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Practical adaptive control for systems with flexible modes, disturbances, and time delays

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Practical adaptive control for systems with flexible modes, disturbances, and time delays

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Content PRACTICAL ADAPTIVE CONTROL FOR SYSTEMS WITH FLEXIBLE MODES, DISTURBANCES, AND TIME DELAYS by Jason M. Levin A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2009 Copyright 2009 Jason M. Levin ii Table of Contents List of Tables List of Figures Abstract Chapter 1 : Introduction 1.1 Problem 1.2 Existing work 1.3 Contribution Chapter 2 : Adaptive Mode Suppression for SISO Systems 2.1 Introduction 2.2 Adaptive mode suppression 2.2.1 Known parameter case 2.2.1.1 Estimation of plant parameters 2.2.2 Adaptive control law 2.3 Stability analysis 2.4 Special Cases 2.4.1 Filter with high frequency poles 2.4.2 Filter with low frequency poles 2.5 Numerical Examples 2.6 Discussion Chapter 3 : Adaptive Mode Suppression for MIMO Systems 3.1 Introduction 3.2 Plant Overview 3.3 Adaptive Mode Suppression Scheme 3.3.1 Adaptive Notch Filter 3.3.2 Robust Online Estimator 3.3.3 Pointwise Stability 3.4 Discussion v vi xiv 1 1 4 8 12 12 14 16 20 23 25 36 37 40 42 49 50 50 52 54 54 61 65 71 iii Chapter 4 : Disturbance Rejection Scheme 4.1 Introduction 4.2 Adaptive Disturbance Rejector 4.2.1 Repeatable Runout Disturbance Rejection 4.2.2 Neural Modeled Disturbance Rejection 4.2.3 Stability Analysis of Neural Model 4.3 Discussion Chapter 5 : Adaptive Time-Delay Compensation 5.1 Introduction 5.2 Plant Model 5.3 Adaptive Smith Predictor 5.3.1 Smith Predictor Control Scheme 5.3.2 Estimation of Unknown Parameters 5.3.3 Adaptive Control Law 5.3.4 Stability analysis 5.4 Numerical Simulations 5.5 Discussion Chapter 6 : Applications 6.1 Adaptive Mode Suppression and Disturbance Rejection Scheme with application to Disk Drives 6.1.1 Introduction 6.1.2 Problem statement 6.1.3 Integrated adaptive mode suppression and disturbance rejection scheme 6.1.3.1 Adaptive Mode Suppression Scheme 6.1.3.2 Adaptive Disturbance Rejector 6.1.4 Simulations 6.1.5 Discussion 6.2 Adaptive Notch Filter Applied to a Flexible Laser Pointing System 6.2.1 Introduction 6.2.2 General Adaptive Notch Filter 6.2.2.1 Known parameter case 6.2.2.2 Estimation of plant parameters 6.2.2.3 Adaptive control law 6.2.3 Control design 6.2.4 Simulations and Experiments 6.2.4.1 Simulations 6.2.4.2 Experiments 6.2.5 Discussion 6.3 Experimental Results of a Neural-Network based Adaptive Disturbance Rejection Scheme for Disk Drives 6.3.1 Introduction 6.3.2 Description of the Experiment 73 73 75 77 79 84 85 87 87 89 90 90 92 95 95 106 109 111 111 111 113 115 115 128 136 148 151 151 153 154 158 161 163 170 174 176 180 182 182 185 iv 6.3.3 Baseline Control 6.3.3.1 Controller Design 6.3.3.2 Closed-loop Model 6.3.4 Disturbance Rejection 6.3.4.1 Adaptive Feed Forward RRO Disturbance Rejection 6.3.4.2 Repetitive Control 6.3.4.3 Neural Modeled Disturbance Rejection 6.3.4.4 Parameter Tuning for Neural Model 6.3.5 Experimental Results 6.3.6 Discussion 6.4 Adaptive Mode Suppression Scheme for a Hypersonic Cruise Vehicle 6.4.1 Introduction 6.4.2 CSULA-GHV Mathematical Model 6.4.3 Adaptive Mode Suppression Scheme 6.4.3.1 Robust Online Estimator 6.4.3.2 Adaptive Notch Filter 6.4.3.3 Rigid body controller 6.4.4 Simulations 6.4.5 Discussion Chapter 7 : Future Work References 187 187 188 189 189 191 195 200 201 205 217 217 220 225 228 233 241 246 252 254 256 List of Tables 2.1 Simulation results 46 6.1 3¾ Value of the Position Error Signal (PES) as a Percentage of the Track Width 209 v List of Figures 2.1 Feedback system diagram for the mode suppression scheme. 16 2.2 Feedback system diagram with the notch filter and mode expressed as an uncertainty¢(s). 19 2.3 Feedback system diagram for the adaptive mode suppression scheme. 24 2.4 Time series plots from ANF whena ¤ = 10. Top plot: Plant outputy p , which is the measured signal. Bottom plot: Complex pole frequency, the dashed red line is the actual value and the solid blue line is the estimate ^ ! d . 47 2.5 Bode plot of the sensitivity functions for ANF when a ¤ = 10 at the start and end of the simulation. 48 2.6 Bode plot of the complementary sensitivity functions for ANF when a ¤ =10 at the start and end of the simulation. 49 3.1 Overall diagram of the adaptive mode suppression scheme for the generic hypersonic vehicle. Here y 2 < 5 is all the measurable out- puts , u2< 2 is the control inputs, and r 2< 5 is the reference input shown for demonstration purposes. 55 3.2 Bode plot of two notch filters designed with the optimization scheme presented. Both filters are designed with constraints 4, 7, and 8 ne- glected, ¹ ³ = 0, and ¹ ³ 2 = 0. Blue line: A narrow notch filter design with ¹ ! = 0:01! and ¹ ! 2 = 0:01! 2 . Green line: A wider notch filter design with ¹ ! =0:05! and ¹ ! 2 =0:05! 2 . 62 3.3 M¢-structure used for the stability analysis. 68 vi 4.1 Closed loop system used for disturbance rejection. y m (k) is the output to be tracked,y p (k) is the measurable output,d 0 (k) is the disturbance, C(z) is a LTI controller, andP(z) is a LTI plant. 77 4.2 System used for disturbance rejection design. y c (k) is the output to be tracked plus disturbance rejection signal, y p (k) is the measurable output, d(k) is the output disturbance, and G(z) is the closed-loop system. 78 4.3 Simple example of how ^ d NN (k) is computed from previous values of the disturbance. Here± =2 andL=3. 83 5.1 Overall diagram of the Smith predictor control scheme. Herey p is the plant output,y m is the reference signal, andu p is the control signal. 94 5.2 Overall diagram of the adaptive Smith predictor control scheme. Here y p is the plant output,y m is the reference signal, andu p is the control signal. 98 5.3 Time series of the reference signaly m in red and the outputy p in blue. 112 5.4 Time series of the estimate of the time delay, which is ½ = ^ T . The correct value of the time delay is 8 seconds. 112 6.1 Overall diagram of the integrated adaptive mode and disturbance sup- pression scheme. Here y(t) is the sample-and-hold version of the HDD PES signal andr(t) is the reference track input. 119 6.2 Closed loop system used for mode suppression. The sampling fre- quency of each block is shown in parenthesis. Here y(k) is the real sampled HDD output, u(t) is the continuous input to the HDD, and r(k) is the reference input. 122 6.3 Simulation results of estimated mode frequencies for different estima- tion speeds. The actual mode of the plant is 5.00 kHz. Solid line: Estimator running atf s . Dashed line: Estimator running at2f s . Dot- ted line: Estimator running at4f s . 124 6.4 HDD simulation to compare the deadzone parameters. The simula- tion is run using the parameters that are described later in this paper, with a step reference input. Solid line: ¯ E , the minimum sum squared diagonal element. Dashed line: ¯, the minimum eigenvalue. 127 vii 6.5 Block diagram of the system used for disturbance rejection. G a (z) = closed-loop system with adaptive controllers; d(k) = disturbance; r(k)= disturbance rejection signal and reference input. 132 6.6 Simple example of how ^ d NN (k) is computed from previous values of the disturbance. Here± =2 andL=3. 137 6.7 Bode plot of the HDD plant,G(s). 140 6.8 Bode plot of the multirate notch filters. Solid line: Adaptive scheme. Dashed line: Non-adaptive scheme. 142 6.9 Simulation results for various mode frequencies. x: Adaptive scheme. o: Non-adaptive scheme. 145 6.10 Estimated mode frequency during simulation Solid line: Estimated frequency. Dashed line: Actual frequency . 146 6.11 Bode plot of open loop system with the adaptive notch filter and adap- tive bandwidth controller. Solid line: Start of simulation, before adap- tation occurs. Dashed line: End of simulation, after adaptation occurs. 146 6.12 Bode plot of open loop system with the non-adaptive notch filter and fixed bandwidth controller. 147 6.13 Time series simulation data with the mode decreased to 5.00 kHz. Top plot: Non-adaptive scheme. Bottom plot: Adaptive scheme. 148 6.14 PSD simulation data with the mode increased to 6.20 kHz. Solid line: Adaptive scheme. Dashed line: Non-adaptive scheme. 148 6.15 Simulation results for various mode frequencies and adaptive distur- bance rejection schemes when combined with the adaptive mode sup- pression scheme. x: No adaptive disturbance rejection. o: K RRO . +: BothK RRO andK NN . 151 6.16 PSD simulation data with the mode decreased to 5.00 kHz. Top Plot: PSD with no adaptive disturbance rejection. Bottom Plot: PSD with adaptive disturbance rejection. (K RRO andK NN ) 152 viii 6.17 PSD simulation data with the mode decreased to 5.00 kHz. Solid line: Adaptive disturbance rejection on. Dashed line: Adaptive disturbance rejection off. 152 6.18 Closeup of PSDs in Fig. 6.17. Top Plot: PSD with no adaptive dis- turbance rejection. Bottom Plot: PSD with adaptive disturbance rejec- tion. (C,U,K RRO , andK NN ). 153 6.19 Feedback system diagram for the mode suppression schemes. 159 6.20 Feedback system diagram with the notch filter and mode expressed as an uncertainty¢(s). 162 6.21 Bode plot of the open loop plant. A single decoupled axis of the FSM experimental setup. 168 6.22 Closed loop diagram for the experimental setup. For the non-adaptive scheme the adaptive notch filter is replaced by a fixed notch filter and the online estimator is removed. Here r(t) is the reference signal which is equal to zero andy(t) is the measured output. 169 6.23 Bode plot of the notch filters. The narrower adaptive notch filter (ANF) adds less phase lag than the non-adaptive notch filter (NA). The ANF is one where the center frequency is frozen at the same value as the non-adaptive notch filter, therefore it can be treated as LTI. 171 6.24 Bode plot of inverse of the filters W A (s) and W NA . The narrower adaptive notch filter (ANF) has a looser constraint on the rigid body control design, when compared to the non-adaptive notch filter (NA). 173 6.25 Bode plot of the closed loop sensitivity functions. The narrower adap- tive notch filter (ANF) allows for better disturbance rejection than the non-adaptive notch filter (NA) due to the increase in bandwidth of the rigid body controller. The ANF can be treated as LTI since we fix the value for the center frequency to be the same value used in the non-adaptive notch filter, which is the nominal plant modal frequency. 