filter. Let then all the high frequency terms will be remove d and only the DC term remains. 0lw w "2 f a (2.9) This signal then passes through an integrator 2 f ak s (2.10) This gives 2 f ak s (2.11) From equation (2.2) assuming f is fixed, then (2.12) From equations (2.11) and (2.12) one can obtain the firstorder differential equation "2 f ak (2.13) with the solution 47 PAGE 48 "2 0 f ak te (2.14) Since f is assumed to be positive and and are positive constants, the estimated error a k will exponentially decay to zero. System Identification Algorithms System identification (ID) uses measured sign als (i.e., inputs and output s of the system) to identify (or estimate) the unknown parameters of an assumed dynamical systems model. It thus provides the necessary system information for control algorithms. System identification algorithms can be divided into two categories: offline (or batch) and online (or recursive). Offline algorithms first digitize a data record and then try to estimate the system using these data offline, usually via a least s quares method. Conversely, onlin e algorithms identify systems recursively while acquiring data in realtime. Online system identification is also known as adaptive filtering. In this research, three system ID algor ithms will be investig ated: ARMARKOV/LS, ARMARKOV/LS/ERA and recursiv e ARMARKOV/Toeplitz algorithms. They are all based on the ARMARKOV representation, which explicitly contains Ma rkov parameters (i.e., pulse response) of the system. The well known AR MA representation cont ains only one Markov parameter and is a special case of the ARMARK OV representation. The main advantage of these algorithms is their robustn ess with respect to low signa ltonoise ratios (Akers and Bernstein 1997 A, B). The ARMARKOV/LS algor ithm is an offline algorithm and implements an overparameterized realization of the sy stem. The ARMARKOV/LS/ERA algorithm uses the same procedures to identify the system pa rameters as the ARMARKOV/LS algorithm, but implements a minimal realization of the system. The recursive ARMARKOV/Toeplitz 48 PAGE 49 algorithm is an online algorithm. The advantage of using an online algorithm is that it can adapt to the changing system. ARMARKOV/LS Algorithm Consider the discretetime finitedimensional linear timeinvariant system: (1)()( ()()() ) x kAxkBu k y kCxkDuk (2.15) where and and are the number of inputs and outputs, respectively, of the system. For a singleinput/singleout put (SISO) system, The algorithm is derived below for a SISO system. ,,,nnnilnliARBRCRDRil1il The Markov parameters are defined by jH (2.16) j=1 j0j j jHD HCAB Next, define the ARMARKOV regressor vector 2()nkR : () (1 () () (1 yk ykn k uk ukn ) ) (2.17) where is the order of the system and n is the number of Markov parameters. Here, y and u denote measured input and output of the system described in equation (2.15) respectively. Next define the estimated output of the system () ykW (2.18) where the ARMARKOV weight matrix is defined by W 12 [ WHH ] (2.19) and 49 PAGE 50 1 ,1, 1 ,1,[] []n n n n R R (2.20) The expression of the weight matrix W is then determined to minimize the output error cost function defined below. First, define the output error kykyk (2.21) and the output mean squared error cost function 2 111111 ()()() 22NN T kkJk NN kk (2.22) where is the number of measurements. NSubstituting equations (2.17) (2.18) (2.19) and (2.21) into equation (2.22) results in 111 2N T kJykWkykWk N (2.23) 111 2N TTT kJykkWykW N k (2.24) 111 ( 2 )N TTT k TT TJykykkW N yk y kWkkWWkk (2.25) Because and TTkWyk T y kWk are transposes of each other and are also scalars, they are equal to each other. So, 111 2 2N TT TT T kJykykkWykkWWk N (2.26) To find the to minimize the output error cost function defined in equation W(2.22) we set the partial derivative of with respect to equal to zero. So, JW 0 T TJJ WW (2.27) From matrix calculus, we derive each term of in equation J(2.26) first, 50 PAGE 51 ()()0 T Tykyk W (2.28) 222 T T T TT T T TkWykkykkyk W (2.29) () () 2 2T TT T T TT TT T TT TkWWk dWW kWWk WW W kkW Wkk Td W (2.30) Thus, 111 022 2T N TT T kJ kykWkk N W (2.31) 1111 NN TT kkWkkkyk NN (2.32) Finally, the expression of the weight matrix to minimize the output error cost function is given by W 1 1111 NN TT kkWk y kk NN k (2.33) After extracting the coefficients and from via equation jH W(2.19) we can obtain the system transfer func tion of the ARMARKOV represen tation, which is defined as follows 11 12 1 1 11 ,1 ,()nnn n nn nHzHzz Gz zz (2.34) This is called the ARMARKOV/LS identification algorithm and this algorithm assumes the numerator has the same order of the denom inator for simplicity. For the systems whose numerators and denominators do not have the same order, some parameters described in equation (2.34) will be identified to be approximately zero. 51 PAGE 52 The wellknown ARMA represen tation only has one explicit Ma rkov parameter and it is a special form of the ARMARKOV representation with =1 in (2.34) 1 11,11 2 1 1,1 1,()nn n nn nHzz Gz zz, (2.35) For system identification problems, the orde r of the system is usually not known in advance, so we adjust and n to improve the performance of the system identification algorithm. ARMARKOV/LS/ERA Algorithm The ARMARKOV/LS/ERA algorithm obtains a minimal realization of the transfer function of the system from the Markov parame ters. It uses the same algorithm as the ARMARKOV/LS algorithm to obt ain the weight matrix by equation W(2.33) Then the Markov parameters can be extracted from equation jH (2.33) by using equation (2.19) Next, define the Markov block Hankel matrix for a SISO system: (2.36) ,, jj rsj jr jrsHH H HH s Qwhere are any positive integers. In this research, is set to be equal to for convenience. ,rsrsThen, we apply the singular value decomposition as describe in Akers and Bernstein (1997A) (2.37) ,,0, T rsrsHPS where and TTPPQQI,diagonal matrix of singular values.rsSFrom the Eigenvalue Realization Algor ithm (ERA), (Juang 1994, pp. 133137) 52 PAGE 53 (2.38) 1/2 1/2 ,, 1, 1/2 1/2 1 T rs rsrs T rss T rrsASPHQS BSQE CEPS DH where The ERA also requires 11 0i iE ,1rsnThis arrives at the minimal state space realization of the system 1 3() GzCzAIBD (2.39) This is called the ARMARKOV/LS/ERA algorithm It is a minimal realization because the system order can be chosen as a minimal valu e when using the singular value decomposition in equation (2.37) However, an important drawback of this algorithm is that the singular value decomposition in equation (2.23) is very computational intensive. Theoretically, the rank of the matrix should be the rank of the system. However, in practical applications, the singular value decompos ition will return more singular values than the system order due to measurement noise, and so th e extra singular values should be small. So, only the largest n singular values obtained by the singular value decomposition will be used. ,rsSRecursive ARMARKOV/Toeplitz Algorithm First, define the ARMA RKOV regressor vector222()npkR : () (2 () () (2 yk ykpn k uk ukpn ) ) (2.40) 53 PAGE 54 where is the order of a system, n is the number of Markov parameters, and a new parameter p determines the averaging window of inputoutpu t data that appears in the above regressor vector. It follows that () ykW (2.41) where the ARMARKOV/Toeplitz weight matrix (222)pnpWR is the blockToeplitz matrix defined by 12 1200 00 00 00 0000 AHHB W AHH B (2.42) and 1 ,1,[]n n R 1 ,1,[]n n R and j H are the Markov parameters. As before, define the output error k and the output e rror cost function Jk (2.43) ()()() kYkYk 1 2TJkkk (2.44) Next, the gradient of with respect to can be calculated by Jk () Wk () ()() ()TJk Ukk Wk (2.45) where denotes the Hadamard product (i.e. elementwise matrix product) and (222) pnpUR is defined by (2.46) 11 ( ) 110000 00 00 0000nn nnII U II ( ) Finally, the recursive update law for the weight matrix is given by W 54 PAGE 55 () (1)()() () Jk WkWkk Wk (2.47) In equation (2.47) () k is the adaptive step size The optimal adaptive step size ()optk is defined as 2 2 2 2() () () ()optk k Jk Wk (2.48) where 2 denotes the spectral norm. The computationally efficient step size ()effk (namely, it is more computational efficient since it only needs to calculate the normal ARMARKOV regressor vector () k ) is defined as 2 21 () ()effk k (2.49) In order to assure convergence, () k should satisfy ()()optkk or ()()effkk where(0,2) After matrix is obtained by W(2.47) we can extract the coefficients and from jH (2.19) Then, we can obtain the system transf er function of the ARMARKOV representation form, which is defined in equation (2.34) Since the and coefficients are updated every iteration, this algorithm is called as the recursive ARMARKOV/To eplitz algorithm. jHAdaptive Disturbance Rejection Algorithms Disturbance rejection controll ers have been widely used in active noise control applications (Kuo and Morgan 1996). A block di agram of a standard disturbance rejection problems is shown in Figure 23 where w is the disturbance, is the control signal, u y is the reference signal, z is the performance signal and is the disturbance rejection controller. The cG 55 PAGE 56 goal for the controller is to generate a control signal u to minimize some cost function of the performance signal. The four transfer matrices, namely, the primary path the secondary path the reference path and the feedback path are standard terminology in the noise control literature (Kuo and Morgan 1996). The feedforwardtype disturbance rejection algorithms, such as FXLMS and FXRLS, assume that zwGzuGywGyuGyuG 0 (no feedback path) and ywGI On the other hand, the ARMARKOV disturban ce rejection algorithm does not make these assumptions. All the disturbance rejection algor ithms require identifying the secondary path by online or offline system identification methods. zuGARMARKOV Disturbance Rejection Algorithm Consider the linear discrete time tw oinput/twooutput system (shown in Figure 23 ) given by ()()zwzkGwk ( uk )zuG (2.50) ()()yw y kGwk (yu) Gu k (2.51) where the disturbance the control the reference () wk () uk () y k and the performance are in () zkwm R um R y l R and z l R respectively, and m and l denote the number of inputs and outputs, respectively. The system transfer matrices (primary path), (secondary path), (reference path) and (control path) are in zwGzuGywGyuG z wm l R z ulm R y wlm y ulm and R R respectively. The objective of the active noise or vibration c ontrol problems is to determine a controller uyml cGR that produces a control signal such that the performance measure is minimized (Sane et al. 2001). A measurement of is used to adapt ( uk )cG () yk () zk () zkcG The ARMARKOV form of (2.50) (2.51) is 56 PAGE 57 (2.52) ,2 111 ,2 11()(1)(1)(1 (1)(1)nn jz w jz w j jjj n zuj zuj jjzkzkjHwkjBwkj HukjBukj ) ) (2.53) ,2 111 ,2 11()(1)(1)(1 (1)(1)n n jy w jy w j jjj n yuj yuj jjykykjHwkjBwkj HukjBukj where j R ,,, z wlm zwjzwjBHR, ,,, z ulm zujzujBHR, ,,, y wlm ywjywjBHR ,,, y ulm yujyujBHR is the order of the system, and n is the number of the Markov parameters. Then, we define the extended performance vector () Z k the extended measurement vector and the extended control vector as () Yk () Uk ()()(1)TZkzkzkp (2.54) ()()(1)TYkykykp (2.55) ()()(1)T cUkukukp (2.56) where p is an averaging or windowing parameter and (1)cpnp The ARMARKOV regressor vectors zw and yw are defined by ()(2)()(2)T zwzkzkpnwkwkpn (2.57) and ()(2)()(2)T ywykykpnwkwkpn (2.58) Then (2.52) and (2.53) can be written as () ()zwzwzu Z kWBUk (2.59) () ()ywywyuYkWBUk (2.60) 57 PAGE 58 where zwWzu B and zwWyu B are the ARMARKOV weight matrices. Only zu B will be used in the control algorithm (shown later), and it will be obtained using the ARMARKOV/Toeplitz system identification algorithm. The ARMARKOV control matrix zu B is given by (2.61) ,1,2,1, ,1,2,100 0 0 0 zuzu zu zu zuzu zu zuzunlmlm lm zu lm lm zu zuzu zuHHBB B HHBB n where 0 z ulm is the zero matrix. Next, the ARMARKOV adaptive disturbance rej ection algorithm is derived. The control signal is given by () uk (2.62) ,, 2, 111()(1)(1)(1)cccn n cjc cj cjc jjjukukjHykjBykj where ,cj R and ,,,uyml cjcjHBR Similarly, the delayed versions are (2.63) ,, 1, 111(1)()()(cccnn cjc cj cjc jjjuk ukjHykjBykj )2) u y (2.64) 1 ,1 11(1)(2) (2)(c ccn cc jcc j n cj c cjcc jjukp ukjp HykjpBykjP Substituting all these equations in (2.56) and reordering gives (2.65) 1()(1)()cp ii iUkLkiRkwhere (2.66) ,1 ,0 ,2 ,1 ,()()()()()()()ucu c ccmcnmc c c cnkkIkIHkHkBkBk 58 PAGE 59 ()()( 2)()( 1)T uy c ccc ccckukuknpykyknp (2.67) (1) ()0 0uu u cuuimm im pimmLI (2.68) and (2.69) 11 1 111 2 1 22 1 222 2 2(1) ()(1) () (1) ()(1) ()00000 00000uc uy uc u yqimqqqpimqilqqqpil i qimqqqpimqilqqqpilI R I c y c y)ywith and 1cuqnm2(1ccqnl Thus from (2.59) and (2.65) we obtain 1()()(1)()()()cp zwzwzui iuy zwzwzu i Z kWkBLkiRkWkBUk (2.70) Next, evaluate the performan ce of the current value of ()k based upon the behavior of the system during the previous p steps to result in the definition of the estimated performance () Z k by 1 ()()()()cp zwzwzuiiuy i Z kWkBLkRk (2.71) Substituting (2.70) into (2.71) we obtain the estimated perf ormance in terms of known and measured variables 1 ()()()()()cp zu iiuy i Z kZkBUkLkRk (2.72) Using (2.72) we define the estimated performance cost function 1 ()()() 2TJkZkZk (2.73) The purpose for the ARMARKOV adaptive controller is to obtain the controller parameters ()k such that the performance cost function is minimized. Using matrix derivative formulae, the gradient of with respect to ()Jk()k()Jk is given by 59 PAGE 60 1() ()() ()cp TTTT izuuyi iJk LBZkkR k (2.74) The gradient is used in the update law () (1)()() () Jk kkk k (2.75) where () k is the adaptive step size. An implementable adaptive step size ()impk is used 2 2 21 () ()imp czuuy Fk p Bk (2.76) where F and 2 denote the Frobenius norm and th e spectral norm (Golub and Van Loan 1996), respectively. The steps involved in implementing the ARMARKOV adaptive disturbance rejection algorithm are summarized as follows: Obtain the matrix zu B (eqn. (2.61) ) by using the recursive ARMARKOV/Toeplitz system identification algorithm (eqn. (2.42) ) or the offline ARMARKOV/LS (eqn. (2.33) ). Calculate the control signal from the controller parameter matrix () uk () k and the vector (eqn. ()uyk (2.65) ). Use the signals ()uk() z k and to update the vectors ()yk () Z k (eqn. (2.72) ) and ()uyk (eqn. (2.67) ). Calculate the gradient () () Jk k (eqn. (2.74) ). Calculate the implementable adaptive step size ()impk (eqn. (2.76) ). Update the controller parameter matrix ()(1)kk (eqn (2.75) ). 