ADAPTIVE CONTROL METHODS FOR EXCITED SYSTEMS

Transcription

1 ADAPTIVE CONTROL METHODS FOR NON-LINEAR SELF-EXCI EXCITED SYSTEMS by Michael A. Vaudrey Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in artial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN MECHANICAL ENGINEERING GRADUATE COMMITTEE William Saunders, Chair William Baumann Uri Vandsburger Harry Robertshaw Donald Leo August 28, 2001 Blacksburg, Virginia Keywords: Adative Feedback, Thermoacoustic Instability, Neural Network, Time Averaged Gradient, Least Mean Square Algorithms, Combustion Control

2 ADAPTIVE CONTROL METHODS FOR NON-LINEAR SELF-EXCITED SYSTEMS Michael A. Vaudrey ABSTRACT Self-excited systems are oen loo unstable lants having a nonlinearity that revents an exonentially increasing time resonse. The resulting limit cycle is induced by any slight disturbance that causes the resonse of the system to grow to the saturation level of the nonlinearity. Because there is no external disturbance, control of these self-excited systems requires that the oen loo system dynamics are altered so that any unstable oen loo oles are stabilized in the closed loo. This work examines a variety of adative control aroaches for controlling a thermoacoustic instability, a hysical self-excited system. Initially, a static feedback controller looshaing design and associated system identification method is resented. This design aroach is shown to effectively stabilize an unstable Rijke tube combustor while reventing the creation of additional controller induced instabilities. The looshaing design method is then used in conjunction with a trained artificial neural network to demonstrate stabilizing control in the resence of changing lant dynamics over a wide variety of oerating conditions. However, because the ANN is designed secifically for a single combustor/actuator arrangement, its limited ortability is a distinct disadvantage. Filtered-X least mean squares (LMS) adative feedback control aroaches are examined when alied to both stable and unstable lants. An identification method for aroximating the relevant lant dynamics to be modeled is roosed and shown to effectively stabilize the self-excited system in simulations and exeriments. The adative feedback controller is further analyzed for robust erformance when alied to the stable, disturbance rejection control roblem. It is shown that robust stability cannot be guaranteed because ii

3 arbitrarily small errors in the lant model can generate gradient divergence and unstable feedback loos. Finally, a time-averaged-gradient (TAG) algorithm is investigated for use in controlling self-excited systems such as the thermoacoustic instability. The TAG algorithm is shown to be very effective in stabilizing the unstable dynamics using a variety of controller arameterizations, without the need for lant estimation information from the system to be controlled. iii

4 ACKNOWLEDGEMENTS I must extend endless thanks to Will Saunders and Bill Baumann, my wife Mandy, and my arents Walt and Sandra. Without any of whom, none of this would have been ossible. iv

5 TABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES ii iv v viii xi INTRODUCTION Research Goals Contributions Document Overview 7 REVIEW OF GERMANE LITERATURE Combustion Instability Modeling and Control Alication of Adative Control to the Self-Excited Combustion Instability Neural Networks in Combustion Control LMS Based Adative Control in Combustion Control Gradient Based Adative Methods in Combustion Control Adative Control Literature 17 FEEDBACK CONTROL OF A THERMOACOUSTIC INSTABILITY Limit Cycle Control and System Stability Feedback Control Review Rijke Tube Exerimental System System Identification Feedback Controller Design 33 v

6 3.6 Tyical Exerimental Results Review 41 ARTIFICIAL NEURAL NETWORK CONTROL Neural Network Training Data Neural Network Design and Training Feedback Controller: Adative Udate Neural Network Conclusions 54 ADAPTIVE CONTROL Introduction Adative Feedback Control Classical Disturbance Suression Self-Excited Systems The Correct Plant Estimate Simulation General Least Mean Squares Algorithm Recursive Least Squares Algorithm Exerimental Results Actuator Authority as a Source of Intermittency Conclusions 95 ADAPTIVE FEEDBACK CONTROL ANALYSIS Filtered-E Control Overview Feedback Loo Instabilities Algorithm Divergence Practical Alications and Considerations Interrelated Instabilities Filtered-U LMS Algorithm Online System Identification Conventional Identification Methods 112 vi

7 A New On-Line Identification Technique Extension to Self-Excited Systems Simulation Feedback Loo Instabilities Algorithm Divergence Conclusions 128 TIME AVERAGED GRADIENT CONTROL Introduction Time-Averaged Gradient Control Simulation Exerimental Results Conclusions 148 CONCLUSIONS Summary and Conclusions Future Work 155 REFERENCES 158 APPENDIX 163 A Neural Network DSP C-File 163 B Adative Feedback Simulation M-Files 172 C Time Averaged Gradient Simulation M-Files 180 VITA 185 vii

8 LIST OF FIGURES FIGURE 1.1 GENERIC SELF-EXCITED SYSTEM 1 FIGURE 1.2 GENERIC THERMOACOUSTIC INSTABILITY CONTROLLER IMPLEMENTATION 2 FIGURE 1.3 GENERIC SELF-EXCITED SYSTEM CONTROL IMPLEMENTATION 3 FIGURE 3.1 STABLE AND UNSTABLE SYSTEM DYNAMICS 22 FIGURE 3.2 STABLE AND UNSTABLE SYSTEM CONTROL 23 FIGURE 3.3 SCHEMATIC OF TUBE COMBUSTOR RIG 25 FIGURE 3.4 PICTURE OF RIJKE TUBE COMBUSTOR RIG 26 FIGURE 3.5 OPEN LOOP COMBUSTOR CONTROL SYSTEM 27 FIGURE 3.6 SYSTEM IDENTIFICATION DIAGRAM 29 FIGURE 3.7 PROBE GAIN AND LIMIT CYCLE GAIN 32 FIGURE 3.8 FEEDBACK CONTROLLER DESIGN - MAGNITUDE 35 FIGURE 3.9 FEEDBACK CONTROLLER DESIGN - PHASE 36 FIGURE 3.10 SPILLOVER AMPLITUDE PREDICTION 37 FIGURE 3.11 FEEDBACK CONTROLLER DESIGN RESULT 39 FIGURE 3.12 FEEDBACK CONTROLLER PERFORMANCE (DASHED-UNCONTROLLED, SOLID- CONTROLLED, DOTTED-EXCESS CONTROLLED GAIN) 40 FIGURE 3.13 FEEDBACK CONTROLLER SEARCHING BEHAVIOR 41 FIGURE 4.1 ALL TRAINING DATA SETS 45 FIGURE 4.2 COMPLEX NETWORK 47 FIGURE 4.3 MAX ERROR OVER EPOCH 48 FIGURE 4.4 NEURAL NETWORK PERFORMANCE 49 FIGURE 4.5 OPEN LOOP PLANT 50 FIGURE 4.6 ANN CONTROL 51 FIGURE 4.7 NARROWBAND ANN CONTROL PERFORMANCE 52 FIGURE 4.8 BANDPOWER AT 130 CC/SEC, INCREASING EQ. RATIO 53 FIGURE 4.9 BANDPOWER AT 140 CC/SEC, DECREASING EQ. RATIO 54 FIGURE 5.1 ADAPTIVE FEEDBACK EXTERNAL DISTURBANCE 58 FIGURE 5.2 ADAPTIVE FEEDBACK SELF-EXCITED 60 viii

9 FIGURE 5.3 REDRAWN CONTROLLED SYSTEM 61 FIGURE 5.4 PERFECT IDENTIFICATION DIAGRAM REDRAWN 64 FIGURE 5.5 SYSTEM IDENTIFICATION OPEN LOOP 66 FIGURE 5.6 SIMULATION IDENTIFICATION RESULTS 67 FIGURE 5.7 LOW HEAT RELEASE LMS SIMULATION 68 FIGURE 5.8 LOW HEAT RELEASE LMS PATH TO OPTIMAL 69 FIGURE 5.9 LMS INTERMITTENT STABILITY SIMULATION 71 FIGURE 5.10 LMS INTERMITTENT PATH TO OPTIMAL 72 FIGURE 5.11 RLS INTERMITTENT STABILITY SIMULATION 74 FIGURE 5.12 CONVERGED RLS CONTROLLER NYQUIST DIAGRAM 75 FIGURE 5.13 EXPERIMENTAL MODEL DATA AND FIT 76 FIGURE 5.14 LOW HEAT RELEASE EXPERIMENTAL LMS CONTROL 77 FIGURE 5.15 FIXED GAIN AND ADAPTIVE CONTROLLER COMPARISON 78 FIGURE 5.16 LOW AND HIGH HEAT RELEASE LMS CONTROL 79 FIGURE 5.17 RLS CONTROL HIGH HEAT RELEASE 80 FIGURE 5.18 FIXED GAIN FEEDBACK CONTROL HIGH HEAT RELEASE 81 FIGURE 5.19 CONTROLLED SYSTEM POLE LOCATIONS 84 FIGURE 5.20 PROBE WITH INTERMITTENCY 85 FIGURE 5.21 PROBE INTERMITTENCY SPECTRA 86 FIGURE 5.22 PROBE AMPLITUDE EFFECTS ON INTERMITTENCY RECOVERY TIME 87 FIGURE 5.23 CONTROLLER GAIN INTERMITTENCY PERIOD 88 FIGURE 5.24 VIRTUAL POLE TRANSIENT RESPONSE 89 FIGURE 5.25 ACTUATOR AMPLITUDE RESPONSE 91 FIGURE 5.26 ACTUATOR VOLTAGE FOR HIGH HEAT RELEASE 92 FIGURE 5.27 SUB-OPTIMAL CONTROL 93 FIGURE 5.28 GAIN SCHEDULED CONTROL 94 FIGURE 6.1 ADAPTIVE FEEDFORWARD CONTROL 98 FIGURE 6.2 FILTERED-E CONTROL 100 FIGURE 6.3 FILTERED-E CONTROL REDRAWN 101 FIGURE 6.4 FILTERED-U CONTROL 110 FIGURE 6.5 FILTERED-E ADAPTIVE IIR CONTROL 111 FIGURE 6.6 ONLINE SYSTEM IDENTIFICATION 113 FIGURE 6.7 ONLINE SYSTEM IDENTIFICATION 115 ix

10 FIGURE 6.8 MODIFIED ONLINE IDENTIFICATION 117 FIGURE 6.9 SELF-EXCITED SYSTEM FILTERED-E CONTROL 119 FIGURE 6.10 FEEDBACK LOOP INSTABILITY AND BODE PLOT PREDICTION 122 FIGURE 6.11 CONTROL-TO-ERROR PATH AND ESTIMATE 123 FIGURE 6.12 LOOP INSTABILITY SEARCHING BEHAVIOR 124 FIGURE 6.13 POLE MOVEMENT WITH TIME 125 FIGURE 6.14 GRADIENT AND STABILITY IN THE WEIGHT SPACE 127 FIGURE 6.15 GRADIENT DIVERGENCE FOR 4 DEGREES OF PLANT ESTIMATION ERROR 128 FIGURE 7.1 TAG CONTROL OF SELF-EXCITED PLANT 132 FIGURE 7.2 HIGH HEAT RELEASE STABILITY MAP 137 FIGURE 7.3 MSE PERFORMANCE SURFACE 138 FIGURE 7.4 SLOW AND FAST TAG CONVERGENCE 139 FIGURE 7.5 LOCAL MINIMUM AND OPTIMIZED CONVERGENCE 140 FIGURE 7.6 LOW HEAT RELEASE RIJKE TUBE TAG CONTROL 143 FIGURE 7.7 SUBHARMONIC TAG CONVERGENCE 145 FIGURE 7.8 HIGH HEAT RELEASE CONVERGENCE COMPARISON 146 FIGURE 7.9 LSU LIQUID FUEL COMBUSTOR 147 FIGURE 7.10 LSU LIQUID FUEL COMBUSTOR, TAG CONTROL 148 x

11 LIST OF TABLES TABLE 5.1 FRF AMPLITUDE AS A FUNCTION OF PROBE GAIN 90 xi

12 Chater 1 Introduction Self-excited systems are oen-loo unstable lants whose time resonse to any arbitrary inut increases exonentially until a hysical limit is reached. Once that hysical limit is reached, tyically in the form of a saturation non-linearity, the system will oscillate by entering into a stable limit cycle. Because the system is oen-loo unstable, an external inut is not the source of disturbance; the disturbance is self-generated from within the lant. Figure 1.1 illustrates a simlified block diagram of a generic self-excited system. G 1 G 2 NL FIGURE 1.1 GENERIC SELF-EXCITED SYSTEM In this case, G1 and G2 may be stable systems themselves, but when their oututs are connected to their inuts, the resulting dynamics generate at least one unstable ole. Because time resonses of hysically realizable systems do not increase exonentially forever, a nonlinearity (NL) is included to limit the loo gain. One hysical examle of a self-excited system is a thermoacoustic instability. Most recently, attemts to reduce ollutant emissions from land-based gas turbines using lean fuel/air ratios have exacerbated this instability roblem. Interaction between lightly damed acoustics and unsteady heat release rate generate the closed loo system of Figure 1.1. For relatively moderate levels of heat release, this self-excited system will exhibit nonlinear limit cycling behavior, releasing energy in the form of sound ressure. For higher levels of heat release this sound ressure can induce vibrations that can hysically damage 1

13 the combustor. It is therefore necessary to imlement control rocedures to ermit lean fuel/air ratios while reventing the instability from reaching unaccetable levels. Figure 1.2 illustrates 1 a generic control aroach where acoustic actuation and sensing are the resective outut and inut to a feedback controller H. In the case of the thermoacoustic instability illustrated in Figure 1.2, GA reresents the acoustic transfer function between unsteady heat release rate and acoustic velocity and GF is the flame dynamics transfer function whose outut is the unsteady heat release rate. This feedback loo can become unstable and is limited by a non-linearity (NL) deending on combustor oerating conditions. G A NL H G F FIGURE 1.2 GENERIC THERMOACOUSTIC INSTABILITY CONTROLLER IMPLEMENTATION To the controls engineer it will be clear that limit cycle oscillations cannot be globally suressed without using some form of feedback control. Feedforward control can only modify the zeros of a lant. In the case of the self-excited system, the limit cycling oscillations are generated from an unstable ole which must be relocated to the left half lane to reduce those oscillations. System oles can only be affected by using feedback control. Because of the wide range of oerating conditions for gas turbines, assive solutions are hard to achieve and active combustion control is a romising alternative. This research is focused on the design, analysis, and imlementation of a variety of active control aroaches for designing H in Figure 1.2 to achieve stabilizing, or attenuating control of the limit cycling lant. It is also imortant to note that the controller designs 1 Note that all summing junctions in figures throughout this document are assumed to be ositive unless otherwise noted with a negative sign on the aroriate terminal. 2

14 examined herein are strictly designed to stabilize an already unstable system. This is a significantly different roblem from the goal of keeing a stable system stable, or actively reventing an instability from occurring. The control methods, simulations, exeriments, and conclusions resented in this work do not directly address this roblem and generally ignore transients between stable and unstable combustor conditions. The control methods and analysis resented here enjoy a wider berth than the restricted alication to thermoacoustic instabilities. The thermoacoustic instability is a hysical examle of a self-excited, or oen loo unstable, system. In fact, the controller design methods resented here are also alicable to the generic self-excited system as shown in Figure 1.3. G H FIGURE 1.3 GENERIC SELF-EXCITED SYSTEM CONTROL IMPLEMENTATION Figure 1.2 is easily reresented by Figure 1.3 by closing the self-excited loo so that 1 G = 1 G G A F NL [1.1] where NL is a DC gain that can be obtained from the describing function analysis deendent uon the form of the non-linearity. Since the hot acoustics and flame dynamics are both resent in the feedback loo when the combustor is oerating, the lant of [1.1] is oen-loo unstable. 1.1 RESEARCH GOALS Classical disturbance rejection can add daming to a linear system in the form of a feedback comensator. Such a comensator is designed to have high oen loo gain between hase crossover frequencies to ensure low closed loo amlitude where required. This concet also 3

15 alies to a self-excited system, but the gain required to comletely stabilize the system cannot be known without knowing the exact (linear) location of the unstable ole of [1.1]. This is a difficult task for the combustion modeling community, and is an area of ongoing research. Further comlications in controlling the self-excited system are twofold. First, the oerating conditions or ower load of the combustor may change causing the lant dynamics to change. If this occurs, the limit cycle frequency may also change, requiring a new controller design for each oerating and load condition. Second, the resence of the feedback controller itself may cause changes in the heat release dynamics which will cause changes in the acoustics, and ultimately the system ole locations. Essentially, the static feedback controller designed for the limit-cycling system may not be a stabilizing controller solution for the system after the control loo is closed. This effect resents a very daunting modeling task where the exact behavior of the nonlinearity must be known to redict the controller s imact on the hysical system. The fixed gain feedback controller (dynamic comensator that is not continually udated) has not been found to be a ractical solution to this time varying lant. The variation in lant dynamics that occurs over a range of combustor oerating conditions revents even a robust fixed gain controller from achieving stabilizing erformance over all of those oerating conditions. The next logical ste taken by researchers has been the use of adative controllers that vary their control arameters in resonse to changes in the lant. While a relatively large number of researchers in this field have imlemented adative controllers both in simulation and exerimentally, few have areciated some of the comlexities that may be encountered during control. Some of the deficiencies in rior research and the lack of a robust algorithmic aroach to thermoacoustic instability control have motivated this research. Beginning with static feedback control, the robust imlementation issues discussed above revent it from being a viable solution to the control roblem. However, as mentioned earlier, all control aroaches that hoe to stabilize the self-excited system must close the loo using feedback control. Previous researchers have met with some success in establishing attenuating control at secific oerating conditions using hase shifting feedback controllers. However, many research efforts 4

16 described secondary eaks when the loo was closed, that revented significant control. The first goal of this research was to identify the source and solution to these secondary eaks so that stabilizing feedback control could be enabled. The second main goal of this research was to examine in detail, Filtered-X least mean squared (LMS) adative control aroaches to control of the thermoacoustic instability. The lack of an uncontrollable reference signal makes this roblem non-trivial, and necessarily feedback in nature. Previous investigations of adative control aroaches have met with varying degrees of success, with nearly all describing anomalous behavior. Because of the time varying nature of the lant, it is clear that the successful controller will adat to maintain stabilizing control. With classical adative control considered as a viable solution, it was the goal of this research to fully characterize the erformance of (feedback) LMS based controllers and rovide any alternate solutions or recommendations. In addition to solving roblems of rior research, an oortunity exists for develoing new and innovative controller designs to attenuate the limit cycle instability. Therefore the final goal of this research was to develo new and intelligent controllers for stabilizing selfexcited systems. Artificial neural networks and time averaged gradient algorithms are offered as new solutions to the active control roblem resented by the thermoacoustic instability. 1.2 CONTRIBUTIONS There have been a variety of contributions made by this research while addressing the aforementioned goals. The most significant contributions to the state-of-the-art are summarized here. As mentioned above, fixed gain feedback controller design (à la hase shifting feedback control ) was lagued by secondary eaks and the inability to achieve stabilizing control. This henomenon was more atly renamed controller induced instabilities as a result of this work. A system identification rocedure was develoed that resulted in the ability to design a feedback controller a riori that revented the formation of controller induced instabilities. While it is not ossible to ensure stabilizing control (because the gain of the nonlinearity is 5

17 unknown), a secific controller frequency resonse can be evaluated for effectiveness at attenuation while reventing additional instabilities. More effective controller designs then become aarent as a result of heuristic examination of the system identification. Building on the feedback controller and system identification technique develoed early in this work, an artificial neural network was constructed that redicted the oen loo lant dynamics based on combustor oerating conditions. The resulting oen loo frequency resonse is then used by a rule based design algorithm to continually udate the feedback controller. While the real-time imlementation demonstrated effective control over a wide range of oerating conditions, some researchers may erceive the rocess to be too cumbersome for ractical alication. In addition the ANN is not scalable to different combustor rigs without new training data sets. Nevertheless, this control aroach was not reviously investigated and it effectively solved many of the roblems lagued by manual and adative feedback controller designs. The secific method of adative feedback control examined in this work has not been addressed by rior researchers investigating self-excited system control. The redominant mode of revious adative control aroaches has been the alication of an adative IIR LMS controller. Using an aroriate lant identification technique not reviously investigated, it is shown that an FIR filter can achieve stabilizing control. A variety of stability issues associated with adative feedback control are also identified and exlained in detail. Two distinct instability mechanisms are identified: feedback loo instabilities and algorithm divergence. Both are shown to generate unexected loss of control for arbitrarily small lant estimation errors. Yet another intermittent instability mechanism is identified and exlained with resect to actuator authority and control of the self-excited system. Finally, the time averaged gradient (TAG) algorithm is alied to the self-excited lant. The TAG offers a new solution for adating both the magnitude and hase of a feedback controller without the need for an estimate of the lant. All LMS based adative aroaches, as well as neural network aroaches, rely on the ability to have an accurate estimate of the control-toerror ath. A solution is offered here that can be alied instantly on any combustor 6

18 indeendent of oerating conditions. Additional rule based arameters are also investigated to imrove the erformance of the algorithm for the general combustion lant. 1.3 DOCUMENT OVERVIEW Chater 2 summarizes the relevant literature on thermoacoustic instabilities and their control. In general this reresents the most seminal aers and most recent work in the areas of adative control, neural network control, and time averaged gradient control. There are also sufficient references resented for the reader to obtain an understanding of the thermoacoustic instability and how it can be modeled. This literature is relevant for the model used throughout this work in various simulations. Chater 3 is rimarily offered as background material to introduce the control of the thermoacoustic instability using fixed feedback controllers. However, it is imortant that the reader is familiar with the system identification aroach described in this chater since it is referred to throughout the remainder of the work in reference to both adative feedback control and the artificial neural network controller. In addition, the fixed gain feedback controller described in Chater 3 is tyically used as a erformance baseline to which other control aroaches are comared. This is also an imortant chater for the reader because it introduces the thermoacoustic instability simulation model used throughout the remainder of the text. Portions of this chater were ublished in [16] and [17]. The fourth chater describes the artificial neural network control aroach in detail. The feedback controller design emloyed by the neural network is described in detail in Chater 3. The discussion of the neural network is self contained in Chater 4, resenting the training, design and fuzzy controller udate as well as the exerimental results. The majority of this chater was taken from [19]; with the excetion of additional material exlaining the limitation of controllable oerating conditions. Aendix A rovides the real-time C-code used for imlementing the artificial neural network and associated controller design algorithm. Chater 5 resents the alication of adative feedback control to the thermoacoustic instability. The results of Chater 5 show stabilizing control using an aroriate lant 7

19 estimate roosed by this work. In addition, an intermittent searching condition is identified and shown to be due to actuator authority limitations. Although a majority of this chater has been submitted in [57], additional evidence of the searching henomenon is included here. Aendix B rovides the MATLAB simulation code that was used to demonstrate the effectiveness of the roosed identification technique for adative feedback control. The detailed stability analysis of the adative feedback controller used in Chater 5, is resented in Chater 6. Analysis of the loo instability and algorithm divergence is resented as a function of magnitude and hase error in the control-to-error ath estimate required in Filtered-X formulations. Although robust stability can never be guaranteed, a new method for a weighted online system identification is resented that can reduce some of the errors encountered in initial system identification. The majority of Chater 6 is taken from the submitted article [58]. In general, Chaters 5 and 6 can be read indeendently from each other and from the other chaters herein. Finally, Chater 7 resents the design and imlementation of the time averaged gradient control algorithm. Simulation and exerimental results are rovided that illustrate its effectiveness on a variety of latforms. Aendix C rovides a MATLAB code that can be used to simulate the TAG algorithm for control of a self-excited system. Since the majority of Chater 7 was taken from [59], it can be read indeendently from other chaters in this text. Although somewhat modular, this document resents a variety of different controller designs that are all used to accomlish the same goal: adative control of a self-excited system. The TAG algorithm is shown to rovide the most robust and effective erformance of all control methods examined. The neural network aroach rovided accetable control over a variety of oerating conditions, but requires a training rocedure that is difficult and time consuming, and is not ortable to combustors of different geometries. Adative feedback control methods utilizing the filtered-x LMS algorithm were shown to be effective at controlling a single oerating condition, but have significant stability limitations reventing stabilizing erformance over a wide range of oerating conditions. Because of inherent controller design differences, some of the chaters can be read alone, without loss of generality. However, taken as a whole, 8