174 6.26 Photograph of the laser beam system 175 6.27 Simulation results for the non-adaptive scheme when the notch filter is placed at 93% of the plant’s modal frequency. Top plot: Position outputy p . Bottom plot: Tracking error. (ANF). 179 ix 6.28 Simulation results for the adaptive scheme when the initial value of the estimated modal frequency ^ ! d is placed at 93% of the plant’s modal frequency. Top plot: Position outputy p . Bottom plot: Tracking error. (ANF). 180 6.29 Simulation results for the adaptive scheme. the actual modal fre- quency is 127.26 Hz. Top plot: Estimation error. Bottom plot: Es- timated modal frequency. 180 6.30 Time series from the FSM experiment, the adaptive notch filter is ini- tially placed at 95% of the actual modal frequency of the plant, how- ever the adaptive notch filter will update the center frequency online. The control loop is closed at 5 seconds. 181 6.31 PSDs computed from the error signal of the FSM experiment. The data used is collected from the 10 second mark until the 25 second mark. Top plot: Open loop system, only disturbance. Bottom plot: Non-adaptive scheme and adaptive notch filter (ANF). 182 6.32 Data from the adaptive scheme when the loop is closed at 5 seconds, the actual modal frequency is 127.26 Hz. Top plot: Estimation error. Bottom plot: Estimated modal frequency. 183 6.33 PSDs computed from experimental data comparing a non-adaptive notch filter with an incorrect center frequency (97% of nominal) and one which is correct. 184 6.34 Schematic idealization of the hard disk drive (HDD) system. 190 6.35 Diagram of the experiment. 191 6.36 Block diagram of the control system. P(z)= open-loop plant;C(z)= simple LTI controller;U(z) = converged inverse QR-RLS controller; y = position of the head;d 0 = aggregate disturbance;y ref = position reference; ~ y = PES; u = control signal; G 1 (z) = closed-loop plant withC(z);G(z)= closed-loop plant withC(z) andU(z). 192 6.37 Block diagram of the system used for disturbance rejection. G(z) = closed-loop system with baseline controllers; d = disturbance; u = control signal. 212 6.38 Bode plot of identified close-loop system ^ G 212 x 6.39 Bode plots for the computed output sensitivity transfer functions. 213 6.40 Bode plots for the computed mappings ^ N C , ^ N C¡U , and ^ N C¡U¡U RRO fromy ref toy, for controllersC,C¡U, andC¡U¡U RRO respectively. 213 6.41 Top plot: Sufficient stability condition for different values of °. Bot- tom plot: Estimated sensitivity function from d to y, 1¡ GK o , for different values of°. 214 6.42 Bode plots for the computed output sensitivity transfer functions. 214 6.43 Bode plots for the computed mappings ^ N C , ^ N C¡U and ^ N C¡U¡U REP fromy ref toy, for controllersC,C¡U, andC¡U¡U REP respectively. 215 6.44 Simple example of how ^ d NN (k) is computed from previous values of the disturbance. Here± =2 andL=3. 215 6.45 Experiment performed on head 0 and track 15;000. Top Plot: PSD with no adaptive disturbance rejection (C andU). Bottom Plot: Red Line: PSD with adaptive feedforward RRO disturbance rejection (C, U, and U RRO ). Blue Line: PSD with repetitive control (C, U, and U REP ). 216 6.46 Experiment performed on head 0 and track 15;000. Top Plot: PSD with no adaptive disturbance rejection (C andU). Bottom Plot: Red Line: PSD with adaptive feedforward RRO disturbance rejection (C, U, and U RRO ). Blue Line: PSD with repetitive control (C, U, and U REP ). 217 6.47 Closeup of PSD in Fig.6.46. Top Plot: PSD with no adaptive dis- turbance rejection (C and U). Bottom Plot: Red Line: PSD with adaptive feedforward RRO disturbance rejection (C, U, and U RRO ). Blue Line: PSD with repetitive control (C,U, andU REP ). 218 6.48 Experiment performed on head 0 and track 15;000. Top Plot: Red Line: PSD with repetitive control and neural modeled disturbance re- jection (C,U,U REP , andK NN ). Blue Line: PSD with repetitive con- trol. (C, U, and U REP ). Bottom Plot: Red Line: PSD with adaptive feedforward RRO and neural modeled disturbance rejection (C, U, U RRO , and K NN ). Blue Line: PSD with adaptive feedforward RRO disturbance rejection (C,U, andU RRO ). 219 xi 6.49 Time series data from experiment performed on head0 and track15;000. At 5 seconds the adaptive disturbance rejection is switched on (C,U, U RRO , andK NN ). 220 6.50 3¾ value of the position error signal as a percentage of the track width for various head and track locations. 220 6.51 Overall geometry of the CSULA-GHV used for simulation. 227 6.52 Bode plot of the complete aeroelastic linearized GHV transfer func- tions. Blue line: Transfer function from the throttle to altitude,G 31 (s). Green line: Transfer function from the elevon to altitude,G 32 (s). 230 6.53 Overall diagram of the adaptive mode suppression scheme for the generic hypersonic vehicle. Here y 2 < 5 is all the measurable out- puts from the GHV , u 2 < 2 is the control inputs, and r 2 < 5 is the reference input shown for demonstration purposes. 232 6.54 Bode plot of two notch filters designed with the optimization scheme presented. Both filters are designed with constraints 4, 7, and 8 ne- glected, ¹ ³ = 0, and ¹ ³ 2 = 0. Blue line: A narrow notch filter design with ¹ ! = 0:01! and ¹ ! 2 = 0:01! 2 . Green line: A wider notch filter design with ¹ ! =0:05! and ¹ ! 2 =0:05! 2 . 244 6.55 Plot showing an optimized notch filter parameter ! R ! Z as a function of ! 2 ! and³. 245 6.56 Plot showing an optimized notch filter parameter ³ R as a function of ! 2 ! and³. 246 6.57 Plot showing an optimized notch filter parameter ³ Z as a function of ! 2 ! and³. 246 6.58 Plot showing an optimized notch filter phase lag as a function of ! 2 ! and³. 247 6.59 Velocity and altitude for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. Blue line: Command response. Green line: Actual response. 252 6.60 Control inputs for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. 253 xii 6.61 Vehicle structural modal frequency ! s1 and estimated complex pole frequency! D , which isµ 16 , for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. 253 6.62 Estimate frequencies for the complex zeros inG 11 (s) andG 12 (s), for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. 254 6.63 Normal acceleration at the nose for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. 255 6.64 Control inputs for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is off. 256 6.65 Normal acceleration at the nose for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is off. 256 xiii Abstract Control systems are used in a variety of applications where tracking and regulation are required, although there exist numerous problems which can cause detrimental effects for standard controllers. The field of adaptive control has been researched and applied to solve some of these difficulties, usually through the use of complete adaptive control strategies. This research proposes adaptive controllers which may be added to fixed non- adaptive controllers to form a single unified scheme. This allows the traditional control designer some intuition in the development while reaping the benefits of a controller which can compensate for unknown or changing dynamics. There are three specific problems tackled in this research: flexible modes, disturbances, and time delays. Since each of these may be uncertain or may change over time, the addition of an adaptive controller may allow the system to meet performance requirements or retain stability. This research will contribute the design and analytical proof of adaptive mode sup- pression schemes to deal with unknown flexible modes. These modes may be uncertain for a variety of reasons, such as variation between production units, changes in the sur- rounding environment, degradation over lifetime, or wear from use. The adaptive mode suppression schemes include an adaptive notch filter which is able to track and suppress xiv a varying flexible mode. However, disturbances can also reduce the performance of a control system, specifically one where precise tracking is paramount. For this, we con- tribute an adaptive neural modeled disturbance rejector which can be added to a stable control system to enhance tracking performance. Lastly, many systems in the field of process controls contain dominant time delays which can change with time, thus affect- ing the stability and robustness of the closed loop system. To solve this problem, we propose an adaptive Smith predictor which is able to estimate and compensate for the uncertain time delay. To show the benefit of the adaptive controllers, several applications are presented, such as a hard disk drive (HDD) where unknown high frequency flexible modes inhibit the achievable bandwidth and dynamic disturbances degrade the tracking quality. Experimental results of the adaptive mode suppression scheme and disturbance rejection scheme are included. Also simulations of an integrated scheme with the adap- tive mode suppression scheme and neural modeled disturbance rejector are done to show an improvement in performance. Simulations are also conducted on a nonlinear hyper- sonic aircraft model which experiences changing elastic modes, where an adaptive mode suppression scheme is shown to retain stability. xv Chapter 1 Introduction 1.1 Problem Control system design has progressed rapidly over the last several decades, in particu- lar, research in adaptive control with various applications has been pursued. Adaptive control as defined by [36] is the combination of a parameter estimator, which generates parameter estimates online, with a control law in order to control classes of plants whose parameters are completely unknown and/or could change with time in an unpredictable manner. This type of control can be applied to systems to solve problems which cannot be solved by non-adaptive controllers. Problems such as unknown or changing flexible modes, unknown and/or dynamic disturbances, and unknown and/or varying time delays can wreak havoc on traditionally designed controllers. Since a practical control system should be designed to ensure robust stability while still meeting performance require- ments for a particular application, there exists a need for