60 PAGE 61 Figure 21. Flow chart of downhill simplex algorithm. 61 PAGE 62 62 Figure 22. Block diagram for the extremum seeking control. Figure 23. Block diagram of disturbance rejection control. PAGE 63 CHAPTER 3 SIMULATION AND VALI DATION EXPERIMENTS Before the algorithms are used for closed loop separation control in the wind tunnel experiments, they are tested by using Matlab/Sim ulink simulations or validation experiments. The purpose of this chapter is to ensure that the algorithms work as desired. Optimization Simulations Downhill Simplex Simulation Results The downhill simplex algorithm is programme d in Matlab. The performance of the algorithm is illustrated by a 1dimensional and a 2dimensional simulation cases. This algorithm can be easily extended to higher dimensions. In the 1dimensional case, the cost function () f x is chosen to be an 8th order polynomial function of x which has a local minimum at 14.2 x and a global minimum at as shown in 67.3 x Figure 31 Two initial conditions are selected to demonstrate that this algorithm can be trapped by a local minimum. The first initial condit ion is at about 45 x One should notice that for this 1dimensional problem, there should be two independent points (a simplex) as the initial condition. As shown in Figure 31 the downhill simplex algorithm crawls down to the global minimum (red trace). On the other hand, the second initial condition is at about 30x which leads the algorithm to the local minimum (b lue trace). This is di ctated by the inherent downhill nature of the algorithm. Another example is to demonstrate how the downhill simplex algorithm works in twodimensional space. The cost function is obtaine d in Matlab by the peaks command. The formula for the cost function is as follows: 22223 5(1 Z = 3(1x)(1)10() 532 2 21 ) x xyxyx eyxyee (3.1) 63 PAGE 64 This function has two local minima and a global minimum as shown in Figure 32 Similar as the 1dimensoinal case, the optimization algorit hm converges to either a local minimum (blue trace) or the global minimum (red trace) de pending on the initial condition. Although each iteration of the algorithm requires several steps (Cha pter 2), it only takes 9 iterations to find the global minimum. This result is encouraging and suggests that it can be fast for some cases. One can also adjust the termination to lerance to control the time consum ption. On the other hand, the time consumption of the separation control experiments is also de pendent on other factors, such as data acquisition. This will be discussed further in Chapter 5. Extremum Seeking Simulation Results The extremum seeking algorithm is implemented in Simulink. Figure 33 shows the simulation block diagram for the extremum seeki ng control. In the simulation, the algorithm seeks a maximum instead of a minimum. One can easily modify the program to search for a minimum by adding a negative sign to the cost func tion. Two numerical m odels are tested. The first model is a quadratic function 2 *ff* which has a single maximum f at as shown in Figure 34 In this case, f is set to 10 and is set to 5. The second model is a double hump model, which is fitted by a 8th order polynomial function, which is the same as the model shown in Figure 31 with a opposite sign. It has a local maximum and a global maximum as shown in Figure 35 Table 31 summarizes all the parameters that are us ed in the simulations. Recall that the detailed derivation of algorithm is given in Chapter 2. Note that the parameters and are the main factors that affect the convergence rate a nd stability. Thus, they are varied in the simulations to understand how they affect the performance of the algorithm. a w 64 PAGE 65 Figure 36 demonstrates how affects the conve rgence rate while is fixed to be 50 Hz. Clearly, the convergence rate increases when decreases. This is consistent with the analytical solution shown in eqn. a wa(2.14) for 21 k a where the convergence rate ( "2 f ak ) is dependant on 1a. However, when is too small, the algorithm becomes unstable. a wFigure 37 shows how affects the convergence rate while a is fixed to be 0.001. Apparently, the convergence rate increases with When is too large, the algorithm again becomes unstable. w wFigure 38 and Figure 39 show the results of the double hump model. Clearly, the extremum seeking algorithm drives the cost function f to the local minimum. Vibration Control Testbed Setup Figure 310 shows a detailed sketch of the whole vi bration control testbed setup. A thin aluminum cantilever beam with one piezocerami c plate bonded to each side is fixed on a block base and connected to the ground. The two piezo ceramic plates are used to excite the beam by applying electrical field across their thickness. The piezoceramic plate bonded to the upper side of the beam is called the "distu rbance piezoceramic" because it is used to apply a disturbance excitation to the beam. The piezoceramic plate bonded to the lower side of the beam is called the "control piezoceramic" because it is supplied with the controller output signal to counteract the disturbance actuator. The beam system has a natural frequency of about 97 Hz. The goal of the disturbance rejection controller is to mitigate the vibration of the aluminum beam generated by an unknown disturbance signal. The controller tries to generate a signal to counteract the vibration of the aluminum beam generated by the disturb ance piezoceramic. The performance (or the residue) signal of the cont roller is measured at the center of the tip of the beam by a laseroptical displacement sensor. The performance signal is filtered by a high 65 PAGE 66 pass filter with 1cz f H to filter out the dc offset of the displacement sensor and then amplified by an amplifier with a gain of 10. The disturbance and control signals are generated by our dSPACE (Model DS1005) DSP system with 466MHz PowerPC CPU and amplified by two separate channels of an amplifier by a same gain of 50. The types and conditions of the signals are discussed in details in the next section. The dSPACE system has a 16bit A/ D and a 16bit D/A board. The computer can acquire data using Mlib/Mtrace programs in MATLAB through the dSPACE system. The whole system was a twoinput/twooutput system. One input was the control signal and the other input was the unmeasured distur bance signal. The two outputs are termed a reference output and a perfo rmance output. For this vali dation test, the reference and performance outputs were identica l. The disturbance rejection algorithm was implemented in the Simulink environment and compiled and downloade d to the dSPACE system. The disturbance signal was bandlimited white noise with frequency of 0150 Hz. The disturbance rejection algorith m runs in one of the followi ng two modes: 1) ID, then control (shown in Figure 311 ): the system (control model) is identified by the ARMARKOV/Toeplitz system ID algorithm and the identified system weight matrix zu B is transferred to the ARMARKOV co ntrol algorithm; then the controller is turned on and the control signal is switched to the controller output. 2) ID and control (shown in Figure 312 ): the ID and control ar e turned on simultaneously. The input ( ) to the system for ID can be either bandlimited white noise or a repetitive linear chirp signal. The controller uses the identified system to achieve maximum suppressi on of the vibration of the beam, subject to constraints on the maximum allowable actuator si gnal. The ID and control mode is better when the system is a time variant system becau se this mode updates the system information IDu 66 PAGE 67 during every iteration. However, the ID and control mode adds an additional signal to the control signal all the time and this certainly aff ects the performance of the disturbance rejection controller. The tradeoff between the adaptation ability and effects on the performance should be kept in mind. IDuResults of the Vibr ation Control Tests Computational Tests For realtime control applications, the turnaroun d time (defined as the time for the program to execute one iteration) is required to be le ss than the sampling time. Complex algorithms are computationally intensive and have large turnaround time, which requires choosing a corresponding larger sampling time (or a smaller sampling frequency s f ). From the Shannon sampling theorem, the sampling frequency must be larger than twice th e highest frequency of interest to avoid aliasing. Thus, algorithms with high computational complexity may not be feasible in flow control applications. The tradeoff between choosing a large s f to satisfy the sampling theorem and choosing a small s f to allow a large turnaround time must be considered. This section analyzes the effect s of varying the parameters of the ID and control algorithms on the computational intensity. This serves as a reference for choosing the parameters with regard to the computational intensity. The sampling frequency was 1024 Hz fo r the computationa l tests. In Figure 313 and Figure 314 the turnaround time of the system ID algorithm by varying either or n p is plotted, while the other two parameters are fixed at unity. It is shown th at the turnaround time is approximately linearly proportional to both and while the slope for is approximately twice of that for n n The dependence of the turnaround time on p is approximately quadratic. Clearly the averaging window number p has much more significant impact on the 67 PAGE 68 computational intensity than the other two parameters. Figure 315 investigates the effects of varying p on the computational intensity with respect to n As shown, the computational intensity is proportional to p From these results, it is suggested to hold p to be a small number and increase to improve the system ID performance. Figure 316 shows the effects of varying or cnc on the computational intensity of the ARMARKOV disturbance rejection algorithm. The results are si milar to those of the system ID algorithm. The turnaround time is a pproximately linearly proportional to and cnc while the slope for is approximately twice that for cnc System Identification The ARMARKOV/Toeplitz system ID algorithm requires the following three parameters: the order of the system the number of Markov parameters n and a parameter p that determines the size of the averag ing window. The SNR is also a parameter that can affect the performance of the system ID algorithm. When the number of Markov parameters is unity, the ARMARKOV model reduces to an ARMA model. The values of the parameters are limited by the requirement that the turnaround time must be less the sampling time. As shown in the last section, has the smallest effect on the turnaround time; thus in this sectio n the performance of the system ID algorithm with varying is compared. The offline nonparametric fit of the freque ncy response of the beam system is also implemented as a comparison and shown as green dot lines in Figure 320 Figure 323. The nonparametric fit uses the invfreqz comma nd in MATLAB and implements a second order approximation. The invfreqz command returns the system matrices and A B of the state space representation. The zeropole map of the nonparametric fitted system is shown in Figure 317. As shown, the beam system is a low damp ing system because it has two poles that are 68 PAGE 69 very close to the unit circle. The controllability matrix is 11.655;01BAB (for a second order system), which has full rank 2. This means that the system is controllable. The sampling frequency was 1024 Hz. The input signal used for the system ID was a periodic chirp signal. The frequency response shown in Figure 320 Figure 323 was implemented with NFFT=1024, no overl ap, and a rectangular window. Figure 318 shows a very good match between the measured and fitted outputs of the system with the system ID parameters of 2pn 10 and SNR=20 dB. Meanwhile, as shown in Figure 319 the weight tracks of the system ID converge at about 0.5 seconds. Figure 320 to Figure 323 show the comparison between the measured and fitted frequency response with the system ID parameters of 2pn SNR=20 dB and varying A significant improvement of the system ID is obtained when changes from 1 (ARMA case) to 10. Figure 324 compares the mean square error (MSE ) verse time of the system ID with varying Surprisingly, it is found that the case with 10 has the best performance. This indicates that for the beam system, increasi ng the number of Markov parameters does not necessarily improve the system ID performance. This also suggests that for a certain system, there may exist an optimal number of Ma rkov parameters. It is also shown in Figure 324 that for the case with 40 the convergence rate is sl ower than the case with 1 although the final MSE is better. This indicates that too many Markov paramete rs may be detrimental to the performance of the ARMARKOV/ Toeplitz system ID. Figure 325 compares the mean square error (MSE ) verse time of the system ID with varying SNR. The SNR is computed by using the formula: 22 1010log() s nSNR where 2 s is the variance of the control signal and 2n is the variance of the disturbance signal. It is clear that 69 PAGE 70 the system ID performs better with a