20 the reader can better areciate the difficulties encountered in designing self-adative controllers for unknown self-excited systems. 9

21 Chater 2 Review of Germane Literature This chater rovides a review of the state-of-the-art in the areas of adative control and control of the self-excited combustion lant; and more secifically, the use of adative control methods in controlling the thermoacoustic instability. These areas of the literature are (and have been) very active for several years. Well known researchers continue to examine different adative techniques in hoes of finding a robust and ractical solution to the roblem of controlling the self-excited system. Because of the large amount of literature existing in these research areas, only the most relevant articles are addressed directly while others are included as references. In order to rovide the reader with an areciation of the available literature and the rogression of adative control techniques alied to thermoacoustic instabilities, a brief review of the general literature is resented first. This will address acceted ractices in combustion instability modeling and feedback control considerations used throughout this work. The remainder of the literature review is divided into two sections that searately cover the alication of adative control methods to combustion instabilities and adative control theory as it alies to this work. Throughout this discussion the deficiencies of the rior art are ointed out with resect to the rimary contributions made by this work. 2.1 COMBUSTION INSTABILITY MODELING AND CONTROL Thermoacoustic instabilities in combustors occur as a result of a self-excited feedback loo between unsteady heat release rate and acoustic ressure. For relatively moderate levels of heat release, this self-excited system will exhibit non-linear limit cycling behavior, releasing energy in the form of sound ressure. There have been a multitude of aers ublished and resentations given on develoing models of this self-excited feedback loo through couling 10

22 of an acoustic model with that of an unsteady heat release rate forcing function [1,3-7]. Most of these aroaches generally result in a reduced order model that can be used to reresent the ressure/heat release rate interaction. In [2], Annaswamy develoed a feedback model of the combustion instability rocess that was reresented by a lightly damed acoustic mode in a feedback loo with a low ass filter. With an aroriate cutoff frequency and gain, this closed loo system will generate an unstable ole which can be limited by some arbitrary saturation non-linearity. This establishes the recedent for the system model used exclusively in simulations throughout this work. As long as combustion instabilities have resented vibration and acoustic roblems, researchers have sought to control them. While both assive and active control techniques are viable solutions, the bulk of research has focused on active control due to the otential for limited sized comonents and tracking through changing oerating conditions. Dowling (among others) in [10] rovides a brief overview of the concet of feedback control to suress the instabilities encountered in the combustion rocess. The fundamental concet of feedback control (and ultimately all control methods for this system) is to aly enough negative gain at (generally) a single frequency so that the unstable ole is stabilized. Barring that, enough daming is added to at least reduce the limit cycle amlitude to accetable levels. Fixed gain feedback control has been accomlished with varying degrees of success in [13-15]. Earlier exeriments were often lagued with what some researchers called secondary eaks. From the literature, it was clear that many exerimentalists did not realize that these were actually controller induced instabilities caused by excess gain at out-of-band hase crossover oints when the controller formed a second feedback loo (the first being the unstable combustion loo). This was later exlained in detail in [16-18]. Early feedback control design techniques used a fixed (or sometimes manually adjustable) gain and a manually adjustable hase delay to drive an actuator. These hase shift controllers are actually very simle feedback controllers, using a measurement of the unsteady heat release rate or ressure fluctuation to drive the controller. More sohisticated feedback controller designs were emloyed in [11] and [12]. However, comared to an intelligent looshaing design technique as in [16], it is difficult to justify the more comlex (and inaccurate) model based 11

23 techniques for single-inut single-outut control systems. (Further background information on the design of feedback controllers for the self-excited system is rovided in the following chater). The relevance of the tye of actuator used in these control systems should be addressed. There are tyically two tyes of actuation that are used in control of combustion instability: acoustic (or air flow) actuation and fuel flow actuation. In ractice it is considered more ractical to use fuel flow actuation because fuel injectors require less drive ower and sace than acoustic actuators delivering equivalent control ower. In ractice, acoustic drivers (seakers) cannot reasonably suly the required ower to control the otentially high sound ressure levels in a relatively confined sace; acoustic ressures inside limit cycling combustors will tyically exceed 150 db SPL. There are many discussions of the alication of fuel flow modulation to the suression of combustion instabilities including [8] and [9]. However, the recise mechanism of the interaction of the modulated fuel with the rimary flame is still a toic of current and future research. Simulation of acoustic actuation versus fuel actuation only requires a change in the inut location of the control signal. This will not significantly affect the controller designs or their erformances as resented in this work. The contributions of this work include the develoment of new controller designs and analyses that will be useful in the control of combustion instabilities. While actuator ower and ractical imlementation of that actuator is always a consideration in control system design, that is not a focus of this research. The simulations used to test the control techniques assume acoustic actuation. Although the majority of the exeriments described in this work use an acoustic actuator couled directly to the combustion chamber, a number of tests have been erformed using a fuel injector as the actuator. The erformance characteristics of the various control strategies that have been develoed should not vary as a function of the actuator, since all are designed to reduce the ressure to an (ideally) stable oerating oint. 12

24 2.2 APPLICATION OF ADAPTIVE CONTROL TO THE SELF- EXCITED COMBUSTION INSTABILITY Within this work there are three general adative aroaches examined: artificial neural network based feedback control, adative filtered-x based control methods, and time averaged gradient control methods. Therefore, there are three main areas of the literature that are relevant to the resective techniques investigated. It is interesting to note that very little rior work has been done in either the area of neural network control of combustion instabilities or the use of time averaged gradient control. In fact, these are the two methods that hold the most romise for roviding reeatable and comrehensive control of the self-excited system NEURAL NETWORKS IN COMBUSTION CONTROL Blonbou and Laverdant in [20] rovide the only notable alication of a neural network toward the control of combustion instabilities. Their discussion is not detailed enough to fully understand the exact imlementation of the multi-network aroach. It aears that one neural network is used as a model of the lant transfer function while the other is used to rovide a control outut. The control results aear to stabilize the combustor over a variety of oerating conditions. The secific structure of Blonbou s alication of a neural network is very different than the structure resented herein. This research resents a single neural network used to generate a comlex valued frequency resonse function reresenting the unstable combustor FRF. This is then used with a rule based algorithm that designs a feedback controller using a looshaing technique [19]. It is interesting to note that the results from the technique of Blonbou and the neural network alication resented here, were made ublic at recisely the same time in two different arts of Germany. The erformance characteristics of stabilizing control over a variety of oerating conditions are consistent between both methods of controller imlementation; this is a characteristic that is not tyically seen in other tyes of adative control LMS BASED ADAPTIVE CONTROL IN COMBUSTION CONTROL Given the oularity and ease of imlementation of Widrow s LMS adative algorithm [49], it is not surrising that LMS based control methods are most commonly alied to combustion 13

25 instabilities. Many times however, these algorithms are alied blindly without regard to the alication because it is simly assumed that adative feedforward control is synonymous with stable control. This section rovides references and brief discussions of the rior art in the area of alying adative control to the unstable combustion lant. All of these aroaches are based on a variation of the commonly known filtered-x LMS algorithm, called the filtered- U LMS algorithm (or adative IIR filtering) [48]. Billoud s [22] alication of the filtered-u algorithm to the combustion instability roblem is referenced by most adative imlementations that follow it in the literature. Kuo in [48] rovides a discussion of the alication of the filtered-u algorithm used by Billoud to noise control roblems. He states that because of the inherent ability of the adatation scheme to adat the oles of the filter, stability cannot be guaranteed. In addition, because of the lack of ersistent excitation and the diminishing reference signal due to feedback interaction, it cannot be stated that the filters aroach their otimum values. Given this, it is not surrising that Billoud finds divergent behavior for some combustion control exeriments using the filtered-u LMS. Adding to the uncertainty of his alication, he does not attemt to find an accurate system identification, required by the adative filter to ensure stable convergence. Instead he uses a gain and hase delay to reresent the control-to-error dynamics. (This is also the case with [30]). The method of adative control examined in the resent research is shown to be convergent and stable for cases where no error exists in the lant estimation. It differs from the filtered-u because the reference signal is derived directly from removing the influence of the control signal from the error signal. As the lant estimate aroaches the actual lant, this is identical to urely feedforward control. Other researchers following Billoud s work have incrementally added features to the filtered-u aroach. Kemal in [25] resented very similar results to that of Billoud using almost the same control architecture. He comares the IIR erformance to that of a simle FIR imlementation in a urely feedback system. Again it is not surrising to find that neither controller worked admirably (IIR because of adating oles and FIR because the feedback loo was not accounted for). Further advancements in the automation of the system identification rocess have been discussed in [27-29]. The results of these studies are mixed: 14

26 some are model based simulations giving excetional control and others are ractical imlementations exhibiting some amount of controller induced instability. There are inherent roblems in this articular control aroach that are either ignored or acceted by each of the studies: Potential instability of feedback loos created when control system is engaged Inherent instability of adative IIR filter from adatation of oles Sensitivity to the error in the lant estimate Inaroriateness of online ID of a closed loo system when only the control-to-error ath is required Each of these issues are addressed in this work. A notable class of adative filtering techniques has dearted slightly from the standard filtered- X LMS based aroaches. Self tuning regulators as discussed in [24-26] are model based control aroaches that attemt to adat to a stable control solution using certain model arameters of the combustion rocess. As the name imlies, these are model based aroaches and hold little ractical aeal until combustor models accurately redict thermoacoustic instabilities. Several exerimentally motivated researchers have reorted varying results of controlling combustion instabilities using adative controllers [30-33]. These ublications are rimarily focused on reorting the results of tests on actual combustion systems rather than exlaining the details of the controller design. In [30], Mohanraj and Zinn use the same adative IIR structure roosed by Billoud, using a tunable delay as the lant estimate. They also reinforce statements made by Billoud concerning the inability of an FIR filter to control the combustion lant, but neither author discusses why this is the case. In fact, the FIR filter can be (and in this work is) used to stabilize the combustion lant if roer conditioning of the error signal occurs. In [31] Ziada and Graf use a commercially available black box feedforward controller from Digisonix to stabilize a household burner instability. They have no discussion of the details of the controller design but it exhibits excetional results, clearly stabilizing the 15

27 lant. Finally, Johnson et. al. in [33] resents results of instability control using an adative controller and real time observer that is stated to be rorietary. Two statements clearly differentiate this aroach from the gradient search (to which it most closely seems to relate): A first ste towards the develoment of an adative scheme was an online identification rocess that utilized the observer to determine the otimal control hase in an already unstable combustor and Future develoment of the adative algorithm will include adatation of the control gain in addition to the control hase. It is clear that they are not adating both gain and hase and that they still require a lant estimate. The most recent literature in the area of alying adative control to the unstable combustion lant continues to ignore some critical elements of erformance. Most researchers focus on the alication of the filtered-u LMS algorithm (an adative IIR filter) to the combustion lant. Some ublications describe effective model based control aroaches, but rely on inherent knowledge of the combustion rocess. All researchers in this area seem to have ignored several key elements that are contributions made by this work: The feedback loos generated from closing the loo with a controller can become unstable during adatation, indeendent of the LMS algorithm convergence. An adative feedback aroach using a measured estimate of the lant can stabilize the combustion system using an FIR adative filter. No one has alied this aroach to the combustion lant. The system identification that is tyically carried out can result in large errors between the estimated lant and the actual control-to-error ath. These errors are significant in both ensuring algorithm convergence over a wide band and reducing the effect of otentially unstable feedback loos. On-line system identification during closed loo control, using the filtered-u aroach is generating an incorrect reresentation of the lant dynamics. Stable convergence using the filtered-x based adative feedback controller cannot be guaranteed because of the inability to accurately know the actual control-to-error dynamics for the closed loo system. 16

28 2.2.3 GRADIENT BASED ADAPTIVE METHODS IN COMBUSTION CONTROL The time averaged gradient aroach develoed as art of this work ermits adatation of both magnitude and hase without the need for an estimate of the lant dynamics. It is a simle algorithm based on erturbing weights and measuring the resulting gradient. There is only one instance in the combustion control literature where a similar algorithm has been develoed. Presented in May 2000, Banaszuk et. al. in [34] discussed results of a triangular search algorithm as a self-exciting adatation algorithm, not requiring external robing as in extremum seeking controllers. They reference the descrition of the algorithm to an article by Zhang [36]. The triangular search algorithm udates the control signal directly and requires no external robing to identify the direction of the gradient. The algorithm is not designed to alter different feedback controller arameters because it udates the control signal and not a control filter; this limits its alicability to multivariable and arameterized control. Similar to extremum control, this algorithm has not yet been altered to adat both hase and magnitude. The time averaged gradient algorithm resented here accomlishes a similar goal of minimizing the error without a lant estimate, but can do so through a variety of arameterizations and control design structures. 2.3 ADAPTIVE CONTROL LITERATURE Distinct from the alication based literature, the stability and erformance characteristics of adative controllers in general have been studied. However, as rigorous as many of these ublications can get, only one mentions the otential instabilities associated with feedback loos during adatation of adative controllers alied to feedback systems. Leboucher et.al. in [53] describes the existence of feedback loo instabilities for filtered-u adative controllers. However, she attemts to rovide a well-defined measure to analyze the stability of any system as a function of the adative arameters. The difficulty here is that all systems are different and loo stability cannot be categorized by a uniform index. Most studies rovide varying aroaches to examine the effect of choosing the aroriate convergence arameter on fast adatation, or the effect of model errors in simle feedforward arrangements. 17

29 In [37], Feintuch resents the first adative recursive filter structure in the literature. The stability of the IIR adatation is not addressed until later ublications such as [42] and [51]. (Other stability analyses and algorithm modifications of the filtered-u aroach exist in the literature but are not resented here). There are multile analyses available including [39-41] that resent the commonly known constraint on control-to-error ath identification error of ninety degrees. Each derive the solution using different methods but come to the same conclusion. However, the 90 degree stability criteria has only been roven for sinusoid inuts and secial samling constraints. Broadband algorithm stability has not been shown to adhere the oular 90 degree rule although simulations and exeriments continue to suort it. Because each aroach focuses on the standard filtered-x form of the LMS algorithm, no feedback loos are resent. Therefore, in the urely feedforward sense, only the algorithm can lead to exonentially unstable behavior. This can be caused by an error in the lant estimate transfer function (as noted above), or by an inaroriately large choice of convergence ste size. This is addressed in detail in [47] and again in [43]. Neither address the creation of feedback loo instabilities because they are dealing with the adative filter erformance itself rather than the controlled system. A convergence analysis of the filtered-u algorithm is resented in [42]. This holds little relevance to the current work because the assumtions made by the analysis revent ractical alication to non-minimum hase systems. Research involving the time averaged gradient algorithm and its erformance on dynamic systems is limited. The most significant contributions are by Clark et. al. in [52], Gibbs and Clark in [44], and Kewley et. al. in [45]. The rimary focus is on the descrition of the algorithm itself and its alication to harmonic control. It is exlained in [45] that the desired erturbation size is a function of the desired mean squared error, since a erturbation from the otimal generates a time varying error away from the otimal. In addition, the erturbation must be large enough to rovide an accurate measure of the gradient. These are rimarily heuristic arguments because the uncertainty of a measurement cannot be known a riori for a hysical system. A new gradient based algorithm is discussed in [55]. Presented as a least 18

30 squares gradient aroach to adatation, the algorithm is nearly identical to the time averaged gradient aroach based on the method of steeest descent. The literature resented in this section covered the state-of-the-art in adative control of thermoacoustic instabilities. It is aarent that there are still contributions to be made in the analysis and alication of alternative, and otentially more useful, adative control strategies. The neural network control strategy resented herein has not been discussed rior to [19], ublished by this author. The otential for erformance has been confirmed simultaneously and indeendently using an alternative aroach in [20]. The filtered-u algorithm holds distinct disadvantages in control because of the adatation of otentially unstable oles. Alication of this to an unstable or lightly damed lant enhances the ossibility of an unstable ole being generated. Nevertheless, combustion engineers continue to aly it to the combustion instability roblem with widely varying results. The adative feedback aroach resented here offers a more direct alication of the filtered-x concet that can be analyzed in terms of the lant estimation error. Nevertheless, it has distinct disadvantages discussed in this work that limit it s otential for ractical imlementation. Time averaged gradient descent aroaches may be the next logical ste for combustion instability control, since they do not require a lant estimate and are easy to imlement. To date, this aroach has not been discussed in the combustion control literature. 19

31 Chater 3 Feedback Control of a Thermoacoustic Instability Static feedback control of thermoacoustic instabilities (also referred to as hase shifting control ) is a common choice among researchers for demonstrating effective control over the self-excited system. As a result, some of the material resented in this chater is considered to be rior art, though unique contributions have been made in the areas of system identification and redictable and robust design of feedback controllers for the thermoacoustic instability. In addition to the new contributions, the motivating factors for the inclusion of this background material are threefold: This section attemts to familiarize the reader with the most rudimentary form of control that is tyically emloyed in self-excited system control. The fixed gain feedback control technique serves as a reference oint for measuring the relative erformance and effectiveness of the various adative methods resented in subsequent chaters. The effective design of a feedback controller for the self-excited system must be well understood before a) the neural network design can be areciated and b) the loo stability analysis of the adative feedback controller can be exlained. Feedback control system design, for general single-in-single-out (SISO) alications, requires accurate knowledge of the loo transfer function. Active combustion control design is usually imlemented using such SISO architectures, but is quite challenging because the thermoacoustic resonse results from a relatively unknown, self-excited system. In addition there are non-linear rocesses that must be understood before learning the gain/hase relationshi of the system recisely at the instability frequency. However, exeriments and analysis resented here have shown that it is ossible to obtain accurate measurements of the 20

32 relevant loo frequency resonse functions at frequencies adjacent to the instability frequency. Using a simle tube combustor, oerating with a remixed, gaseous, burner-stabilized flame, these loo frequency resonse measurements have been used to develo a methodology that leads to test-based redictions of the absolute hase settings and best gain settings for a roortional, hase-shifting controller commanding an acoustic actuator in the combustor. The contributions of this methodology are twofold. First, manually searching for the required hase setting of the controller is no longer necessary. In fact, this technique allows the exact value of controller hase to be determined without running the controller. To the author s knowledge, this has not been reviously reorted in the literature. Second, the best gain setting of the controller, based on this new design aroach, can be defined as one that eliminates or reduces the limit cycle amlitude as much as ossible without generating controller-induced instabilities. Controller induced instabilities result from excessive gain alied by the controller at oen loo hase crossover frequencies resulting in amlification in the closed loo system. Excessive gain can cause this amlification to become unstable. It is shown that the tradeoff in limit cycle suression and avoidance of controller-induced instabilities is actually the well-known tradeoff in the sensitivity/comlementary sensitivity function for feedback control solutions. 3.1 LIMIT CYCLE CONTROL AND SYSTEM STABILITY Determining the stability of a hysical system can be difficult in ractice. Without an accurate model, it cannot be stated with any certainty whether a system is unstable, or very lightly damed. The time and frequency resonses between a lightly damed (stable) comlex ole and a linearly unstable, limit-cycling instability can look identical. Whether a system is limit cycling or oscillating due to lightly damed dynamics, does not govern the decision to aly control. Instead, an accurate measure of a thermoacoustic system requiring control is based on an unaccetable noise level or vibration level generated from a articular oerating condition. In general, this work has assumed that an unstable lant exists when harmonics of a fundamental frequency aear in the ressure sectrum. In fact, this is only an indication of a non-linearity, and still does not confirm the exact linear system ole location. 21

33 Ultimately, the question of stability is not critical if the system requires control. The controller designs resented throughout this work are not strictly deendent on whether the system is unstable or simly lightly damed. This can be more easily understood by considering two linear systems. The first has a stable comlex ole in the left half lane, but very near the imaginary axis, that dominates the time resonse and generates unaccetable acoustic or vibration levels. The second (linear) system has a comlex ole in the right half lane. A nonlinearity exists in this system such that the time resonse will not increase without bound. We can assume that this nonlinearity limits the oscillatory resonse to the same magnitude as that of the lightly damed stable system. The oen loo dynamics of these two systems are illustrated below in the comlex lane Imag. Imag. X X Real Real FIGURE 3.1 STABLE AND UNSTABLE SYSTEM DYNAMICS The time resonses and ower sectra of the two systems (after a brief transient) will aear almost identical with the ossible excetion of harmonics in the limit cycling system. Now consider a rudimentary controller design for the stable system that has the goal of reducing the daming, and thus the oscillation amlitude. Such a controller might have a comlex zero in the left half lane to generate a more heavily damed closed loo ole. The same controller design alied to the unstable system (with the roer gain adjustment) will stabilize the unstable ole and create the desired daming on the closed loo ole. The root locus of each system is shown below for clarification of these oints. (Note that two real oles were added to generate a causal controller and only the ositive frequencies are illustrated). 22

34 Imag. Imag. O X O X X Real X Real FIGURE 3.2 STABLE AND UNSTABLE SYSTEM CONTROL The same feedback controller design alied to a stable and unstable system can accomlish the same desired goal of adding daming and reducing the oscillation. For the thermoacoustic instability, a single comlex ole tyically dominates the time resonse and this tye of control strategy is emloyed. It is therefore irrelevant to the feedback controller design as to whether or not the system is stable or limit cycling, as long as the design goal remains: to reduce the oscillations. It should be noted that feedforward control cannot alter system dynamics. Feedforward control is tyically well suited to disturbance rejection for stable lants. It can add an out of hase signal to an existing disturbance, thereby reducing the oscillation and altering the zeros of the lant. However, strictly feedforward control is not caable of altering system oles and therefore is not well suited for adding daming to either stable or unstable systems where the disturbance is generated from within the lant. As will be shown, all ractical imlementations of adative schemes in the control of thermoacoustic instabilities are necessarily feedback in nature. 3.2 FEEDBACK CONTROL REVIEW There are several linear control analyses that can be used to redict closed-loo erformance from the oen-loo transfer function [50]. One very useful tool for frequency-domain feedback controller design is the frequency resonse function (FRF). The frequency resonse of the oen-loo transfer function, which includes both lant dynamics and the controller 23

35 dynamics, can be used to redict both closed-loo stability and erformance by examining the gain and hase margins. It may be successfully argued that Nyquist lot reresentations, which suly identical stability/erformance information for a feedback system, could be used for the following discussions. However, the author believes that the combustion community will benefit from the relative simlicity in understanding the FRF lots and Bode gain/hase rules versus the more challenging Nyquist diagrams. In general, for negative feedback, a designer attemts to ensure that the oen-loo gain (in log units) is less than -10dB at all hase crossover oints (±180 multiles) and the oen-loo hase is greater than 20 degrees away from the ±180 hase crossovers when the gain equals 0 db. If these criteria are not satisfied at every frequency, an increase in amlitude of the closedloo frequency resonse is redicted at those frequencies. This is sometimes termed sillover, but is more atly named controller induced instabilities in the following discussions [16]. In fact, a controller induced instability is not truly unstable, but has the otential to become unstable if the controller gain is continually increased; nevertheless this terminology has been adoted and is used throughout this work to describe increased closed loo frequency resonse amlitude due to controller dynamics. (It should be noted that this discussion also alies to ositive feedback control, requiring only a shift of 180 in the stated hase margins). For negative feedback systems, suression of the noise entering the loo is achieved when the oen-loo FRF gain is ositive (decibel levels greater than zero, i.e. greater than unity magnitude) at frequencies having 0 hase values. The more ositive the gain, the higher the suression of those frequencies in the closed-loo frequency resonse. For lants with significant amounts of hase delay, closed-loo suression will be limited to very narrow frequency regions. Fortunately, active combustion control only requires suression of a very narrow bandwidth, making the feedback controller design a relatively easy task. The most common method of designing acoustically actuated feedback controllers for combustors has been to use a bandass filter in conjunction with a manually oerated hase shifter. Clearly if the lant were known, a fixed-gain controller could be designed a riori to suress the 24

36 instability. However, until now, many researchers have used the hase shifting controller because the oen-loo frequency resonse function has been unattainable; and thus a fixedgain controller design was imossible. A measurement method that leads to rediction of the hase setting for the controller for any oerating condition is discussed following a brief descrition of the exerimental system used throughout this work. 3.3 RIJKE TUBE EXPERIMENTAL SYSTEM The system used for this study as well as most of the subsequent adative exeriments was a simle closed-oen tube combustor oerating with remixed air-methane. Figure 3.3 illustrates the test aaratus. Thermocoules T 8 T 7 T 6 T 5 T 4 T 3 T 2 T 1 P 8 P 7 P 6 P 5 P 4 P 3 P 2 P 1 Pressure transducers Flame stabilizer Premixed Methane-Air FIGURE 3.3 SCHEMATIC OF TUBE COMBUSTOR RIG Figure 3.4 shows a icture of the actual Rijke tube combustor exerimental setu. The silver cylinder on the side of the tube is the lenum to which the acoustic driver (3 seaker) attaches. This is shown in greater detail as the insert in the uer right corner of Figure