adaptive controllers which can 1 be readily applied. However, adaptive schemes are not always accepted in industrial ap- plications due to the inability to verify and validate the robustness to a satisfactory level for a marketable product and the lack of intuition for traditional control designers. There is a need to create a smooth transition in industry from traditional to adaptive controllers. High frequency flexible modes occur in numerous mechanical systems where control systems are desired. These flexible dynamics need to be accounted for to achieve good closed loop performance. One method of suppressing these modes is through the use of a notch filter. However, in many applications, the modal frequencies are often uncertain and can even vary over time. For example, in the hard disk drive (HDD) the high- frequency flexible modes may vary between production units and may even vary during the lifetime of the HDD [13, 69]. Hypersonic aircraft, another system with varying flexible modes, will travel at very high velocities through the atmosphere on a variety of missions. These missions will require payloads that vary in terms of mass and location on the aircraft, which will affect the center of gravity and the flexible modes. Variations in the flexible dynamics of this kind are common in military aircraft, and flight control systems generally incorporate notch filters to suppress the structural modes [73, 22, 16, 11]. However, hypersonic vehicles will also experience extreme heating of the fuselage during flight. The thermal properties of the aircraft will cause the flexible modes to change in terms of mode shapes, damping ratios, and natural frequencies [57, 56, 59, 60, 30]. Both HDDs and aircraft may require the control system to include a wide notch filter to account for the varying modal dynamics so as to retain stability. As the notch 2 filter becomes wider it also induces greater phase lag at lower frequencies resulting in a lower bandwidth system. This reduced closed loop bandwidth can diminish the ability to meet the performance requirements set forth for the system. Unknown disturbances may also account for a reduction in the performance of a con- trol system where precise tracking is required. In applications such as the HDD, where tracking is crucial, every minute amount of tracking error that can be reduced may in- crease the desirability of the control system. In the HDD, if the tracking of the read/write head can be more precise then more data can be stored on a single disk, resulting in higher storage capacity drives, which may lead to greater profits for the HDD manufac- turer [13]. The disturbance causing the tracking error may be dynamic, or non-stationary, and possibly nonlinear making attenuation of the disturbance very difficult for traditional methods. Time delays are a significant problem in the field of process control as they limit the bandwidth, and hence performance, of control systems. These time delays arise from the presence of fluid travel times, recycle loops in chemical processes, and also long computation times [84, 8, 75]. Robustness and stability will be affected by the addition of known time delays, but unknown or changing time delays present an even greater problem. The time delay may vary from the congestion of pipes, changing of filters, or variation in the properties of the chemicals in the process being controlled. With variation in the time delays, a decrease in performance is usually realized and instabilities can 3 occur. For this reason, unknown and varying time delays in process control systems are problems that requires ongoing research. 1.2 Existing work There has been research into the areas of controlling systems with flexible modes, dis- turbances, and time delays. Some solutions for these problems simply incorporate robust designs that are able to deal with varying or uncertain parameters and other solutions incorporate the use of parameter estimators, and therefore become adaptive schemes. A robust control scheme for a flexible system must adequately deal with the flexible modes. Mode suppression in hypersonic vehicles has been studied and reported in the literature [15, 64, 29]. A linear parameter-varying (LPV) synthesis approach to account for the changing flexible mode due to atmospheric heating has been studied in [57, 56]. Mode suppression schemes for conventional aircraft usually incorporate notch filters to suppress the flexible modes [73, 22, 16, 11]. The notch filters are added on top of the rigid-body controller in a complete aircraft control scheme. However, researchers have also applied integrated techniques, which do not explicitly use structural filters. This is accomplished through robust controller design using ¹-synthesis [40], dynamics inver- sion [26], or adaptive dynamic inversion [10]. A conventional notch filter is ineffective when the frequencies and damping ratios of the structural modes involve large uncertain- ties and/or may change during flight, as is the case with airbreathing hypersonic vehicles. Similar statements can be made for the HDD where conventional control schemes will 4 incorporate fixed notch filters, however they must be wide to deal with varying modal frequencies and therefore induce undesirable phase lag [13, 37]. One possible solution to the problem of varying flexible dynamics is the adaptive notch filter, a notch filter whose center frequency varies online to track the modal fre- quencies of the system. The adaptive notch filter has been studied in signal processing research [87, 78, 20] as well as various applications, such as the HDD [69], launch vehi- cles [21], aircraft [61], and space structures [54]. The adaptive notch filter presented in [21] is used on the model of a booster from the Advanced Launch System (ALS) program. The least squares estimator in the publication uses a simple undamped resonator as the model for estimation and functions well since the resonant mode is very pronounced. However, in other applications, full plant param- eterization is necessary as the flexible mode may not be as significant. Another strategy for the estimation of the center frequency can be found in [69], where frequency weight- ing functions are used. The downside is that there are several known failure modes, and avoidance requires some modal information a priori. A stochastic state space algorithm for mode frequency estimation is presented in [61]. However, it relies on the injection of a probe signal which is not needed in the scheme presented here. The indirect adaptive compensation (IAC) scheme in [94] also requires a probe signal to complete the estima- tion. The adaptive mode suppression scheme in [54] uses an LMS algorithm to update filter coefficients and then the modal parameters are extracted from the filter. This is the 5 opposite of what will be presented in this research, where the modal parameters are first estimated and then used in the adaptive notch filter. Numerous disturbance rejection algorithms for tracking systems have been researched, each with their own advantages and downfalls [93, 79, 72, 71, 33, 42, 94, 52, 95, 39, 62]. Adaptive feedforward disturbance rejection schemes have been developed to cancel sin- gle frequency disturbances where the frequency is known [79, 80]. There has been work done in using neural networks for feedforward disturbance rejection in various control systems [55, 74, 66]. Rejection of the disturbance torque for missile seekers using neural networks is presented in [55]. A multilayer neural network uses the measurable load disturbance to cancel the degrading effects through a feedforward controller. Many sys- tems, such as the conventional HDD, do not have the luxury of extra sensors for the disturbance, so the only possible input to a neural network is the position signal. In [74], a general approach with simulations show the benefit of adapting a dynamic neural network to cancel an unknown disturbance. Here, the idea of passing the disturbance estimate through an estimated plant inverse is used. The methods proposed in [66] use multilayer neural networks to model disturbances as outputs of dynamical systems and then expand the plant model to try to reduce the adverse effects. Radial basis functions (RBF) are sometimes utilized in neural network schemes and have been used to model sea-clutter noise in radar applications [31]. The approach uses RBFs to remove the noise 6 from radar signal data, since the clutter noise has been shown to be chaotic, thus pro- viding the ability to detect small targets in the clutter. Training data is used to adapt the neural parameters before being implemented on actual test data. There has also been significant research into solving the problem of time delays in process control systems. One of the oldest solutions is the so called Smith predictor [85], which uses a model of the plant and time delay to allow for an increase in overall system gain and performance. The Smith predictor has since been modified and improved to include steady-state tracking [91] and then further modified for ease of tuning and in- creased robustness [5, 58]. However these methods require a well known model of the plant and time delay, so work has been done to improve control systems while estimat- ing parameters online [68]. A simple adaptive smith predictor with some analysis was completed in [8], but there have been other efforts to create algorithms to estimate the time delay during a realtime process [44, 28]. One method of estimation incorporates an overparameterization of the plant in the discrete domain and then an RLS algorithm to estimate the plant parameters. An algorithm is performed to extract the plant parameters and the time delay, which are then used in an extended minimal variance controller and a deadbeat controller. However, this method has the disadvantage of poor convergence and a growing number of parameters with an increase in sampling rate. An adaptive Smith predictor is compared to a non-adaptive robust predictive controller in [4], where the benefit of the adaptive controller is evident with an unknown time delay, however no analysis of the adaptive scheme is given. The idea of using a rational representation of 7 a time delay was used for estimation in [24] and then furthered in [3, 90]. This polyno- mial representation and estimation was used for control in [77, 76], however no stability results are presented. A fully analyzed and simulated adaptive controller for time delay systems was published in [86], where a polynomial representation of the time delay is used in an RLS algorithm to estimate the plant parameters, which are then incorporated into a pole placement controller. More advanced neural network approaches have been used for systems with unknown time delays, however the controllers are more complex and not necessarily intuitive for a classical control designer [25, 14]. 