higher SNR. This suggests that when the system ID is conducted with unknown disturbance, it is better to apply a large system input within the maximum allowable range. Adaptive Disturbance Rejection The ARMARKOV disturbance rejection al gorithm requires the following three parameters: the order of the controller the number of Markov parameters of the controller cnc and the adaptive step size constant that controls the convergence rate of the controller. The controller uses the system informa tion identified by th e ARMARKOV/Toeplitz system ID with the parameters of 2 pn 10 and SNR=20 dB (shown in Figure 321 ). All cases of the controller design use the same identified system. Bandlimited white noise with frequency of 0 150 Hz and variance of 0.09 is used to excite the disturbance piezoceramics. Figure 326 and Figure 327 show the time data of the performance and control signals with the parameters of 2cn 20c and The system ID is off for the whole period and the controller is off initially and turned on at t = 20 sec. In 1 Figure 328 the power spectra of the performance signa l with control off and on are compared. The power spectra were calculated by using the time data of 20second duration with NFFT=1024, 50% overlap and a hanning window. The control on case is taken after the controller is turned on for 30 seconds. The performance of suppression is calculated by 22 1010logoffon and this case gives 11.7 dB s uppression. Interestingly, in Figure 328 the power around 70 Hz and 120 Hz of the control on case is higher than that of the control off case. This is generally defined as spill over (Hong and Bernstein 1998). Hong and Bernstein (1998) used the Bode integral c onstraint to analyze th e spillover problem a nd concluded that the spillover is inevitable if the reference and perfor mance signals are collocated or the disturbance 70 PAGE 71 and control actuators are collocated. For th is vibration control test, the reference and performance signals are collocated, t hus the spillover is unavoidable. Figure 329 Figure 330 and Figure 332 show the performance of the disturbance rejection algorithm with varying cnc and respectively, while other parameters are held constant. From Table 32 and Table 33 it is interesting to find that there is not much difference of the suppression performance for varying and cnc However, the step size parameter does play a significant role. Larger gives much faster convergence and better performance. However, if is too large, it is possible for the controller to become unstable. This tradeoff should be kept in mind when choosing The disturbance rejection controller can also be run at the ID and control mode. In this mode, the bandlimited white noise with frequency of 0 150 Hz and variance of 0.09 is used to excite the disturbance piezoceramics. Meanwhil e, the bandlimited white noise with frequency of 0 150 Hz and variance of 0.01 is added to th e controller output. The c ontrol signal is shown in Figure 333 and the performance signal is shown in Figure 334 Comparing with Figure 327, the control signal of the ID and control case is significantly larg er at the beginning because of the additive signal and the evolution of the controller output is buried under it. IDu Figure 335 compares the power spectra of the performance signal of the two different modes. It is surprising that th e ID and control mode results in lower power around the natural frequency of the beam. This is hard to see in Figure 334 because the ID, then control mode seems much better. However, it is not surprising that the ID and control mode results in higher power at other frequencies than the natura l frequency because of the additive signal In addition, IDu 71 PAGE 72 Table 35 shows that the ID, then control m ode gives better suppres sion performance of the overall power. Unfortunately, for this setup, it is not feasible to test the adaptability of the two modes. However, this will be done in the wind tunnel experiments. As a summary, the computational tests are condu cted first to determine how the parameters affect the computational complexity of the system ID and control algorithm s. It is shown that the averaging window number p has much more significant impact on the computational intensity than the other two parameters for th e system ID algorithm. The dependence of the computational complexity vs. is approximately twice of that for n Similarly it is found that the turnaround time of the c ontrol algorithm is approximate ly linearly proportional to and cnc while the slope for is approximately twice of that for cnc The ARMARKOV/Toeplitz syst em ID algorithm successfu lly identifies the system (control model) and results in very good fre quency response approximations. A significant improvement of the performance of the AR MARKOV system ID over the ARMA (when 1 ) system ID is found. However, too many Markov parameters of the ARMARKOV system ID may be detrimental to the performance. Higher SNR improves the performance, thus when the system ID is conducted with unknown noise, the input signal should be chosen as large as possible within the maximum allowable level. The order of the controller and the number of Markov parameters cnc do not play significant roles on the performance of the ARMA RKOV controller for th is vibration control test. However, this conclusion may vary with different systems and remains to be investigated. The step size constant significantly affects the converg ence rate of the controller. should be chosen as large as possible before it makes th e controller unstable. Th e spillover effect is 72 PAGE 73 identified in this vibration control test. This effect is unavoidable becau se the reference signal and performance signal are colloc ated (Hong and Bernstein 1998). Table 31. Parameters for the simulations. Fs (Hz) 500 Perturbation amplitude a = 0.001, 0.002, 0.005 Adaptation gain 21 ka Perturbation frequency ( ) Hzw = 30, 40, 50 High pass filter cutoff frequency ( ) Hz 10hw Low pass filter cutoff frequency ( ) Hz10lw Table 32. Suppression performance of the disturbance rejecti on algorithm with 20c and varying 0.1 cn 20c 0.1 1 5cn cn 10cn Suppression (dB) 8.9 8.6 9.0 Table 33. Suppression performance of the disturbance rejecti on algorithm with 2cn 0.1 and varying c 2cn 0.1 1c 20c 40c Suppression (dB) 8.4 8.7 7.9 Table 34. Suppression performance of the disturbance rejecti on algorithm with 2cn 20c and varying 2cn ,2c0 0.01 0.1 1 Suppression (dB) 3.9 8.9 11.7 73 PAGE 74 Table 35. Suppression performance of the dist urbance rejection algorithm at different modes with 2cn 20c and 0.1 2cn ,2 0c and 0.1 ID, then control ID and control Suppression (dB) 7.1525 4.9994 0 10 20 30 40 50 60 70 80 90 100 100 80 60 40 20 0 20 xCost function Two different initial conditions f (x) Figure 31. Onedimensional exampl e of the downhill simplex algorithm. 74 PAGE 75 Figure 32. Twodimensional exampl e of the downhill simplex algorithm. Figure 33. Simulation block diagra m for extremum seeking control. 75 PAGE 76 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 ff = f* = 10 when = = 5 Figure 34. Single glob al maximum test model: 2 *ff* where and *10 f *5 0 10 20 30 40 50 60 70 80 90 100 20 0 20 40 60 80 100 f Figure 35. Double hump model with one lo cal maximum and one global maximum. The function is fitted by a polynomial: 1281077655 3423121.2e1.6e+1.4e2.3e +1.5e4.4e+3.8e2.35.4 f 76 PAGE 77 0 10 20 30 40 50 60 70 80 90 100 5 0 5 10 t (sec) a=0.001 a=0.002 a=0.005 0 10 20 30 40 50 60 70 80 90 100 20 10 0 10 t (sec)f a=0.001 a=0.002 a=0.005 Figure 36. converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while *f 10 a 50 w 0 10 20 30 40 50 60 70 80 90 100 10 5 0 5 10 t (sec) w=30 Hz w=40 Hz w=50 Hz 0 10 20 30 40 50 60 70 80 90 100 150 100 50 0 50 t (sec)f w=30 Hz w=40 Hz w=50 Hz Figure 37. converges to the optimal input *5 and f converges to the global maximum at various convergence rates due to various values of while *f 10 w 0.001 a 77 PAGE 78 0 50 100 150 200 250 300 350 400 2 0 2 4 6 8 10 12 14 t (sec) Figure 38. converges to the local optimal input (see Figure 35 ) *14 ( ). 0.001,50 aw 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 40 45 t (sec)f Figure 39. f converges to the local maximum (see Figure 35 ) 40 f ( ). 0.001,50 aw 78 PAGE 79 Figure 310. Vibrati on Control Testbed. Figure 311. Block diagram of vibration control with ID, then control. 79 PAGE 80 Figure 312. Block diagram of vibra tion control with ID and control. Figure 313. Effects of varying or n p on the computational intensity of the ARMARKOV/Toeplitz system ID. 80 PAGE 81 Figure 314. Effects of varying p on the computational intens ity of the ARMARKOV/Toeplitz system ID. Figure 315. Effects of varying p on the growth rate of the co mputational intensity of the ARMARKOV/Toeplitz system ID with respect to n 81 PAGE 82 Figure 316. Effects of varying or cnc on the computational intensity of the ARMARKOV disturbance rejection. 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Real PartImaginary Part Figure 317. Zeropole map of the nonparametric fit of the frequency response. 82 PAGE 83 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 1.5 1 0.5 0 0.5 1 1.5 t (sec) Measured ARMARKOV/Toeplitz ID Figure 318. Measured output a nd fitted output by the ARMARKOV/Toeplitz system ID with 2 pn 10 and SNR=20 dB. 0 1 2 3 4 5 6 7 8 9 10 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t (sec)ARMARKOV/Toeplitz ID weight tracks Figure 319. Weight tracks of the ARMARKOV/Toeplitz system ID with 2pn10 and SNR=20 dB. 83 PAGE 84 0 50 100 150 0 10 20 Magnitude Frequency response Nonparametric fit ARMARKOV/Toeplitz ID 0 50 100 150 200 0 200 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 320. Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2pn 1 and SNR=20 dB. 0 50 100 150 0 10 20 Magnitude Frequency response Nonparametric fit ARMARKOV/Toeplitz ID 0 50 100 150 200 0 200 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 321. Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2pn 10 and SNR=20 dB. 84 PAGE 85 0 50 100 150 0 10 20 Magnitude Frequency response Nonparametric fit ARMARKOV/Toeplitz ID 0 50 100 150 500 0 500 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 322. Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2pn 20 and SNR=20 dB. 0 50 100 150 0 10 20 Magnitude Frequency response Nonparametric fit ARMARKOV/Toeplitz ID 0 50 100 150 200 0 200 400 Phase (deg.) 0 50 100 150 0 0.5 1 Frequency (Hz)Coherence Figure 323. Comparison of measured frequency response, nonparametric fit and the ARMARKOV/Toeplitz system ID with 2pn 30 and SNR=20 dB. 85 PAGE 86 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 x 103 t (sec)MSE =1=10=20=30 Figure 324. Comparison of MSE of th e ARMARKOV/Toeplitz system ID with SNR=20 dB with different 2pn 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 x 103 t (sec)MSE SNR=40dB SNR=20dB SNR=0dB Figure 325. Comparison of MSE of th e ARMARKOV/Toeplitz system ID with 2pn20 with different SNR. 86 PAGE 87 0 5 10 15 20 25 30 35 40 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 t (sec)Performance signal Control off Control on Figure 326. Performance signal of the ARMA RKOV disturbance rejection to bandlimited white noise (0150 Hz) with 2cn 20c and 1 0 5 10 15 20 25 30 35 40 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 t (sec)Control signal Figure 327. Control signal of the AR MARKOV disturbance rejection with 2cn 20c and 1 87 PAGE 88 0 50 100 150 100 80 60 40 20 0 20 Frequency (Hz)Power spectrum (dB) Control off Control on Figure 328. Power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 20c and 1 0 50 100 150 100 80 60 40 20 0 20 Frequency (Hz)Power spectrum Control off Control on nc=1 Control on nc=5 Control on nc=10 Figure 329. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection with 0.1 20c and different cn 88 PAGE 89 0 50 100 150 100 80 60 40 20 0 20 Frequency (Hz)Power spectrum Control off Control on c=1 Control on c=20 Control on c=40 Figure 330. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 0.1 and different c 0 5 10 15 20 25 30 35 40 1 0 1 =0.01 0 5 10 15 20 25 30 35 40 1 0 1 Performance signal =0.1 0 5 10 15 20 25 30 35 40 1 0 1 t (sec) =1 Figure 331. Comparison of convergence of the ARMARKOV disturbance rejection with 2cn 20c and different 89 PAGE 90 0 50 100 150 100 80 60 40 20 0 20 Frequency (Hz)Power spectrum Control off Control on =0.01 Control on =0.1 Control on =1 Figure 332. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection with 2cn 20c and different 0 5 10 15 20 25 30 35 40 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 t (sec)Control signal Figure 333. Control signal of the ARMARKOV di sturbance rejection at ID and control mode with 2cn 20c and 0.1 90 PAGE 91 91 0 5 10 15 20 25 30 35 40 1 0.5 0 0.5 1 ID, then control 0 5 10 15 20 25 30 35 40 1 0.5 0 0.5 1 t (sec)Performance signal ID and control Figure 334. Comparison of convergence of the ARMARKOV disturbance re jection at different modes with 2cn 20c and 0.1 0 50 100 150 100 80 60 40 20 0 20 Frequency (Hz)Power spectrum (dB) Control off ID, then control ID and control Figure 335. Comparison of power spectra of the performance signals of the ARMARKOV disturbance rejection at different modes with 2cn 20c and 0.1 PAGE 92 CHAPTER 4 EXPERIMENTAL SETUP AND DATA ANALYSIS METHOD The separation control experime nts are conducted in an open return lowspeed wind tunnel with a 30.48 cm (1 ft) by 30.48 cm test secti on. The wind tunnel has two antiturbulence screens, an aluminum honeycomb and a 9:1 contr action ratio. The airspeed is controlled by the variable frequency of the motor fan. A twodimensional NACA 0025 airfoil that is e quipped with synthetic jet actuators, Kulite dynamic pressure transducers and a lift/drag balance is used as the test model. A Particle Image Velocimetry (PIV) system is used for flow visualization and quantitative flow field measurements. A Dantec CTA hot wire system is used to measure instantaneous velocity. This chapter describes each part of the experime ntal setup in detail. A brief description of the Higher Order Statistical Analys is (HOSA) is also presented in this chapter because it may be used for nonlinear flow instability analysis. NACA 0025 Airfoil Model A twodimensional NACA 0025 airfoil with chord le ngth of 15.24 cm (6 in.) is built as a test bed for flow separation control ( Figure 41 ). The span of the airf oil model is 29.21 cm (11.5 in.), which allows for a slight gap on either end to accommodate a sidewallmounted straingauge sting balance. The boundary layer is tripped at the lead ing edge region using No. 60 sand grit. Two pairs of synthetic jets are embedded in the airfoil at approxim ately 3% chord and 30% chord, respectively. Six ports near the rear of the airfoil at th e midspan location are available for dynamic pressure transducers. The six ports are located at approx imately 44.0%, 52.5%, 61.0%, 69.5%, 77.9% and 86.4% chord. A preamplifier PCB board for the dynamic pressure transducers can be also installed in the airfoil. The detailed side view of the airfoil is shown in Figure 41 92 PAGE 93 Synthetic Jet Actuators The airfoil is fitted with two pairs of synthetic jet arrays (each with 0.5 mm wide slots separated by 2.4 mm), which ar e located in the central 1/3rd spanwise region of the airfoil (see Figure 41 ). The first pair is located near the leading edge of the airfoil, at approximately 3% chord, while the second is placed near the point of maximum thickness at about 30% chord. The first array is fixed, while the second array can be translated be tween 25% chord and 37% chord. The detailed design procedures of the synthetic jet actuators that ar e used in this research can be found in Gallas et al. (2003) and Gallas (2005). The primary goal of the design is to maximize the magnitude of the volume flow rate th rough the orifice per a pplied voltage (i.e. outacQV, where is the volume flow rate and is the applied ac voltage) over a frequency range of while the size of the syntheti c jet actuators are limited by th e geometry of the airfoil. As mentioned, the frequency response of the synt hetic jet actuators is another important design criterion. The frequency response of the actuators must be chosen appropriately to effectively control (via amplitude and burst modulation te chniques) the flow sepa ration over a range of frequencies, ranging from the lo w frequency shedding in the wake to the high frequency shear layer instability. The side and top views of the synthetic jet actuators are shown in outQHzacVOkFigure 43 The cavity is 151 mm long, 28 mm high and 2 mm wi de. Five piezoceramic disks are attached to one side of the cavity. They are driven in phase using a single amplified drive signal to achieve maximum flow rate. A thin slot (0.5 mm wide by 101.6 mm l ong) at the top of the cavity permits oscillatory fluid flow. Two closely spaced synthetic jets are obtained by introducing a rigid wall to se parate them, as shown in Figure 43 The two synthetic jet actuators are nominally identical. See the detailed characterization of the actuators in CHAPTER 5 93 PAGE 94 Experimental Methods Flow Visualization Flow field velocity data over the surface of th e airfoil and in the wake are acquired using Particle Image Velocimetry (PIV). The PIV system consists of a pair of New Wave Minilase 15 Hz, 50 mJ per pulse, Nd:YAG lasers with appropriate light sheet opt ics. The width of the light sheet is approximately 12 mm at the plane of measurement. A TSI model 630157 Powerview Plus 2MP 10bit CCD camera is used to acquire images. This camera contains 1600 x 1200, 7.4 m square pixels. A series of Nikon lenses (60 mm, 75240 mm, 200 mm) are available. The flow is seeded with waterbased fog fluid by a LeMaitre G150 seeder and the seeding density is adjusted to insure uniform seeding density. The laser pulse generator and the camera are synchronized by a TSI Model 610032 Synchronizer which is configured to acquire a pair of images using TSI INSIGHT Software version 6.1.1. The computation of the velocity fi eld begins by dividing the image into a grid of interrogation windows overlapped in space by 50%. These windows typically range from 32 x 32 pixels to 64 x 64 pixels. The velocity is determined by the know n distance that a particle is displaced during the known time dT. The INSIGHT Software utilizes an FFT crosscorrelation process in conjunction with a Gaussian peak search algorithm to cal culate the average velocity of the particles in the interrogation widow. A numb er of validation schemes are available in the software, such as range outlier rejection and median filtering. Lift/Drag Balance A straingauge balance is designe d to measure lift and drag forces of the airfoil test bed. The detailed design procedure can be found in Griffin (2003). Two pairs of strain gauges are attached to the cantilever that supports the airfoil to measure th e normal and axial forces on the airfoil, respectively ( Figure 44 ). The layout of the strain gauges and Wheatstone bridge 94 PAGE 95 configuration are shown in Figure 45 and Figure 46 The output of the Wheatstone can be calculated by the following equation outVR V R (4.1) From the above equation, we can see that the output is linearly depe ndant on the change of resistance R The output of the strain gauges is measur ed by a highresolution HP34970A DAQ system and is averaged over 2 power line cycles to el iminate 60 Hz noise. The lift and drag are calculated from the normal and axial forces together with the angle of attack ( Figure 44 ) via the following equations: (4.2) cos()sin() LNAOAAAOA (4.3) sin()cos() DNAOAAAOA where and L D stand for lift and drag, respectively, while and N A stand for normal and axial force, respectively. Before the balance can be used for the wind t unnel experiments, it is calibrated by adding known weights on the balance and measuring the output from the normal and axial strain gauges. Figure 47 and Figure 48 show typical normal and axial for ce calibrations vs. balance output. Very good linear relationships between the bala nce output and the forces on the balance are achieved. The coefficients of the linear equations are used to back out the forces on the airfoil from the voltage output of the strain gauges. The balance is also validated by comparing with the lift and pressure drag coefficients measured by integrating the static pressure around the airfoil. The pressure taps are located at the center span of the airfoil and the static pressure is measured via a Heise static pressure gauge. Figure 49 shows the static pressure distributions on the airfoil su rface at different AOAs when 95 PAGE 96 Re150,00020. From Figure 49 it can been identified that the flow is separated at AOA=1 and The suction zones on the upper surface shrink dramatically. This is generally referred to as pressure loss due to flow separation and is primarily res ponsible for deteriorating lift to drag ratio. 5The lift and drag coefficients are calculated by integrating the static pre ssure around the airfoil surface, assuming surface friction is negligible compared with pressure forces and the flow is twodimensional. Figure 410 and Figure 411 show the comparison of the lift and drag coefficients calculated by the two different methods at Re100,000 and R The uncertainty was calculated at %95 confidence interval (i.e. e150,000 2 uncertainty N where is standard deviation and is number of measurements). As shown, they agree reasonably well considering measurement uncertainties. This vali dates that the balance works as desired. The main reason for the differences is the threedimens ional effect as the pressure taps only measure at the center span. NDynamic Pressure Transducers To measure the pressure fluctuat ion on the airfoil surface, it is required that the pressure sensors must be compact so that they can be instal led within the limited space in the airfoil. It is also desired that they have large enough bandwidth to capture the characteristics of the oscillations of the flow above th e airfoil, and their response is lin ear with respect to the pressure load within the range of interest. For these r easons, a number of commercially available MEMS Kulite LQ1255A dynamic pressure transducers ( Figure 412 ) are used to obtain dynamic pressure response on the upper surface of the airfoil. The transducers can be flush mounted in the six available locations on the upper surface. A preamplifier/filter board for the transducers 96 PAGE 97 is designed to eliminate dc response ( 1.5cutoff f Hz ) and amplify the outputs by a gain of 100. The preamplifier/filter board can be installed insi de the airfoil so that the airfoil acts like an electronic enclosure. Before the transducers can be used in the experiments, they are dynamically calibrated in a 2.54 cm (1 in.) by 2. 54 cm plane wave tube (PWT). A speaker was used as a source, and a Brel & Kjr (Mode l 4318) microphone was used as a reference transducer. Figure 414 shows the linear response of a typica l Kulite sensor that is obtained by fixing the frequency and increasing the input amp litude of the speaker. The frequency response is measured using a periodic chirp signal ( Figure 414 ). As shown, the frequency response does not vary up to approximately 3000 Hz, whic h is sufficient for this research. Hot Wire Anemometry A Dantec constanttemperature hot wire anem ometry system (CTA module 90C10) is used to measure timeresolved velocity in the unsepar ated flow above the airfoil. The CTA system includes A/D converter and all the signal conditioners needed. Befo re the measurements, a static calibration is performed by the calibration m odule and the flow unit (90H01 and 90H02). A typical calibration cu rve is shown in Figure 415 Since the output of the hot wire system usually drifts due to temperature changes, connections, etc, the calibration should be done before each measurement. Two algorithms are commonly used for curve fitting. One is a polynomial that is used here, and the other is Kings law (power law): 1 2 nUEAB where is the voltage output of the hot wire and U is the flow velocity (Jrgensen 1996). The difference between the temperatures at calibration and measurements should also be corrected by means of E 0 1 1 w corr wTT EE TT where is the wire temperature, is the temperature at calibration, is the raw wire voltage, is the temperature during measurement and is the corrected wT0T1E1TcorrE 97 PAGE 98 voltage (Jrgensen 1996). During experiments, the hot wire probe (55P11) is mounted on a 2dimensional Velmex traversing system wh ich has spatial resolution of about 1.6 / mstep in both directions. Control System Hardware and Software The control systems for the separation contro l experiments are implemented by a dSPACE (Model DS1005) DSP system with a 466MHz PowerPC CPU. The dSPACE system has a 5channel 16bit A/D board (DS2001) and a 6chan nel 16bit D/A board (DS2102) as the data acquisition equipments. The range of the data acq uisition boards can only be 10 to +10 V, 0 to 10 V or 5 to +5V. The control algorithms are first programmed in Matlab/Simulink and C programs (cmex sfunction) and then compiled and downloaded to the dSPACE system. The compiled programs together with the data acquisition boards are able to run the experiments in real time. The computer is also able to acquire data into Matlabs workspace through the dSPACE system via the mlib pr ograms provided by the dSPACE. Higher Order Statistical Analysis (HOSA) Higher order spectral anal ysis is used to uncover the nonlinea r interactions in signals or to identify nonlinear systems (Nikias and Mendel 1993). As discussed in CHAPTER 1 there are three characteristic frequencies a nd nonlinear interactions between them are inherent in separated flow. Unfortunately, the power spectrum alone is incapable of providing a ny conclusive proof of the nonlinear interactions. The power spectrum only provides proof of presence of power at certain frequencies. On the other hand, highe rorder spectral method can quantify quadratic coupling between frequency pairs. For exampl e, it can provide the information that the generation of power at a certain frequency is the result of quadrat ic coupling of other frequencies. The autobispectrum uses third order cumulants and is defined as 98 PAGE 99 *1 ,limxxxij ijij T B ffEXfXfXff T (4.4) and the autobicohere nce is defined as 2 2, ,xxxij xxxij x xixxjxxijBff bff PfPfPff (4.5) where X f denotes the Fourier transform of x t, denotes the complex conjugate and denotes the autospectrum of xxPf x t. The autobicoherence is bounded by zero and un ity. Disturbances with frequencies i f j f and ij f f are quadratically coupled if 2,ijbff 1 not quadratically coupled if bfijf2,0 and partially coupled if 20,ijbff 1 Just as the autospectrum has the crossspectrum as its counterpart for signals x t and y t, the autobispectrum has the crossbispectrum as its counterpart, which is defined as: *1 ,limxxyij ijij T B ffXfXfYff T (4.6) Similarly, the crosscoherence is obtained by normalizing the crossbispectrum and defined as follows: 2 2, ,xxyij xxyij x xixxjyyijBff bff PfPfPff (4.7) Some examples that illustrate the applications of the HOSA are given in the APPENDIX 99 PAGE 100 Figure 41. NACA 0025 airfoil model with actuators and pressure tran sducers installed. (Adapted from Holman et al. 2003) Instantaneous Timeaveraged Figure 42. Schematic of a synthetic jet actuator. 100 PAGE 101 text text text text text side view top view piezoceramic disks splitter plate (2.4 mm thick) slots (0.5 mm wide) Figure 43. Synthetic jet array. (Adapted from Holman et al. 2003) Figure 44. Forces on NACA0025 airfoil. 101 PAGE 102 Figure 45. Closer view of the strain gauges. Figure 46. Wheatstone bridge configuration of the balance. 