37 FIGURE 3.4 PICTURE OF RIJKE TUBE COMBUSTOR RIG A ceramic honeycomb flame stabilizer was situated in aroximately the center of the tube. The Rayleigh criteria [4] redicts interaction of the unsteady heat release rate from the flame and the acoustics of the combustor, resulting in an instability occurring for the second acoustic mode (180 Hz) of the tube. Using the same stability criteria discussed above, one could show that the oen-loo gain of the self-excited system is greater than 0 db at the hase crossover oint 0 for ositive feedback. Exerimentally, this is not easily verified since the acoustics cannot be searated from the flame dynamics without breaking the self-excited loo, ultimately 26

38 altering both indeendent transfer functions. The controller loo, however, can be broken in order to examine the oen-loo transfer function of the controlled system. The oen-loo transfer function (OLTF) block diagram is shown in Figure 3.5. In -H(s) P C 1 1-K H G H (s)g A (s) P T Out FIGURE 3.5 OPEN LOOP COMBUSTOR CONTROL SYSTEM H(s) reresents the digital feedback controller consisting of the following comonents in series: an anti-alias filter, A/D converter, digital hase delay, D/A converter, smoothing filter, ower amlifier and a seaker. The seaker is attached to a small fitting that connects directly into the tube combustor at a location just below the flame holder (Figure 3.4). The ressure outut of this actuator is considered to be PC. This controller ressure then drives the tube which is itself a closed-loo system reresented by the second block in Figure 3.5; where KHGH reresents the dynamics of the flame transfer function and GA reresents the acoustic lant. (Generally, the recise dynamics of the self-excited closed-loo FRF are still largely unknown to the accuracy required for generalized frequency domain controller design because KHGH is not known). A ressure sensor at the bottom of the tube is used to measure the outut of the ath shown in Figure 3.5, consisting of the ressure generated by the self-excited oscillations lus the action of the controller. Therefore, the lant dynamics shown in Figure 3.5 must be identified before the control design can roceed. It is shown next that the effects of the nonlinear thermoacoustic resonse are band-limited to a narrow range of frequencies sanning the instability frequency. The inut for the system identification rocess was chosen to be a sine wave for the urose of frequencyby-frequency identification. Ideally, a white noise source could be used in this case to identify all frequencies simultaneously, simlifying data collection. However, the limited ower density available from the actuated random signal and the high amlitude instability from the self- 27

39 excited combustor, revent adequate coherence between the inut and outut at multile frequencies. Many simulations of the thermoacoustic lant were also carried out to investigate various controller designs throughout this work. The general form of the simulated lant was similar in all cases. The acoustic transfer function was taken to be a lightly damed (ξ=0.02) comlex ole at aroximately 175 Hz (the second mode of the hot acoustics in the exerimental setu). The flame dynamics were reresented by a single ole low ass filter at aroximately 200 Hz. Different levels of gain were added to the loo to simulate higher and lower heat release conditions requiring more or less control authority to stabilize. The nonlinearity was aroximated by a hyerbolic tangent function. When simulated in the closed loo, deending on the loo gain, the system would reach a stable limit cycle well within one second. The controllers that were investigated in simulation (and exeriment) were not engaged until the limit cycle was established. For the remainder of this work, this simulation format will be assumed unless otherwise noted. The details of several MATLAB simulations are also shown in the Aendix. 3.4 SYSTEM IDENTIFICATION Because of the uncertainty in revious work concerning the lant to be estimated, the toic of system identification has not been addressed adequately. In [22] Billoud discusses using the Filtered-X algorithm where the lant estimate is either generated from a single tone identification near the limit cycle frequency or by using a gain and hase delay. Kemal and Bowman [23] discuss obtaining an oen loo frequency resonse but do not rovide details regarding the exact nature of the transfer function obtained. Ultimately they note that using a gain and delay aears sufficient for tonal control. Koshigoe et. al. [28] among others, have investigated using an online system identification rocedure. Because of the feedback nature of the self excited system, the online identification will necessarily include the controller and feedback dynamics. If GACT is the correct estimate (as will be shown in a subsequent chater), online identification rocedures cannot roduce the correct result after the loo has been closed. 28

40 The identification roblem is to find a stable reresentation of GACT when we have access to the inut to GACT but the only measurable outut is PT as shown in Figure 3.6. NL G A G F Multi-Tone Source G ACT P T FIGURE 3.6 SYSTEM IDENTIFICATION DIAGRAM Extinguishing the flame so that the hysical feedback loo disaears is not an otion since the hot acoustics, which are an integral art of GACT, are very different from the cold acoustics and are directly influenced by the flame temerature. Thus, the identification must take lace in the resence of the thermoacoustic limit cycle. The aroach to identifying the oen-loo lant relies on a robe signal consisting of lowamlitude sinusoids at frequencies near the limit-cycle frequency and within the assband of any bandass filters used to filter the ressure signal before control. Using a Fourier transform of the outut signal, the frequency resonse of the lant at a small number of frequencies can be determined. Using a least-squares aroach, a low-order, discrete-time model can be fit to this data and is used as the model of the stable lant. Since the oen-loo lant is in a steady limit cycle, it is not immediately clear whether such an identification aroach will roduce a reasonable linear model or whether such a model will be stable or unstable. The exerimental results have shown that very low-order linear models can account accurately for the frequency resonse data and that these models are always stable. To understand this, consider the block diagram in Figure 3.6. From a describing function analysis, the gain of the limit cycle through the static nonlinearity is such that the total gain around the loo is unity. When a robe signal is injected into the system, we exect that the 29

41 frequency resonse at the robe frequency will be that of the linear system made u of the linear blocks in the diagram and with the nonlinear block relaced by a suitable linear gain. To determine the value of this gain, consider an inut to the nonlinearity of the form x = A1 sinω 1t + A2 sin( ω 2t + θ ) [3.1] where ω1 is the frequency of the limit cycle, ω2 is the frequency of the robe and A first order aroximation of the outut of the nonlinearity, f(x), is given by [3.2]. >> A. A 1 2 f A sinω t + A sin( ω t + θ )) = f ( A sinω t) + f ( A sinω t) A sin( ω t + ) [3.2] ( θ and will be valid for A 2 sufficiently small. The gain of the nonlinearity at the limit cycle frequency is given by [3.3]. 2 A T 1 [ f ( A sinω t) f ( A sinω t) A sin( ω t θ )] sinω tdt T [3.3] 1 where T is the length of a eriod of the overall waveform. If no eriod exists, then the limit as T can be taken. By arguing that the integral of incommensurate frequencies will vanish, the second term in the integral disaears and the linear gain associated with the limit cycle frequency, which is equal to the describing function of the nonlinearity, is g 2 T = f ( A1 sinω1t)sinω tdt A T [3.4] 0 LC 1 1 The gain of the nonlinearity at the robe frequency is given by [3.5]. 30

42 2 A T 2 [ f ( A sinω t) + f ( A sinω t) A sin( ω t + θ )] sin( ω t + θ ) dt T [3.5] 0 Arguing as before, the first term in the integral will go to zero. In addition, the only art of the second term which will contribute to the integral is the constant comonent of f ( A sin ω t) times the constant comonent of A given by [3.6] sin ( ω t + θ). Thus, the gain at the robe frequency is 2 g Pr 1 T = f ( A1 sinω1t) dt T [3.6] 0 Note that the gain of the robe signal is indeendent of the amlitude and frequency of the robe signal, subject to the restriction that the amlitude of the robe is small. For the tanh nonlinearity considered here, the limit-cycle and robe gains can be comuted numerically and are shown in Figure 3.7. The gain of the robe signal is less than the gain of the limit cycle signal through the nonlinearity. Since the gain of the limit cycle frequency is just that value needed to make the closed-loo system marginally stable, the lower robe gain will cause the closed-loo system to aear stable. Since the robe gain is not a function of the robe frequency, the system identified by considering the frequency resonse at the robe frequency will aear linear and stable. 31

43 Gain through Tanh Nonlinearity Limit Cycle Gain Probe Gain Amlitude of Limit Cycle FIGURE 3.7 PROBE GAIN AND LIMIT CYCLE GAIN Given the above analysis and in view of Figure 3.7, the identified transfer function of Figure 3.6 will be PT = G G ˆ ACT = G ACT [3.7] 1 G AGF g Pr The reason that this transfer function is aroximately equal to GACT over the bandwidth of interest is because the denominator is very close to one at the robe frequencies. There are two reasons for this. First, the factor GAGFgPr is always less than one at all frequencies in the bandwidth. Secondly, GA contains lightly-damed acoustic oles that will have a high gain very near the instability frequency. The gain will be significantly lower a small distance from this frequency, where the robing is actually erformed. Thus, at the robe frequencies, the factor GAGFgPr will be significantly less than one, resulting in accurate measurements of GACT. This lant estimate is stable at all robe frequencies, thereby avoiding the issue of attemting to use an unstable system identification, or a time-delay lant, as other investigations have discussed. In addition, it aroximates GACT, which will be shown to be the desired transfer 32

44 function of the lant estimate. This system ID method also makes it ossible to design a feedback controller a riori that will aly the correct hase to the system while avoiding secondary eaks induced from the controller feedback loo [16]. This rocess is described next. 3.5 FEEDBACK CONTROLLER DESIGN The oen-loo frequency resonse function that is the redictor for controller erformance includes all comonents of the feedback loo. The examle controller considered here has been designed as a bandass filter with a lowass comonent having a cutoff frequency of 178 Hz and a high ass comonent with a cutoff frequency of 200 Hz. With overlaing cutoff frequencies, the bandass filter has a eak magnitude that is less than unity and a very narrow bandwidth. Because each of the filters have high rolloff rates of 160 db er decade, the filter dominates the magnitude of the oen-loo FRF. (This is evident by examining the shae of the frequency resonse data of Figure 3.8, which resembles a narrow bandass filter). Changes in the lant will cause the magnitude and hase to shift left and right as well as u and down. However, the general bandass shae of the controller rovides enough gain to stabilize the lant without inducing sillover at the lower and higher frequency hase crossover oints. When the oen-loo data was collected, the hase shifter had been adjusted to a value that seemed to work well in a variety of oerating conditions, including the one shown in Figure 3.8. (This exlains why the crossover oints at 0 and 360 have high gain margins as desired). Changes in the lant, however, may cause the overall magnitude to decrease or shift, or cause the hase to shift or change shae slightly. If the controls engineer has the oen-loo shae at his disosal, he can alter the gain and hase delay of the controller (and thus the oen-loo lant) to regain the desired gain and hase margins. And since the general shae of the controller and lant does not change with widely varying oerating conditions, the two sillover locations will be in the same vicinity of the FRF for most conditions (i.e. the controls engineer/search algorithm will know where to look to redict the feedback induced instabilities). In addition, it should be noted that the magnitude of the oen-loo FRF decays raidly below 150 Hz and above 200 Hz because of the bandass filtering. The gain margins 33

45 at all other hase crossover oints (both higher and lower in frequency) will be sufficiently high to ensure stability at those frequencies. The high gain of the ositive feedback controller at 180 Hz causes the unstable oles of the lant to travel into the stable left half lane. There are two comlex conjugate oles (arox. 166 Hz and 194 Hz) from the control filter (or other lant dynamics) that will move into the right half lane as gain is increased. Consequently, there is a range of gain where the unstable lant oles move into the left half lane before the stable controller oles move into the right half lane. Deending on how far into the right half lane the unstable lant ole begins, it is certain that sufficient gain exists in the feedback loo for the small scale combustor to be stabilized (at nearly any oerating condition) without inducing the sillover at the adjacent frequencies. (As the heat release increases, the unstable oles move further into the right half lane, requiring more gain to stabilize them. The gain at 180 Hz only needs to reach zero to stabilize the unstable oles, leaving sufficient gain margin at the adjacent hase crossover locations to ensure stability). However, the actuator (seaker) used in this examle cannot rovide the sound ressure level necessary to move the oles back into the left half lane for some high gain oerating conditions. This limitation is not an imortant consideration for this study (since the condition examined is controllable), and can be remedied by a more owerful actuator. This discussion was intended to motivate the Bode gain/hase relationshi from a root locus ersective. The added dynamics from the controller induced instabilities for high gain at two locations adjacent to the self excited instability. These locations are redicted from the oen-loo transfer function. Once the oen-loo FRF is determined, a series of rules are examined that rovide a deterministic measure of the redicted stability of the closed-loo system. These rules are summarized below using several illustrations of the oen-loo transfer function to assist in the exlanation. 1. First, the hase crossover frequency oints for ositive feedback (0 and -360 ) are determined from the oen-loo hase. In this examle they are found to be 166 Hz and 194 Hz as shown by dotted vertical traces in the hase lot of Figure

46 2. Next, the gain margin at these frequencies is determined by examining the oen-loo gain value relative to 0 db. Referring again to Figure 3.8, the gain margin at 166 Hz is aroximately 20 db and the gain margin at 194 Hz is 24 db. This means that the gain can be raised by 20 db before an instability occurs at 166 Hz. (Since the gain margin is less at 166 Hz, the gain margin at 194 Hz is ignored). 3. A secific gain margin must be established to meet the desired closed-loo design requirements. A gain margin of 10 db ensures that eak values of feedback induced instabilities are limited to 3 db. Therefore, the gain that can be alied to this system is limited to 10 db to satisfy this design requirement Mag. (db) FIGURE 3.8 FEEDBACK CONTROLLER DESIGN - MAGNITUDE 4. The gain is then alied to the oen-loo transfer function, shifting the magnitude u or down to satisfy the design criteria. Figure 3.9 illustrates this new FRF. 5. The hase margin determination requires a slight modification from the traditional definition due to the limited data set. Tyically, the hase where the oen-loo gain is zero is secified as the hase margin. In this case it should be obvious that the data does not afford the luxury of gain crossover oints (which might occur at 175 Hz and 35

47 183 Hz through extraolation). Therefore, instead of using 0 db, the more conservative value of -10 db (the gain margin) is chosen as the critical gain crossover oint. The frequencies where this occurs are 166 Hz where the hase is 0 (as exected) and 190 Hz where the hase is These oints corresond to hase margins of zero and 35 degrees resectively Mag. (db) Phase Freq. (Hz) FIGURE 3.9 FEEDBACK CONTROLLER DESIGN - PHASE 6. Stability has been ensured by the gain margin calculation. Otimum erformance in terms of the two sillover eaks can be created by adjusting the hase margins to be identical at the gain crossover oints. A simle way to do this is to average the two calculated hase margins and aly the necessary digital delay. The number of digital delay samles needed deends on the samle rate and frequency of interest, and will increase with increasing frequency. Therefore the comuted value will not exactly rovide the correct delay for both hase margins, although the error will be quite small since the bandwidth is so narrow. One final source of error in this comutation will occur as a result of the discrete data oints. Interolation between the data oints is quite time consuming for each gain margin and hase margin calculation, and is not necessary to ensure aroriate accuracy. For this data set, the 36

48 largest jum between magnitude values is 5 db and the largest hase jum is 15. The desired gain margins and hase margins can be easily adjusted to accommodate this uncertainty by recalculating the actual gain and hase margins after the gain and hase delay have been alied to the oen-loo data. The design rocedure resented above can be used in conjunction with Figure 3.10 to create a feedback controller design that will be stable. In addition, the feedback induced instabilities can be limited to a reset value (determined from Figure 3.10), while ensuring that both instabilities will exhibit equal amlitude amlification. Figure 3.10 shows the redicted closedloo magnitude (in db) as a function of hase margin and gain margin for ositive feedback systems. Note that at a hase margin of 0, the gain margin must be greater than 10dB to ensure less than 3dB of closed-loo amlification. This lot can be used to establish the desired design criteria used in Ste 3 of the design rocess described above Closed Loo Magnitude Phase Margin Gain Margin FIGURE 3.10 SPILLOVER AMPLITUDE PREDICTION Given that these margins are equally saced about the instability, maximum control can be ensured at the instability for a given value of gain since the hase at that frequency is otimal for ositive feedback control (i.e ). It should be noted that careful choice of the bandass shae was made to ensure that the most control of the instability could be achieved. 37

49 This design aroach will certainly work for any oen-loo transfer function/fixed gain controller shae; however, other fixed-gain controller shaes (such as low ass or high ass) often have margins that cause the feedback induced instabilities to become high in amlitude before achieving areciable control of the combustor instability. This frequency domain based design rocess will reveal to combustion/controls engineers that the frequency resonse shae of the hase shift controller and oen-loo lant is critical to ensuring good erformance. Using the narrow bandass filter controller shae (very simlistic, but high order filters) and the design rocedure resented above with the oen-loo FRF of Figure 3.8, the controller design shown in Figure 3.11 results. The unmodified oen-loo FRF (lant and controller) is shown as circles in Figure By first selecting the closest gain margin (164 Hz) from both hase crossover frequencies, the gain needed to just satisfy the desired gain margin of 10 db was found to be The trace shown as + signs illustrates the FRF after this amount of gain is alied. Next, the new hase margins are determined and the hase is adjusted through a digital delay so that both gain crossover oints have the same hase margin. The x signs indicate the oen-loo FRF after it has been modified by both the gain and the hase. With the same hase at both gain crossover oints (at -10 db) the amlitudes of the controller induced instabilities as these oints will be the same. In addition, since the gain is less than -10 db at the hase crossover oints, no more than 3 db of added amlitude will occur. This design results in stabilizing control of the thermoacoustic instability as will be shown in the following section. Furthermore, continual udate of the controller gain and hase using the above rocedure was shown to rovide stabilizing control over a variety of oerating conditions as discussed in the next chater. 38

50 0 os regular, +s gain, xs both 5 10 Mag.(dB) Phase Freq.(Hz) FIGURE 3.11 FEEDBACK CONTROLLER DESIGN RESULT 3.6 TYPICAL EXPERIMENTAL RESULTS As motivated earlier, the inclusion of this material is intended to rovide a baseline of erformance for fixed gain feedback controllers from which other control methods can be referenced. The Rijke tube combustor rig described above was used most frequently for exerimental testing of the algorithms develoed herein. Although it is considered by many to be an imractical exeriment (due rimarily to the acoustic actuation method) it is effective for examining the erformance characteristics for various control algorithms. In general, the erformance of the control method itself should be indeendent of the actuator, and scalable to any self-excited system. Power sectrum data was tyically collected to establish the amount of attenuation achieved by a tyical control design. In general, if a ower sectrum is shown without an accomanying time trace, the system exhibited the illustrated behavior for all time. However, as will be shown for a variety of aroaches, transient searching behavior was aarent in many control exeriments. These data are tyically reresented with a time trace of aroriate duration to characterize the behavior. 39

51 Figure 3.12 illustrates tyical behavior of the tuned feedback controller described in the revious section. The dotted trace shows the uncontrolled ower sectrum of the limit cycling combustor described above. The solid trace shows a closed loo feedback controlled system where the gain has been manually otimized to revent secondary eaks. The third trace illustrates the same feedback controller where too much gain and has been alied, generating two eaks at the hase crossovers on either side of the instability. It should be noted that using the design methodology described above (and ultimately automated by the neural network control design), the frequency and amlitude of these secondary eaks is redicted, and revented Mag. (dbvrms) Freq.(Hz) FIGURE 3.12 FEEDBACK CONTROLLER PERFORMANCE (DASHED-UNCONTROLLED, SOLID- CONTROLLED, DOTTED-EXCESS CONTROLLED GAIN) For low heat release (low gain in the combustion instability loo) this is tyical erformance that does not vary over time. The question of stable versus unstable and linear versus nonlinear arises again when examining Figure It is aarent that the control is being exercised on a nonlinear lant. Daming is only being added at a few Hertz surrounding the limit cycle frequency, but we see significant suression over the entire bandwidth. The amount of gain alied by the controller may be used to determine if the suression of that 40

52 instability is a linear function of the amount of daming being alied. While this is not the focus of this work, it is an interesting rosect for future contributions to this field. For high heat release conditions (where the unstable ole is resumably further into the right half lane, thus requiring more control gain), stabilizing control is not so readily achieved. Figure 3.13 illustrates a time trace exhibiting a searching behavior after initiation of the fixed gain feedback controller Voltage Time (s) FIGURE 3.13 FEEDBACK CONTROLLER SEARCHING BEHAVIOR This searching behavior at higher heat releases is a recurring theme resent not only in this fixed gain controller, but also in the adative controllers, although in the TAG controller scheme to a lesser extent. The source of this behavior is analyzed and exlained in detail in Chater REVIEW In this chater fixed gain feedback control of combustion instabilities was resented. The significant contributions of this work focus on the system identification method and analytical techniques for designing fixed gain feedback controllers for the thermoacoustic instability. 41

53 These two concets are carried throughout this work. The system identification method is an integral art of the neural network controller design as well as the key for develoing an accurate lant estimate for use in Filtered-X based adative controllers. The fixed gain feedback controller is used as art of the neural network control method discussed next and is also used as a baseline for comarison to the adative control methods. Although similar fixed gain control aroaches have been used in the ast by many researchers, detailed discussion of the combustion system identification and feedback controller design has not been reviously addressed. 42

54 Chater 4 Artificial Neural Network Control As with each of the adative control techniques resented in this work, the motivating factor behind using a neural network is to rovide a suressing or stabilizing control that automatically tracks changes in the lant. Changes in the lant frequency resonse resulting from changes in oerating conditions or the inclusion of the controller itself, may render nonadative controller designs useless. A neural network has been designed to redict those changes and rovide the redicted frequency resonse to a rule based looshaing controller design algorithm. During the course of designing the neural network and feedback controller algorithm, it became aarent that a secondary benefit would result from this aroach. As discussed earlier, rior art has not successfully redicted the severity, location, or even existence of controller induced instabilities. However, the controller design algorithm imlemented here as art of the neural network control aroach, requires that limits are laced on the gain of the resulting controller design to revent controller induced instabilities. Therefore the resulting design not only tracks changes in the lant and modifies the controller accordingly, but also revents controller induced instabilities from forming by generating an otimal gain for given design constraints and controller shae. This chater describes how a neural network was trained to redict the frequency resonse of the thermoacoustics in the Rijke tube combustor and how that neural network outut was used by the feedback controller design rocess to control combustion instabilities. The neural network training was erformed using comlex valued, oen-loo frequency resonse function data as the desired signal with the oerating conditions and sarse temerature rofile as the 43

55 inut signals 2. Once the neural network was trained, it was used to redict the aroximate hase and gain margins as a function of temerature and flow conditions. The margins were then used to automatically udate and design a fixed shae feedback controller having the roer hase and magnitude to ensure stability and control in the resence of changing oerating conditions. 4.1 NEURAL NETWORK TRAINING DATA In order for the network to effectively interolate, it was imortant to choose inuts that affect the desired outut. In this case the desired outut is the comlex valued oen-loo frequency resonse function (OLFRF) at frequencies surrounding the limit cycle frequency. It should be clear that one variable that will obviously affect the acoustic comonent of the thermoacoustic lant is the temerature in the tube (due to the deendence of acoustic ressure on temerature). Four temeratures in the tube were used to generate a rofile for each oerating condition tested. However, reliminary testing indicated that the variation in temerature was not enough to exlain the limitations found in the feedback control erformance at various oerating conditions. For this reason, flow rate and equivalence ratio were added as inuts since they reresent the only user controllable comonents of this combustion rocess. A range of tyical flow rates and equivalence ratios was chosen in an attemt to san a small sace of combustor oerating conditions. Eleven sets of oerating conditions were chosen using three equivalence ratios from 0.47 to 0.82 and four total volume flow rates from 109 cc/sec to 145 cc/sec. All combinations of these conditions were tested (flow rates : 110, 120, 130, and 140 cc/sec and equivalence ratios: 0.47, 0.6, and 0.8) with one excetion at 110 cc/sec and 0.47 where the flame could not be sustained. At each of these eleven conditions, four temerature measurements and 40 sine dwell frequency resonse values were recorded (comlex values at 20 frequency oints). Figure 4.1 illustrates the eleven frequency resonse magnitude data sets used in training the neural network. 2 The oen loo data was collected for a narrow frequency range surrounding the limit cycle instability by erforming a sine dwell at discrete frequencies. The argument in favor of this aroach to lant identification was resented in the revious chater. 44