1.3 Contribution The problems of unknown flexible modes, disturbances, and time delays have been re- searched and applied in the past, however this report will attempt to improve upon the previous research by contributing adaptive schemes which possess non-adaptive and adaptive components. There are adaptive components which will address the specific problems of flexible modes, disturbances, and time delays while still retaining classical control elements for the remaining part of the system. This may aid in increasing the usage of adaptive controllers in industry, by allowing specific components to be added to a traditional scheme to increase the performance and/or robustness. Specifically the contribution of this report is: 8 ² The design and analytical proof of adaptive mode suppression schemes for contin- uous time SISO systems which incorporate an adaptive notch filter, making them able to track and suppress the effects of a single unknown or changing flexible mode [46]. The scheme incorporate an online estimator which uses full plant pa- rameterization, therefore enabling the use of various parameter estimation schemes. The adaptive part of the scheme is strictly meant to suppress the modal dynamics and work in conjunction with another controller which is designed for the rigid body system, that is the system without the flexible modes. This enables vari- ous types of rigid body controllers to be designed somewhat independently of the flexible dynamics, which has numerous advantages in many applications. Each of these schemes has the ability to improve the performance of the system, although the measurable benefit is application dependent. ² The design and analytical proof of a neural modeled disturbance rejection scheme. This disturbance rejection scheme uses a neural model similar to [31] with the addition of an extra term in the neural model to account for extra delays and there is no training set of data; the adaptive neural disturbance rejector is both adapted and implemented in real-time. Also the adaptation of the neural parameters uses a deadzone modification not present in [31], which allows adaptation to cease once the performance begins to degrade. This disturbance rejector may be added to any control system which stabilizes the system. 9 ² An adaptive Smith predictor control scheme, with stability analysis, for systems with unknown or changing time delay. These types of systems arise in the area of process control and the need for a control system that is able to cope with such problems is presented and simulated. The adaptive Smith predictor uses an esti- mate of the time delay which come from an online estimator based on the para- metric model of the time delay system, while approximating the time delay as a polynomial transfer function. A non-adaptive base controller is used to meet performance requirements and the adaptive Smith predictor is added to gain the benefits of removing the time delay from the characteristic equation. ² An integrated adaptive mode suppression and disturbance rejection scheme with particular application to a HDD, which is presented in [45]. Through simulation, this scheme is shown to improve the tracking performance of a disk drive servo system. The scheme is comprised of a multirate adaptive notch filter that is able to track resonant modes of the plant, even if they exist near of above the nyquist frequency, without the addition of a probing signal. This is done through the plant parameterization and a novel deadzone modification. An adaptive bandwidth con- troller is added to maintain stability and performance requirements as the multirate adaptive notch flter changes online. Lastly, the neural modeled disturbance rejec- tion scheme is added to create a single integrated scheme. Each component is an individual contribution of this report, as well as the integration strategy. 10 ² Empirical verification of the adaptive mode suppression scheme which utilizes an adaptive notch filter presented in [49]. The scheme is experimentally tested on a laser beam pointing system. This setup possesses a single lightly damped flexible mode which makes control difficult in the frequency region around the resonance. The control design methodology is included to show how the bandwidth of the closed loop system can be increased with the addition of a narrow adaptive notch filter. ² The experimental results of the neural modeled disturbance rejection scheme pre- sented in [51, 50]. The scheme developed and analyzed previously is shown to in- crease performance through empirical tests on a commercially available disk drive. The neural scheme is added to several different control strategies to demonstrate the effectiveness as being an add-on disturbance rejector. ² An adaptive mode suppression scheme, presented in [48], for a hypersonic aircraft is developed and simulated. This scheme is slightly different from those developed above as the plant for this system is nonlinear and MIMO, contributing to the need to a new algorithm. Several robust online estimators are used to estimate the flexible dynamics of the aircraft while in flight, again without the use of a probing signal. These estimated parameters are then used to tune numerous notch filters, which are design offline through an optimization algorithm. 11 Chapter 2 Adaptive Mode Suppression for SISO Systems 2.1 Introduction High frequency flexible modes occur in numerous mechanical systems where control systems are desired. These flexible dynamics need to be accounted for to achieve good closed loop performance. One method of suppressing these modes is through the use of a notch filter. However, in many applications, the modal frequencies are often uncertain and can even vary over time creating the need for a wide notch filter. As the notch filter becomes wider it also induces greater phase lag at lower frequencies resulting in a lower bandwidth system. One solution to such a problem is the adaptive notch filter, a notch filter whose center frequency varies online to track the modal frequencies of the system. The adaptive notch filter has been studied in signal processing research [87, 78, 20] as well as various applications, such as the hard disk drive (HDD) [69], launch vehicles [21], aircraft [61], and space structures [54]. 12 The adaptive notch filter presented in [21] is used on the model of a booster from the Advanced Launch System (ALS) program. The least squares estimator in the publica- tion uses a simple undamped resonator as the model for estimation and functions well since the resonant mode is very pronounced. However, in other applications, full plant parameterizations is necessary as the flexible mode may not be as significant. Another strategy for the estimation of the center frequency can be found in [69], where frequency weighting functions are used. The downside is there are several failure modes that are known and avoidance requires some modal information a priori. A stochastic state space algorithm for mode frequency estimation is presented in [61]; however it relies on the in- jection of a probe signal which is not needed in the scheme presented here. The indirect adaptive compensation (IAC) scheme in [94] also requires a probe signal to complete the estimation. The adaptive mode suppression scheme in [54] uses a LMS algorithm to up- date filter coefficients and then the modal parameters are extracted from the filter. This is opposite to what is being presented in this paper, where the modal parameters are first es- timated and then used in the adaptive notch filter. However, research into the closed-loop stability of control schemes which incorporate adaptive notch filters is lacking. Also the design methodology for such control systems is not clear in the literature, that is to say how does the rigid-body control interact with the adaptive mode suppression techniques. This paper will present the design and stability analysis of an adaptive mode suppres- sion scheme that uses a second order adaptive notch filter. The adaptive filter is strictly meant to suppress the modal dynamics and work in conjunction with another controller 13 which is designed for the rigid body system, that is the system without the flexible modes. This enables various types of rigid body controllers to be designed somewhat indepen- dently of the flexible dynamics, which has numerous advantages. It may be useful for systems such as aircraft where the modes are not precisely known early in the control design process or for systems such as the HDD with modal parameters that vary between units and the tracking performance of the controller is critical. Numerical simulations are conducted to show the benefit of the adaptive notch filter. In the simulations the rigid body control is designed using pole placement techniques, although these same adaptive filters have been augmented with a LQ controller in [48] and a classical phase lead design in [47, 45]. The actual design methodology for the rigid controller will vary based on the performance requirements of the system, so the numerical simulations are meant to show one simple example of how the design is com- pleted and may be useful for an increase in performance. The adaptive mode suppression scheme is presented in Section 2.2 along with the stability analysis in Section 2.3. Two special cases of the adaptive scheme are given in Section 2.4 and numerical examples are presented in Section 2.5. Finally conclusions are drawn in Section 2.6. 