102 PAGE 103 Normal Force ( N ) Figure 47. Normal fo rce vs. balance output. Axial Force ( N ) Figure 48. Axial fo rce vs. balance output. 103 PAGE 104 0 10 20 30 40 50 60 70 80 90 100 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x/c (%)CpAOA = 0 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 1.5 1 0.5 0 0.5 1 x/c (%)CpAOA = 5 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 2.5 2 1.5 1 0.5 0 0.5 1 x/c (%)CpAOA = 10 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x/c (%)CpAOA = 15 Upper surface Lower surface 0 10 20 30 40 50 60 70 80 90 100 1.5 1 0.5 0 0.5 1 x/c (%)CpAOA = 20 Upper surface Lower surface Figure 49. Static pressure distributions on the airfoil surf ace at different AOA at Re150,000 104 PAGE 105 5 0 5 10 15 20 25 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 AOA (degree)CL CL from pressure CL from balance 5 0 5 10 15 20 25 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 AOA (degree)CD CD from pressure CD from balance Figure 410. Comparison of lift and drag coeffici ents measured by the static pressure and the balance at Re100,000 105 PAGE 106 5 0 5 10 15 20 25 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 AOA (degree)CL CL from pressure CL from balance 5 0 5 10 15 20 25 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 AOA (degree)CD CD from pressure CD from balance Figure 411. Comparison of lift and drag coeffici ents measured by the static pressure and the balance at Re150,000 106 PAGE 107 Figure 412. Picture of a Kulite transducer. Sensor out p ut () mV Figure 413. Linear response (at 500Hz ) of a typical Kulite transducer. 107 PAGE 108 108 Fre q uenc y res p onse Figure 414. Frequency response of a typical Kulite transducer. Figure 415. Hot wire calibration curve. PAGE 109 CHAPTER 5 RESULTS AND DISCUSSION The experimental results of the dynamic feedback control an d nonlinear control approaches are presented in two parts. Dynamic Feedback Control Experimental Configuration Figure 51 shows the complete experimental confi guration. The system ID and control algorithms run on the dSPACE controller system in realtime (~4 KHz). The controller generates control signal that is amplified by an amplifier. Th e pressure fluctuation signals measured by Kulite sensors are amplified and filter ed before sending to the dSPACE controller. System Identification Coherent flow structures Unlike PODbased approaches (Holmes et al 1998; Tadmor et al. 2007; Ausseur et al. 2007), the unsteady surface pressure si gnals are used exclusive of th e velocity field for feedback in this research. Although system ID and PODbased met hods of modeling the flow are different, they all attempt to capture the signat ure of the separated flow the coherent flow structures. In this section, we show that th e surface pressure signals indeed represent the footprint of the cohere nt flow structures. The following experiment is devised to show this. A continuous pulse train (repetition rate at 1 Hz and amplitude at 50 V) is fed to the ac tuator A1 at Re=120,000. The pressure signals are phase averaged relative to th e pulse signal. Four thousand av erages are taken to obtain the statistically converged pressure signal profiles. As shown in Figure 52 a vortex (produced by the actuator pulse) propagates downstream indica ted by the surface pressure fluctuations. The vortex reaches the sensor S1 first and then S2 S6. After the vortex passes by, the averaged 109 PAGE 110 surface pressure fluctuating approaches zero. Th is is because the random pressure fluctuations that are not correlated with th e pulse input from the actuato r possess random phase and are averaged out. The convective velocity is much slower than the free stream velocity, with a nominal lifetime of more than 5 airfoil chord lengths. With the link between the flow structure and the surface pressure clearly esta blished, the goal of the system ID approach is to correlate the actuator input and the correspondin g surface pressure fluctuations and utilize the re lationship to model the coherent flow structure wi th linear dynamical equations. Linear prediction The control approach is based on the assumption that the coherent flow structures may be modeled by linear dynamical equations. This sec tion demonstrates that the linear model is capable of predicting the downstr eam evolution of the flow dynami cs (measured by the pressure sensors) subject to the actuati on upstream (provided by the ZNMF actuators). In fact, previous studies in turbulent boundary layer (Rathnasingham and Breuer 2003) and cavity flows (Cattafesta et al. 1999) have suggested that linear approximations can reasonably predict inherently nonlinear flow structures. The system ID algorithm is applied first to demonstrate this. The computational requirements are demanding for both the system ID and control algorithms. When implementing the algorithms, the algorithm parameters were chos en due to hardware limitations and were not optimized. The sampling frequency is chosen to be 4096 Hz. In Figure 53 z denotes the pressure signal measured by th e #6 pressure sensor shown in Figure 51 subject to a bandlimited random input provided to actuator A1. Using the input and z, one can fit a model to represent the flow structures using the a pproach described in earlier sect ions. One should be aware that this model actually includes actuator and sensor dynamics and other hardware in the loop, e.g. amplifiers and filters. Here, in z Figure 53 is the estimated using the model mentioned z 110 PAGE 111 above, and the data show a reasonable match ove r a range of time or frequency scales. The errors are primarily due to the random turbulent st ructures that are uncorrelated with the actuator input, and they cannot be modeled by the system ID algorithm. An indication of the convergence of the system ID to the actual flow model is the expected value of the Mean Squared Erro r (MSE), which is defined as 2 ()() MSEEzkzk As shown in Figure 54 at the beginning the MSE has a relatively large value. This is because the model parameters are initialized to zero. Then the model parameters are trained by the ARMARKOV system ID algorithm targeting th e objective of minimizing the error. Frequency response and the performance of system ID To further evaluate the performance of syst em ID approach, we compare the frequency response of the flow system determined using conventional FFT methods for singleinput/singleoutput systems described in Chapter 6 of Bendat and Piersol (2000) with that determined using the converged ARMARKOV system ID model parameters. Wh en computing the frequency response, the parameters are s f =4096 Hz, NFFT=1024, 75% overlap, a Hanning window and 320 effective averages. Figure 55 shows that the system ID does not perf ectly match the frequency response, but it does capture the essential characte ristics over a broad frequency range It is also clear that the coherence between the input and out put is close to zero at frequenc ies less than 600 Hz, which is a characteristic of the present piezoelectri c zeronet mass flux actuators, which posses a resonance near 1200 Hz (Holman et al. 2003). The low coherence renders the FFTbased frequency response estimate uncertain and highlight s the difficulty of desi gning a control system using classical frequency domain approaches. 111 PAGE 112 Acoustic contamination In all of the discussions above we have ignored an important potential issue related to acoustic contamination. It is well known that zeronet mass flux actuators can produce significant sound. In the present control problem, the pressure sensors are intended to capture the hydrodynamics of the coherent flow st ructures. However, as demonstrated by Figure 56 the pressure sensors do not discrimi nate between acoustic and hydrodyna mic pressure fluctuations. Since the pressure measurements contain both components, the disturba nce rejection control algorithm will try to suppress th e acoustic power as well as the hydrodynamic power, possibly resulting in an undesirable reduction in the actu ator amplitude. Furthermore, from a control standpoint, the acoustic and hydrodynamic paths have significantly different propagation speeds, leading to significant phase lag differences and an unstable controller. One way to address this problem is to es timate a frequencywavenumber spectrum using Fourierbased methods, but such a method is not amenable to a realtime control system. A second approach, adopted here, incorporates a di gital filter to predict and remove the acoustic signal. We design this digital filter using the same system ID method described in the earlier section, i.e. using the system ID method to pred ict the acoustic signal with the actuators on and the flow off and then subtracting the com puted acoustic component from the sensor measurement with both the actuators and flow on (see Figure 56 ). The ID parameters were p=1, n=100, =1. Note this filter has much higher order than that used in Figure 53 However, this will not add significant computational intensity b ecause after the filer is designed by the system ID algorithm the filter parameters are fixed during the closedloop control. Figure 57 shows a comparison between the actual m easured acoustic noise with wi nd tunnel off and the predicted acoustic noise by the digital filter. Good agreement is achieved. To te st this digital filter further, 112 PAGE 113 we used the same digital filter in Figure 57 but reduced the actuator amplitude by 50%. Figure 58 shows the digital filter works well when the input signal is changed, indicating the linear behavior of the acoustic signal produced by the actu ator for typical excitation levels used in the experiment. Next, the digital filter is applied to the m easurements with the wind tunnel running. Figure 59 shows the power spectra comparison of the pressure measurements before and after applying the digital filter for a system ID cas e. The power spectrum with the digital filter applied clearly shows lower power at the frequency band 500 Hz to 1500 Hz, where the piezoelectric actuator generates most of its acous tic noise due to the ac tuator resonance as indicated in Figure 56 The digital filter is thus able to mitigate the acoustic noise component and is used for all results presented below. Disturbance Rejection Closedloop control As described earlier, the dist urbance rejection algorithm re quires both reference (used for feedback) and performance signal measurements. As described ear lier, the goal of the algorithm is to minimize the fluctuations of the pe rformance signal. A ccording to Venugopal and Bernstein (2000), the reference and performance signals can be the same. Herein, different reference and performance transducer combinations are tested for comparison. Since S1 is the closest to the leading edge and S6 is the closest to the trailing edge, it is reasonable to investigate the extremes shown in Table 51 The disturbance rejection algorith m is applied to all four ca ses and the ID and controller parameters are summarized in Table 52 Note that higher values increase the nu mber of adjustable parameters in the system ID and the controller, which may increase pe rformance. However, th is is not guaranteed. 113 PAGE 114 First, to make sure the control is inducing a global effect, we examine the lift/drag performance that is the aerodynamic objective of the separation control system. The control objective is to attach the sepa rated flow and thereby reduce th e fluctuating pressure spectrum associated with the convection of the vertical stru ctures over the airfoil su rface. The lift and drag are measured after the closedl oop control algorithm converges by the balance, which is only capable of providing mean or timeaveraged data. The lifttodrag ratios fo r all the cases are also summarized in Table 51 and include uncertainty estimates that account for calibration and random errors (Tian 2007). All four closedl oop control cases give similar L/D improvement, L/D of the uncontrolled baseline case. Note that the lift is increased while the drag is decreased. Close inspection via tu ft and smoke flow visualization reveals that, in all 4 cases, the controller is able to full y attach the separated flow. Since th e four closedloop control cases give the same L/D within experimental uncertainty, th is indicates that the ch oice of the performance and reference sensor locations does not have a significant imp act on the integrated lift/drag performance for this flow condition. In the fo llowing sections, we choose to study case #2 using S6 more closely. ~7Effect of control on surface pressure signals Recall that we use the actuator and surface pr essure signals to model the plant dynamics, and the disturbance rejection algorithm attempts to suppress the surface pressure spectra. Figure 510 shows the time traces of the performance surface pressure (measured by S6) and control input signals for case #2 before and after the contro l is turned on. The re sults clearly show that before the closedloop control is initiated, the perfor mance pressure signal has relatively large amplitude. After the ID and control is initiated from a zero initial condition for all parameters, the performance pressure signal starts to decr ease driven by the control input from the ZNMF actuators generated by the disturbance reject ion algorithm. The entire process takes 114 PAGE 115 approximately 79 convective time scales to lear n the dynamics and optimize the controller. After a steady state is achieved, the pressure sign al stays at the lower level corresponding to an attached low oscillation flow, as will be shown in the next section. Figure 511 shows the comparison of the power spect ra for baseline and case #2 measured by the performance transducer S6. For the clos edloop control case, the power spectra are based on the surface pressure signal af ter the disturbance rejection algorithm converges. The noise floor of the transducer is also plotted for