56 Magnitude Data Set Frequency (Hz) FIGURE 4.1 ALL TRAINING DATA SETS The ga in the continuous FRF reresents the locations where the limit cycle amlitude revented accurate system identification. Nevertheless, mental interolation indicates that a lightly damed comlex ole exists at 180 Hz. This agrees with the exected stabilizing action of the controller when the gain is raised at this frequency. In addition, we see that the resulting identifications are stable systems, which is commensurate with the system identification discussion in the revious chater. One final note regarding the utility of this choice of inuts and the neural network should be made. Once the feedback control loo is closed, there is no efficient method for obtaining the oen-loo frequency resonse. This is critical because the self-excited system dynamics have been known to change under the action of the controller. When this haens, a fixed-gain controller is not udated and may become unstable. Therefore the controller that was designed for a single oerating condition, may not even work for that oerating condition because its own resence influences the lant enough so that the design is no longer valid 3. While the flow rate and equivalence ratio are not affected by the controller, the temerature rofile inside the tube is. Therefore, by using temerature as an inut to the neural network, 3 This henomenon is exlained in further detail at the end of Chater 5. 45

57 the estimate of the OLFRF can be udated even when the control loo is closed. Any affect that the controller itself may have on the oen loo transfer function (and thus the temerature), can be reresented and redicted by the neural net, which in turn allows the controller to be udated in real time. This is an imortant observation made by this work. Other on-line ID aroaches cannot redict the oen loo frequency resonse during control because the control loo is always closed during such rocedures. 4.2 NEURAL NETWORK DESIGN AND TRAINING Many iterations and modifications to the network were investigated before the final design was selected. These variations included: one and two hidden-layer networks, biolar sigmoid and hyerbolic tangent squashing functions, static and adative udate arameters, and comlexvalued and real-valued backroagation algorithms. Without examining the details of all design iterations, the final design is resented here with training results. Once the eleven frequency resonse functions and inut data vectors were collected, they were converted into MATLAB formatted arrays for use in designing and training the neural network. The frequency resonse data (desired network outut) consisted of 20 comlex values, each comonent having a value less than one. This is imortant to note because no scaling of the network outut was required to avoid saturation of the squashing functions whose maximum values were set at unity. The network inuts, however, did require scaling. The thermocoule measurements of temerature ranged from 106 F to 604 F. These four inuts received a scaling of 1/610 before entering the network. In an effort to maximize the dynamic range of the squashing function (and ultimately the resolution of the DSP), the flow rate received a different scaling. The minimum value of the flow rate inut was 109 cc/sec and the maximum value was 145 cc/sec. The scaling chosen for the flow rate variable was 1/150. The final inut (the equivalence ratio) was not scaled, since its values ranged from 0.47 to The best solution for the converged neural network was obtained from a network using a fixed udate arameter of 0.95 for both layers, a comlex-valued backroagation algorithm, and the biolar sigmoid squashing function that did not take into account any secial handling for comlex-valued numbers. The converged weights for this system were 46

58 determined from the training rocess conducted using original MATLAB. MATLAB facilitates the handling of a comlex-valued number as a single array element, whereas C rogramming requires that the real and imaginary comonents be held as searate values. Realizing the otential difficulty for DSP imlementation and the excessive comutation required for comlex multilication s of four array elements, the network was redesigned to accommodate real and imaginary comonents simultaneously and comletely indeendently. The resulting network can be thought of in three dimensions where the layers are reresented from left to right, the nodes aear vertically in each layer, and one network sits on to of another into the age; one side is used for real comonents and one for imaginary comonents. This is shown grahically in Figure 4.2. Network Inuts Imaginary Outut Real Outut Inut Layer Nodes Hidden Layer Nodes Outut Layer Nodes FIGURE 4.2 COMPLEX NETWORK The convergence behavior for the comlex backroagation algorithm versus the real backroagation used searately for real and imaginary values, rovided almost identical results. Clearly, the synatic weights converged to different values because the behavior of the squashing function changes drastically for real-valued versus comlex-valued inuts. The network configuration shown in Figure 6 essentially doubles the number of weights used but simlifies the task of imlementing the comlex network in C code. The forty oututs of the network are then combined to form the twenty comlex numbers reresenting the comlex FRF. From that, magnitude and hase can be easily determined. Figure 4.3 illustrates the eoch convergence of the network shown in Figure 4.2 for the maximum error over each eoch for 1,000 eochs. (i.e. the maximum error over all training data sets for each eoch). It is clear that by the 800 th eoch, there is little imrovement in the error 47

59 and the network has converged. This lot reresents the sum of the maximum error of real and imaginary comonents. The maximum error of every other training data set and oint was, in fact, well below 0.1, aroaching zero Maximum Error Value Eoch*Training Data Set FIGURE 4.3 MAX ERROR OVER EPOCH Once the network converged, the weights were saved and used to check the erformance before imlementing the network on the DSP. As mentioned earlier, a data set was collected that did not match any of the training data, and was used for checking the erformance of the converged neural network. Figure 4.4 illustrates the actual value of the magnitude and hase of the oen-loo transfer function data as comared to the neural network outut. The actual data is shown as o and the neural network outut is shown as +. The errors are well within the tolerance needed to erform an accurate feedback controller design for these oerating conditions. 48

60 Mag (db) Phase (Rad) Freq. (Hz) FIGURE 4.4 NEURAL NETWORK PERFORMANCE After the network convergence was tested, the converged weights were saved into arrays. The first ste in imlementing the network in real time was to create a C code (included in the aendix) that simly did the necessary matrix multilication and nonlinear squashing that constitutes the forward ass of the neural network. The inuts were designed to be collected either externally from the A/D or inutted manually from a grahical user interface. The only remaining task for imlementing this in terms of an adatable feedback controller is to now use the outut of the neural network (the oen loo frequency resonse function) and redesign the fixed-gain feedback controller using the feedback controller design rocess described earlier. 4.3 FEEDBACK CONTROLLER: ADAPTIVE UPDATE The oen-loo frequency resonse function that is the neural network outut and the redictor for control includes all comonents of the feedback loo. The controller shae itself has been designed as a bandass filter with a lowass comonent having a cutoff frequency of 178 Hz and a high ass comonent with a cutoff frequency of 200 Hz. With overlaing cutoff frequencies, the bandass filter has a eak magnitude that is less than unity with a very narrow bandwidth. Because each of the filters have high rolloff rates of 160 db er decade, 49

61 the filter becomes the dominant magnitude comonent in the OLFRF. (This is evident by examining the shae of Figure 4.4 which resembles a narrow bandass filter). Although, all of the dynamics of the combustion system are still reresented by the frequency resonse data collected (the oen loo FRF contains the lant and the controller). Changes in the lant will cause the magnitude and hase to shift left and right as well as u and down. However, the general shae of the controller rovides enough gain to stabilize the lant without inducing sillover at the lower and higher frequency hase crossover oints for many oerating conditions. In addition, some amount of feedback gain has been included in the loo as was required by the system identification rocess. Therefore any changes redicted by the neural network/feedback controller design rocess will be included in addition to the hase and gain already resent in combustion system. Finally, note that due to the stee rolloff of the bandass controller design, no other crossover oints in the system bandwidth can generate instabilities. All of the comonents in the feedback control loo were included in creating the neural network training data. The FRF outut of the neural network included all of the comonents shown in Figure 4.5. H G H Amlifier Seaker Self Excited Combustor Mic HP Filter LP Filter Anti-Alias Delay/DSP LP Filter In Out FIGURE 4.5 OPEN LOOP PLANT The amlifier used to ower the actuator was set at a gain of 0 db and the delay in the DSP corresonded to aroximately 274 degrees. This hase setting rovided moderate control at a wide range of oerating conditions and was determined by manual trial and error. This is not a necessary ste in order for the neural network control to be effective. The feedback controller design algorithm will automatically determine the aroriate gain and hase for accetable control at any condition. 50

62 Figure 4.6 illustrates the configuration of the neural network combined with the combustor feedback control loo. The unstable, self-excited combustor system (now reresented by G) is controlled by the feedback controller H. Measurements of the combustor oerating conditions and temerature rofile are continually rovided to the artificial neural network. The outut of the network (the oen-loo FRF) was used in the controller design rocedure to automatically modify the gain and hase of the fixed shae controller, H. The digital feedback controller oerated at a samle rate of 11 khz on a TI C40 DSP, simultaneously with the neural network that udated once every second on the same DSP under searate timer interrut subroutines. The C-code that erformed this control oeration is included in the Aendix. G H ANN Control Predictor FIGURE 4.6 ANN CONTROL Figure 4.7 illustrates three conditions of control for oerating conditions of a total flow rate of 140 cc/sec and an equivalence ratio of

63 Mag. (dbvrms) Freq.(Hz) FIGURE 4.7 NARROWBAND ANN CONTROL PERFORMANCE The higher magnitude (dashed) trace reresents the RMS magnitude of the voltage for the ressure sensor at the bottom of the tube in the case where no acoustic feedback control is alied to the system. The lowest (solid) trace indicates the control rovided by the neural network/feedback udate algorithm. Although the free acoustic resonse of the combustor is not shown in Figure 4.7, the neural network clearly rovides control that aroaches the lightly damed modal shae of the tube. The arameters determined by the neural network for this controlled case were a hase of 256 degrees and a linear gain of 1.8. It should be clear that since all data oints are less than the uncontrolled resonse, the neural network does not induce any new instabilities from excessive gain and oor hase margins. The dotted trace (shown just above the solid trace) in Figure 4.7 reresents the control action that could haen as a result of a manual controller not governed by the neural network. The hase of the manual controller was chosen (based on subjective erformance of the acoustic outut) as 274 degrees while the gain was chosen to be This excessive gain was more than enough to suress the rimary instability at 178 Hz but was too high to revent controller induced instabilities at frequencies slightly higher and lower than 178 Hz. The neural network is quite useful in redicting changes necessary to ensure a stable and effective feedback controller design in the face of changes in oerating conditions. Figures

64 and 4.9 illustrate this effectiveness for a variety of oerating conditions. Figure 4.8 shows the uncontrolled (dotted) and controlled overall acoustic bandowers (0-800 Hz) for a flow rate of 130 cc/sec and steadily increasing equivalence ratio s from 0.48 to dbvrms Band Equivalence Ratio FIGURE 4.8 BANDPOWER AT 130 CC/SEC, INCREASING EQ. RATIO Since the temerature changes will also affect the lant, the controller must udate the feedback design to track these changes. With an average of 50 db of control over the stated bandwidth, the controller is quite effective over all oerating conditions. Figure 4.9 illustrates a similar data set with a flow rate of 140 cc/sec and a decreasing equivalence ratio from 0.56 to 0.47 similar erformance is observed. 53

65 dbvrms Band Equivalence Ratio FIGURE 4.9 BANDPOWER AT 140 CC/SEC, DECREASING EQ. RATIO The neural network feedback control strategy does not use the neural network directly in the actual control of the thermoacoustic instability. Instead it is used to redict the oen loo frequency resonse of the lant to be controlled by a rule based feedback controller design routine. The result is a control system that redicts changes in the lant that occur either as a result of changing oerating conditions or the control action itself. 4.4 NEURAL NETWORK CONCLUSIONS This ortion of the research illustrated that an artificial neural network can rovide an effective system identification in the frequency domain. The outut of the neural network was a function of system arameters that were measured in the oen-loo, indeendent of the control actuation rocess. In the case examined here, the redicted frequency resonse function was used to udate the gain and hase of a fixed shae feedback controller based on gain and hase crossover oints that were deterministic. The same frequency resonse function could also be used to rovide a control-to-error ath estimate for an adative feedforward controller, although this was not exlicitly examined by this work. This aroach holds many advantages over revious attemts at active noise control of thermoacoustic instabilities in combustors. Prior controllers have been redominantly manual, 54

66 requiring constant adjustment of the gain and hase of the controller to ensure stability for different oerating conditions. Successes at using the hase shifting feedback controller have met with equal amounts of failure from controller induced instabilities (sillover). Without an accurate lant estimate, the controller is often oorly designed and insufficient gain or hase margins elsewhere in the sectrum revent adequate control of the rimary instability by causing a new feedback controller induced instability. By erforming a careful system identification, a controller can be designed that satisfies the Bode gain/hase relationshis and still rovides adequate control. The ANN facilitates this by allowing real-time udate of the oen-loo frequency resonse in the face of changing combustor conditions as well as during closed-loo control. The examle shown here illustrated how a neural network can rovide as much as 50 db of control over a range of oerating conditions without manually udating the feedback controller. While the network udated feedback controller is effective at controlling the instability over a wide range of conditions, an exhaustive system identification rocedure is required to build a database of training data for the neural network. In real world alications this may be an imractical solution to the roblem because of the difficulty in creating such a database. 55

67 Chater 5 Adative Control 5.1 INTRODUCTION During the ast several years, there has been a trend away from mostly emirical or exerimentally-based active combustion control (ACC) methods in favor of control systems that rely on more accurate understanding of the dynamic rocesses involved in the thermoacoustic limit cycling resonse. Early demonstrations of hase-shifting ACC designs relied only on a measurement of the acoustic ressure and actuation of the unsteady heat release rate after aroriate delay (i.e. hase shift) relative to the measured ressure signal. The hase-shifting controllers roved to be effective in many situations but were lagued by inadequate knowledge of how to redict the required hase, and gain, of the controller for varying oerating conditions of the combustor. These roblems naturally led to investigations of adative control methods. A number of ACC researchers recognized that adative signal rocessing methods offered a ossible alternative to the manually-tuned, hase-shifting controller. These adative filtering methods, such as the LMS algorithm roosed by Widrow and Stearns [49], had been highly successful in the active noise and vibration control community for narrowband disturbance rejection. The most relevant alication of LMS control for noise or vibration control was the so-called Filtered-U algorithm [48] which required feedback comensation for the reference sensor. The ACC imlementation is similar in its need for comensation because the reference sensor is often identical to the error sensor. A number of combustion researchers investigated the usefulness of these LMS algorithms for ACC [22,23,30,25,28]. However, most of the results from those exeriments and numerical simulations indicated an uncertainty about the roer imlementation with regard to emloying a lant estimate. As a result, the literature 56

68 shows that the LMS controlled systems often diverged, sometimes minutes after the combustor s ressure signature seemed to have been reduced to accetable levels. This ortion of this work investigates filtered-x LMS control alied to the self-excited thermoacoustic instability. Previous research has not carefully recognized that an aroriate system identification is required to ensure stable and redictable convergence. Together with the system identification rocedure and argument resented in Chater 3, this discussion exlains why the stable control-to-error ath is the aroriate (and only) choice for lant identification. Previous research has not identified this as an aroriate or even accetable aroach for system identification and control imlementation. The analysis of adative feedback control alied to a general system is resented in the following chater. Those results show that adative feedback control is not a ractical controller design because of inherent instability and algorithm divergence roblems when the lant estimate does not exactly match the actual control-to-error ath. This chater assumes that a erfect identification is available. It is then shown that only one arguable choice for the system identification exists. It is also shown through simulation and exeriment that stabilizing control is achievable when the aroriate system identification method is used. An intermittent erformance condition was also identified during exerimental control of higher heat release oerating conditions. A similar intermittency was seen in the simulation develoed to model controller erformance. The simulation illustrates an intermittency caused by feedback loo instabilities which could not be stabilized by the memoryless LMS algorithm. However when the RLS algorithm was emloyed on the same system, the system was stabilized. Ultimately, exerimental results showed that feedback loo instabilities were not causing the intermittency witnessed when controlling higher heat release conditions. Instead, actuator authority was shown to limit the controllability as the lant changes under the influence of the controller. The following section introduces the structure of adative feedback control for both exogenous disturbances and the self-excited system. The most aroriate choice for the lant 57

69 estimate is roosed and analyzed in the next section. Next, simulation and exerimental results of the filtered-x forms of the LMS and RLS algorithm using the roosed system identification rocedure of Chater 3 are resented. The intermittent erformance condition is then exlained in detail. The final section rovides concluding remarks. 5.2 ADAPTIVE FEEDBACK CONTROL CLASSICAL DISTURBANCE SUPPRESSION The adative feedback controller considered in this work can be viewed as a secial case of Filtered-U control where the control-to-reference ath and the control-to-error ath are identical [48]. Figure 5.1 shows the adative feedback control block diagram for the case of an external disturbance and stable lant. It is instructive to briefly examine this system as a recursor to controlling the self-excited system. - G ^ n r W c G e G ^ LMS FIGURE 5.1 ADAPTIVE FEEDBACK EXTERNAL DISTURBANCE In a urely feedforward control system, a searate reference signal that is correlated with the disturbance is used as the inut to the adative filter. It is well-known that correct estimation of the dynamic hase, within ninety degrees, is sufficient to revent divergence of the LMS gradient search [39]. In adative feedback control, the reference signal is derived directly from the error sensor by removing the comonent of the error signal that is due to the control signal, leaving only a measure of the disturbance. If we let the control signal be the outut of the FIR adative filter (W), we note that it must go through the lant dynamics (G) before acting on the external disturbance (n). The dynamics reresented by the lant include everything resent in the control signal to error signal ath, including the A/D and D/A. 58

70 Therefore, removing the control signal comonent from the error requires subtracting the outut of the adative filter, filtered by an estimate of the lant, from the measured error signal. Hence, if the lant estimate is not erfect, a non-zero feedback ath is introduced. The closed loo transfer function between the error and the disturbance can then be written as: e n 1+ Gˆ W = 1+ Gˆ W G W [5.1] Unlike the self-excited system, if the lant estimate is exact ( Ĝ = G), there is no feedback loo since the denominator of the above transfer function vanishes and the system behaves as a strictly feedforward system where Ĝ and W are both stable systems. For the case where Ĝ G, it is clear that the oles of [5.1] will change with the adatation of the filter W and reresent a otential source of controller instability SELF-EXCITED SYSTEMS Figure 5.2 illustrates a simlified block diagram of an adative feedback controller alied to a self-excited system. Although the hysical self-excited system is extremely comlex and a subject of ongoing research, a simlified model consisting of linear acoustics, GA, linear flame dynamics, GF, and a nonlinear couling is used in the Figure. In the analysis and simulation, the nonlinearity is treated as a static, saturating-tye nonlinearity. Though crude from a broadband modeling oint of view, the extremely tonal nature of the system makes this a useful aroximation. 59

71 - NL r W G ^ c G ACT d G A G F P T G ^ LMS FIGURE 5.2 ADAPTIVE FEEDBACK SELF-EXCITED The actuation signal asses through a linear system, GACT, that incororates both the actuator dynamics and the acoustic dynamics. Since the oututs of GA and GACT are both acoustic ressures, they can be suerosed to roduce the total combustor ressure variation, PT. In the controller, the outut of the adative filter is filtered by the lant estimate and is used to generate the reference signal from the error signal as before, although the aroriate lant estimate is less obvious now as a result of the inclusion of the self-excited system. The controlled system consists of two main loos, the hysical feedback loo and the control loo, as shown in Figure

72 NL G A G F P T G ACT W 1+G ^ P W FIGURE 5.3 REDRAWN CONTROLLED SYSTEM The otimal controller is defined to be that which comletely nullifies the hysical feedback loo. Equating the to loo to the bottom loo (with a minus sign) and solving for the otimal adative filter (WOPT), results in W ot A F = [5.2] G ACT G G + G G Gˆ A F By substituting [5.2] into the block diagram of Figure 5.3 as the adative filter W, it is obvious that the lower loo (feedback controller) exactly cancels the uer loo GAGF resulting in a comletely stabilized closed loo system. Equation 5.3 illustrates this result for the robe inut shown in Figure 5.3: PT GACT (1 + Gˆ W ) = (1 G G )(1 + Gˆ W ) G A F ACT W W = Wot = G ACT [5.3] Discussion of the otimal adative filter weights is more interesting for this self-excited lant alication because of the quiescent state that the combustor will reach as soon as the oles have crossed back into the left-half Lalace lane under the action of the controller. Because the oen-loo self-excited system of [5.3] does not contain dynamics that are on the imaginary 61

73 axis (i.e. marginally stable), there is a set of gains (between the imaginary axis and the otimal solution) that will stabilize the system without requiring the adative filter to reach its otimal solution. This system will be more lightly damed than the oen-loo system but still stabilized. If the adatation causes the system to stabilize, the error signal will go to zero and the adatation will sto, never reaching the otimal gain. 5.3 THE CORRECT PLANT ESTIMATE Initially, the system will be examined in the linear range so that the nonlinearity reduces to a simle gain that can be lumed with the flame dynamics, GF. The total ressure (PT) serves as the error signal to be reduced and is used by the adative feedback structure to udate the weights and create the reference signal. The exression for the transfer function between the robe inut of Figure 5.3 and PT results in: PT = 1 G A G F GACT (1 + Gˆ W ) G G Gˆ W + Gˆ W A F G ACT W [5.4] In a urely feedforward situation, the dynamics from the control-to-error ath are clear, and can be identified directly, often in the absence of the disturbance, and a model can be generated. The aroriate choice for the control-to-error ath estimate is not as clear when the self-excited system is addressed. The self-excited system of Figure 5.3 offers two logical choices for the lant estimate. Following the standard rocedure for feedforward roblems, one choice for the lant estimate is the oen loo dynamics of the control-to-error ath, which yields PT = G G ˆ ACT = 1 GAG [5.5] F The acoustic dynamics have been moved from the forward ath of the self excited loo and aear searately as art of the actuator dynamics and the self-excited feedback loo. (The same 62

74 acoustics influence the self-excited loo and the actuator ath). Substituting this choice for the lant estimate, [5.5], into the denominator of [5.4] yields 1 G A G F G G Gˆ W A F + Gˆ W G ACT W G Gˆ ACT = 1 GAGF = 1 G A G F [5.6] This result guarantees that the adative filter cannot influence the oles of the closed loo system, and they will remain in the unstable right half lane. Therefore, this choice of the lant estimate [5.5] is not considered to be a valid solution and can never robustly stabilize the selfexcited system. An alternative roosal is to use the actuator ath alone as the lant estimate such that Ĝ = GACT. Assume the outut of the acoustic ortion of the self-excited loo is the external disturbance to be canceled at the error sensor. In this case, the dynamics between the control outut and the error sensor reresent the actual control-to-error ath. In addition, the artificial reference signal now becomes an estimate of the exact disturbance signal at the error sensor. This can be seen by recognizing that: P T r = P = d + G T act c Gˆ c = d + G act c Gˆ c [5.7] in view of Figure 5.2 where d is the outut of the acoustic lant, c is the outut of the controller, PT is the measured error signal and r is the derived reference signal. Not only is the reference signal accurately reresenting the disturbance to be cancelled (when the lant estimate is GACT), but the required lant estimate is of a strictly stable system. Using GACT as the control-to-error ath estimate, we can again examine the closed loo system of [5.4]. 63

75 PT = 1 G A G F GACT (1 + Gˆ W ) G G Gˆ W + Gˆ W A F G ACT W Gˆ = Gact G = 1 G G A ACT F (1 + G G A ACT G F W ) G ACT W [5.8] It is clear from [5.8] that the adative filter can now influence both the zeros and the oles of the closed loo control system, allowing for the ossibility of stabilizing the self-excited system. If we assume that the lant estimation error is zero and diagram of Figure 5.2 can be redrawn as shown in Figure 5.4. Ĝ is exactly equal to GACT, the block G A NL G F W c G ACT P T G ACT LMS FIGURE 5.4 PERFECT IDENTIFICATION DIAGRAM REDRAWN It is clear that when using the aroriate lant estimate (assuming erfect identification), the system is identical to the feedforward filtered-x structure and can theoretically behave as a feedforward system. In view of Figure 5.3, it is easy to see how the adative filter imarts the needed gain and hase to control the limit cycling system through the feedback loo. The significance of the control-to-error ath estimate (Ĝ) in the controller transfer function is also evident from Figure 5.3. In ractice, achieving a erfect system identification is not always ractical. Here it is shown that if such a erfect identification was achievable, the aroriate lant estimate is the actuator dynamics. (The following chater addresses the effects of errors in the lant estimate on controller stability). It is not a trivial task to obtain the actuator dynamics since they are influenced by the limit cycling system. However, the system identification rocedure resented 64