2.2 Adaptive mode suppression The setup for this problem is a rigid plant with a single flexible modes which may include a complex pole or a complex dipole. Our analysis will deal with the case of a complex dipole, which is a complex pole and zero, however special cases of controllers will be 14 presented for the complex pole system in Section IV . A controller is designed to achieve good performance in the presence of disturbances, which means shaping the sensitivity and complementary sensitivity functions appropriately. The flexible dynamics must be suppressed to acquire stability and performance, this will require the use of a notch filter. The control objective includes tracking a certain class of reference signal y m 2 L 1 by using the internal model principle. The controller designed for this problem will include Q m (s), which is an internal model ofy m and is a known monic polynomial of degreeq with all roots in<[s] · 0 and with no repeated roots on the j!-axis. The analysis will be similar to that of adaptive pole placement completed in Chapter 6.3 of [36]. The plant takes the form y p =G p (s)M(s)u p = Z p (s) R p (s) Z m (s) R m (s) u p ; (2.1) where Z m (s) Rm(s) is the flexible mode of the plant and Z p (s) Rp(s) is the non-modal part of the plant. We will assume that the rigid plant, or non-modal part of the plant, is stabilizable and a non-adaptive controllerC(s) can be designed. We also have the following assumptions about the rigid plant (P1) R p (s) is a monic polynomial of degreen (P2) Z p (s) is a polynomial of degreem·n. The flexible part of the plant takes the form Z m (s) R m (s) = s 2 +2³ n ! n s+! 2 n s 2 +2³ d ! d s+! 2 d (2.2) 15 ) (s G p ) (s M ) (s C ) (s F p u m y p y Figure 2.1: Feedback system diagram for the mode suppression scheme. where ³ n ;³ d > 0, ³ n 6= ³ d are the damping of the numerator and denominator of the mode, respectively, and! n ;! d >0. For this type of system we will present a methodology for designing adaptive notch filters which use an online estimator. The adaptive notch filter may be made very narrow and centered at the natural frequency of the system’s complex pole. Once this is done, the rigid body controller is designed separately, while treating the flexible dynamics and notch filters as uncertainty that must be accounted for in the design process. This type of adaptive mode suppression scheme is similar to those in [69, 21, 61, 54], where an online estimator is used to vary the adaptive notch filter. The control scheme is presented in the following subsections, where the known parameter case, the estimator, and the adaptive control law are described for each. 2.2.1 Known parameter case The control scheme includes a narrow adaptive notch filter centered at the natural fre- quency of the flexible pole in (2.1), and a compensator designed for G p (s) using any design technique while completely neglecting the flexible dynamics. Adaptive notch fil- ter schemes of this nature have been presented in the literature as described previously, 16 however the interaction between the filter and rigid-body control is not always explicit. The control loop is seen in Fig. 2.1 and we have u p =¡F(s)C(s)(y p ¡y m ) (2.3) C(s)= P(s) Q m (s)L(s) (2.4) F(s)= Z f (s) R f (s) : (2.5) We assume that the rigid-body controllerC(s) is designed such that the polynomial equa- tion L(s)Q m (s)R p (s)+P(s)Z p (s)=A ¤ (s); (2.6) gives a Hurwitz A ¤ (s), which are the desired closed loop poles when neglecting the flexible dynamics. Since we are interested in the notch filter and flexible dynamics, we place assumptions on the rigid control (P3) L(s) is a monic polynomial of degreen l ¸0 (P4) P(s) is a polynomial of degree0·n p ·n l +q. The filterF(s) in (2.3) is Z f (s) R f (s) = s 2 +2³ z ! d s+! 2 d s 2 +2³ r ! d s+! 2 d (2.7) 17 where! d is the same as in (2.2),³ z ;³ r >0 and³ z <³ r . We would like to design the filter to fully suppress the flexible modes at the resonant frequency, this gives us a condition that must be met: ¯ ¯ ¯ ¯ ! 2 d ! 2 n Z m (j!)Z f (j!) R m (j!)R f (j!) ¯ ¯ ¯ ¯ ·k m (2.8) where k m · 1 is the desired margin. Now treating the modes and notch filter as un- certainty the system can be put into the form of Fig. 2.2, whose characteristic equation is 1+C(s)G p (s)(1+¢(s))=0 (2.9) which, due to the stable roots of1+C(s)G p (s)=0, implies 1+ C(s)G p (s) 1+C(s)G p (s) ¢(s)=0: (2.10) For stability of the system the following must be satisfied ° ° ° ° P(s)Z p (s) A ¤ (s) ¢(s) ° ° ° ° 1 <1 (2.11) ¢(s)=F(s)M(s)¡1: (2.12) 18 m y p y ) (s G p ) (s C ) (s ' Figure 2.2: Feedback system diagram with the notch filter and mode expressed as an uncertainty¢(s). The above comes from neglecting the flexible dynamics and notch filter and treating them as an uncertainty ¢(s) and then applying the small gain theorem. It can be shown that the tracking errore 1 =y p ¡y m is e 1 =¡ L(s)R p (s) Q m (s)L(s)R p (s)+P(s)Z p (s)(1+¢(s)) Q m y m (2.13) which is a proper stable transfer function since (2.11) is satisfied. Therefore the control law will causee 1 ! 0 ast!1 exponentially fast. It should be noted this result is for the system when all the parameters are known and the requirement in (2.11) is met. Now we will assume the parameters of the flexible mode are unknown and must be estimated online. 19 2.2.1.1 Estimation of plant parameters The adaptive mode suppression scheme that is used when the flexible dynamics are un- certain or changing will now be designed. Starting with the system in (2.1) we have y p = Z p (s) R p (s) Z ¤ m (s) R ¤ m (s) u p ; (2.14) where Z ¤ m (s) R ¤ m (s) is the unknown mode of the plant Z ¤ m (s) R ¤ m (s) = s 2 +2³ ¤ n ! ¤ n s+! ¤2 n s 2 +2³ ¤ d ! ¤ d s+! ¤2 d (2.15) and Zp(s) R p (s) is the known part of the plant. Z p (s);R p (s);Z ¤ m (s); andR ¤ m (s) follow the all the same assumptions made in the known parameter case. The parametric model to estimate the unknown modal frequency is as follows: z =µ ¤T Á (2.16) z = s 2 R p (s) ¤ p (s) y p ¡ s 2 Z p (s) ¤ p (s) u p (2.17) Á= · ® T 1 (s)Zp(s) ¤p(s) u p ¡ ® T 1 (s)Rp(s) ¤p(s) y p ¸ T (2.18) ® n (s)= · s n ¢¢¢ s 1 ¸ T (2.19) µ ¤ = · µ ¤T n µ ¤T d ¸ T (2.20) 20 R ¤ m (s)=s 2 +µ ¤T d ® 1 (s) (2.21) Z ¤ m (s)=s 2 +µ ¤T n ® 1 (s) (2.22) and ¤ p (s) is a monic Hurwitz polynomial of degree n+2. Our goal is to estimate the modal frequencies and dampings. A wide class of adaptive laws can be used to estimate the unknown parameters, but we adopt the gradient algorithm with parameter projection. Allow Á = [Á 1 ;Á 2 ;Á 3 ;Á 4 ] T ;µ = [µ 1 ;µ 2 ;µ 3 ;µ 4 ] T , and also some a priori known bounds on the dampings and natural frequencies such that 1¸³ u n ¸³ ¤ n ¸³ l n >0 1¸³ u d ¸³ ¤ d ¸³ l d >0 ! u n ¸! ¤ n ¸! l n >0 ! u d ¸! ¤ d ¸! l d >0 (2.23) are satisfied. The gradient update equations become _ µ 1 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 1 "Á 1 if(2³ u n ! u n >µ 1 >2³ l n ! l n ) or(µ 1 =2³ l n ! l n and"Á 1 ¸0) or(µ 1 =2³ u n ! u n and"Á 1 ·0); 0 otherwise (2.24) 21 _ µ 2 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 2 "Á 2 if((! u n ) 2 >µ 2 >(! l n ) 2 ) or(µ 2 =(! l n ) 2 and"Á 2 ¸0) or(µ 2 =(! u n ) 2 and"Á 2 ·0); 0 otherwise (2.25) _ µ 3 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 3 "Á 3 if(2³ u d ! u d >µ 3 >2³ l d ! l d ) or(µ 3 =2³ l d ! l d and"Á 3 ¸0) or(µ 3 =2³ u d ! u d and"Á 3 ·0); 0 otherwise (2.26) _ µ 4 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 4 "Á 4 if((! u d ) 2 >µ 4 >(! l d ) 2 ) or(µ 4 =(! l d ) 2 and"Á 4 ¸0) or(µ 4 =(! u d ) 2 and"Á 4 ·0); 0 otherwise (2.27) "= z¡µ T Á m 2 s (2.28) m 2 s =1+Á T Á: (2.29) In the update equations, the bounds³ l n ;³ u n ;³ l d ;³ u d ;! l n ;! l n ;! l d ;! l d are constants determined a priori, and the adaptive gains ° 1 ;° 2 ;° 3 ;° 4 > 0 are also design parameters chosen a priori. The above estimation law guarantees that (i) µ2L 1 22 (ii) ";"m s ; _ µ2L 2 \L 1 . 2.2.2 Adaptive control law The adaptive control law is formed by replacing the notch filter in (2.3), which has the form of (2.7), with an adaptive notch filter. The diagram for the scheme is in Fig. 2.3. The online estimates used in the adaptive notch filter come from the online estimator and are µ = · 2 ^ ³ n ^ ! n ^ ! 2 n 2 ^ ³ d ^ ! d ^ ! 2 d : ¸ T : (2.30) The adaptive control law becomes u p =¡ ^ Z f (s) ^ R f (s) P(s) L(s)Q m (s) (y p ¡y m ): (2.31) In the above control law, the notch filter ^ Z f (s) ^ R f (s) is designed to cancel the unknown mode of the plant Z ¤ m (s) R ¤ m (s) . This is done by using the estimate of the modal frequency as the center frequency thereby making it an adaptive notch filter. The filter becomes ^ Z f (s) ^ R f (s) = s 2 +2³ z ^ ! d s+ ^ ! 2 d s 2 +2³ r ^ ! d s+ ^ ! 2 d (2.32) where ^ ! d is the estimate of the modal frequency and the damping ratios are set a priori using (2.8) as a reference. 23 ) (s G p ) (s M ) (s C ˆ ( ) F s p u m y p y Estimator Figure 2.3: Feedback system diagram for the adaptive mode suppression scheme. We also must make sure that (2.11) is satisfied at every frozen timet, which leads to ° ° ° ° P(s)Z p (s) A ¤ (s) ¢¢(s;t) ° ° ° ° 1 <1 (2.33) ¢(s;t)= ^ F(s;t)¢ ^ M(s;t)¡1: (2.34) By frozen time we mean that the time-varying coefficients of the polynomials are treated as constants when two polynomials are multiplied. Therefore the controller C(s) must be designed such that (2.33) is always satisfied. This implies a priori knowledge of the bounds on the unknown parameters which leads to a convex setµ2S, that the estimator designed must use for projection. These bounds, through the updating of the parameters in the adaptive notch filter, create a convex set of possible¢(s;t) which we use to obtain a weightW(s) used for control design. Now, denote ¹ F(s;µ); ¹ M(s;µ) as the frozen time versions of the transfer functions ^ F(s;t); ^ M(s;t). That is to say, the overbar versions have estimated parameters that come from µ 2 S but are frozen in time, and therefore LTI systems. We have l(!)=max µ2S ¯ ¯¹ F(j!;µ) ¹ M(j!;µ)¡1 ¯ ¯ (2.35) 24 and a rational transfer function weight jW(j!)j¸l(!); 8!: (2.36) This weight can be substituted in (2.33) to acquire the LTI stability requirement as ° ° ° ° C(s)G p (s) 1+C(s)G p (s) W(s) ° ° ° ° = ° ° ° ° P(s)Z p (s) A ¤ (s) W(s) ° ° ° ° 1 <1: (2.37) This requirement for stability can be achieved offline from knowledge of the parameter bounds and the adaptive notch filter design. Since the adaptive notch filter is narrower, it will give less phase lag at lower frequen- cies, which may lead to increased phase margin for the rigid body controller. Also, the requirement in (2.37) gives some insight into why the narrow notch filter may benefit the performance of the mode suppression scheme. As F(s) becomes wider so will W(s), and it will put more limitations on the controllerC(s) in (2.37). 