comparison to verify th at the pressure signals for all cases are well above the noise floor. Figure 511 clearly shows that th e disturbance rejection algorithm is able to lower the sp ectrum of the surface pressure si gnal compared with the baseline case at all frequencies. Quantitative flow visualization Preliminary experimentation shows that A1 and A2 give similar results but A3 and A4 are ineffective because they are located downstream of the separation location. Thus, the results with A1 are studied closed. Normalized stream wise velocity and vorti city contours (obtained using 500 PIV image pairs) over th e airfoil for the base line and closedloop control case #2 are shown in Figure 512 and Figure 513 For the closedloop control case, the images are taken after the disturbance rejection al gorithm converges. The actuator A1 and pressure sensors S1 S6 are shown as circles on the airfoi l surface. For the baseline case in Figure 512 (a), the flow separates from the leading edge just downstream of actuator A1 and all six pressure sensors are located inside the separated region. Instantane ous PIV data reveal th at the separated flow features large coherent vortices sweeping over the airfoil upper surface, which results in highly unsteady pressure signals on th e airfoil upper surface. The di sturbance rejection algorithm senses the pressure fluctuations and genera tes actuation signals to negate the pressure fluctuations. This process ultimately organizes th e unsteady flow into an attached turbulent flow 115 PAGE 116 in a closedloop (smart) fashion. As shown in Figure 512 (b), the flow is fully attached for the closedloop control case. This ma y explain why the four cases in Table 51 result in similar lift/drag performance, i.e. they share similar information about the flow before and after the closedloop control is initiated. Control input In order to gain physical insight into the actuator control signal that the disturbance rejection algorithm generates to attach the flow, we examine the input voltage power as well as the corresponding input electrical power to the actuator A1. The first quantity has units of V2, while the second quantity has an SI unit of W. Note that these two quantities are different because piezoelectric actuators are capacitive devi ces that draw higher current as frequency is increased by virtue of the derivative operation, iCdVdt where denotes the capacitance of the piezoelectric actuators. The electrical pow er is calculated by multiplying the input voltage by the current of the actuator. The current is measured by a standalone current probe (Tektronix TCP A300). CThe voltage and rms electrical power spectra are shown in Figure 514 The total rms electrical power sums up to 12.7 mW. It is clear from Figure 514 (a) that the disturbance rejection algorithm generates a broadband contro l input to the actuator with spectral peaks in both the low (20 Hz 80 Hz) and the high (around 1 kHz) frequency ranges. The emphasis at low frequencies is due to the larger scale cohere nt flow structures described earlier, while the emphasis at higher frequencies corresponds to the smaller scale shear layer structures. More detailed discussions on the two types of flow stru ctures can be found in Ti an et al. (2006) and Wu et al. (1998). Since the flow is most recep tive at these inherent frequency scales, the 116 PAGE 117 disturbance rejection algorithm attempts to util ize the two characterist ic frequency scales by energy addition at these frequencies. On the other hand, actuator characterization e xperiments reveal that the actuator produces very small output at low frequencies (< 500 Hz ) (Tian et al. 2006). This is clear from Figure 55 and may be deduced from Figure 514 (b). The electrical power is concentrated at higher frequencies near the actuator resonance (than th e voltage signal power) while remaining almost flat at low frequencies. This implies that although the contro ller attempts to control the low and high frequency instabilities associated with the wake and shear layer, respectively, the dynamic response of the actuator significantly influences the control system dynamics. It is important to recall that the plant in cludes the actuator. Inspection of Figure 514 also brings into question whet her the voltage power should be used as a penalty function. The electrical power may be a better choice for the penalty function since it reflects the actual power consumption by the actuator. Such questions must await future studies. Discussion To examine how the adaptive controller pe rforms under different flow conditions, the angle of attack is varied continuously from 12 to 20, with a fixed free stream Re of 120,000. The lifttodrag ratios for the baseline and controlled cases are plotted in Figure 515 Clearly, the baseline flow separates starti ng from 12. The adaptive contro ller gives the best performance at AoA=12. The performance deteriorates as AoA increases. The improvement in lifttodrag ratio becomes very small at AoA=20, which means the controller is ineffective at this AoA. The comparison of the power spectra for baseline and case #2 measured by the performance transducer S6 is shown in Figure 516 It is clear that unlike the 12 counterpart (shown in Figure 511 ), the power spectrum of the closedloop control case is higher than that of 117 PAGE 118 the baseline case. This defeats the purpose of the closedloop c ontroller and ther efore makes the controller ineffective. The key question is why the pressu re spectral characteristics look conversely different at different AoAs. Our hypothesis is that the flow can only be partially attached at higher AoA in the mean sense. Inst antaneously the partially attached flow contains highly unsteady flow vortices that cause the increase in the pressure spectra. Possible solutions include usi ng alternate surface sensors, such as MEMSbased direct shear stress sensors for feedback instead of pres sure sensors. This can also solve the acoustic contamination issue. However, we believe that the key limitation of the present scheme, as evidenced by its failure at higher angles of at tack, include the assumption of linearity in the system identification and disturbance rejection algorithm. The nonlinearity, especially in the partially attached unsteady flow, has strong impact on the performance of the closedloop controller. Future improvements to the current approach include exploration of nonlinear control methods, which is the subject of th e next section. By using the modulated signals as input (Tian et al. 2006), the nonlinear interactions in the unsteady flow is promoted. On the other hand, nonlinear dynamical approaches are desirable co mpared to the quasistatic approach in our companion study. The nonlinear system iden tification algorithms are demonstrated in simulations by Pillarisetti and Ca ttafesta (2001). Future di rection includes implementing a nonlinear dynamical system identifi cation and control scheme in the wind tunnel experiments. Nonlinear Control Experimental Configuration As shown in Figure 517 dualtiming control loops are configured to implement the optimization algorithms (described in the next section). The first loop synchronously controls the actuators and measures the lowpass filtered and amplified balance signal, while the second 118 PAGE 119 loop averages the balance output and performs optimization in an asynchronous fashion. The second loop acts as a supervisory cont roller that updates the control parameters in the first loop. The sampling rate of the first loop is 40 kHz, while the second loop runs on a host PC at O(Hz). The optimization algorithm is programmed in Matlab and communicates directly with the dSPACE system to adjust the actuator signal parameters. Flow Instabilities In the present flow conditions (AoA=2 and 0Re120,000c ), the baseline uncontrolled flow is massively separated and doe s not reattached before the trai ling edge (i.e. post stall). As mentioned earlier, this type of poststall flow is characterized by leadingedge shear layer rollup and vortex shedding in the wake (Wu et al. 1998). These two types of flow structures are clearly visible in the instan taneous snapshot of the flow shown in Figure 518 The shear layer rollup structures in the left figure have a much smalle r length scale than the vo rtex shedding structures in the wake shown in the right figure. To characterize the velocity fluctuations in the two flow structures a hotwire anemometer is used. The hotwire is traversed vertically ac ross the shear layer (near the separation point) and the wake vortices (1 chord aft of the trailing edge). The maximal location in the attached subregions is then determined at the two streamwise locations. rmsu Figure 519 shows a plot of the power spectral density (PSD) of the wake and the shear layer at the respective peak rms locations. The PSD was estimated using a 4096 point FFT, a Hanning window with 75% overlap, and 320 effective blocks. The plots zoom in on two intere sting regions. The left plot clearly shows the dominant wake frequency wake f ~ 40 Hz. The right plot shows the much higher shear layer frequency at SL f ~2040 Hz. The plot also provid es evidence for the nonlinear coupling between the shear layer and wake inst abilities via the presen ce of the shear layer 119 PAGE 120 frequency SL f and the sum/difference frequencies SLwake f f Note that SL f is much higher than wake f in accordance with classical scaling arguments that ~SL sepfU and ~wake wakefUW. Based on the definition FfcU SL f f gives and ~( FO30)wa ke f f give This evidence supports our hypothesis th at more than a single characteristic frequency exists, perhaps explai ning the wide range of effectiv e forcing frequencies reported in the literature. ~( FO 0.6) To study our hypothesis about the nonlinear quadr atic coupling between the instabilities, higherorder spectral analysis of the same velocity data in Figure 519 is performed (Nikias and Petropulu 1993). The autobicoherence contour plot shown in Figure 520 is bound between 0 and 1 and is only nonzero due to nonlinear quadr atic phase coupling (lockon). The autobicoherence thus quantifies the fraction of power in a random signal as a function of triad between two frequency components and their sum or difference 1 f 2 f 12 f f A close inspection of the contour plot reveals distinct features at (in particular near 2 w f f 2 ,SLSL,1.5SL f ff), 2 SL f f and 2 SL w f ff (in particular between SL f and 1.5SL f ), and especially along the lines 12 SL f f f and 12 SL w f ff f. These data conclusively show the presence of nonlinear quadratic coup ling between the KelvinHelmholtz and wake instabilities. Actuator Calibration Frequency response ZNMF actuator dynamics is a critical issue for the control of a separated flow. A typical ZNMF device contains a cavity a nd a vibrating diaphragm to dr ive oscillatory flow through a small orifice on the cavity. The synthetic jet represents a coupled electromechanicalacoustic system with frequency dependent properties de termined by device dimensions and material 120 PAGE 121 properties (Gallas et al. 2003). Gallas et al. use the lumped element modeling approach to model these actuators. In this research, although modeling our ZNMF devices is not needed, we do need to characterize the frequency response of the actuator A1 (that is used in this research). Since the ZNMF actuator is an inherently nonlinear device, the traditional approach that uses a swept sine as an input signal is not appropriate. Instead, a single sine wave is used as input, and the anemometer signal is recorded. The frequency is then in creased in a loop, while the forcing amplitude is held constant. Figure 521 shows the rms velocity per input voltage in the frequency band from 500 Hz to 2500 Hz. Three different input leve ls are used. The peak output occurs at approximately 1200 Hz, and significant output is apparent ly limited to a bandwidth of 5001500 Hz. The output level is very low for fr equencies less than 500 Hz and larger than 1500 Hz. This precludes the possibility of directly forcing either the lowfrequency (~40 Hz) wake or high frequency (~2020 Hz) shear layer instabilities via sinusoidal excitation. It also highlights the preferential output of the actuator near its resonance frequency. Furthermore, the nonlinear nature of the actuators is revealed, since the frequency response function is not independent of the input voltage. (If the actuator were linear, th ese curves would collapse.) However, that is this nonlinear behavior that is leveraged to en able forcing at low a nd high frequencies, as explained below. Types of actuation waveforms Three typical multimodal waveforms are studied to take advantage of the multiple instabilities of a separated flow. They are shown in Figure 522 : (a) amplitude, (b) burst and (c) pulse modulation. In (a) and (b ), the lower plot in the figure is the result of a pointbypoint product of the top two waveforms. Such forcing is, in general, a modula tion of a (usually) high frequency carrier signal, (e.g., a sine wave with frequency c f ) by a low frequency modulation 121 PAGE 122 signal (either a sine wave or square pulse with frequency m f ). In addition, a parameter is multiplied to determine the amplitude. For the BM signal, there is an additional parameter, the duty cycle, which determines how many sine wave periods occur in each burst In this research, the duty cycle is adjusted such that only one peri od occurs in each burst. The last case in part (c) is a pulse train, which can be interpreted as th e modulation of a constant signal by square pulse. Similar to the BM signal, the duty cycle can be an additional parameter. In this research, it is kept to be the shortest achievable width on th e dSPACE control, i.e. one discrete sample At In this case, there is only one waveform parameter m f to vary. As one moves from