76 in Chater 3 illustrates how the actuator dynamics can be closely aroximated by low level robing near the limit cycle frequency followed by a least squares model fit. In the following simulations and exeriments, this system identification aroach was emloyed and shown to generate models accurate enough to achieve stabilizing control. 5.4 SIMULATION GENERAL A simulation has been designed to ermit raid and easy investigation of the erformance of the adative feedback controller in conjunction with the self-excited, limit cycling system. Results from the simulation agree closely with the behavior of the actual exeriment. This is undoubtedly due to the fact that the dynamics which dominate the system are a single air of lightly-damed acoustic oles. The two-art simulation described below is included in the Aendix. Figure 5.2 reresents the general form of the adatively controlled self-excited system that was simulated. In the simulation, the self-excited loo consisted of a low ass filter to reresent a model of the flame dynamics (GF), a static nonlinearity reresented by a hyerbolic tangent function, and a single-mode lightly-damed (2%) acoustical model (GA) at aroximately 175 Hz. The loo gain and nonlinearity gain were adjusted to yield a steady limit cycle after aroximately 2 seconds at a samling rate of 1600 Hz. The actuator ath consisted of the same acoustical model lus some amount of hase delay. For the exeriment, this delay reresents all comonents in the control-to-error ath. Equation 5.9 illustrates the actual transfer functions used in the simulation. G A = z G 2 f = Gact = s s e6 s z z z [5.9] The simulation runs in two searate modes, as does the exerimental setu. After the limit cycle is established, a multi-tone robe signal is alied to the oen loo lant as shown in Figure

77 NL G A G F Multi-Tone Source G ACT Inut Outut FIGURE 5.5 SYSTEM IDENTIFICATION OPEN LOOP The inut to outut relationshi at the robe frequencies establishes the magnitude and hase of the linear lant as described in the system identification section from Chater 3. In the simulation, the limit cycle is established rior to executing the identification with the multi-tone source inut. As shown in Chater 3, the resulting identification closely aroximates GACT for the case where the inut sinusoids are sufficiently lower than the limit cycle amlitude. The discrete frequency resonse data reresenting GACT is then used to generate a least squares infinite imulse resonse (IIR) transfer function fit in the z-domain, tyically of order less than 6, with a ole very near the unit circle reresenting the hot acoustic mode. This fit is then used as the lant estimate, Ĝ, during the second mode of the simulation. Figure 5.6 illustrates the results of the system identification ortion of the simulation. The blue asterisks indicate the frequency resonse data collected with the multi-tone source in the resence of the limit cycle. The solid line illustrates the IIR transfer function fit to that data while the dotted line indicates the actual frequency resonse of the actuator dynamics. As redicted by [3.7] the actuator dynamics are closely aroximated by the system identification rocedure described in Chater 3. 66

78 Mag Phase Freq.(Hz) FIGURE 5.6 SIMULATION IDENTIFICATION RESULTS After the limit cycle has reached a steady state, and the robe frequencies have been turned off, the lant model shown in Figure 5.6 is used in the adative feedback control loo shown in Figure 5.2; both as the Filtered-X art of the LMS and RLS algorithms and the lant estimate used to derive the reference signal LEAST MEAN SQUARES ALGORITHM Imlemented as shown in Figure 5.2, the Filtered-X LMS algorithm was used to control the self-excited system after it had sufficiently reached a stable limit cycle. To first simulate a low heat release oerating condition, the lant dynamics shown in [5.9] were slightly altered to result in a lower loo gain. This was accomlished by reducing the gain of the flame dynamics from 1500 to 500. The lant estimate obtained from the system identification rocedure (and shown in Figure 5.6) was used for both the feedback comensation and gradient adjustment of the Filtered-X algorithm. The FIR adative filter was chosen to have 2 weights for controlling the limit cycle frequency. This was determined to be adequate for the low heat release condition examined here. The adatation stabilized the system within aroximately 1 second using a convergence arameter of 1e-5. Figure 5.7 illustrates the ower sectra of the uncontrolled and controlled systems. 67

79 db Freq.(Hz) FIGURE 5.7 LOW HEAT RELEASE LMS SIMULATION As mentioned earlier, it is not required that the adative filter achieve the otimal gain in order to drive the error signal to zero. Figure 5.8 clearly illustrates this by showing the ath of the magnitude and hase of the actual adative filter during adatation, as comared to the otimal magnitude and hase as comuted from [5.2], at the limit cycle frequency. 68

80 Phase Mag Iteration/10 FIGURE 5.8 LOW HEAT RELEASE LMS PATH TO OPTIMAL A second simulation was investigated in an effort to mimic the intermittent control witnessed in exeriments at higher heat release conditions. The rimary difference in the simulation and exeriment was a bandass filter that was used to filter out-of-band frequency comonents. This filter adds significant dynamics which ultimately affect adatation and require additional adative filter weights to avoid instability. Using the dynamics of [5.9] to reresent the higher heat release oerating conditions, adding a bandass filter, and increasing the adative filter size from 2 to 10 resulted in an intermittent searching condition not unlike the conditions seen in the exeriment. In the controlled environment of a simulation, it is easier to identify the culrit of the intermittent behavior because every system arameter is available. By investigating the feedback loos generated during adatation, it was determined that the intermittency in the simulation resulted from the generation of unstable oles near the limit cycle frequency. Essentially, the added dynamics from the bandass filter and the adative filter in the feedback loo, created unstable feedback loos within the structure of Figure 5.2. This is most easily exlained by considering the Nyquist diagram at various oints in the adatation. First consider Figure 5.3. The oen (control) loo with resect to the robe inut and ressure outut is exressed as 69

81 OLTF GACT = 1 G G A F W 1+ Gˆ PW [5.10] where the bandass filter is absorbed into the adative filter and the minus sign ermits Nyquist analysis of a negative feedback system. It is imortant to note here that the Nyquist lot can only be evaluated at a secific oint in time, since the adative filter is continually changing. Therefore, the number of encirclements required to achieve stabilizing control may vary since the number of oen loo unstable oles is a function of the adative filter. Figure 5.9 illustrates the intermittent time resonse of the LMS control of the high heat release condition. In addition, the Nyquist diagrams for various moments in time are also resented with associated stability criteria in the titles. Inherent in the evaluation of the Nyquist (and Bode) criteria is the assumtion of system linearity as well as time invariance. The gain of the nonlinearity within it s cliing region can be accounted for in order to evaluate While examining instantaneous Nyquist lots in this time varying system is not strictly accetable, it can rovided insight into the stability of the system at a secific oint in time. If W is assumed to be held fixed at the moment of examination of the Nyquist criteria, we can establish the stability of the system at that secific moment in time. At the very least, this rovides a tool for examining the movement of the oles over time and ermits instantaneous stability information. No assumtion is made for any of the Nyquist or Bode analyses, that stability will be guaranteed for all time, as long as W continues to adat. Therefore, the Nyquist lots in the following figure do not reresent controller stability redictions but instead rovide instantaneous stability information during adatation. 70

82 2 Nyquist Diagrams Nyquist at 7240, P=2, N=2, Z=4 1 Nyquist Diagrams Nyquist at 6060, P=2, N=0, Z=2 1.5 Nyquist Diagrams Nyquist at 7650, P=2, N=2, Z= Imaginary Axis Imaginary Axis Imaginary Axis Real Axis Real Axis Real Axis Mag n 0.5 Nyquist Diagrams Nyquist at 6500, P=2, N=0, Z=2 1.5 Nyquist Diagrams Nyquist at 7500, P=2, N=2, Z=4 20 Nyquist Diagrams Nyquist at 7800, P=2, N=2, Z= Imaginary Axis Imaginary Axis Imaginary Axis Real Axis Real Axis FIGURE 5.9 LMS INTERMITTENT STABILITY SIMULATION The Nyquist lots shown above illustrate the stability of the closed loo system for different moments in time. In this case, the number of oen loo unstable oles remains at 2 for the entire duration of the intermittency. The first two lots at 6060 and 6500 do not generate any encirclements of the 1 oint and therefore the closed loo system has 2 unstable oles. At iteration 7240 two clockwise encirclements are generated during the adatation. Unfortunately two counterclockwise encirclements are required to stabilize the system, and the result is 4 unstable Real Axis 71

83 closed loo oles. The same result is seen at iterations 7500, 7650 and 7800 where the adative filter is unable to effectively generate the required encirclements. The rocess reeats starting at iteration It is interesting to examine the ath of the frequency resonse of the controller as comared to the otimal solution at the limit cycle frequency as shown in Figure Phase Mag n FIGURE 5.10 LMS INTERMITTENT PATH TO OPTIMAL It is clear that the otimal magnitude and hase at the limit cycle frequency is reached by the adative filter and it cycles within the surrounding areas in conjunction with the intermittency. The difficulty arises due to the inability of the LMS algorithm to control frequencies that are not ersistently excited. The unstable loos generated during adatation create controller induced instabilities at frequencies other than the limit cycle frequency. Once these tones become redominant in the mean squared error, the LMS algorithm attemts to control these frequencies, while forgetting about the limit cycle frequency. This lack of memory inherent in the LMS algorithm is the cause of the inability to maintain stabilizing control in this articular simulation. 72

84 5.4.3 RECURSIVE LEAST SQUARES ALGORITHM The memoryless LMS algorithm can be relaced by the more comutationally intensive Recursive Least Squares (RLS) algorithm that retains knowledge of rior disturbances [47]. Emloying the RLS algorithm in the same controller structure of Figure 5.2, the system is stabilized within one second of initiating control. Figure 5.11 is a similar reresentation to that of Figure 5.9, of the changing stability of the RLS controlled system. 73

85 6 Nyquist Diagrams Nyquist at 5100, P=4, N=0, Z=4 20 Nyquist Diagrams Nyquist at 5290, P=4, N=0, Z=4 5 Nyquist Diagrams Nyquist at 5500, P=2, N=0, Z= Imaginary Axis Imaginary Axis Imaginary Axis Real Axis Real Axis Real Axis Mag n 50 Nyquist Diagrams Nyquist at 5250, P=4, N= 2, Z=2 30 Nyquist Diagrams Nyquist at 5400, P=4, N=0, Z=4 4 Nyquist Diagrams Nyquist at 6000, P=2, N=0, Z= Imaginary Axis Imaginary Axis 10 0 Imaginary Axis Real Axis Real Axis FIGURE 5.11 RLS INTERMITTENT STABILITY SIMULATION Real Axis As the RLS algorithm begins to converge at iteration 5100, there are 4 unstable oen loo oles requiring 4 counter clockwise encirclements to ensure stability. As the algorithm adats through iteration 5250 two counterclockwise encirclements are made but the system is still not stabilized. Through iteration 5290 and 5400 the frequency resonse begins to change and by iteration 5500 we see that only two oen loo oles are unstable, requiring two 74

86 counterclockwise encirclements for stability. By iteration 6800 shown in Figure 5.12, the system has been stabilized 4. 4 Nyquist Diagrams Nyquist at 6800, P=2, N= 2, Z=0 3 2 Imaginary Axis Real Axis FIGURE 5.12 CONVERGED RLS CONTROLLER NYQUIST DIAGRAM This set of simulations illustrates one ossible mechanism for instability that may occur when emloying an adative controller as shown in Figure 5.2. More detailed analysis of this and other instability roblems is resented in the following chater. While the RLS algorithm clearly offers significant benefits in avoiding controller induced instabilities, the intermittent behavior controlled in this simulation was not remedied in the exeriment by the RLS algorithm. 5.5 EXPERIMENTAL RESULTS The exerimental rocess was identical to that described in the simulation section above. A multi-tone FRF was erformed on the self-excited lant and a least squares fit was alied to 4 For the benefit of recreating this simulation, it should be noted that the hyerbolic tangent function generated a less than unity gain in its linear region in the lant feedback loo. This gain must be taken into account when dislaying the oen loo frequency resonse data. 75

87 the data to generate an IIR filter model that reresented the control-to-error ath. This model was then used in the structure shown in Figure 5.2. With the heat release at a relatively low gain (controlled by adjusting the equivalence ratio to a value of 0.51 and a total flow rate of 120 cc/sec), the Rijke tube combustor instability at 175 Hz was stabilized indefinitely with a two weight adative filter emloying the LMS adatation algorithm. Figure 5.13 shows (in asterisks) the actual magnitude and hase data collected from the tube along with the 6 th order model frequency resonse shown as the solid line. Mag. (db) Freq.(Hz) 0 Phase (degrees) Freq.(Hz) FIGURE 5.13 EXPERIMENTAL MODEL DATA AND FIT Figure 5.14 shows the uncontrolled (dotted) and controlled ower sectra of the total ressure in the tube (converted from voltage units). The second harmonic at 350 Hz disaears under control, revealing the shae of the third acoustic mode of the tube. The natural daming of the second acoustic mode is greater than that which is shown in Figure 5.14, but as discussed earlier, the system can be stabilized by having a ole in the left half lane that is more lightly damed than the natural acoustic mode. 76

88 Mag. db Freq.(Hz) FIGURE 5.14 LOW HEAT RELEASE EXPERIMENTAL LMS CONTROL A manually adjustable gain and hase shift controller was also alied to the same limit cycling system. Referring to Figure 5.3, it is aarent that the fixed gain controller relaces the transfer function: W 1+ Gˆ W [5.11] Examining the magnitude and hase of the converged equation 5.11 at 175 Hz as comared to the magnitude and hase of the fixed feedback controller, it is seen in Figure 5.15 that the hase of both controllers is nearly the same at 175 Hz. The magnitude, however, is significantly lower for the adative system and will not increase with time because the error signal has been driven below the 1-bit noise floor of the A/D. It is known that excessive gain can roduce controller-induced instabilities[16]. The adative controller can revent controller induced instabilities by changing its shae and adjusting its magnitude to minimize the mean squared error. 77

89 Mag. (db) Freq.(Hz) 0 Phase (degrees) Freq.(Hz) FIGURE 5.15 FIXED GAIN AND ADAPTIVE CONTROLLER COMPARISON The results discussed above are indicative of the results obtained at a number of oerating conditions at low equivalence ratios. As the equivalence ratio, and hence heat release, was increased, a oint was reached where intermittent behavior was observed. Figure 5.16 comares the convergence behavior in the time domain of the adative feedback LMS algorithm for low and high heat release conditions. The uer time trace reresents the convergence corresonding to the control erformance in Figure 5.14 whereas the lower trace exhibits the intermittent searching behavior resent at the higher heat release oerating conditions. 78

90 4 2 Mag Mag Time(s) FIGURE 5.16 LOW AND HIGH HEAT RELEASE LMS CONTROL As discussed above, the simulation redicted that the intermittency could be caused by the inability of the LMS controller to effectively control multile disturbances occurring over different time eriods. Also shown in simulation was the successful imlementation of an RLS control algorithm that retained memory of revious disturbances and was therefore able to stabilize the high heat release system. The RLS controller was exerimentally alied to the higher heat release oerating conditions (equivalence ratio = 0.6 and total flow = 135 cc/sec) in the Rijke tube combustor. Figure 5.17 illustrates the corresonding time trace of that exeriment. 79

91 Mag Time(s) FIGURE 5.17 RLS CONTROL HIGH HEAT RELEASE It is clear that the RLS algorithm does not solve the intermittency roblem at the higher heat release conditions. This rovides convincing evidence that the intermittency is caused by a source other than feedback loo instabilities. To better understand the intermittency henomenon, a fixed gain feedback controller was alied to the combustor at the same oerating conditions. This removed the variability inherent in the time varying (adative) controllers (both RLS and LMS). As shown in Figure 5.18, the fixed gain, controlled system exhibited the same cyclic intermittency. 80

92 Voltage Time (s) FIGURE 5.18 FIXED GAIN FEEDBACK CONTROL HIGH HEAT RELEASE Since the fixed gain feedback controller has no time varying comonents, it was determined that the flame dynamics must be changing in a eriodic manner to cause the intermittent behavior. The following section rovides amle exerimental evidence to suort this intermittency hyothesis. 5.6 ACTUATOR AUTHORITY AS A SOURCE OF INTERMITTENCY Researchers have frequently witnessed intermittent behavior of self-excited systems when controlled by adative aroaches. Quite often this behavior is attributed to algorithmic anomalies or controller induced instabilities. While each of these roblems can contribute to searching or marginally stable behavior (as discussed in the following chater), a third reason for intermittent behavior is resented here which is not related to time varying control. It is imortant to distinguish this form of intermittency from other instability mechanisms because the controller design itself is not at fault. The self-excited combustion system is unique in that the lant dynamics change under the action of the controller. For a given set of limit cycling oerating conditions described by a fixed equivalence ratio and flow rate, the combustor oerating oint can be described by the dynamic 81

93 gain of the flame dynamics, unsteady heat release rate that is a function of temerature, heat transfer rates to the combustor walls, and other unmodeled, low frequency dynamics. The ressure fluctuations from the limit cycle influence these low frequency dynamics. Consequently, when the ressure fluctuations are reduced by the feedback control, the oerating oint changes. This new, controlled, oerating oint can have enough increased gain and altered dynamics so that the actual control effectiveness may be limited by the original controller gain, controller hase, or actuator ower. This effect can also be more detrimental at higher heat release conditions where actuator authority limitations become more relevant, required controller gain in the closed loo is higher, and the hase required for otimal alication of controller gain has changed significantly. After exerimentally describing the source of intermittency as a function of changing lant dynamics, a controller design solution is resented here where both controller hase and magnitude changes favorably correct the intermittency roblem. The hysical behaviors causing the increased transfer function gain due to the lack of flame oscillation are described in some detail in [60]. Although the reasons for this changing gain as a function of flame oscillation are not clearly roven in terms of hysical dynamics, there are several hyotheses. When the flame is disturbed by acoustic oscillations, fewer reactants can be effectively burned, thus reducing the amount of unsteady heat release that can occur. Another ossible reason is that when the flame is not oscillating, it is sitting down on the flame holder and is distributed over a much smaller dimension in the direction of the flow. With the majority of the heat release concentrated in a more localized area, more gain can coule into the acoustic mode at a secific location. Deending on the combustor geometry, this may be a stabilizing or destabilizing effect. A third hysical exlanation of reduced flame transfer function gain is that dynamic oscillation of the flame (by the acoustic ressure fluctuations) causes more heat to be effectively transferred to the flame holder and surrounding walls rather than into the acoustic self excited loo. Whatever the reason for the altered gain, the following exerimental evidence shows that the effect is real and reeatable. This effect can adversely affect the control erformance and will be considered when redesigning the controller. 82

94 Based on the above observation, a hyothesis was develoed and roven exerimentally to define the searching henomenon seen for fixed gain feedback control. The hyothesis can be exlained most effectively in a multi-ste rocess reresenting the transient characteristics seen in Figure It is helful to first recognize that the linear reresentation of the unstable ole in the s-lane moves further into the right half lane as the heat release is increased; this corresonds to an increase in self-excited loo gain. This indicates that more control authority/gain is required at higher heat release conditions to stabilize the system. The core of the hyothesis is that the oerating oint changes, causing the self-excited loo gain to increase, when the flame is not being excited by oscillating ressure. If we begin with the uncontrolled system at a high heat release oerating condition and refer to Figure 5.19, the following hysical henomena occur: Without control, the self-excited linear system ole resides in the right half lane at oerating oint T1. Recognize that the oerating oint may change while the oerating conditions remain constant. When the feedback controller stabilizes the system, the controlled system ole moves to location 2, still at the T1 oerating oint. Because the flame is no longer oscillating, the oerating oint begins to change causing gain in the self excited loo to increase. The controlled system ole gradually moves to location 3. When the self excited loo gain reaches oerating oint T2, the gain alied by the feedback controller is no longer sufficient to stabilize the system, and the flame begins to oscillate as the limit cycle grows. After sufficient oscillation occurs, the oerating oint moves back to T1, the selfexcited loo gain is reduced, and the feedback control gain can again stabilize the system. The rocess reeats indefinitely. 83

95 Imag X ➂T2-CON X ➃T2-Uncon X ➁T1-CON X ➀T1-Uncon Real FIGURE 5.19 CONTROLLED SYSTEM POLE LOCATIONS It is clear that this occurs because there is insufficient gain in the feedback controller to kee the system stable after initial control raises the self-excited loo gain. Indefinite stabilization only occurs when the controller has enough gain to stabilize the self-excited loo when the flame is not oscillating. Position 4 reresents the uncontrolled self-excited dynamics if the flame were not oscillating, at the original oerating condition and the new oerating oint. Because a stable flame increases the loo gain of the self-excited system, the controller must be caable of stabilizing the ole as it exists in location 4. For the exerimental Rijke tube examined here, this only occurs in the range of lower equivalence ratios, based solely on the ower of the acoustic actuator. A series of exerimental tests was devised to illustrate the above hyothesis. As a corollary to the hyothesis, it is assumed that oscillation of the flame induced by acoustic ressure fluctuations reduces the gain in the self-excited loo. To illustrate this effect, the Rijke tube oerating conditions were set so that the fixed gain feedback control generated the undesirable intermittency behavior similar to Figure After the intermittency was established, an external acoustic robe signal was injected into the combustion chamber at a high enough amlitude to oscillate the laminar flame. The time resonse is shown in Figure

96 6 4 2 PROBE ON Mag Time (s) FIGURE 5.20 PROBE WITH INTERMITTENCY Under the conditions of the hyothesis, the robe oscillates the flame and subsequently reduces the gain in the self-excited lant. As a result, the feedback controller should have enough gain to exercise control over the new lant whose gain has been reduced by the external robe. Beginning at aroximately 3 seconds, the robe is turned on and the intermittency stos. This is due to the fact that the controller has laced the ole in a new stable limit cycle commensurate with the available actuator ower. This will fall somewhere between ositions 2 and 3 in Figure When the robe signal is turned off at aroximately 10 seconds, the intermittency behavior resumes as exected. Figure 5.21 illustrates the ower sectra of a single average for three locations on Figure The dotted trace deicts the eak sectrum during a high intermittency cycle whereas the solid sectrum shows the minimum sectrum during a low intermittency cycle. The dashed trace reresents the oint in time when the robe frequency (at 130 Hz) was on. 85

97 Mag. (db) Freq. (Hz) FIGURE 5.21 PROBE INTERMITTENCY SPECTRA By injecting the robe at an amlitude equal to the minimum oscillating ressure seen during the intermittency cycle, the overall RMS ressure was limited to that of the minimum oscillation ressure. This may be viewed as a form of controller design where the intermittency is eliminated and the RMS ressure is reduced by 20 db. But more imortantly, this exeriment illustrates that the loo gain is reduced by oscillating ressure, and conversely it is increased by the lack of oscillation. A second exeriment was conducted to analyze the effect of the amlitude of the robe signal on the self-excited loo gain. During intermittency, the control actuator is exercising it s maximum authority and it is still unable to maintain stabilizing control. As shown in Figure 5.20, the oscillation of the external robe reduced the gain of the self excited loo enough to allow the maximum available control authority to be sufficient to maintain a stable limit cycle. This imlies that the oscillation has moved the ole ositions of Figure 5.19 further into the left half lane by reducing the gain in the self excited loo. Consequently, higher amlitude oscillations should result in higher reductions in self excited loo gain. This effect is illustrated in Figure

98 0.26 Am Am Am Time (s) FIGURE 5.22 PROBE AMPLITUDE EFFECTS ON INTERMITTENCY RECOVERY TIME Each of three robe amlitudes (0.26V, 0.5V and 0.75V) was initiated at time 0 and ket on for aroximately 10 seconds. Uon release of the external robe at 10 seconds, the intermittency returned after increasingly long durations. This imlies that the controlled ole location of osition 2 in Figure 5.19 exerienced a longer transient before returning to the familiar cycle between osition 2 and osition 3. The higher amlitude robe signals reduced the selfexcited loo gain thereby causing the existing control authority to be more stabilizing. Restated, the time required to return to the intermittent behavior increased with robe gain because the controlled system ole was further into the left half lane after the robe was released. The eriod of the intermittency also holds some information regarding the location of the oles on Figure A transient resonse occurs each time the ole shifts from osition 2 to osition 3 and back, which is a function of the distance between the two ole locations. We have some control over the ole locations by imarting more or less gain to the feedback controller within a small window surrounding the limitations of the actuator. Figure 5.23 illustrates the effects of increasing the controller gain on the intermittency eriod for a given oerating condition. 87