2.3 Stability analysis The adaptive mode suppression scheme with the rigid-body control and adaptive notch filter has stability properties which are defined in the following theorem: 25 Theorem 2.3.1 The system described by (2.14), (2.15), (2.24) - (2.29), (2.31) has the property that all signals in the closed loop are uniformly bounded and the tracking error converges to zero asymptotically with time. Proof: The proof is carried out using the properties of the adaptive law and the following four steps: Step 1. Write the input and output in terms of the estimation error The objective of this step is to get the inputu p and outputy p in terms of the estimation error". The control law in (2.31) and normalized estimation error can be written as ^ R f LQ m 1 ¤ u p =¡ ^ Z f P 1 ¤ (y p ¡y m ) (2.38) "m 2 s = ^ R m R p 1 ¤ p y p ¡ ^ Z m Z p 1 ¤ p u p (2.39) where¤; is a monic Hurwitz polynomial of degreen l +q+2 ifn l ¸n¡1 andn+q+1 if n l < n¡1. Without loss of generality for the rest of the proof we will assume that n l < n¡ 1 and also assume m = n¡ 1, this is possible since the order of any of the polynomials can be expanded by adding parameters with zero coefficients. Now we design¤ = ¤ p ¤ q where¤ q is an arbitrary monic Hurwitz polynomial of degreeq¡1 if q¸2 and¤ q =1 forq <2. Now the following are defined u f , 1 ¤ u p ; y f , 1 ¤ y p (2.40) 26 and write (2.38) and (2.39) as ^ Z f Py f + ^ R f LQ m u f =y m1 (2.41) ^ R m R p ¤ q y f ¡ ^ Z m Z p ¤ q u f ="m 2 s (2.42) wherey m1 , ^ Z f P 1 ¤ y m 2L 1 . The polynomials may be expressed as ^ R m R p ¤ q =s n+q+1 + ¹ µ T 1 ® n+q (s) (2.43) ^ Z m Z p ¤ q = ¹ µ T 2 ® n+q (s) (2.44) ^ Z f P =p 0 s n+q+1 + ¹ p T ® n+q (s) (2.45) ^ R f LQ m =s n+q+1 + ¹ l T ® n+q (s) (2.46) ® i (s),[s i ;s i¡1 ;:::;s;1] T (2.47) where order of some of the polynomials are increased by adding parameters with zero coefficients. Now defining the state x, h y (n+q) f ;:::; _ y f ;y f ;u (n+q) f ;:::; _ u f ;u f i T (2.48) 27 the system _ x=A(t)x+b 1 (t)"m 2 s +b 2 (t)y m1 (2.49) is formed, where A(t)= 2 6 6 6 6 6 6 6 6 6 6 4 ¡ ¹ µ T 1 j ¹ µ T 2 I n+q ¹ O j O (n+q)£(n+q+1) p 0 ¹ µ T 1 ¡ ¹ p T j ¡p 0 ¹ µ T 2 ¡ ¹ l T O (n+q)£(n+q+1) j I n+q ¹ O 3 7 7 7 7 7 7 7 7 7 7 5 (2.50) b 1 (t)=[1;0;:::;0; | {z } n+q+1 ¡p 0 ;0;:::;0 | {z } n+q+1 ] T (2.51) b 2 (t)=[0;:::;0; | {z } n+q+1 1;0;:::;0 | {z } n+q+1 ] T : (2.52) and O (n+q)£(n+q+1) is an (n + q) by (n + q + 1) matrix, ¹ O is a 1 by (n + q) matrix, both with all elements equal to zero. Becauseu = ¤u f = u (n+q+1) f +¸ T ® n+q (s)u f and y =¤y f =y (n+q+1) f +¸ T ® n+q (s)y f where¤=s n+q+1 +¸ T ® n+q (s), we have u=[0;:::;0 | {z } n+q+1 1;0;:::;0 | {z } n+q+1 ]_ x+[0;:::;0 | {z } n+q+1 ¸ T ]x (2.53) y =[1;0;:::;0 | {z } n+q+1 0;:::;0 | {z } n+q+1 ]_ x+[¸ T 0;:::;0 | {z } n+q+1 ]x: (2.54) Step 2. Establish exponential stability (e.s.) of the homogeneous part 28 For the matrixA(t), at each frozen timet we have det(sI¡A(t))=¤ q ³ Q m LR p ¢ ^ R f ¢ ^ R m +PZ p ¢ ^ Z f ¢ ^ Z m ´ : (2.55) A(t) is a stable matrix at each timet since¤ q is Hurwitz and the roots ofQ m LR p ¢ ^ R f ¢ ^ R m +PZ p ¢ ^ Z f ¢ ^ Z m are stable due to (2.37) being satisfied. (2.55) can be verified from (2.38) and (2.39) where the output at each frozen timet becomes y f = 1 K 2 ³ ^ R f LQ m ("m 2 s )+ ^ Z m Z p ¤ q y m1 ´ (2.56) K 2 =¤ q ³ Q m LR p ¢ ^ R f ¢ ^ R m +PZ p ¢ ^ Z f ¢ ^ Z m ´ (2.57) whose state space realization is (2.42). By frozen time we mean that the time-varying coefficients are treated as constants when two polynomials are multiplied. We represent frozen time multiplication of two polynomials by the operatorX(s;t)¢Y(s;t), which is to distinguish it from the operationX(s;t)Y(s;t) wheres is a differential operator and the product involves time derivatives of the coefficients of the polynomials. The adaptive law guarantees that µ 2 L 1 and _ µ 2 L 2 . The elements of the matrix A(t) are ¹ µ 1 , ¹ µ 2 , ¹ p, and ¹ l which are linear combinations of the parameters inµ and p µ 4 2 L 1 , sinceµ 4 is always greater than zero due to projection. It then follows thatkA(t)k2 L 1 . Also the elements of _ A(t) are linear combinations of _ µ 2 L 2 and 1 2 p µ 4 _ µ 4 2 L 2 29 which implies thatk _ A(t)k2L 2 . Using Theorem A.8.8 in [36], the homogeneous part is uniformly asymptotically stable which is equivalent to exponentially stable. Step 3. Establish boundedness The exponentially weightedL 2± norm is defined as kx t k 2± , µZ t 0 e ¡±(t¡¿) x T (¿)x(¿)d¿ ¶ 1 2 ; (2.58) where±¸0 is a constant. We say thatx2L 2± ifkx t k 2± exists. For clarity of presentation we will denote theL 2± norm ask¢k. From (2.49) and using Lemma A.5.10 from [36] combined withA(t)2L 1 andb 1 (t)2L 1 we have kxk·ck"m 2 s k+c; (2.59) jx(t)j·ck"m 2 s k+c (2.60) for some± >0. Define a fictitious normalizing signal as m 2 f ,1+kyk 2 +kuk 2 : (2.61) Using Lemma A.5.10 in [36] and (2.18) and (2.29) it follows that Á m f ; m s m f 2L 1 . Taking (2.49), (2.53), (2.54), and (2.59) with Lemma A.5.10 in [36] we have m 2 f ·c+ck"m 2 s k 2 ; (2.62) 30 and now using the fact that m s m f 2L 1 , m 2 f ·c+ck"m s m f k 2 ; (2.63) which by definition of theL 2± norm is m 2 f ·c+ Z t 0 e ¡±(t¡¿) " 2 (¿)m 2 s (¿)m 2 f (¿)d¿: (2.64) Since "m s 2 L 2 is guaranteed by the adaptive law and using the Bellman-Gronwell Lemma (Lemma A.6.3 in [36]) withc 0 = 0;c 1 = c;c 2 = c;® = ±;k(¿) = " 2 (¿)m 2 s (¿), and y(t) = m 2 f (t) it is established that m f 2 L 1 . The boundedness of the rest of the signals follows from this fact. Step 4. Establish tracking errore 1 converges to zero Starting with (2.38) above we have "m 2 s = ^ R m R p 1 ¤ p y p ¡ ^ Z m Z p 1 ¤ p u p (2.65) and then filtering each side with ^ R f LQ m 1 ¤q we obtain ^ R f LQ m 1 ¤ q ("m 2 s )= ^ R f LQ m 1 ¤ q µ ^ R m R p 1 ¤ p y p ¡ ^ Z m Z p 1 ¤ p u p ¶ : (2.66) 31 Now we will use the fact that¤=¤ q ¤ p and use Swapping Lemma 1 (Lemma A.11.1 in [36]) to obtain the following equation LQ m ^ R m R p ¤ y p = ^ R m R p LQ m ¤ y p +r 1 (2.67) where r 1 ,W c1 (s) ³ ¡ W b1 (s)® T n+1 (s)y p ¢ _ ¹ µ 5 ´ (2.68) ^ R m R p =s n+2 + ¹ µ T 5 ® n+1 (s) (2.69) and W c1 (s);W b1 (s) are defined as in Swapping Lemma 1 with W = LQm ¤ in (2.68). Since _ ¹ µ 5 is a linear combination of _ µ2L 2 it follows that _ ¹ µ 5 2L 2 . Combining this with y p 2L 1 we have thatr 1 2L 2 . Now using Swapping Lemma 3 (Lemma A.11.3 in [36]) on the other term on the right side of (2.66) ( ^ R f LQ m )( ^ Z m Z p 1 ¤ u p )=( ^ Z m Z p )¢ µ ^ R f LQ m 1 ¤ u p ¶ +r 2 (2.70) where r 2 ,[1; ¹ l T ]D n+q (s) · ® n+q (s) µ ® T n+1 (s) 1 ¤ u p ¶ _ ¹ b ¸ (2.71) ^ R f LQ m =s n+q+1 + ¹ l T ® n+q (s)=[1; ¹ l T ]® n+q+1 (s) (2.72) ^ Z m Z p = ¹ b T ® n+1 (s) (2.73) 32 and D n+q (s) is as defined in Swapping Lemma 3. Since _ ¹ b is a linear combination of _ µ 2L 2 it follows that _ ¹ b2L 2 . Similarly ¹ l is calculated from the parameters inµ, so we have ¹ l 2 L 1 . Combining this with u p 2 L 1 we have that r 2 2 L 2 . Using (2.67) and (2.70) in (2.66) we obtain ^ R f LQ m 1 ¤ q ("m 2 s )= ^ R f ^ R m R p LQ m ¤ y p ¡ ( ^ Z m Z p )¢( ^ R f LQ m ) ¤ u p + ^ R f r 1 ¡r 2 (2.74) Again Swapping Lemma 3 will be used on the first term on the right side of (2.74) ^ R f µ ^ R m R p LQ m ¤ y p ¶ = ^ R f ¢ µ ^ R m R p LQ m ¤ y p ¶ +r 3 (2.75) where r 3 ,[1; ¹ µ T r ]D 1 (s) · ® 1 (s) µ ® T 2 (s) R p LQ m ¤ y p ¶ [0; _ µ T d ] T ¸ (2.76) ^ R f =s 2 + ¹ µ T r ® 1 (s)=[1; ¹ µ T r ]® 2 (s) (2.77) ^ R m =s 2 +µ T d ® 1 (s)=[1;µ T d ]® 2 (s) (2.78) and D 1 (s) is as defined in Swapping Lemma 3. Again, since ¹ µ r is computed from the parameters in µ 2 L 1 and each parameter in µ is bounded to be greater than zero by 33 projection, we have that ¹ µ r 2L 1 . Combining this with _ µ d 2L 2 andy p 2L 1 we have thatr 3 2L 2 . Plugging back into (2.74) ^ R f LQ m 1 ¤ q ("m 2 s )= ^ R f ¢ ^ R m R p LQ m ¤ y p ¡ ( ^ Z m Z p )¢( ^ R f LQ m ) ¤ u p + ^ R f r 1 ¡r 2 +r 3 (2.79) Using the identity ^ R f LQ m = ^ R f ¢LQ m , now ^ R f LQ m ¤ u p = ^ R f ¢LQ m ¤ u p =¡ ^ Z f ¢P ¤ e 1 (2.80) and combining this with Q m y p = Q m (e 1 + y m ) = Q m e 1 , ^ R m R p = ^ R m ¢ R p , and ^ Z m Z p = ^ Z m ¢Z p , now (2.79) becomes ^ R f LQ m 1 ¤ q ("m 2 s )= ^ R f ¢ ^ R m ¢R p LQ m ¤ e 1 + ( ^ Z m ¢Z p )¢ ^ Z f ¢P ¤ e 1 + ^ R f r 1 ¡r 2 +r 3 (2.81) Solving for the tracking errore 1 we obtain e 1 = ¤ B " ^ R f LQ m ¤ q ("m 2 s )¡ ^ R f r 1 +r 2 ¡r 3 # (2.82) 34 B = ^ R f ¢ ^ R m ¢R p LQ m + ^ Z m ¢Z p ¢ ^ Z f ¢P: (2.83) The roots of (2.83) are stable due to (2.37) being satisfied a priori. Since all the transfer functions in (2.82) are proper stable transfer functions with inputs bounded and inL 2 , it follows thate 1 2L 2 \L 1 . Now starting with y p = Z p (s) R p (s) ^ Z m (s) ^ R m (s) u p ; (2.84) we can write ¤ p y p =(R p ^ R m ¡¤ p )y p =Z p ^ Z m u p (2.85) which becomes y p =¡ (R p ^ R m ¡¤ p ) ¤ p y p + Z p ^ Z m ¤ p u p (2.86) and _ y p =¡ s(R p ^ R m ¡¤ p ) ¤ p y p + Z p ^ Z m ¤ p u p : (2.87) Since we defined ¤ p to be a monic hurwitz polynomial of degree n+2 all the transfer functions in the above equation are proper and stable. Also because y p ;u p 2 L 1 then _ y p 2L 1 . This then implies that _ e 1 2L 1 . Using this and thate 1 2L 2 \L 1 with a form of Barbalat’s Lemma implies thate 1 !0 ast!1. ¥ 35 2.4 Special Cases The adaptive notch filter scheme presented is a general methodology which involves an online estimator and an adaptive notch filter which uses a parameter estimate as the center frequency. However, if the system were to possess a single complex pole instead of a dipole, a special variation of the adaptive filter may be designed using a similar methodology and analysis. We will replace the mode in (2.2) with the form Z m (s) R m (s) = ! 2 d s 2 +2³! d s+! 2 d (2.88) where³ > 0 is the damping and! d > 0 is the natural frequency of the mode. There are two different adaptive filter variations that are designed for this type of flexible system, however the adaptive notch filter presented earlier can also be used, this is left as a design decision. The first variation involves an adaptive filter that cancels the complex poles and replaces it with very high frequency real poles. The rigid body controller for this scheme is designed while completely neglecting the flexible dynamics and adaptive filter. The second variation contains an adaptive filter that will cancel the complex pole and replace it with a damped low frequency resonance, which the rigid body controller will use in the design process. 