amplitude to burst to pulse modulation, the modulation process re sults in an increasingly rich signal spectrum with broader spectral content, which improves the likelihood that the excitation waveform will excite an inherent instability. Furthermore, the required actuator power reduces as will be shown in the following section. Next we study the output of the actuator subject to the AM waveform excitation as an example. Figure 523 shows the velocity output of the ZNMF actuator A1 measured by the hotwire anemometry subject to an AM excitation, where A 50 Vpp (peaktopeak voltage), Hz and Hz. The wire was placed at a sufficient distance (~1mm) above the actuator slot so that there is no reverse flow occurs and, hence, no signal rectificat ion is needed. The temporal record shown in (a) clearly shows the low frequency oscillations at 50 Hz and the high frequency oscillations at 1180 Hz. The corresponding power spect ral density of the velocity signal shown in (b) reveals that the high power is at the low modulation frequency 50 Hz while the second highest power is at the hi gh carrier frequency, with additional harmonic distortion peaks. Recall that th e response to the sinusoidal exc itation at frequencies less than 500 Hz is very low, as shown in 50mf 1180cf Figure 521 The modulation enables the actuator to generate high 122 PAGE 123 power signals at low frequencies, while the si nusoidal response is limited by the actuator dynamics. This allows an actuator operating at or near its resonant freque ncy via a carrier signal at c f to generate significant disturbances at charac teristic frequencies of the flow that are far from the natural frequency of the device. This char acteristic is attributed to the nonlinear nature of the actuator system. Similar beha vior is observed for BM and PM. and electrical power calibration C Figure 521 showed that the actuator response varies significantly with actuation frequency when subjected to sinusoidal excitation. This is typical for a ZNMF actuator. When the performance of separation control is studied, the actuation frequenc y is always of prime importance. However, when the actuation frequency is varied, the actuator response is also varied even for constant amplitude. As menti oned earlier, it has been shown that sinusoidal authority varies monotonically with rmsu /JVU up to some maximum value (Seifert et al. 1993, 1996, 1999; Glezer and Amitay 2002; Mittal and Ra mpunggoon 2002). In other words, when the actuation frequency is varied, th e two performancedetermining parameters (namely frequency and amplitude) are varied simultaneously. Unfortuna tely, this can lead to misleading results. For example, one may find that the performance is the best at the peak response frequency of the actuator, which could be simply because the actuator is providing higher output as opposed to the flow being more receptive as that frequency. The same problem above still exists even when multimodal waveforms are used; the actuator response varies when Am f and c f are varied. To separate amplitude forcing effects, the following calibration is performed. For th e AM and BM signals, a twodimensional grid in the ,mc f f space is generated, and C and rms electrical power c onsumed by the actuator are measured for various excitation amplitudes. This timeconsuming task takes days to complete 123 PAGE 124 and verify repeatability. The results are shown in Figure 524 and Figure 525 The profiles for five amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are recorded and shown. The lowest profile is for the 30 Vpp case and the highest profile is for the 50 Vpp case. For the PM signal, there is only one frequency parameter m f ; thus a contour plot in the ,mAf space is shown in Figure 526 The amplitudes are again 30 Vpp to 50 Vpp with 5 Vpp increment. There are several important observations in th e results. For the AM signal, the response ( C and electrical power) is approximately independent of the modulation frequency m f but strongly dependent on c f The C at a fixed m f is similar to the sinu soidal velocity response shown in Figure 521 In addition, the shape of the el ectrical power surf aces are somewhat different than the shapes of the C and the peak frequencies are different too. This means that when the actuator provides the maximum velocity output, the electrical power consumption is not maximized. For the BM signal, the C and electrical power resp onses are dependent on both m f and c f They monotonically increase with m f as expected. As with the AM signal, the shapes of the C and electrical power surfaces are different and possess different peak frequencies. The profiles are simpler for the PM signal since there is only one frequency parameter. Both the C and electrical power responses increase with m f and as expected. Furthermore, typical levels of AC are and for the AM, BM and PM signals, respectively. 410510610With this information on C and electrical power, additional constraint functionality is added to the adaptive optimization program to hold C or electrical power constant, while m f and c f are varied. This is done by adjusting in accordance with AFigure 524 Figure 525 and 124 PAGE 125 Figure 526 in each iteration of the optim ization algorithm. Thus, it is possible that at certain ,mc f f combinations, constant C or electrical power cannot be achieved. In this case, m f and c f are set to the values at th e boundary such that the moment um or power constraint is maintained. This will hopefully become clearer when the optimization results are discussed in the next section. Adaptive Control Results First, the constrained optimiza tion is carried out at AoA=12 and Re=120,000. At this AoA, the dynamic control approach in our compani on study works well. The constraint is set at = 7.15 x 106. This is a relatively low level, which can be achieved in most portions of the C ,mc f f space for the AM and BM signals and all values of m f for the PM signal. To protect the actuator from physical damage, the maximum amplitude is limited to 50 Vpp. The results are summarized in Table 53 The lifttodrag ratios are comp arable with the results using the dynamic control approach (within uncertainties). This is not surprising because the flow is completely attached by both approaches at AoA=12. On the other hand, when the AoA is 20 where the dynamic control a pproach fails to attach the flow, the present nonlinear control shows its clear performance improvement. Therefore, a more detailed study is performed at this AoA. In addition to the constrained cases for C = 7.15 x 106, the constrained optimization is also carried out with constant electrical power of 0.0005 W. The constrained optimization results are summarized in Table 54 Typical search paths are plotted in Figure 527 to Figure 529. The shaded areas in Figure 527 and Figure 528 denote the achievable parameter space, inside which the optimization is c onstrained. It is clear that th e achievable areas cover most of the parameter space for the AM cases while lim ited space for the BM cases. Since there are two 125 PAGE 126 parameters: m f and c f three initial points are needed in each optimization run as mentioned in earlier sections. The symbols denote the search paths fo llowing each initial point during the optimization. On the other hand, for the PM signal, only two initial points are needed. Different sets of initial points are chosen to cover most of the opera tional space to achieve a global optimum. In Table 54 the first column indicates if the constraint is active in the optimization process. The last three columns summarize th e optimal values for converged results. The experimental uncertainties are also added to the LD values. By simply observing the L/D values, we find that the performance for the AM case (a) gives the best performance. The AM and BM signal are superior to the PM signal with constant C = 7.15 x 106. For the AM and BM cases (a), note that the converged c f is near the shear layer frequency SL f On the other hand, the modulation frequency m f assumes a value near the wake frequency wake f and its superharmonics. Most importantly, effective separation control is achieved by using the multimodal waveforms with C of at least an orderofmagnitude smaller than typical values reported in the literature (Table 2 in Gree nblatt and Wygnanski 2000). Selected sinusoidal excitations are also tested on our airfoil model to compare with the results using the multimodal waveforms. The voltage amplitude for these tests is held at 50 Vpp. The results are summarized in Table 55 It is clear that sinusoida l control (even with higher C ) gives poorer performance than the AM and BM excitations. In addition, except the PM signal, the nonlinea r control is able to achieve much better performance than the dynamic contro l in our companion study at AoA=20. The nonlinear 126 PAGE 127 control approach benefits from the nonlinear co upling of the instabilities and the integrated performance (L/D) measurements instead of local unsteady pressure measurements. Table 51. Case descriptions and performance for disturbance re jection experiments. (AoA=12 and Re=120,000) Case # Reference transducer yPerformance transducer z L C D C LD Baseline 0.21 0.02 0.21 0.09 1.01 0.08 1 S1 S1 0.84 0.01 0.12 0.01 6.97 0.37 2 S6 S6 0.83 0.01 0.12 0.01 7.21 0.46 3 S1 S6 0.84 0.01 0.12 0.01 7.11 0.40 4 S6 S1 0.84 0.01 0.12 0.01 7.09 0.43 Table 52. Summary of parameters in disturbance re jection algorithm. p n cpcnc 1 2 10 1 pn 2 20 Table 53. Constrained optimi zation results using the AM, BM and PM signals. (Baseline L/D=1.01 at AoA=12 and Re=120,000) Signal type Constraint Constraint Active? Converged m f Converged c f Converged LD AM C = 7.15 x 106 No 74 1667 7.47 0.45 BM C = 7.15 x 106 Yes 48 1305 7.63 0.36 PM C = 7.15 x 106 No 16 NA 7.14 0.25 127 PAGE 128 Table 54. Constrained optimi zation results using the AM, BM and PM signals. (Baseline L/D=1.1 at AoA=20 and Re=120,000) Signal type Constraint Constraint Active? Converged m f Converged c f Converged LD AM: Case (a) C = 7.15 x 106 Yes 61 2405 2.18 0.07 AM : Case (b) Power=0.0005 No 202 1005 1.77 0.05 BM : Case (a) C = 7.15 x 106 Yes 55 1979 1.95 0.05 BM : Case (b) Power=0.0005 Yes 56 1318 1.52 0.04 PM : Case (a) C = 7.15 x 106 No 16 NA 1.49 0.04 PM : Case (b) Power=0.005 No 29 NA 1.48 0.04 Table 55. Results using sinusoidal excitations. /sepF f LU C LD 0.5 ~0 1.14 0.02 15 3.16 x 104 1.76 0.03 26 7.79 x 106 1.77 0.03 128 PAGE 129 Figure 51. NACA 0025 airfoil m odel with actuators, sensors a nd closedloop control system. Figure 52. Phase averaged pulse response m easured by six pressure sensors. The slow propagation velocity of the coherent flow structures is clearly visible. 129 PAGE 130 Figure 53. Comparison between measured signal from the pressure sensor (#6 in Figure 51 ) and the fitted output by ARMARKOV system ID algorithm for long and short time intervals. Results show a reasonable match at low frequencies between measured and fitted outputs. For ARMARKOV ID: p=1, n=2, =10. Figure 54. Mean Squared Error (Running MSE) between measured and fitted outputs. Results show that the ARMARKOV ID algorithm converges, i.e. error being minimized. 130 PAGE 131 Figure 55. Comparison between frequency re sponse (FR) and fitted response by ARMARKOV ID algorithm. Parameters for FR: s f =4096 Hz, NFFT=1024, 75% overlap and Hanning window. For ARMARKOV system ID: p=1, n=2, =10. Figure 56. Dual signal paths from the actuator to the pr essure sensor (acoustic and hydrodynamic). A digital filter is introduced to remove the acoustic component by turning off the flow to isolate the acoustic path. Figure 57. Actual measured and predicted acoustic noise using a bandlimited random signal to the actuator. Figure 58. Actual measured and predicted acoustic noise using the same filter as in Figure 57 but with one half of the input amplitude. 131 PAGE 132 Figure 59. Power spectra of the sensor signa ls (with wind tunnel runn ing) before and after applying acoustic filter. Figure 510. Performance surface pressure (S1) and control input signals (in Volt) before and after the ID and control is initiated for case #2. Control is established within 1 second or <100 convective time scales. Figure 511. Power spectra of the pressure transducer output for the baseline and the closedloop control cases measured by S6 (performance). 132 PAGE 133 Figure 512. Contours of streamwise velocity uU for (a) baseline and (b) closedloop control case #2 at AoA = 12 and Rec=120,000. Figure 513. Contours of vorticity for (a) baselin e and (b) closedloop control case #2 at AoA = 12 and Rec=120,000. Figure 514. (a) Voltage and (b) electrical power spectra of the actuator A1 input signal for the closedloop control case. 133 PAGE 134 Figure 515. Performance comparison at different AoA. Figure 516. Power spectra of the pressure transducer output for the baseline and the closedloop control cases measured by S6 (performance) at AoA = 20 and Rec=120,000. Figure 517. NACA 0025 airfoil model with actuators, sensors and closedloop control system. 134 PAGE 135 135 Figure 518. Flow structures in separated flow. Figure 519. Wake (1 chord aft of TE) and shea r layer (near separation) power spectral density functions at peak rms location. PAGE 136 Figure 520. Autobicoherence of the same velocity signal analyzed in Figure 519 using the same parameter settings. The autobicohe rence is zero except where nonlinear phase quadratic phase coupling occurs due to inte ractions between the shear layer and wake instabilities. Figure 521. Frequency response of ZNMF actuator A1. 