99 Time(s) FIGURE 5.23 CONTROLLER GAIN INTERMITTENCY PERIOD With each increase in controller feedback gain, the intermittency eriod increased. Two distinct behaviors occur when increasing controller gain that contribute to this increased eriod. First, higher control gain moves the stabilized osition 2 of Figure 5.19 further into the left half lane, as much as can be tolerated by the given actuator ower. Second, increased stabilized time (lower amlitude acoustic oscillations) contributes to greater gain in the self excited loo, moving the unstable osition 3 (of Figure 5.19) further into the right half lane. Therefore as the control gain is increased, the distance between the two oerating oints grows and the transient resonse between conditions increases 5. Evidence of the existence of the virtual ole location of osition 4 in Figure 5.19 is also readily available. When the feedback control is immediately released after intermittent oscillation, the oerating oint is briefly at osition 4 and ultimately decays to osition 1. This is seen in the transient time resonse of Figure 5.24 where the feedback control is abrutly released. 5 Heat transfer rates do not change in the system and therefore do not contribute to changing transients. 88

100 Mag Time (s) FIGURE 5.24 VIRTUAL POLE TRANSIENT RESPONSE While these exerimental studies rovide suorting evidence toward the hyothesis, they do not rove that it is correct. The difficulty in directly illustrating the effects of oscillations on the flame dynamics transfer function lies in the fact that the nonlinearity in the self-excited loo is unknown. Therefore, the flame dynamics transfer function cannot be directly analyzed in the absence of the acoustics. However, a searate test rig was constructed that resented a burner stabilized flame similar to that in the Rijke tube, without the acoustic chamber. Lacking the acoustic interaction, the system is not unstable and therefore does not contain a nonlinearity. To rove the deendence of loo gain on flame oscillation, this burner stabilized flame was excited at various robe amlitudes while the gain of the flame transfer function (acoustic velocity to OH*) was recorded. The following table shows the results. 89

101 TABLE 5.1 FRF AMPLITUDE AS A FUNCTION OF PROBE GAIN Probe Gain 130 Hz FRF Amlitude 170 Hz The FRF amlitude of the flame dynamics transfer function translates directly into self excited loo gain once the acoustic dynamics are resent. It is therefore clear that changes in acoustic ressure oscillations affect the gain of the self excited loo. It should be noted that this effect can also be observed in the low equivalence ratio cases that were stabilized. For examle, in the uer trace of Figure 5.16, near the zero second oint, a change in the level of the controlled oscillation is observed. This is undoubtedly due to the changing of the flame dynamics following the dramatic change in oscillation amlitude after control is first alied. The revious discussions illustrate that the intermittency behavior is caused by changing lant dynamics. It should also be clear that the dynamics reresented by Figure 5.19 are an accurate reresentation of the actual behavior of the controlled system. Actuator authority has been stated to be the reason for the inability of the control system to imart enough gain to control the higher heat release oerating condition. This is confirmed by examining Figure

102 10 1 Outut Voltage Inut Voltage FIGURE 5.25 ACTUATOR AMPLITUDE RESPONSE Figure 5.25 illustrates the ressure outut (exressed in voltage) as a function of the inut voltage to the seaker. The ressure voltage (y-axis data) is resented on a log scale because the ability to control sound ressure is most commonly exressed in db (log units). Areciable increases in outut ressure for similar increases in inut voltage continue until between 15 and 20 V of inut voltage. At this oint, increases in the ressure outut of the seaker are significantly reduced, and no further areciable actuation authority can be rovided. For the high heat release oerating condition considered above, when the flow rate is 135 cc/sec and the equivalence ratio of 0.6, the actuator did not have enough ower to achieve effective (stabilizing) control (Figure 5.18). Figure 5.26 illustrates the early transient voltage alied to the actuator over a 2.5 second eriod at the incetion of control for an even higher equivalence ratio of 0.64 corresonding to higher unsteady heat release rate and self-excited loo gain. 91

103 Voltage Time (s) FIGURE 5.26 ACTUATOR VOLTAGE FOR HIGH HEAT RELEASE During the first second of control, the actuator is driven with nearly 20 volts (eak). Referring to Figure 5.25, this is clearly in the nonlinear range and the actuator has difficulty roviding the SPL required to achieve control. The feedback control system began intermittency after the first 2.5 seconds. Acoustic actuators caable of delivering higher amlitude SPL s in a more linear fashion than that illustrated in Figure 5.25, will facilitate control at a broader range of oerating conditions, eliminating the intermittency for higher heat release conditions. In this articular case the actuator does not have the ower to stabilize the system indefinitely, therefore no controller design can rovide stabilizing control. However, it is ossible to redesign the feedback controller such that a lower amlitude limit cycle is created, without generating an intermittency. A new, lower amlitude, stable limit cycle is clearly more desirable than high amlitude intermittencies, although the actuator may be oerating near its ower limitations. Figure 5.27 illustrates the alication of a controller with a certain magnitude and hase setting designed to stabilize T1 flame condition discussed above. 92

104 6 4 2 Mag Mag Time (s) FIGURE 5.27 SUB-OPTIMAL CONTROL The uer lot of Figure 5.27 illustrates the fast alication (turning on) of the feedback controller designed for this flame condition. After aroximately seconds the intermittency begins. Alying the same gain gradually to the self-excited system, we see that the intermittency occurs regardless of the seed of alication of the gain. Although it is (exerimentally) clear from the above discussions that actuator authority limits controllability of the system and hels cause the intermittency, the controller can be redesigned to revent the intermittency if it is designed to exercise the maximum amount of control for the T2 flame condition. In order to design a controller (magnitude AND hase) for a secific condition, we must first generate that condition. Since the controller itself affects the lant and generates the condition requiring control, we mush slowly aroach that condition while continually searching for a better controller design. First assume that the controller designed for condition T1 is accetable for controlling the T2 condition. We slowly increase the feedback gain on the controller while examining different hase settings to achieve the minimum error for a given gain setting. This rocess reeats until either 1) further increases in gain and subsequent hase adjustment do not result in detectable decreases in the error signal (indicates actuator is 93

105 saturated) or 2) the system is stabilized. The uer lot of figure 5.28 illustrates this iterative manual control design rocess Mag Mag Time (s) FIGURE 5.28 GAIN SCHEDULED CONTROL This new controller magnitude and hase has now been designed for the flame condition that it will be controlling because that condition was created by the existence of the controller in the loo. Now, when the controller is turned on instantaneously as shown in the lower lot of Figure 5.28, the system doe not oscillate intermittently. Instead, a lower amlitude stable limit cycle is created at the limits of the actuator authority at a different hase and gain setting that might be selected to control the T1 flame condition. For adative controllers, this can be imlemented using an aroriately large and scheduled leakage arameter so that the adative weights cannot reach their otimal solution immediately. The adative filter would then gradually adat to an otimal solution, allowing the controlled flame condition to be taken into account during slow adatation. Another ossible solution is to add a secondary robe signal to the control signal to eliminate the searching behavior as seen in Figure This intentionally oscillates the flame so that the self excited loo gain is low enough that the controller can maintain stabilizing control at the limit 94

106 cycle frequency. Each of these techniques strives to reduce the limit cycle amlitude as much as ossible using all of the available actuator authority, without allowing intermittency. 5.7 CONCLUSIONS Previous research aimed at alying filtered-x LMS techniques to control combustion instabilities has met with limited success. One reason for this limited success is that the selfexcited roblem is fundamentally different from the stable, exogenous-disturbance roblem. This chater shows that the most aroriate choice for the lant dynamics in a filtered-x LMS structure is the control-to-error dynamics, which include the hot acoustics but not the hysical feedback loo of the self-excited system. Furthermore, it is shown that this stable transfer function can be accurately estimated when the self-excited loo is closed. Exerimental results on a Rijke tube combustor with acoustic actuation have shown that an LMS-based adative controller, using the identification technique discussed in Chater 3, can stabilize the combustor indefinitely. For oerating conditions that did not achieve stability, a searching behavior was observed. Although simulations indicated that unstable feedback loos can generate such behavior, the intermittency was ultimately shown to be due to limitations in actuator authority and not the adative nature of the controller. Exeriments showed that the gain of the hysical feedback loo was related to the RMS velocity fluctuations in the combustor and this gain increased as control reduced the fluctuations. At the higher heat release conditions, the actuator was not owerful enough to stabilize the system after the loo gain increased following an initial reduction in oscillations. Although the adative structure roosed in here erformed reliably in exeriments, this structure introduces additional feedback loos and differs fundamentally from feedforward LMS imlementations, which have strong stability guarantees. The following chater rovides a detailed analysis of the stability and convergence of the adative feedback structure. 95

107 Chater 6 Adative Feedback Control Analysis Adative feedforward control is inherently stable and can effectively attenuate and often cancel disturbances. However, the requirement for a coherent reference signal that is unaffected by the control signal revents ractical imlementation for many alications. The rimary reason for this limitation is that contamination of the reference by the control signal invariably creates a feedback loo, thus rendering the stability advantage of feedforward control nonexistent. When the required reference signal is contaminated by the control signal, the roblem can be converted to a feedforward form by adding an additional signal ath that cancels the effect of the control signal on the reference. When the reference signal is identical to the error signal, it is termed filtered-e control. Filtered-U control is a more general version where the reference and error signals are hysically different, but are both influenced by the control actuator. Adative IIR filtering can be used to try to adatively cancel the feedback ath created by the control signal s influence on the reference signal. Adative IIR and filtered-u are often used interchangeably because fixed gain comensation of the feedback ath is less common in ractice. The majority of the discussion in this chater focuses on filtered-e control but the conclusions are extended to filtered-u forms. The filtered-e aroach was used exclusively throughout Chater 5. Adative LMS-based feedback controllers have been imlemented in a variety of forms on a variety of systems including thermoacoustic instabilities. While some success has been realized in secific alications, soradic behavior has lagued broad accetance. This chater examines two distinct mechanisms that may exlain these unstable behaviors: feedback loo instabilities and algorithm divergence. Unlike feedforward adative control, feedback loo instabilities can be created when adative controllers are used in structures containing feedback 96

108 loos. This mechanism is examined as a function of lant estimation error for both stable and unstable systems. A second mode of instability can occur when the lant estimate is inaccurate, causing the filtered-x LMS algorithm to diverge. This discussion shows why the conventional interretation of accetable lant estimation errors is incorrect in a feedback setting and resents a method for comuting the correct gradient filter. The chater is organized as follows. A brief introduction to filtered-e control is rovided in the context of filtered-x LMS control. Feedback loo instabilities and algorithm divergence are examined analytically in terms of the lant estimate for stable systems. This exlanation of instabilities exerienced during adatation resents some exected causes for soradic behavior in exerimental alications. Next, the results are extended to filtered-u control structures as well as the control of unstable, self-excited lants. In addition, the deendence of the instabilities on each other, and the use of online system identification techniques are examined for both adative feedback and filtered-u control. Finally, two simulations are resented. The first illustrates a feedback loo instability indeendent of algorithm divergence. The second simulation shows how a conventionally accurate lant estimate can yield an algorithm divergence indeendent of a loo instability. This is shown to agree with the analytical results. 6.1 FILTERED-E CONTROL OVERVIEW Adative feedforward control has received considerable attention over the last few decades, articularly with alication to active noise control [46]. The oular filtered-x variant shown in Figure 6.1 has clear advantages over comarable feedback structures. 97

109 r P n W c G y G ^ LMS e FIGURE 6.1 ADAPTIVE FEEDFORWARD CONTROL d Due to its feedforward architecture, the control system is inherently stable when W is an FIR filter and the adatation is sufficiently slow. In addition, the filtered-x LMS algorithm ensures convergence for hysical systems whose adative filter outut encounters a dynamic system before being sensed by the error signal. The LMS algorithm udates the future weights of the adative filter in resonse to a scaled, instantaneous measurement of the local gradient (derivative of the squared error) as W k + = W k µ 1 [6.1] k The filtered-x moniker indicates that the reference signal must be filtered by an estimate of the control-to-error dynamics in order to accurately estimate the correct gradient at the error signal. This can be seen by comuting the gradient directly as the derivative of the squared error with resect to the weight vector as shown below. 98

110 E( z) = N( z) G de( z) E( z) = = G dw ( z) r e = G * k k = e k ( z) W ( z) R( z) 2 r k = 2[ ek ] ek = 2( G * k ( z) R( z) ) e k [6.2] where * denotes the convolution oeration in the samle domain. It is clear that the reference inut (r) must be filtered by the lant before being multilied by the error to comute the instantaneous gradient. Since the lant itself is not available in hysical alications, an estimate of that lant transfer function is tyically used. It is well known [39, 48, 40] that this estimate must be accurate to within 90 of hase when comared with the actual lant to ensure convergence of the algorithm. If the estimate is incorrect by more than 90, the algorithm will search in the wrong direction and eventually diverge 6. One of the rimary limitations of ractical imlementation of the adative feedforward controller of Figure 6.1 is the requirement for an uncontrollable, coherent reference signal. The LMS algorithm assumes that r is highly correlated with n (the disturbance to be canceled) and that the outut of the adative filter (c) does not influence r. If the former is violated the control erformance suffers; if the latter is not satisfied, a feedback loo is introduced that might become unstable during adatation. In active noise control alications, it is often difficult to obtain a reference signal that is both coherent with the disturbance and not influenced by the controller. Adative feedback control attemts to remedy this roblem. Figure 6.2 illustrates a tyical filtered-e controller arrangement [56, 48]. 6 This result has been roven using a variety of tehcniques [39,48,40]. However, the literature only contians roofs for single sinusoids and/or secial samling conditions. Although no broadband roof of the 90 rule exists, all exeriments and simulations confirm the criteria in a broadband sense. Proving this criteria for generalized systems may be a source of future work. 99

111 - G ^ r x W c G y G ^ LMS e - d FIGURE 6.2 FILTERED-E CONTROL This arrangement emloys the filtered-x LMS algorithm with an estimate of the actual controlto-error ath, Ĝ. The external disturbance (r) enters at the error sensor along with the filtered control outut as in Figure 6.1. Since there is not another reference signal available, the error is used to estimate the reference signal (x) by subtracting the influence of the controller from the error signal sensor. x = ( r + G c) Gˆ c [6.3] In view of [6.3], if Ĝ = G the reference signal is exactly equal to the disturbance to be canceled and is therefore coherent and uncontrollable. Significant difficulties can arise when Ĝ deviates from G. Before continuing, it should be noted that Figure 6.3 illustrates the same system of Figure 6.2, only redrawn. 100

112 r W c G 1-(G -G ^ )W y ^ G 1-(G -G ^ )W LMS e d FIGURE 6.3 FILTERED-E CONTROL REDRAWN This reformulation of the filtered-e block diagram into a feedforward, filtered-x style system, clearly shows the inherent deendencies that the system has on the adative filter. Here we see that the adative filter is a art of the control-to-error ath and can also adat oles. The effects of these deendencies on the stability of filtered-e controllers are the focus of this chater FEEDBACK LOOP INSTABILITIES Simly using an adative filter in a control system does not imly global stability. In fact, if it is emloyed in an arrangement as in Figure 6.2, stability can not be guaranteed during adatation. Consider the system of Figure 6.2 at a moment in time when the adatation is slow, or has stoed. A transfer function exressing the system inut to outut relationshi is y r 1+ Gˆ W = 1+ Gˆ W G W [6.4] If Ĝ is exactly equal to G the denominator vanishes, leaving a strictly feedforward system. Assuming W is an FIR filter and the control-to-error ath is a stable system, the system dynamics are guaranteed to be stable regardless of W. If however, Ĝ is not equal to G at any frequency, the system oles can become unstable and are defined by the denominator of [6.4]. 101

113 There are many classical control techniques that can be used to analyze the stability of linear systems including root locus, Routh Hurwitz, Nyquist lots, and Bode lots [50]. For exerimental systems where dynamics are not exlicitly known, the most convenient and intuitive of these is the Bode analysis. The Bode stability criteria uses the oen loo frequency resonse function to redict closed loo erformance. The oen loo transfer function of Figure 6.2 can be exressed as G W 1+ Gˆ W [6.5] At frequencies where the magnitude of [6.5] is greater than unity for hase values equal to 360 degree multiles (for ositive feedback), the loo will be unstable. It should be noted that the Bode analysis assumes that the oen loo transfer function is stable, i.e. that the roots of (1+ Ĝ W) are in the left half lane. If they are not (determined by examining the oen loo frequency resonse of Ĝ W), the Nyquist criteria rovides a similar stability analysis for the closed loo system assuming the number of unstable roots is known exlicitly. 7 It should be clear from [6.4] and [6.5] that any loo instability resulting from unstable oles of [6.4] is a function of both the adative filter and the error between the lant and its estimate. This is a system deendent henomenon which cannot be redicted by assuming W only reaches a fraction of the otimal solution as suggested by [53]. The otimal adative filter (found when the error is set to zero) is 7 Inherent in the use of Bode and Nyquist stability criteria is the assumtion of linearity and time invariance. While the systems considered here are tyically linear, or oerating within the bounds of any resent non-linearities, they are not strictly time invariant because of the resence of the adative filter W in the oen loo lant. Nevertheless, these stability tools can be alied assuming that the adative filter is held fixed at the oint in time at which they are examined. This gives no guarantees concerning future stability, but rovides an indication of the instantaneous stability assuming adatation has stoed. 102

114 W ot 1 = [6.6] Gˆ Assuming a lant estimate error exists, and the adative filter has reached the otimal, [6.4] can still become unstable at a secific frequency even though the numerator is zero. Likewise, it is ossible that the denominator of [6.4] is stable, and the system has driven the error to zero with no unstable roots. The stability of the system must be examined on a case by case basis. It will be a function of the frequency deendent error between the lant and lant estimate as well as the size and shae of the adative filter, regardless of whether it has reached the otimal solution. Equation [6.6] illustrates that the otimal solution is only a function of the lant estimate, which we have obtained a riori. This raises imortant questions about the urose of adating if the otimal solution is already re-determined. For broadband disturbance suression, [6.6] must be satisfied at every frequency. Tyically this is an imractical modeling task for nonminimum hase systems or finite duration FIR filters. Adatation of W will result in the best comromise for control by inverting at the secific frequencies contributing most to the mean squared error. Tracking changes in those disturbance frequencies also resents a good argument in favor of adating W which will ensure the otimal is maintained ALGORITHM DIVERGENCE Conventional interretation of the lant estimation error for the filtered-x LMS algorithm indicates a 90 error between the control-to-error ath and it s estimate is tolerable. This discussion illustrates that this interretation is not valid for the filtered-e system of Figure 6.2. First consider that x = y Gˆ c = r + ( G Gˆ c [6.7] ) follows from Figure 6.2 and [6.3]. 103

115 Assuming slow adatation, it is ossible to rearrange Figure 6.2 such that the disturbance (r) is the system inut as well as the exogenous disturbance; Figure 6.3 results. The feedforward structure of Figure 6.3 facilitates the understanding of the actual lant error analysis as comared to the conventional lant error. The conventional lant estimate error of this system is the difference in the control-to-error ath and the estimate of the control-to-error ath used for the filtered-x ortion of the algorithm. From Figure 6.3, this can be seen to be G 1 ( ˆ G G ˆ G ) W 1 ( ˆ G G < ) W o 90 [6.8] This is exressed equivalently as o G ˆ G < 90 [6.9] and is hereafter defined as the conventional lant estimation error (which is equivalent to the lant estimation error in view of [6.2]) because of its common usage in all adative controller arrangements. This is consistent with the conclusions reached in [6.2] and reresents the common interretation of the lant error that must be satisfied to ensure convergence for filtered-x LMS control. The filter receding the LMS algorithm (the second term in [6.8]) is the conventional reference signal filter when emloying the filtered-x control strategy of Figures 6.2 and 6.3. Conventional thinking dictates that as long as [6.9] is satisfied, convergence to the otimal solution (minimum MSE) will continue. The instance when Ĝ is exactly equal to G satisfies this constraint because the feedback ath is eliminated from the actual control-to-error ath of Figure 6.3. However, when Ĝ is not recisely equal to G, a feedback ath is introduced that is a function of the adative filter W. We will now consider the case during adatation, when 104

116 105 Ĝ has an arbitrarily small amount of hase estimation error with resect to G, causing the feedback loo to exist. As before, the LMS algorithm udates the weights based on the negative gradient as in [6.1]. For simlicity we examine the error vector gradient which defines the filter used to filter the reference signal in the filtered-x algorithm. The actual error gradient of the filtered-e system is now comuted based on the nonzero feedback ath in the hysical control-to-error transfer function of Figure 6.3. ) ( ) ( )) ( ˆ ) ( ( ) ( )) ( ˆ ) ( 2( 1 ) ( ) ( ) ( ) ( ) ( )) ( ˆ ) ( ( 1 ) ( ) ( ) ( ) ( ) ( ) ( 2 2 z X z W z G z G z W z G z G z G z dw z de z E z W z G z G z X z W z G z X z D z E + = = = [6.10] The filter used to filter the reference signal in [6.10] must now be comared to what is conventionally used by the filtered-x in Figure 6.3 to estimate the gradient (without the exlicit deendence on z) as o 90 ) ˆ ( 1 ˆ ) ˆ ( ) ˆ 2( < + W G G G W G G W G G G [6.11] Equation [6.11] reresents the difference in the filter that should be used to filter the reference based on the actual gradient estimate, and the filter that is conventionally used in the filtered-x formulation of Figure 6.2. The well-known 90 hase limitation between the hysical control-to-error ath and the lant estimate has historically been derived only for a single sinusoid [40, 41, 39]. Assumtions made for these derivations rely on a filtered-x roblem formulation where there is no deendence of the control-to-error ath dynamics on the adative filter. An examination of Figure 6.3 shows

117 that unlike the feedforward situation, where we adat an FIR filter W(z) that linearly affects the control signal, the filtered-e structure is essentially adating an IIR filter H(z) given by G ( z) W ( z) H ( z) = [6.12] 1 ( G ( z) Gˆ ( z)) W ( z) where the arameters to be adated are contained in the FIR transfer function W(z). To minimize the mean square error, the gradient is still comuted as in [6.10] and can be exressed as the artial derivative of H(z) with resect to W(z) H ( z) W ( z) = 1 2( G ( z) Gˆ G ( z) ( z)) W ( z) + ( G ( z) Gˆ 2 ( z)) W ( z) 2 [6.13] Clearly, this is much different than the transfer function used in the filtered-e algorithm, which is the lower left block of Figure 6.3, and shown as the second term in [6.11]. The question then arises as to how much can the gradient used by the filtered-e algorithm differ from the true gradient and yet still roduce convergence to the otimal? It would seem to be extremely difficult to answer this question globally, since the equations describing nominal trajectories in the weight sace are nonlinear. Certainly near the otimum, Wot, the cost will be quadratic and we can aroximate H(z) in a neighborhood of the otimum as a transfer function that is affine in the arameters W as a Taylor s Series exansion about the otimal: H H ( z) z) = H ( z) + ( W W ) [6.14] Wot W ( z) ( ot Wot The first and last terms of [6.14] are moved to the forward (disturbance) ath between the reference and error sensor in Figure 6.3, while the control ath containing W is altered by the linearized control-to-error dynamics of H ( z) / W ( z) W ot. Because this filter is not a function of the changing adative filter, the standard 90 hase error analysis for the filtered-x algorithm 106

118 now alies. The conclusion is that as long as the hase of the gradient filter does not deart from the hase of H ( z) / W ( z) W ot by more than 90 at the frequency of interest, the algorithm will converge to the otimal solution. If the hase difference is greater than 90, then the otimal cannot be aroached, and the algorithm will diverge from the otimal. Since we have no idea a riori as to the value of Wot, it makes sense to use [6.13] as the gradient filter. It should be noted that in the simulations, divergence over a wide range of the weight sace occurred whenever the gradient filter used by the filtered-e algorithm differed from [6.13] by more than 90. This would seem to imly that a more global conclusion could be drawn concerning the accetable lant hase error with resect to broadband inut signals. Thus far such a roof has not been made. (Note that the negative sign comuted in [6.10] cancels with the negative sign of [6.1] when the FX-LMS algorithm is imlemented so that the sign of the filter agrees with the hysical control-to-error ath redicted by [6.14] from [6.13]). Returning to the conventional imlementation of the filtered-e controller, the inequality exressed by [6.11] defines the difference in the actual lant estimation filter and the conventional lant estimation filter that is tyically used. It should be clear that when comaring [6.11] and [6.9], the errors will result in different redictions of algorithmic stability as a function of conventional lant error. It is therefore conceivable that a small lant estimation error in conventional terms that satisfies [6.9], could roduce a large error that violates the actual estimation constraint [6.11], causing an unexected algorithm divergence, but one that is redicted by [6.11]. Given this result, one might assume that since the actual gradient filter is known, it can be used in lace of the conventional filter to ensure that the lant estimate is accurate enough to satisfy [6.11]. Since the actual lant estimation filter of [6.10] is a function of the difference in the actual lant dynamics (G) and the estimate of the dynamics ( Ĝ ), this is imossible. If the actual dynamics were known exactly, they could be used to eliminate the feedback ath resulting in a strictly feedforward system. The tacit assumtion in conventional adative feedback control is that the estimate exactly equals the actual lant and there is no error, thereby turning [6.11] and [6.4] into [6.9] and the numerator of [6.4]. In ractice there always 107

119 108 exists some finite error between the estimate and the actual control-to-error ath. However, this can only be analyzed in a simulation where the error can be exlicitly controlled and the actual lant is known exactly. It is interesting to examine the behavior of [6.11] as a function of the adative filter W. Evaluating [6.11] when W=0, results in o 90 ˆ ) ˆ ( 1 ˆ ) ˆ ( ) ˆ 2( < = + = W G G W G G G W G G W G G G [6.15] Also noteworthy is the result when [6.11] is evaluated when the adative filter reaches the otimal solution of [6.6]. 0 ) ˆ ( 1 ˆ ) ˆ ( ) ˆ 2( 1 ˆ = + = G W W G G G W G G W G G G [6.16] Therefore at the incetion of control, assuming a zero initial condition on the adative filter, the conventional gradient error estimation is valid. In addition, when the adative filter reaches the otimal solution, there is no estimation error, regardless of the choice of Ĝ! As a result, divergence of the adative algorithm due to inaccurate lant estimation should only be exected during adatation. Although, ractically seaking, a finite length FIR filter will not likely have the ability to invert the control-to-error ath estimate at every frequency in the controllable bandwidth. Therefore gradient divergence as redicted by [6.11] will always remain a ractical concern when imlementing adative feedback controllers like the filtered- E.