36 2.4.1 Filter with high frequency poles Going back to the feedback diagram in Fig. 2.1, we replace the filter with F(s)= R m (s) Z m (s) Z f (s) R f (s) : (2.89) This filterF(s) should place the new damped poles such thatZ f (s) = ¯ 2 andR f (s) = (s+¯) 2 where ¯ is much faster than the fastest pole in A ¤ (s) in (2.6). Again we will treat the flexible modes and filter as an uncertainty and for stability of the system the following must be satisfied ° ° ° ° C(s)G p (s) 1+C(s)G p (s) ¢(s) ° ° ° ° 1 = ° ° ° ° P(s)Z p (s) A ¤ (s) ¢(s) ° ° ° ° 1 <1 (2.90) ¢(s)= Z f (s)¡R f (s) R f (s) : (2.91) Similar to before, it can be shown that the tracking errore 1 =y p ¡y m is e 1 =¡ L(s)R p (s) Q m (s)L(s)R p (s)+P(s)Z p (s)(1+¢(s)) Q m (s)y m (2.92) 37 which is a proper stable transfer function since (2.90) is satisfied. Therefore the control law will cause e 1 ! 0 as t ! 1 exponentially fast. The sensitivity function of this control scheme is S 1 (s)= Q m (s)L(s)R p (s)R f (s) Q m (s)L(s)R p (s)R f (s)+P(s)Z p (s)Z f (s) (2.93) which, due to the design of F(s) having much faster poles than A ¤ (s), can be approxi- mated by ¹ S 1 (s)= Q m (s)L(s)R p (s) A ¤ (s) : (2.94) This is the same as the sensitivity function for the system when the filter and flexible modes are neglected. Similarly, the complementary sensitivity function can be approxi- mated by ¹ T 1 (s)= P(s)Z p (s)Z f (s) A ¤ (s)R f (s) : (2.95) This shows that the system, with the filter and flexible modes neglected, will be aug- mented with a pair of high frequency real poles. Both of these approximations may be useful in analyzing the response of a system when neglecting the flexible dynamics. Now the online estimator will need to be changed to reflect the change in the plant structure. We will replace (2.15) with Z ¤ m (s) R ¤ m (s) = ! ¤2 d s 2 +2³ ¤ ! ¤ d s+! ¤2 d : (2.96) 38 The parametric model to estimate the unknown modal frequency is as follows: z =µ ¤T Á (2.97) z = s 2 R p (s) ¤ p (s) y p (2.98) Á= · ¡sR p (s) ¤p(s) y p Z p (s) ¤p(s) u p ¡ R p (s) ¤p(s) y p ¸ T (2.99) µ ¤ = · 2³ ¤ ! ¤ d ! ¤2 d ¸ T (2.100) and ¤ p (s) is a monic Hurwitz polynomial of degree n+2. Allow Á = [Á 1 ;Á 2 ] T ;µ = [µ 1 ;µ 2 ] T , and also some a priori known bounds on the damping and natural frequency such that 1 ¸ ³ u ¸ ³ ¤ ¸ ³ l > 0 and ! u d ¸ ! ¤ d ¸ ! l d > 0 are satisfied. The update equations are _ µ 1 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 1 "Á 1 if(2³ u ! u d >µ 1 >2³ l ! l d ) or(µ 1 =2³ l ! l and"Á 1 ¸0) or(µ 1 =2³ u ! u and"Á 1 ·0); 0 otherwise (2.101) _ µ 2 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 2 "Á 2 if((! u d ) 2 >µ 2 >(! l d ) 2 ) or(µ 2 =(! l ) 2 and"Á 2 ¸0) or(µ 2 =(! u ) 2 and"Á 2 ·0); 0 otherwise (2.102) 39 "= z¡µ T Á m 2 s (2.103) m 2 s =1+Á T Á: (2.104) In the update equations, the bounds ³ l ;³ u ;! l d ;! l d are constants determined a priori, and the adaptive gains ° 1 ;° 2 > 0 are also design parameters chosen a priori. The above estimation law guarantees that (i) µ2L 1 (ii) ";"m s ; _ µ2L 2 \L 1 . The adaptive control law is formed by replacingF(s) in (2.89) with an adaptive notch filter whose parameters are replaced with the online estimatesµ to form F(s)= s 2 +µ 1 s+µ 2 µ 2 Z f (s) R f (s) : (2.105) This makes the adaptive control law u p =¡ ^ R m (s) ^ Z m (s) Z f (s) R f (s) P(s) L(s)Q m (s) (y p ¡y m ) (2.106) where (2.90) is satisfied. It should be noted that (2.90) does not contain any unknown or estimated parameters, but instead can be satisfied offline a priori. The analysis of the adaptive scheme is very similar to that given earlier and ommitted for brevity. 40 2.4.2 Filter with low frequency poles In this scheme the notch filter will exactly cancel the flexible mode and replace it with a highly damped low frequency mode. The rigid control law is designed for the system with the lightly damped mode replaced with a damped mode at a desired frequency. This approach for control design is similar to the MIMO decoupling control design in [53] when the new damped mode is slower than the desired closed loop bandwidth. However, if the new damped mode is placed at the same frequency as the flexible mode, the filter will take the shape of the adaptive notch filter. The advantage of this scheme is that the newly placed poles can help shape the sen- sitivity and complementary sensitivity functions to achieve the disturbance rejection re- quired. Although depending on the particular application, the stability margins and clas- sical performance measures, such as overshoot and settling time, may be worse than the first variation presented above. Same as placing the poles outside the bandwidth we use the filter in (2.89) except for the modification to the new poles which become Z f (s) R f (s) = ! 2 r s 2 +2³ r ! r s+! 2 r (2.107) where³ r ;! r > 0. In this variation, the filterF(s) in (2.89) should place the new pole’s natural frequency ! r slower than the bandwidth of the closed loop rigid system, that is the closed loop system while neglecting the flexible modes and adaptive filter. Also ³ r should be made close to unity so as not to induce a resonance into the system. Now the 41 rigid body controllerC(s) is no longer designed forG p (s) but instead it is designed for G p (s) Z f (s) R f (s) , which leads to the polynomial equation L(s)Q m (s)R p (s)R f (s)+P(s)Z p (s)Z f (s)=A ¤ (s) (2.108) where A ¤ (s) are the desired closed loop poles. It can be shown that the tracking error e 1 =y p ¡y m is e 1 =¡ L(s)R p (s)R f (s) A ¤ (s) Q m (s)y m (2.109) which is a proper stable transfer function. Therefore the control law will cause e 1 ! 0 as t ! 1 exponentially fast. Examining the sensitivity and complementary sensitivity functions points to an result seen in [53], which shows that the sensitivity function will become S 2 (s)= Q m (s)L(s)R p (s)R f (s) A ¤ (s) (2.110) where the slow poleR f (s) has now become a zero ofS 2 (s). This will create a corner in the plot, which may be used for disturbance rejection based on the desired performance requirements. The estimator is the same as in (2.101)-(2.104) and the adaptive control law is the same as in (2.106) except theZ f (s) andR f (s) polynomials are as in (2.107). Again the stability results and analysis of this scheme is very similar to that presented in Section 2.3. 42 2.5 Numerical Examples A simple numerical example is added to further describe the advantages of the adaptive mode suppression scheme presented. The adaptive notch filter scheme (ANF) will be compared to a non-adaptive notch filter (NA) scheme as well as a classical adaptive pole placement (APPC) scheme. In all cases control schemes are designed for various desired closed loop polesA ¤ (s) and should track a constant reference signaly m = 10 leading to Q m (s)=s. The plant is y p = µ 10 s+1 ¶µ s 2 +2³ ¤ n ! ¤ n s+! ¤2 n s 2 +2³ ¤ d ! ¤ d s+! ¤2 d ¶ (2.111) where the nominal values are ! ¤ n = ! ¤ d = 20, ³ ¤ n = 0:3, and ³ ¤ d = 0:03. The online estimation scheme described in Section II is then z =µ ¤T Á (2.112) z = s 2 (s+1) (s+100) 3 y p ¡ 10s 2 (s+100) 3 u p (2.113) Á= · 10s (s+100) 3 u p ; 10 (s+100) 3 u p ; ¡ s(s+1) (s+100) 3 y p ; ¡ (s+1) (s+100) 3 y p ¸ T (2.114) µ ¤ = · 2³ ¤ n ! ¤ n ! ¤2 n 2³ ¤ d ! ¤ d ! ¤2 d ¸ T : (2.115) 43 The bounds on the unknown parameters are given a priori as 0:4 ¸ ³ ¤ n ¸ 0:2, 0:04 ¸ ³ ¤ d ¸ 0:02,25¸ ! ¤ n ¸ 15, and25¸ ! ¤ d ¸ 15. These bounds are used in the projection of the parameters in the following gradient update algorithm _ µ 1 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 1 "Á 1 if(20>µ 1 >6) or(µ 1 =6 and"Á 1 ¸0) or(µ 1 =20 and"Á 1 ·0); 0 otherwise (2.116) _ µ 2 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 2 "Á 2 if(625>µ 2 >225) or(µ 2 =225 and"Á 2 ¸0) or(µ 2 =625 and"Á 2 ·0); 0 otherwise (2.117) _ µ 3 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 3 "Á 3 if(2>µ 3 >0:6) or(µ 3 =0:6 and"Á 3 ¸0) or(µ 3 =2 and"Á 3 ·0); 0 otherwise (2.118) _ µ 4 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 4 "Á 4 if(625>µ 4 >225) or(µ 4 =225 and"Á 4 ¸0) or(µ 4 =625 and"Á 4 ·0); 0 otherwise (2.119) 44 "= z¡µ T Á m 2 s (2.120) m 2 s =1+Á T Á (2.121) where the adaptive gains ° 1 ;° 2 ;° 3 ;° 4 are set offline based on the desired A ¤ and the adaptive control law. The adaptive notch filter is F A4 (s)= s 2 +2(0:015) p µ 4 s+µ 4 s 2 +2(0:321) p µ 4 s+µ 4 : (2.122) The rigid body controllerC(s) will be designed using a pole placement strategy by solv- ing (2.6) with a desired closed loop poles of A ¤ (s) = (s+a ¤ ) 2 where a ¤ is an integer that will be varied for different simulations. For the APPC, the filterF(s) is removed and C(s) becomes an adaptive pole placement controller based on the full estimated plant of G p (s) ^ M(s). Lastly, there is a non-adaptive control strategy (NA) which consists of a wide fixed notch filterF NA (s) and a pole placement rigid body controllerC(s) which is the same as for the adaptive notch filter case (ANF). The wide notch filter becomes F NA2 (s)= s 2 +0:6s+400 s 2 +38s+400 : (2.123) Simulations are conducted with the uncertain flexible mode frequency being ! ¤ n = ! ¤ d =18 and the damping constants remaining unchanged from the nominal values. The desired closed loop bandwidth is varied with a ¤ = 5;10;50 and the overshoot and 2% settling times for the schemes are seen in Table 2.1. These results point to the advantage 45 Table 2.1: Simulation results a ¤ =5 Overshoot (%) Settling Time (s) ANF 12.93 2.53 APPC large large NA 21.71 4.45 a ¤ =10 Overshoot (%) Settling Time (s) ANF 16.75 4.26 APPC 74.08 2.10 NA unstable unstable a ¤ =50 Overshoot (%) Settling Time (s) ANF 11.79 0.12 APPC 205.50 4.06 NA unstable unstable of the adaptive schemes as they all remain stable, even when the plant is uncertain and a high bandwidth is desired. This is only a single numerical situation where the only one advantage of the adaptive notch filter is seen. The narrow adaptive notch filter also allows for the system to have better disturbance rejection and higher bandwidth as seen in [47, 45]. The simulation when a ¤ = 10 will be highlighted. For