136 PAGE 137 ()sin2sin2cmetAftft (a) amplitude modulation (AM) ()sin2square pulsecetAft (b) burst modulation (BM) () square pulse etA (c) pulse modulation (PM) Figure 522. Various waveforms of unit amplitude 1A that can be used to excite multiple instabilities or modes in a separated flow. Figure 523. Velocity response (a ) and its power spectral density (b ) subject to an AM excitation for the ZNMF actuator A1. A 50 Vpp 50mf Hz and 1180cf Hz. Measurements were made outside the region of reverse flow. Figure 524. C (a) and electrical power (b) profile s ubject to AM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. 137 PAGE 138 Figure 525. C (a) and electrical power (b) profile s ubject to BM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. Figure 526. C (a) and electrical power (b) profile s ubject to PM excitation. Five actuation amplitudes (30 Vpp, 35 Vpp, 40 Vpp, 45 Vpp and 50 Vpp) are shown. 138 PAGE 139 139 Figure 527. Constrained search using AM: Ca ses (a) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. Figure 528. Constrained search using BM: Ca ses (a) and (b). The shaded area denotes the constraint area and amplitude is adjusted at each step to satisfy the constraint criteria. AOA = 20 deg. and Re = 120,000. Figure 529. Constrained search using PM: Cases (a) and (b). AOA = 20 deg. and Re = 120,000. PAGE 140 CHAPTER 6 SUMMARY AND FUTURE WORK An adaptive system identification and feedb ack control algorithm is applied to the separation control problem for a NACA 0025 airf oil at nominal angles of attack of 12 and and a chord Reynolds number of 120,000 with a tr ipped boundary layer, corresponding to control of a massively leadingedge se parated flow. In particular, a recursive ARMARKOV system ID algorithm is used to model the flow dynamics a nd provide the informati on required to implement the disturbance rejection algorithm in real time with no prio r knowledge of the system dynamics. Phaselocked PIV and fluctuating surface pressure measurements provided evidence of the link between the separated flow vertical structures and the surface pressure fluctuations. The chosen control objective was thus to suppress the airfoil surface pressure fluctuations. The disturbance rejection algorithm was able to automatically generate contro l input to the ZNMF actuator, emphasizing low (i.e., wake) and high (i.e., shea r layer) characteristic frequencies of the separated flow. The effect of the control is to enhance nearw all mixing and suppress the highly unsteady flow structures. This adaptive control sc heme is able to completely reattach the flow using low (~12.7 mW) power to a single piezoel ectric synthetic jet actuator. The closedloop control results show ~ 7 x improvements in th e lift/drag ratio, with a corresponding increase in lift and reduced drag and concomitant reductions in the fluctuating surface pressure spectra. The present results are, to the best of our knowledge, the first expe rimental demonstration of an adaptive dynamic feedback contro l of a separated flow. The results reveal the tremendous potential of closedloop fl ow control to real aircraft applic ations but also reveal key issues worthy of further study. 20First, in terms of positives, the adaptive cl osedloop control scheme has several attractive features. It is quite general, and no prior knowledge of the system dynamics is required. The 140 PAGE 141 system identification and disturbance rejecti on algorithms are integrated, and the system dynamics can be obtained with minimal a priori user knowledge. The cont roller is implemented using DSP hardware and can be easily incorporated in hardwareintheloop applications. It can be applied to not only flow separation control pr oblems, but also, for example, cavity oscillation control and turbulent bounda ry layer control. Second, in terms of unresolved technical issu es, there remain many concerning the actuator and sensor dynamics. Clearly, better actuators with not just higher out put but flatter dynamic response over wider frequency range are desirabl e. While measuring un steady surface pressure is relatively straightforward, the potential ac oustic contamination issue was highlighted. This difficulty was mitigated with an acoustic digita l filter here but at the cost of additional computational complexity from an already limited DSP. This issue suggests the use of alternate surface sensors, such as MEMSbased direct shear stress sensors for feedback instead of pressure sensors. Or perhaps thermal sensors may be suffici ent despite their sensitivity to more than just shear stress. Ultimately, it is believed that th e key limitation of the present scheme, as evidenced by its failure at higher angles of attack, incl ude the assumption of linearity in the system identification and disturbance re jection algorithm. The second pa rt explores nonlinear control methods. In the poststall separated flow where th e flow does not reattach, there are two characteristic instabilities: the shear layer Kelvin Helmholtz instability and wake instability. Our experiments have the evidence for such instabili ties. In addition, the second order spectral analysis has quantified the quadratic phase coupling between the two instabilities, which indicates the separated flow is a complex multifrequency system. 141 PAGE 142 Three multimodal waveforms (namely amplitude modulation, burst modulation and pulse modulation) are used targeting excitation of the multifrequency separated flow system. A simplex optimization approach for controlling the separated flow has been developed to search for the optimal actuation parameters of the thre e waveforms using the ZNMF devices. It is typical for the C response to vary when the waveform frequency parameters vary for the ZNMF devices. This can potentially lead to misleading results about the optimal forcing frequency. The actuator dynamics is taken into consideration in the optim ization approach. To offset the actuator dynamics implications, a special routine is devised to hold the C at constant during the optimization process utilizing the precalibrated actuator response profiles. This is specifically done by varying the actuation voltages according to the response profiles keep C to the at constant levels. The constrained optimization results seeking to maximize LD are promising and reveal the importance of forcing nonlinear interactions between the shear layer and wake instabilities. Effective separation control is achieved by using oscillat ory momentum coefficients which is more than an orderofmagnit ude smaller than test cases on our model with sinusoidal excitations and typical values reported in the literature (see summary in Greenblatt and Wygnanski 2000). Specifica lly, the optimized carrier frequency 651010 Oc f targets the shear layer frequency while the optimized modulation frequency m f targets the wake frequency and its superharmonics. The nonlinear control is able to achieve similar performance at AoA=12 and much better performance than the dynamic control in our companion study at AoA=20. The nonlinear control approach benefits from the nonlinear coupling of the flow 142 PAGE 143 143 instabilities and the integrated performance (L/D) measuremen ts instead of local unsteady pressure measurements. Future direction includes exploring nonlinear dynamic control strategies and implementing feasible penalties in the cost functions. PAGE 144 APPENDIX HIGHER ORDER SPECTRUM EXAMPLES This chapter gives an illustration of the appl ication of higher order spectral analysis on systems with quadratic a nd cubic nonlinearities. Example 1: Quadratic System Second order systems are tested as examples to analyze nonlinear couplings using the bispectral methods. The two systems that are tested are 2 y xxn and 2 y xn where x is input, y is output and is additive white noise. nCase 1 The quadratic nonlinear system in Figure A1 is used as the test model. In the model () x t is input, () y t is output and is uncorrelated additive white noi se. Note there is a linear part in this model. () nt Input is a pure 15 Hz sinusoid signal: ()sin(215) x tAt (4.8) where is one. A The parameters used to compute the autospectrum are: Fs=128, NFFT=1024, Hanning window, 75% overlap. The parameters used to compute the autobicoherence and crossbicoherence are: Fs=128, NFFT=256, Hanni ng window, 75% overlap. Averaging using 100 blocks is used for all cases. The power spec trum of the output y, autobicoherence of y and crossbicoherence between x a nd y are plotted below. Based on trigonometric identities, the output can be expressed as follows: 144 PAGE 145 2sin(215)sin(215) 1cos(2215) sin(215) 2 11 sin(215)cos(230) 22yttn t t tt n n (4.9) Based on this, there should be peaks at 15 and 30 Hz. The power spectrum plot in Figure A2 confirms this. Figure A3 shows the autobicohere nce of y. There is a strong peak at (15, 15), which means that the 15 Hz signal interacts with itself to generate the 30 Hz signal that is shown in the power spectrum plot. The crossbicoherence shown in Figure A4 also shows this interaction. Furthermore, the cr ossbicoherence recovers that the 30 Hz signal also interacts with the original signal to generate additional 15 Hz signal (i.e. 1215 f fH z ). Case 2 The nonlinear model in Figure A5 is also tested. Compared with the case 1, there is no linear part in this model. The input and additive white noise are the same as the last case, as are the parameters for spectral analysis. The plot of power spectrum is as expected, w ith dc and 30 Hz components. On the other hand, the autobicoherence of y is close to zero at all points ( Figure A7 ). This is reasonable since there is only one frequency com ponent in the output y and from the definition, the autobicoherence represents quadratic coupling between two frequency components. This is also the case for the crossbicoherence as shown in Figure A8 Example 2: Cubic System From the definitions in Chapter 4, the bispectr al analysis calculates quadratic coupling. This works as expected with quadratic systems as shown above. A question naturally arises. Does it work with cubic or higherorder nonlineariti es? This section uses the third order system in Figure A9 as the test model. 145 PAGE 146 All the signals and processing para meters are the same as for the previous cases. Ttrigonometric identities as follows: 3sin(15*2)sin(15*2) 31 sin(15*2)sin45*2 44 ytt tt n n (4.11) The power spectrum of y verifies that ther e are two peaks at 15 Hz and 45 Hz, which agrees with the above equation. It is also clear that the autobicoherence of y and crossbicoherence between x and y are very close to zero ( Figure A11 and Figure A12 ). This result shows that the bispectral analysis is not able to reveal cu bic nonlinearities. Example 3: x x The system in Figure A13 is also tested. By means of least mean square estimation, this system can be approximated by a cubic system: 31632 1515 A y xxxx A (4.12) Substituting eqn. (7.1) into eqn. (7.5) y takes the form as follows: 81 sin(215)sin245 35 y t t (4.13) All the signals and processing parameters are th e same as for the previous cases. In the plot of the power spectrum, there are several peaks. This indicates that the system is much more complicated than the previous cases. The auto bicoherence and the cros sbicoherence are very close to zero ( Figure A15 and Figure A16 ), which indicates that the bispectral analysis does not work for this system. This is expected as the system acts like a cubic system according to eqn. (7.6) 146 PAGE 147 Figure A1. Test model #1 for higher order spectrum analysis. 0 10 20 30 40 50 60 70 107 106 105 104 103 102 101 100 101 Power Spectrum of y f (Hz) Figure A2. Power spectrum of y for test model #1. 147 PAGE 148 0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)AutoBicoherence of y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure A3. Autobicoherence of y for test model #1. f1 (Hz)f2 (Hz)CrossBicoherence between x and y 0 10 20 30 40 50 60 60 40 20 0 20 40 60 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure A4. Crossbico herence between x and y for test model #1. 148 PAGE 149 Figure A5. Test model #2 for higher order spectrum analysis. 0 10 20 30 40 50 60 70 107 106 105 104 103 102 101 100 101 Power Spectrum of y f (Hz) Figure A6. Power spectrum of y for test model #2. 149 PAGE 150 0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)AutoBicoherence of y 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Figure A7. Autobicoherence of y for test model #2. f1 (Hz)f2 (Hz)CrossBicoherence between x and y 0 10 20 30 40 50 60 60 40 20 0 20 40 60 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 Figure A8. Crossbico herence between x and y for test model #2. 150 PAGE 151 Figure A9. Test model #3 for higher order spectrum analysis. 0 10 20 30 40 50 60 70 107 106 105 104 103 102 101 100 101 Power Spectrum of y f (Hz) Figure A10. Power spectrum of y for test model #3. 151 PAGE 152 0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)AutoBicoherence of y 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Figure A11. Autobicoherence of y for test model #3. f1 (Hz)f2 (Hz)CrossBicoherence between x and y 0 10 20 30 40 50 60 60 40 20 0 20 40 60 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 Figure A12. Crossbicoherence be tween x and y for test model #3. 152 PAGE 153 Figure A13. Test model #4 for hi gher order spectrum analysis. 0 10 20 30 40 50 60 70 107 106 105 104 103 102 101 100 101 Power Spectrum of y f (Hz) Figure A14. Power spectrum of y for test model #4. 153 PAGE 154 154 0 10 20 30 40 50 60 0 5 10 15 20 25 30 f1 (Hz)f2 (Hz)AutoBicoherence of y 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Figure A15. Autobicoherence of y for test model #4. f1 (Hz)f2 (Hz)CrossBicoherence between x and y 0 10 20 30 40 50 60 60 40 20 0 20 40 60 0.02 0.03 0.04 0.05 0.06 0.07 Figure A16. Crossbicoherence betw een x and y for test model #4. PAGE 155 LIST OF REFERENCES Akers, J. C. and Bernstein, D. 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Under the guidance of Dr. Louis Cattafesta, he was engaged in research in flow control applications funded by NASA and AFOSR. He obtained his Master of Science in 2004 and his Ph.D. in 2007. 