120 6.2 PRACTICAL APPLICATIONS AND CONSIDERATIONS INTERRELATED INSTABILITIES The two instability modes resented above are connected to each other after an initial transient. Each instability can initiate indeendently, but because of the inherent deendence on W in the gradient estimate, they will eventually influence each other. Suose that a conventional control-to-error ath estimation error exists, but that [6.11] is satisfied and the adative filter is converging toward the otimal solution. Because an error exists, the oles of [6.4] exist and are moving as a function of the error and the adative filter. It is ossible for one of these oles to leave the unit circle during adatation and create a feedback loo instability. This can haen indeendent of violating [6.11]. However, as the loo instability grows in magnitude, the mean squared error will increase and cause the adative filter to resond. In view of [6.11], a changing adative filter will affect the actual gradient filter error, and otentially cause it to exceed 90. Therefore a feedback loo instability can cause the hase error to exceed the 90 criterion established by [6.11] resulting in algorithm divergence. Alternatively, the adatation could otentially re-stabilize the system! It is also imortant to note that because of the deendence of the control-to-error ath on the adative filter, it is ossible that the hase estimation error defined by [6.11] could be selfcorrecting during adatation. In other words initial error in excess of 90 may not guarantee algorithm divergence if the adative filter changes the control-to-error ath in such a manner that the actual hase error is reduced. Although each instability mechanism can initiate indeendent of the other, they are ultimately codeendent through the adative filter magnitude. That is, if one grows without bound, the other will follow. For this reason, it is virtually imossible to ascertain the mechanism of instability in exerimental alications because the time scales are tyically too fast. The simulations to follow will illustrate these henomena. 109

121 6.2.2 FILTERED-U LMS ALGORITHM The filtered-e controller examined above can be viewed as a secial case of the more generalized filtered-u LMS algorithm [42]. The filtered-u algorithm, often emloyed in duct noise control roblems, is illustrated in Figure 6.4. v P F n x A c G y G ^ LMS B LMS G ^ e d FIGURE 6.4 FILTERED-U CONTROL Here there are two control sensors: the ustream reference microhone detects the original source disturbance (v) and generates the inut to the algorithm (x) while the downstream error microhone senses the noise (n) after being altered by the duct acoustics existing between those two sensors, reresented by P. Because the two sensors are in the same duct, when the control signal (c) is alied to the system it influences the error signal through the control-toerror ath dynamics (G) as well as the ustream reference sensor through a feedback ath, F. To see the similarities between the filtered-u and filtered-e structures we need to consider the case when the reference and error signals are identical. When this occurs, hysically, there is only one microhone in the duct and therefore no reference signal v. The transfer function F disaears, P becomes unity, and the reference signal, x, is the same as the error microhone measurement, y. The disturbance now aears as an exogenous inut to the error sensor. These changes and the resulting system are deicted in Figure

122 n x A c G y G ^ LMS B LMS G ^ e d FIGURE 6.5 FILTERED-E ADAPTIVE IIR CONTROL Comaring Figure 6.2 and Figure 6.5, an obvious difference exists in the controller structure. Figure 6.2 (the filtered-e) uses a fixed estimate of the control-to-error dynamics to comensate for the feedback ath created by using the error signal as the reference. Figure 6.5 illustrates a filtered-e controller structure as well, where the feedback ath is comensated adatively by the adative IIR form. Equating the transfer functions of Figure 6.2 and Figure 6.5, we see that the two systems are identical when A = W and B = -WĜ. However, because of the adative IIR structure there is no guarantee that these will be the otimal filters or that they will ever be reached [48]. As with the adative feedback analysis, the filtered-u system of Figure 6.4 has the same loo stability limitations. Because the reference signal is controllable and the adative feedback comonent does not begin at the otimal solution, the ossibility for feedback instabilities exists. In addition, because of the feedback to the reference signal and the existence of an adative feedback filter (B), the conventional gradient estimate cannot be accurate during adatation as shown next. Block diagram algebra of Figure 6.4 reveals: 111

123 G ( z) A( z) E( z) = D( z) P( z) V ( z) V ( z) 1 B( z) A( z) F( z) V ( z) = X ( z) F( z) C( z) [6.17] Examining the actual gradient of the error signal (which is roortional to the gradient of the cost function) with resect to the forward ath filter (A) results in de da BG G G (1 B) FG (1 B) = V = X + C [6.18] ( 1 B AF) (1 B AF) (1 B AF) Conventional alication of the filtered-u algorithm uses an estimate of the control-to-error dynamics ( Ĝ ) to filter the inut of the adative filter for udating the weights. In the case of the forward comonent of the IIR filter emloyed here, the inut is x and it would tyically be filtered by Ĝ. In fact, the actual gradient estimation includes the feedback ath and is also a function of the outut signal y as shown in [6.18]. A similar analysis can be erformed for the artial derivative of the error signal with resect to the feedback adative filter B, yielding similar results. It is therefore clear from [6.18] that the conventional and actual gradient estimation errors can differ quite significantly, and may unredictably result in algorithm divergence even when [6.9] is satisfied ONLINE SYSTEM IDENTIFICATION Conventional Identification Methods Online system identification is tyically emloyed to track changes in the control-to-error dynamics over time. For adative feedforward systems with time varying control-to-error transfer functions, this technique may be required to maintain the correct gradient estimate in the LMS algorithm. Figure 6.6 [48] illustrates one ossible technique for erforming the online system identification for a filtered-x feedforward system. It is assumed that the lant G is time varying and must be estimated by continually udating Ĝ. 112

124 x P W c ^ G LMS G y coy n G ^ d LMS FIGURE 6.6 ONLINE SYSTEM IDENTIFICATION As shown above for filtered-e control, the actual control-to-error ath is a function of the adative filter, and is therefore changing with time. The transfer function coefficient of X in [6.10] reresents the actual control-to-error ath estimate that should be used to filter the inut to the LMS algorithm. The question then arises, if the actual gradient equation is known, why can t it be used in the filtered-x udate algorithm. Uon insection of [6.10], it should be clear that G is never known exactly. Ĝ is used to estimate G; and if the assumtion is that there is no estimation error Ĝ = G and [6.11] becomes [6.9]. However, in ractice there is always an estimation error, no matter how small, that makes [6.11] the correct gradient error criteria. Therefore, without knowing G exactly, the (G - comuted directly. Ĝ ) terms of [6.10] cannot be Another alternative for generating a more accurate estimate of the actual gradient is the online identification of the gradient filter. Once the loo is closed (the adative filter becomes nonzero), an online identification rocedure will take into account the feedback loo and the adative filter. However, it should also be obvious that a linear system identification of any inut/outut relationshi of Figure 6.2, will never result in the required transfer function coefficient in [6.10]. This is shown clearly by considering that the characteristic equation of Figure 6.2 is the denominator of [6.4]. Therefore any attemt at an online linear system 113

125 identification cannot result in the transfer function of the first term in [6.11] which has a quadratic denominator. A similar argument can be made for an online system identification emloyed in the filtered-u system of Figure 6.4. The quadratic denominators of [6.18] revent an online system identification rocedure from ever yielding the roer frequency resonse. [48] and [37] roosed that the feedback terms resulting from F and B in [6.18] were negligible. In this case the gradient estimates for both adative filters become Ĝ = G; a result that ignores the recursion of both B and F. However, an online system identification rocedure cannot avoid identifying the hysical feedback aths, F and B, in Figure 6.4. Therefore the frequency resonse resulting from the online identification rocess (Figure 6.6) of the filtered-u algorithm (Figure 6.4) is G ( 1 B AF) [6.19] which is not an accurate reresentation of their roosed gradient filter aroximation, G A New On-Line Identification Technique There is one ossible solution that can be roosed for reducing the detrimental effects of the (G - Ĝ ) terms on the gradient estimate and the otential for unstable feedback loos. Consider again the block diagram of Figure 6.3 illustrating the deendence of the loo stability and gradient filter on the lant estimate error and the adative filter. Let F 1 = ) [6.20] 1 ( G G W ) Assuming W is an FIR filter and G is a stable lant, the loo stability requirement can be restated as: F must have stable oles. 114

126 The gradient filter error criteria exressed by [6.11] can also be restated as ( G F ( G ( G 2 ) ( Gˆ Gˆ Gˆ ) + ( F F) < 90 ) + ( F) < 90 2 o F) < 90 o o [6.21] so that the hase error to consider is the sum of the hase error in the original lant estimate lus the hase of the feedback term, F. Addressing the stability of F (and thus the closed loo system) first, consider the block diagram of Figure 6.7 below r W A - G ^ c B G n y FIGURE 6.7 ONLINE SYSTEM IDENTIFICATION The transfer function between the outut B and the inut A is equal to [6.20], and thus F. At locations in the frequency resonse of [6.20] where the amlitude is high, or increasing, a ole is nearing the axis. Continual monitoring of this transfer function, identified eriodically during adatation, will indicate stability. Ideally, [6.20] will remain low, or near unity to ensure that no oles are nearing the axis. This is also an ideal scenario for ensuring gradient convergence of the LMS algorithm. In [6.10] we showed that the ideal gradient filter is reresented by GF 2. Conventional imlementation of the Filtered-X LMS algorithm uses Ĝ F as the gradient filter, creating the estimation hase error described by [6.11]. An available alternative is to use the identification of B/A reresented by Figure 6.7 to generate an online estimate of F while the adative filter, 115

127 W is held constant, thereby generating an Fˆ. If we multily our estimate, Fˆ, by the existing conventional gradient filter, the new hase error criterion of concern can then be exressed as ( G ( G ( G FF) ( Gˆ Gˆ Gˆ FFˆ ) < 90 ) + ( F Fˆ ) < 90 o ) + ( F F) + ( F Fˆ ) < 90 o o [6.22] where the first term of the first line is the ideal gradient filter and the second term is the roosed estimate of the ideal gradient filter. Here we can conceivably identify both the oen loo lant and the closed loo control system within enough accuracy to ensure less than ninety degrees of total hase error for the criterion of [6.22]. As the adative filter changes, identification of Fˆ should be eriodically udated to revent algorithm divergence. While this aroach can theoretically hel some of the stability and gradient roblems identified earlier in this chater, continual monitoring is required in order to ensure loo and gradient stability. An alternative online identification technique can also be roosed. Consider that A = 1 = 1 ( G G ) ) W [6.23] B F can be obtained from an inverted system identification of Figure 6.7. When A = B, F = 1 requiring that (G - Ĝ ) = 0 when W is non-zero. An effective way to ensure that A and B are equal is by using an LMS based adative controller as shown in Figure

128 B G P W ^ G W - e = A-B W LMS FIGURE 6.8 MODIFIED ONLINE IDENTIFICATION A significant benefit of this formulation, as seen in [6.23], is that G aroaches Ĝ most effectively at frequencies where W is high; i.e. the lant estimation error is weighted by the adative filter magnitude, corresonding to frequencies requiring significant control. These are also the frequencies where gradient convergence is most critical because they are the frequencies with the highest MSE. Now suose that the gradient filter in the adative feedback LMS algorithm is NOT modified as suggested by [6.22], but instead the online identification of Figure 6.8 is eriodically carried out while W remains fixed. The original system identification of G is used in the conventional Filtered-X formulation and does not require a weighted or biased estimate, since we want to ensure convergence for unknown disturbances over the entire bandwidth of the lant. Therefore that original estimate will be called Ĝ 1and will remain fixed during all adatations. The identification carried out in Figure 6.8 rovides a weighted estimate of the feedback comensation filter which is critical in affecting loo stability and gradient filter errors. That resulting estimate will be known as be modified, the actual gradient error can be exressed as Ĝ 2. Following [6.11], and since the gradient filter will not 117

129 G ˆ ) ( ˆ 1 2( G G 2 W + G G ˆ o ( G F F ) ( G F ) < 90 ( G 2 2 Gˆ 1 1 ) + ( F 2 2 ) + ( F 2 2 ) 2 W F 2 2 ˆ G 1 1 ( ˆ G G ) = ( G Gˆ 1 2 ) + ( F < 90 ) W 2 o ) < 90 o [6.24] where F2 is [6.20] using Ĝ 2 in lace of Ĝ. Now, the hase error amounts to the error in the original estimate lus the hase associated with the feedback ath using the second weighted lant estimate. When F = 1, the hase of F will aroach zero as desired. It can also be easily shown that when the adative filter aroaches its otimal, the angle of F is the difference in the lant and the second online estimate. There will always be some amount of error attributable to a least squares system identification. However, it is now ossible to lace some heuristic bounds on how much error can be tolerated based on For instance, assume that the initial online identification of Ĝ 1 is carried out with an error of less than 45 degrees at every frequency. Based on [6.24] the remainder of the allowable hase will come from F2, containing the secondary online identification rocedure that udates the feedback lant estimate can lace a bound on the magnitude of the term (G - Ĝ 2. Examining [6.20] we Ĝ )W (such that it cannot be above 0.707) to ensure that the hase of F is never greater than 45 degrees. (If it is never greater than unity, the hase angle of F will always be below 90 degrees). Using the weighted least squares adatation of Figure 6.8, this reresents a realistic tolerance for broadband system identification. In ractical alications there will always be some error associated with an adative system identification rocedure. This section rovided some bounds that can be laced on that error to ensure stable convergence. However, eriodic identification as required by this online identification rocedure, may not be an accetable solution for real world alications. 118

130 6.2.4 EXTENSION TO SELF-EXCITED SYSTEMS All revious analyses and discussions in this Chater have focused on systems having exogenous disturbances from stable lants. The thermoacoustic instability can generate disturbances through the nonlinear limit cycling behavior of an unstable lant. Chater 5 discusses the exerimental alication of adative feedback control to the thermoacoustic instability. In Chater 5 it is shown that the most aroriate choice for a control-to-error ath estimate is the stable acoustic dynamics lus actuator dynamics so that Ĝ = Gact; exact estimation caability is assumed throughout Chater 5. Here it is assumed that the lant estimate is not able to exactly model the desired transfer function. Figure 6.9 illustrates the adative feedback controller alied to a self-excited instability in the form of a thermoacoustic instability just as in Chater 5. - r W G ^ c G act n G a G f y G ^ LMS e - FIGURE 6.9 SELF-EXCITED SYSTEM FILTERED-E CONTROL d Adative feedback and filtered-u controllers have been alied to the thermoacoustic instability in an attemt to reduce the ressure oscillation that may change with changing oerating conditions [25, 22, 23, 28]. Unfortunately, the same instability mechanisms for the stable lant control exist for the unstable lant control. The characteristic equation for the adative feedback control system of Figure 6.9 is reresented by 119

131 G G W ( G + Gˆ G G Gˆ ) [6.25] 1 a f act a f There are no guarantees on the stability of the roots of [6.25]. The adative filter could easily cause a root of this equation to become unstable during adatation, regardless of the choice of Ĝ. This is unlike the stable lant control where the feedback loo is canceled if Ĝ = G. Without accurate knowledge of the self-excited system dynamics, it is imossible to limit the roots of [6.25] to strictly stable values. It should be noted that when W = Wot the roots of [6.25] are the roots of the denominator of Gact. But no guarantees can be made on reaching the otimal solution if the gradient estimate is incorrect. The gradient of the cost function is even more comlex for the self-excited case. The filter to which the inut to the LMS algorithm should be alied will be a function of the adative filter, lant estimate and actual lant as before, but will also be influenced by the self-excited system dynamics. Comutation of the actual gradient of the filtered-e self-excited system should not be required in order to recognize that the conventional estimate of Gact will be inaccurate with resect to the actual gradient. 6.3 SIMULATION Three simulations have been designed that illustrate the two secific instability mechanisms indeendently. Each simulates the stable disturbance rejection system of Figure 6.2 using different lants and lant estimates. As set forth in the revious discussions, the conventional lant error refers to the criterion of [6.9] which was shown to be inaccurate for adative feedback controllers. Equation [6.11] reresents the actual estimation error criterion FEEDBACK LOOP INSTABILITIES This simulation illustrates a case where the conventional lant error is in excess of 90 at many frequencies, but no algorithm divergence is observed. Instead, an unstable feedback loo is generated due to a ole of [6.4] leaving the unit circle. 120

132 The lant shown in Figure 6.2 was chosen to have unity magnitude and a linear delay of 25 samles while the lant estimate was chosen to have unity magnitude and 19 samles of delay at a samle rate of 2000 Hz. The exogenous disturbance was chosen to be a single sinusoid at 32 Hz with additive white noise at a lower level at every other frequency. The adative filter was a 2-weight FIR filter with a convergence arameter of The difference in hase between the lant and estimate increased almost linearly, reaching an excess of 1000 of hase error by 1000 Hz, thus allowing for the ossibility of divergence of the weights due to a conventionally inaccurate system identification. Figure 6.10 illustrates the uncontrolled (solid) and controlled (dotted) ower sectra of the tonal disturbance from the stable lant at 32 Hz. 121

133 50 0 db db Degrees Freq.(Hz) FIGURE 6.10 FEEDBACK LOOP INSTABILITY AND BODE PLOT PREDICTION Initially, the tone is suressed with only two adative filter weights but the loo gain that accomanies the otimal adative filter causes a loo instability at 810 Hz. This is accurately redicted by the Bode gain/hase relationshi in that the oen loo frequency resonse magnitude is in excess of 0 db at the 810 Hz hase crossover frequency. This adative feedback controller diverges because of the feedback loo instability, not divergence of the algorithm. Increasing the number of adative filter weights reduces the gain of the adative filter at 810 Hz thus eliminating that loo instability. However, the added 122

134 hase and filter comlexity causes oles at other frequencies, away from the disturbance, to become unstable before convergence is achieved. For this simulation, the conventional hase error between the lant and estimate is still in excess of 90 at many frequencies throughout the control bandwidth but the algorithm never diverges because the actual hase error is less than 90. A second simulation was develoed that illustrates a feedback loo instability indeendent of algorithm instability. The rimary difference in this simulation is that no hase error between the lant and lant estimate was created. Instead, a modest DC gain difference in the two dynamic models was created. With a (digital) zero at and a ole at 0.97, the control-toerror ath had a gain of 1.3 while the estimate of that ath was given a gain of 1.0. Figure 6.11 illustrates the actual and estimated control-to-error ath frequency resonses as well as the difference in hase between them (lowest lot) Mag Phase Phase Delta Freq.(Hz) FIGURE 6.11 CONTROL-TO-ERROR PATH AND ESTIMATE This simulation illustrates two interesting behaviors. First, a feedback loo instability revents convergence to the otimal solution of [6.6] because a ole is leaving the unit circle at 1000 Hz, half the samle rate. Stated differently, the otimal solution to this roblem results in an unstable ole because of the amlitude difference in the estimate creates the denominator of 123

135 [6.4] which is a function of the adative filter. Second, a searching behavior not unlike that described in the revious chater is witnessed. The time resonse of this adative feedback simulation is shown in Figure Mag Iteration x 10 4 FIGURE 6.12 LOOP INSTABILITY SEARCHING BEHAVIOR Because of the adative nature of the controller, once the increase in MSE is detected, the weights are corrected. However, since the otimal solution for a two weight adative filter requires that the loo contain an unstable ole, the algorithm continues to drive toward that solution. A searching behavior begins that is a function of the convergence arameter µ. To see this more clearly, consider the closed loo oles of Equation [6.4]. Since we are in the z- domain, an absolute magnitude of the comlex ole coordinates that is greater than unity indicates an unstable ole. Figure 6.13 shows the ath of the absolute magnitude of the oles of [6.4] over time. 124

136 Absolute Mag Iteration/400 FIGURE 6.13 POLE MOVEMENT WITH TIME It is clear that as the system adats toward the otimal solution, one ole has a trajectory leading out of the unit circle. The adative filter corrects it but the intermittency continues as the ole moves in and out of the unit circle. The otimal solution requires an unstable ole in this situation because of the adative filter order. While any hase and magnitude can be obtained using a two weight FIR filter, this is only true at a single frequency. Because of the limited control over the entire bandwidth, the otimal solution of [6.6] cannot be effectively inverted at every frequency, only the frequency generating the greatest MSE. This results in excess out of bandwidth magnitude and eventually an unstable ole at half the samling rate. This roblem is easily remedied by increasing the adative filter order where more recise inversion of the estimate of the control-to-error ath can occur over the entire bandwidth. Nevertheless, this simulation effectively illustrates the limitations of the adative feedback controller in terms of loo stability even when a very accurate system model is available. 125

137 6.3.2 ALGORITHM DIVERGENCE When the adative filter is at zero or its otimal solution, the actual lant error is either equivalent to the conventional error [6.15] or is degenerate [6.16]. Therefore we are only concerned with the times during adatation when neither of these conditions are satisfied. In addition, we are interested in illustrating a case where the conventional lant estimation error is arbitrarily small but the actual lant estimation error exceeds the constraints of [6.11]. Finally, it is imortant to continue to differentiate the gradient based algorithm divergence from the loo instability resented above. This simulation accomlishes each of these goals. Because the actual lant estimation error [6.11] established herein is a function of the lant estimation error as well as the adative filter, it is imossible to generalize the exected gradient error. For this articular simulation the lant estimate was chosen to be unity so that the otimal adative filter was 1. The control-to-error ath was chosen to be a comlex zero at 200 Hz with a daming ratio of.025 combined with a comlex ole at 300 Hz with a daming ratio of.016. Due to the way the simulation was designed an additional delay was imarted to both the control-to-error ath and it s estimate (with a samle frequency of 2000Hz). These choices resulted in a conventional hase difference in the lant and lant estimate that was less than 4 at the disturbance frequency of 150 Hz. Therefore in a conventional interretation where the tolerable error satisfies [9], there is no chance for gradient divergence at 150 Hz. In a user controlled simulation environment, we have access to both the actual lant and the lant estimate. Therefore it is ossible to comute the hase difference described by [6.11] directly for a variety of adative filters. In order to effectively visualize the weight sace, a two weight adative filter was chosen. This is also tyically sufficient to control a single tone disturbance. Figure 6.14 illustrates the actual gradient error as comuted by [6.11], as a function of the two adative weights, at 150 Hz. The * s reresent the weight combinations that result in a gradient error of greater than 90 at 150 Hz; the absence of * s reresent areas where the actual estimation error is less than 90. Recall that the conventional gradient error for the entire weight sace, at 150 Hz, is less than 4. The otimal adative solution of [6.6] is shown as a 126

138 star at coordinates (0.98,-1.77). Note that the actual gradient error at (0,0) and the otimal, is within the 90 degree secification w w2 FIGURE 6.14 GRADIENT AND STABILITY IN THE WEIGHT SPACE Figure 6.14 also illustrates loo stability. For every weight combination shown, [6.4] was evaluated for unstable roots. The O s reresent areas in the weight sace where the loo has no unstable roots; areas without O s have at least 1 unstable root. By insection, it is ossible to see a union where the feedback loo is stable but the actual gradient is greater than 90 at 150 Hz ([6.11] is not satisfied) while the conventional gradient is less than 4 ([6.9] is satisfied). If we choose an initial condition of the weights in this region (-2.8 and 1.3 for examle), the algorithm diverges at 150 Hz but the feedback loos remain stable for some time. The uer ortion of Figure 6.15 illustrates the error signal with time while the lower ortion shows the increase in amlitude of the 150 Hz tone after control has been alied. 127

139 20 10 Mag n db Freq.(Hz) FIGURE 6.15 GRADIENT DIVERGENCE FOR 4 DEGREES OF PLANT ESTIMATION ERROR This examle illustrates that the conventional interretation of the lant hase error of [6.9] is insufficient to ensure convergence to the otimal solution during adatation. Equation [6.11] accurately redicts the actual hase error that can be exected during adatation. 6.4 CONCLUSIONS Filtered-E control and filtered-u control have been imlemented in the ast with varying degrees of success. Some systems illustrate stable and reeatable convergence while others diverge, even with accurate (in a conventional sense) lant estimates. This chater resents two distinct reasons for such divergence: feedback loo instabilities and inaccurate gradient estimation. Both modes of instability are heavily deendent on the form of the lant, lant estimate, and the adative filter. For this reason it is difficult to redict the onset of instability outside of a simulation environment since the actual lant can never be known exactly. It is obvious that any time an feedback loo exists, it has the otential of becoming unstable. Using an adative controller in a feedback system structure is no excetion. For an inaccurate lant estimation in the control of a stable lant, the controller can become unstable. In addition, in controlling an unstable lant, no choice of the lant estimate will guarantee a stable 128

140 system during adatation. The recognition and rediction of these unstable feedback loos for adative feedback and filtered-u control of both stable and unstable lants has not been reviously discussed. Simlifying assumtions have led researchers to adot a conventional lant estimate for the filtered-x LMS algorithm that estimates the oen loo control-to-error dynamics. It was shown for both stable and unstable lants that this estimate can be insufficient during adatation, and may generate enough error in the actual gradient to cause the algorithm to diverge. Therefore the conventional interretation of the gradient was differentiated from the actual gradient error to rovide a method to accurately redict the correct gradient during adatation. It was then shown that for an arbitrarily small amount of conventional hase error, the actual hase error could exceed the 90 criterion and cause divergence. Desite the difficulties shown here in guaranteeing stable erformance of filtered-u and filtered-e algorithms, they have erformed effectively in certain alications. An exact estimation of the control-to-error dynamics will eliminate feedback loos and result in an accurate gradient estimation for the filtered-x LMS algorithm. Limitations on the tolerable error in the lant estimation and actual lant are entirely system deendent, and cannot be evaluated in general. Therefore it is conceivable that a sufficiently accurate estimate is hysically achievable for certain alications such that feedback instabilities and algorithm divergence are avoided. This can only be determined exerimentally, on a case-by-case basis. 129

141 Chater 7 Time Averaged Gradient Control 7.1 INTRODUCTION Many algorithmic aroaches to active control of thermoacoustic instabilities have been investigated, and they are all hindered by significant limitations. As we have seen reviously, manually adjustable feedback controllers cannot effectively track changes in lant dynamics or combustor oerating conditions. Model-based controller designs such as H-infinity and LQG require accurate knowledge of the lant to be controlled. Prior discussion of filtered-x LMSbased adative controllers has shown them to be an imractical solution to the thermoacoustic control roblem. Some gradient-based methods, such as extremum-seeking control and the triangular search algorithm, do not require system models, but are not easily adated to multiarameter controllers. Here, a simle and versatile algorithm for suressing instabilities is described. The TAG adative control algorithm requires no detailed knowledge of the lant or system dynamics. Generally, time-averaged gradient (TAG) control algorithms automatically determine the direction of the local gradient and adats control arameters to minimize the mean-squared error of the inut signal. Because the gradient is comuted on-line, no a riori information regarding the detailed behavior of the thermoacoustic instability is required. The algorithm searches the erformance surface, which is a function of the control arameters, and finds a solution that locally minimizes the mean-squared error. The algorithm is versatile enough to udate any tye of control arameters (FIR filter coefficients, gain and time-delay arameters, etc.) and oerate with a wide variety of actuation schemes (roortional, on-off, subharmonic, etc.) with few or no algorithmic modifications. 130

142 The efficacy of such a simle control algorithm for the thermoacoustic instability roblem is due largely to the simle nature of this articular control roblem. Although the thermoacoustic feedback loo is extraordinarily comlicated with regard to the detailed hysics, we have seen that the control roblem is dominated in ractice by a single air of acoustic oles crossing the imaginary axis due to the gain of the flame dynamics subsystem. Stabilizing the system can usually be accomlished with extremely simle control structures. To maintain control over widely varying conditions, the controller must be made adative and the general TAG scheme is ideally suited for modifications that allow it to be successfully used for active combustion control. 7.2 TIME-AVERAGED GRADIENT CONTROL Time-averaged gradient control uses a gradient search to udate control arameters so as to reach the minimum mean-squared error of the given inut. The gradient descent algorithms and time-averaged gradient controllers resented here have many notable advantages over reviously investigated algorithms for ACC. The controller does not require any rior knowledge of the lant dynamics and no system identification is required. Since the algorithm continually seeks the minimum mean-squared error, it can adat to changing oerating conditions and revent controller-induced feedback instabilities. Finally, the TAG control method can be alied to any controller arameterization and actuation scheme and is ideally suited to single frequency control because of its inherent simlicity. Figure 7.1 illustrates the TAG controller alied to the combustion lant. The control filter, W, is udated by a gradient descent algorithm (GDA). The outut of the control exeriences the control-to-error ath dynamics indicated here by GACT. This controller design is differentiated from revious embodiments in that it does not require a searate measure of the noise (reference signal) and therefore does not require a control-to-error ath estimate. The control filter, W, does not have any strict imlementation requirements. As will be shown it can take on a variety of arameterizations such as an FIR filter, a constrained IIR filter, or a subharmonic actuation signal generator. The GDA seeks to minimize the error signal in all 131

143 cases, by udating the control arameters to find the minimum on the erformance surface defined by the mean squared error and the control arameters. NL G A G F W c G ACT e GDA FIGURE 7.1 TAG CONTROL OF SELF-EXCITED PLANT Gradient descent algorithms, such as Newton s method and the method of steeest descent, are caable of searching a erformance surface and locally minimizing multi-dimensional functions. Their rimary limitation is that the higher order filters necessary to control large bandwidths require longer adatation times than their LMS driven counterarts of lower order. Recalling that the thermoacoustic instability roblem is very narrowband, the following discussion resents these algorithms. Gradient descent algorithms measure the gradient of a cost function and continually udate the indeendent arameters to move in the direction of the minimum cost. The method of steeest descent is a simle imlementation of this concet and can be reresented by the following weight udate equation: w + = w µξ '( w ) [7.1] n 1 n n where the convergence arameter µ controls the seed of the weight udate. Equation [7.1] reresents the udate for a single weight at iteration n, but can be extended to the multi-weight case by relacing the scalar arameters with vectors. (It is also clarified that the weights of [7.1] can form the coefficients of the adative filter W in Figure 7.1, or some other controller 132

144 arameterization.) The gradient of the cost function for a given udate is comuted using a measurement of the cost at the current weight location and a erturbed weight location by the formula ξ ( w) ξ ( w δ ) ξ '( w ) = [7.2] δ where the cost is given by the mean-squared error (MSE) of some signal e: N 1 2 ξ ( w) = e ( n) [7.3] N This method minimizes the cost as a function of each filter weight so the comutational comlexity increases with the number of weights, resulting in a lengthy convergence time for large filter sizes. In the case of the thermoacoustic instability, it may be sufficient to adat as few as two weight arameters to attain a stable oerating oint and cost minimization. 2N samles are collected before a single weight udate can occur. For a two-weight adative filter, 3N samles are collected after two searate erturbations in order to udate the entire filter. n= 0 Newton s method is an alternative gradient descent method that finds a zero of an arbitrary function. In the case of cost minimization, the minimum mean-squared error may never become zero. However, the gradient of the cost will become zero at the local minimum. Newton s method derives from the definition of the derivative and can be exressed as f ( x) xn+ 1 = xn [7.4] f '( x) In general alications, we are only concerned with the multi-arameter case where the weight udate becomes 133

145 W 1 [ ξ ''( W )] ξ '( ) n+ 1 Wn W = [7.5] and ξ is the matrix of second artial derivatives, referred to as the Hessian matrix. Comuting and inverting this matrix greatly increases the comutational comlexity of the algorithm. The otential for singularity of the Hessian matrix also introduces comlications. Each of the weight-udate algorithms has advantages and disadvantages. Comutationally, the steeest descent is able to udate the weights much faster than Newton s method. However, Newton s method can converge to the otimal solution faster than the steeest descent algorithm. This is because the steeest descent algorithm moves in the direction of the steeest local gradient, which is only in the direction of the otimal solution if the initial condition is on a rincile axis of the error surface. But faster convergence, which imlies larger arameter stes, can be a liability when dealing with the non-quadratic cost surface generated by the feedback nature of the ACC control architecture shown in Figure 7.1. This fact, couled with the increased comutational comlexity and otential singularity roblems of Newton s method, makes the steeest descent technique a better choice for the thermoacoustic instability control solution. There are two limitations inherent to the TAG control structure shown in Figure 7.1: 1) feedback loos can generate instabilities and 2) the cost function may contain local minima. Although the gradient descent algorithm may drive the weights in the direction of the negative gradient, there is no assurance that a global minimum will be reached. There is also no guarantee that the minimum that is found will result in a stabilized system. Even with these constraints, the gradient descent algorithm has outerformed its filtered-x counterart in comarable simulations where the control-to-error ath G may change from trial to trial. It should also be noted that the two limitations are fairly easily dealt with in ractical imlementations. If a arameter ste results in the control feedback loo going unstable, this instability can be sensed, the arameter ste undone, and a smaller ste size chosen. To alleviate the local minimum roblem, the highly scheduled nature of real-world controllers will allow control gains near the global minimum to be swaed in as oerating oints are changed. In addition, some amount of random robing can be incororated in the algorithm when the 134

146 current oerating cost is viewed as too high. This random robing will eventually cause the controller to find a better arameter set and extract itself from a local minimum condition. Widrow and Stearns [49] examine the stability constraints of the steeest descent algorithm in detail. The convergence arameter µ must be less than the inverse of the maximum eigenvalue of the inut correlation matrix to ensure stable convergence. In a ractical setting, the maximum eigenvalue can be aroximated by the ta inut ower. The other arameters that will affect the convergence of the TAG controller are the erturbation size δ and the average duration N. After the weight vector has converged to the otimal solution, the erturbation will ensure continual tracking of lant changes. Small erturbations may cause slow convergence to the otimal but will reduce the minimum mean squared error after convergence. Alternatively, relatively large values of the erturbation may rovide more accurate estimates of the local gradient ermitting faster convergence; this is at the exense of an increased minimum mean squared error after convergence. The aroriate average duration, N, is largely deendent on the noise resent in the system. For non-noisy systems, low average durations can generate accurate estimates of the local mean squared error, and thus accurate comutation of the gradient. Longer average durations will lead to longer convergence times because the weight cannot be udated until the gradient is comuted. More detailed observations on the effects of these arameters on convergence and erformance of the control of a thermoacoustic instability are resented in conjunction with simulations and exeriments described next. 7.3 SIMULATION The thermoacoustic instability and TAG controller using the steeest descent algorithm were simulated using MATLAB. The simulation is included in the Aendix. The acoustic ortion of the combustion lant was modeled as a lightly damed single acoustic mode while the flame dynamics were modeled as a low ass filter. The modeled acoustic and flame dynamics were 135

147 GA = G = 2 f [7.6] s s e6 s A saturation nonlinearity was aroximated using a hyerbolic tangent function. While this is a crude model from a hysical standoint, it is sufficient to illustrate the controller erformance because the sectrum and RMS ressure of the hysical system is tyically dominated by a single frequency limit cycle and its harmonics. The gain for the self-excited loo was chosen such that the limit cycle was established after only one second. The actuator dynamics, GACT in Figure 7.1, include all dynamics associated with A/D and D/A conversion, the dynamics of the actuator, and the acoustic ressure resonse due to actuation. Two different actuator aths were chosen to illustrate different convergence behaviors of the TAG algorithm. Both contained the acoustic dynamics while one was a high gain, delayed version and the other used a lower gain without delays; both are shown here: 0.11z z GACT = G = ACT [7.7] 2 z 1.525z z z 1.525z For most exeriments discussed here, the TAG controller was a two-weight FIR filter whose coefficients were udated using the steeest descent algorithm. Deviations from this are exlicitly noted. Figure 7.2 illustrates how the feedback loo stability varies with a two-weight feedback controller. The stable region (darker) is surrounded by a larger region where at least one closed loo ole is unstable. The TAG controller drives the weights to the stable region as long as it does not encounter a local minimum. In addition, it will adat away from higher amlitude limit cycles that may be resent in the unstable region of Figure 7.2, thereby reventing controller-induced instabilities. 136

148 w w2 FIGURE 7.2 HIGH HEAT RELEASE STABILITY MAP Figure 7.3 illustrates a segment of the normalized erformance surface of the system of Figure 7.1 when the actuator ath contains delays. This corresonds to the same system for which the stability ma of Figure 7.2 was generated. It is aarent that a local minimum does exist on this erformance surface. It is therefore ossible that a small erturbation size might result in the algorithm settling to a local minimum. 137

149 MSE w w FIGURE 7.3 MSE PERFORMANCE SURFACE As discussed above, there are three algorithmic arameters that affect the convergence of the steeest descent algorithm to a stable solution: the convergence arameter µ (controls the seed of the weight udate), the erturbation size δ (controls the ste size used in comuting the gradient), and the average duration N (controls accuracy of cost estimate). Large convergence arameters may result in faster convergence or ossible divergence deending on the shae of the MSE surface. The aroriate erturbation size also deends on the shae of the MSE surface and may add noise to the overall sectrum because it continually changes in a square wave fashion even after convergence. The average duration should be increased as noise ollutes the sectrum. However, simulations indicate that s worth of average time is sufficient to rovide convergence for very high noise levels and a 175 Hz instability samled at 1600 Hz. Figure 7.4 illustrates the simulated TAG control of the limit cycle where the actuator dynamics included hase delay and added gain. For a non-otimized choice of the erturbation size and convergence arameter (uer trace), the error signal is seen to initially increase in resonse to the erturbation. After the weights are corrected by the steeest descent algorithm, the controller attenuates the limit cycle and begins driving it toward the minimum. The cycling amlitude seen in the error signal reresents the weight erturbation and subsequent re- 138

150 evaluation of the gradient and weight udate. The slow convergence and initial misdirection are corrected as seen in the lower trace, by changing the erturbation direction (adding a minus sign to δ) and increasing the magnitude of the convergence arameter µ Mag x Mag Samle x 10 4 FIGURE 7.4 SLOW AND FAST TAG CONVERGENCE As mentioned earlier, the convergence arameter size is limited by the stability of the algorithm. It is generally desirable to have a slightly overdamed resonse for noisy systems and gradient estimates to ensure continual convergence toward a minimum. Therefore the size of the convergence arameter is governed by the inut ower and the system noise and consequently controls the rate of convergence. The erturbation size is redominantly system deendent. The amlitude of the erturbation must be sufficiently large so as to affect the RMS level of the inut signal in a measurable manner. If the erturbation is too small, the system will not be affected, no gradient will be comuted and the weights will not be udated. This is equivalent to finding a local minimum on the gradient surface, where sufficient weight excitation is not available to significantly alter the gradient estimation. Figure 7.5 reresents another simulation result where the actuator dynamics were a scaled version of the acoustic lant without additional delays. Here we see that for a non-otimized 139

151 choice of the convergence arameter and erturbation, a new limit cycle (at a slightly lower frequency) is found. In this case, significant adatation stos and the system has not been stabilized; a local minimum is found on the cost function. To remedy this (result shown in lower trace), the erturbation size was doubled and the convergence arameter was increased. Larger erturbations in the weights ermit more global gradient estimates and the increased convergence arameter amlifies the gradient estimate to affect the weights more significantly. The result is a stabilized system. 4 2 Mag x Mag Samle x 10 4 FIGURE 7.5 LOCAL MINIMUM AND OPTIMIZED CONVERGENCE The effects of the erturbation size and convergence arameter seen in simulation can be categorized into a set of rules that can adatively control the convergence behavior of the TAG controller. These are summarized as follows: If the MSE increases after the first erturbation, change the sign of the erturbation. If the MSE increases after changing the sign, reduce the magnitude of the erturbation. 140

152 If the gradient is zero after the first several erturbations, increase the magnitude of the erturbation until a change in gradient is detected. If the MSE is near the minimum MSE, decrease the magnitude of the erturbation to reduce the misadjustment noise. If the system continually diverges, reduce the convergence arameter. If the gradient is too small to change subsequent MSE calculations, and the erturbation size has been increased, increase the convergence arameter. Clearly these rules will interact with each other and care must be taken when alying them in actual exerimental settings so as to avoid local minima and divergence. A subset of these rules has been imlemented in conjunction with the TAG controller in a quasi-exerimental setting. The rule based TAG was emloyed on a TI-C31 DSP and used to control an analog electronics version of the single mode simulation described above. All of the rules described above result in the desired effect of faster and more reeatable convergence. However, once a erturbation size was chosen, simly changing the direction of the erturbation when an increase in MSE was detected, resulted in stabilization for every trial. A maximum ta inut ower was determined based on the initial limit cycle amlitude and the control actuator authority. This governed the choice of µ to guarantee stable convergence. This value was then reduced to limit the seed of convergence. 7.4 EXPERIMENTAL RESULTS The Rijke tube combustor described reviously was used as the exerimental test bed. Recall that the 2 nd acoustic mode is unstable when the burner is laced at the mid-length of the tube, in accordance with the Rayleigh criterion. At low heat release oerating conditions, the frequency of the 2 nd mode is nominally 178 Hz and the limit cycle frequency is aroximately the same. The control actuator was a three-inch acoustic driver (seaker) couled to the tube through a lenum and attached just above the flame. The actuator ower was sufficient to achieve control at low heat-release conditions, corresonding to low equivalence ratios. At higher equivalence ratios, however, the increased unsteady heat release rate caused the selfexcited loo gain to increase to the oint that no controller could stabilize the system, although 141

153 the limit cycle amlitude could be reduced to some degree. As described in earlier sections, this was due to the actuator ower limitations and eventually resulted in a dynamic searching between stable and unstable regimes. The Rijke tube combustor was set to oerate with an equivalence ratio of 0.54 and a total volume flow of 130 cc/sec. The two-weight TAG controller using the steeest descent algorithm was then emloyed using a TI-C31 DSP control system. The uncontrolled inut level of the ressure signal measurement was adjusted external to the DSP. Based on the inut signal level, the convergence arameter was chosen as 0.4 to ensure gradient convergence. The erturbation value of 0.4 was found to be small enough to revent excessive noise after convergence, yet large enough to rovide a measurable change in the gradient. Tyically this can be determined exerimentally, or algorithmically as described above. Figure 7.6 illustrates the converged TAG controlled ower sectrum of the ressure signal (solid) comared to the uncontrolled ressure sectrum (dotted). The TAG controller converged to a stabilizing solution within aroximately 0.25s and maintained control of the low-heat release case indefinitely. 142

154 Mag. (db) Freq. (Hz) FIGURE 7.6 LOW HEAT RELEASE RIJKE TUBE TAG CONTROL The TAG controller is a very versatile method for controlling the combustion instability and can be alied in a variety of formats and arameterizations. Several different algorithm variants were investigated exerimentally including the multi-weight TAG (FIR filter using four weights) and the direct adatation of gain and hase. The IIR all ass filter form of the gain and hase controller in the z-domain is z + 2az + a H ( z) = b [7.8] az + a z This reresents two first order all-ass filters in series where the control arameter a controls the hase delay and b controls the gain. Two all-ass filter sections ermit 360 degrees of hase adjustment at 175 Hz (the instability frequency) when samled at 1600 Hz. If a is limited to remain between +1 and 1, H(z) is guaranteed to be stable while allowing controller oles to adat. The gain and hase of the controller are erturbed and udated directly without altering the shae of the control filter s frequency resonse. Both methods stabilized the Rijke tube combustor after brief transients, resulting in erformances nearly identical to Figure 7.6. The steeest descent formulation of the weight 143

155 udate equation in no way limits the number of weights that can be udated or the form of the control filter. In the case of the Rijke tube combustor, two control weights were found to be sufficient to achieve stabilizing control. A notable benefit for additional weights is that they can more readily control the frequency resonse, reventing control loo instabilities induced by the feedback controller. There is no secific advantage of choosing the gain and hase arameterization of [7.8] for the control of the Rijke tube combustor, as the erformance was identical to that of the FIR filter control udate. However, there may be instances where it is more advantageous to udate controller arameters other than FIR filter coefficients; the TAG formulation can easily accommodate such changes. To illustrate the versatility of the algorithm, it was also used to adat a subharmonic controller to achieve stabilization of the Rijke tube. The outut of a TAG-controlled two weight FIR filter was frequency divided to roduce an outut with half the frequency of the inut and with a duty cycle of 50% that was used as the control signal. The second harmonic of the subharmonic control signal serves to attenuate the limit cycle in a linear system sense while the cycle stress on the actuator is reduced. The TAG controller requires no a riori information of this change since its only concern is to minimize the RMS ressure of the error signal. The subharmonic alication of the TAG algorithm (transient convergence shown in Figure 7.7) led to limit cycle amlitude reductions that were identical to those shown for the standard TAG imlementation in Figure

156 4 2 Error Control Time (s) FIGURE 7.7 SUBHARMONIC TAG CONVERGENCE Higher equivalence ratios, corresonding to higher self excited loo gains, were not stabilizable using the low-ower acoustic actuator emloyed in these exeriments. However, it is informative to examine the erformance of the TAG controller when comared to other controllers for this oerating condition. Figure 7.8 illustrates three controlled time resonses at the high heat release condition. 145

157 Voltage Phase Shift Voltage LMS Voltage VT TAG Time (s) FIGURE 7.8 HIGH HEAT RELEASE CONVERGENCE COMPARISON The uer trace reresents a manually-tuned fixed feedback controller. The intermittent behavior is a result of the actuator ower limitations and changing lant dynamics in resonse to the control as described earlier. The second time resonse reresents an adative feedback controller of the tye shown in Figure 5.2, emloyed at the same oerating conditions. In both the fixed gain and adative feedback controllers, the intermittency amlitude was aroximately the same as the original limit cycle amlitude. However, when the TAG controller was used at these oerating conditions, the intermittency eriod increased and the amlitude decreased significantly. In this case, the TAG controller reresents a more robust adative control solution, even in the resence of actuator authority limitations. The erformance of the TAG controller was extended to a more ractical setting when it was tested using a liquid fuel combustor actuated with fuel injectors at Louisiana State University; shown in Figure

158 FIGURE 7.9 LSU LIQUID FUEL COMBUSTOR The two weight TAG controller used in the Rijke tube exeriments and sub-harmonic control exeriments, was not significantly altered when emloyed on the liquid fuel combustor. The erturbation size was changed slightly in order to affect the lant enough to measure the local gradient. The outut signal was converted from a sinusoid to a square wave using a threshold detector at a fixed amlitude to aroriately drive the fuel injector valve. The TAG controller erformed very well as shown in Figure 7.10, reducing the limit cycle and its harmonics by aroximately 20 db. 147