this simulation the system output and estimated parameters of interest, the complex pole frequency ! d , are shown in Fig. 2.4. At the beginning of the simulation theµ ¤ vector has incorrect values, thereby causing the adaptive notch filter to be centered at the incorrect frequency, which in turn causes the system to be unstable. As the simulation continues the parameters begin to 46 0 2 4 6 8 10 0 5 10 15 Time (s) Output y p 0 2 4 6 8 10 17 18 19 20 21 Time (s) ω d (rad/s) Figure 2.4: Time series plots from ANF whena ¤ =10. Top plot: Plant outputy p , which is the measured signal. Bottom plot: Complex pole frequency, the dashed red line is the actual value and the solid blue line is the estimate ^ ! d . change and the notch center frequency ^ ! d locks on to the correct value of 20 rad/s. The system becomes stable and the output settles; the estimated parameter also converges as we are in a disturbance free environment. The sensitivity and complementary sensitivity functions for the closed loop system in adaptive notch filter scheme can be plotted. Since the estimated parameters converge, the values are frozen and the adaptive notch filter is treated as LTI. The bode plots for both functions are plotted at the beginning of the simulation, when the estimates are incorrect, and at the end of the simulation, when the parameters converge. The sensitivity bode is seen in Fig. 2.5 and the complementary sensitivity bode in Fig. 2.6. The plots show possibly undesirable disturbance rejection as well as noise rejection at the start of 47 -40 -30 -20 -10 0 10 Magnitude (dB) 10 0 10 1 10 2 -45 0 45 90 135 Phase (deg) Bode Diagram Frequency (rad/sec) Start End Figure 2.5: Bode plot of the sensitivity functions for ANF whena ¤ = 10 at the start and end of the simulation. the simulation and once the adaptive notch filter tracks to the correct frequency these problems are solved. 2.6 Discussion This paper presents an adaptive mode suppression scheme for dealing with systems with unknown or changing flexible dynamics. An online parameter estimate is able to track the frequency of the flexible mode and this estimate is used online to update the param- eters of the adaptive notch filter. Therefore the adaptive notch filter can be designed narrowed which leads to higher closed loop performance and less deterioration due to the phase lag induced from the notch filter. Theoretical analysis is given, which leads to 48 -20 -15 -10 -5 0 5 Magnitude (dB) 10 0 10 1 10 2 -135 -90 -45 0 45 Phase (deg) Bode Diagram Frequency (rad/sec) Start End Figure 2.6: Bode plot of the complementary sensitivity functions for ANF whena ¤ =10 at the start and end of the simulation. certain requirements on the point-wise stability of the adaptive notch filter and rigid body controller. The stability proof that is provided shows that the all signals in the closed loop are bounded and the tracking error of such systems will converge to zero. Simulations are added to demonstrate the benefit of such an adaptive mode suppression scheme. In this paper only a single flexible mode is suppressed and the case where multiple modes are suppressed with adaptive notch filters is currently under investigation. 49 Chapter 3 Adaptive Mode Suppression for MIMO Systems 3.1 Introduction This chapter will take the adaptive mode suppression techniques of the previous chapter to the next level, by incorporating their usage on MIMO systems. MIMO systems differ from that of SISO in that the same adaptive notch filter may not be desired for every input or output of the plant. There may be a better way to optimize the notch filter designs, and this is very application dependent. Here the intended application of such a control scheme is aircraft and other flight vehicles as they are generally nonlinear MIMO systems that possess uncertain or changing flexible modes. Taking the problems and system characteristics associated with aircraft into consideration, the adaptive mode suppression scheme for MIMO systems is designed. Later in this report an application example will be given with the control scheme presented in this chapter. 50 One possible solution to the changing dynamics is an adaptive notch filter, a notch filter that is able to track and suppress the flexible modes online. The adaptive notch filter presented in [21] is used on the model of a booster from the Advanced Launch System (ALS) program. The least squares estimator in the publication uses a simple undamped resonator as the model for estimation and functions well since the resonant mode is very pronounced. However in the GHV , and other applications, full plant parameterizations is necessary as the flexible mode may not be as significant. Another strategy for the estima- tion of the center frequency can be found in [69], where frequency weighting functions are used. The downside is there are several failure modes that are known and avoidance requires some modal information a priori. A stochastic state space algorithm for mode frequency estimation is presented in [61]; however it relies on the injection of a probe signal which is not needed in the scheme presented here. The indirect adaptive compen- sation (IAC) scheme in [94] also requires a probe signal to complete the estimation. The adaptive mode suppression scheme in [54] uses a LMS algorithm to update filter coeffi- cients and then the modal parameters are extracted from the filter. This is opposite as to what is being presented in this paper, where the modal parameters are first estimated and then used in the adaptive notch filter. An overview of the type of plant being controlled will be presented first and then the adaptive mode suppression scheme for such a MIMO plant is given. A point-wise stability argument that will be used in a full stability proof, which is still in the works, will follow. 51 3.2 Plant Overview The dynamics of the given linear MIMO plant are given as y =G(s)u (3.1) where y 2 < 5 is the output vector, u 2 < 2 is the input vector, and G(s) is the transfer function matrix. An element in theith row andjth column ofG(s) can be written asG ij . We will assume some of the parameters in the model are unknown and will be denoted as £ ¤ and in the following sections these parameters will be estimated online and replaced with the estimates£, so the transfer function matrix becomes G(s;£ ¤ )= 2 6 6 6 6 6 6 4 G 1 (s;µ ¤ 1 ) . . . G 5 (s;µ ¤ 5 ) 3 7 7 7 7 7 7 5 (3.2) where µ ¤ i = [µ ¤ i1 :::µ ¤ i6 ] T 2 i ½< 6 are the actual parameters and = 1 £¢¢¢£ 5 and £=[µ ¤ 1 :::µ ¤ 5 ]2½< 5£6 : (3.3) Now consider the transfer functions matrix G i (s;µ ¤ i )= 1 d i (s;µ ¤ i ) £ n i 1 (s;µ ¤ i ) n i 2 (s;µ ¤ i ) ¤ (3.4) 52 where d i (s;µ ¤ i )=D k (s)(s 2 +2µ ¤ i5 s+µ ¤ i6 ) (3.5) n i 1 (s;µ ¤ i )=N k i1 (s)(µ ¤ i1 s 2 +2µ ¤ i2 s+µ ¤ i6 ) (3.6) n i 2 (s;µ ¤ i )=N k i2 (s)(µ ¤ i3 s 2 +2µ ¤ i4 s+µ ¤ i6 ) (3.7) and D k (s)=s r + ¹ ® r¡1 s r¡1 +¢¢¢+ ¹ ® 1 s+ ¹ ® 0 (3.8) N k i1 (s)= ¹ ¯ i 1;r¡1 s r¡1 +¢¢¢+ ¹ ¯ i 1;1 s+ ¹ ¯ i 1;0 (3.9) N k i2 (s)= ¹ ¯ i 2;r¡1 s r¡1 +¢¢¢+ ¹ ¯ i 2;1 s+ ¹ ¯ i 2;0 (3.10) where r is the order of the polynomial D k (s). In the above equations the polynomials D k (s), N k i1 (s), and N k i2 (s) are considered known and encompass both the rigid-body dynamics and any assumed known flexible dynamics. We then have d i (s;µ ¤ i )=s r+2 +® i r+1 (µ ¤ i )s r+1 +¢¢¢+® i 1 (µ ¤ i )s+® i 0 (µ ¤ i ) (3.11) n i 1 (s;µ ¤ i )=¯ i 1;r+1 (µ ¤ i )s r+1 +¢¢¢+¯ i 1;1 (µ ¤ i )s+¯ i 1;0 (µ ¤ i ) (3.12) n i 2 (s;µ ¤ i )=¯ i 2;r+1 (µ ¤ i )s r+1 +¢¢¢+¯ i 2;1 (µ ¤ i )s+¯ i 2;0 (µ ¤ i ) (3.13) 53 3.3 Adaptive Mode Suppression Scheme The adaptive mode suppression scheme will be presented in the following subsections. An overview of the control scheme is shown in Fig. 3.1. Here the plant has measurable outputsy2< 5 and control inputsu2< 2 which will be used by a robust online estima- tor to estimate the flexible dynamics of the plant while in realtime without the use of a probing signal. These estimated parameters of the modal dynamics will be used to tune to the notch filters, thereby creating adaptive notch filters. There are five separate adap- tive notch filters, one for each measured output. The estimated flexible dynamics of the plant are used as inputs to functions which will return parameters that define the shape of each filter. These functions are found offline based on a priori knowledge of possi- ble modal parameters of the plant. The rigid-body controller is designed to control the rigid-body dynamics while ignoring the flexible modes and adaptive notch filters. The adaptive notch filters and flexible dynamics are treated as an uncertainty for the design. Because our objective is to demonstrate the performance of the adaptive notch filter in suppressing flexible modes the rigid-body controller denoted as K 0 (s) is any controller which satisfied performance requirements and a stability condition which will be given later. Each of these systems are further described below. 54 Plant G(s) r Robust Online Estimator Rigid-Body Controller u y Adaptive Notch Filter F(s) Figure 3.1: Overall diagram of the adaptive mode suppression scheme for the generic hypersonic vehicle. Here y 2 < 5 is all the measurable outputs , u 2 < 2 is the control inputs, andr2< 5 is the reference input shown for demonstration purposes. 3.3.1 Adaptive Notch Filter The adaptive notch filter is implemented to suppress the slowest flexible mode of the GHV by filtering each of the outputs from the plant, which may also be considered the outputs from measurable sensors. The frequency of the flexible mode may, for various reasons given earlier, be uncertain or change. The desired shape of the notch filter in each sensor path is determined by solving an optimization problem offline. This approach to designing the notch filters by optimization has been done in a MIMO setting in [11] and a single loop problem in [16]. In both of these references the authors were concerned with meeting the Military Specification for gain margin as it pertains to flexible modes. The design is posed similarly here, except instead of using gain margin we will be at- tempting to suppress the flexible dynamics in each measured output path individually and add some new constraints for the amount of variation tolerated. The parameters for the offline optimization problem will be varied over a range of possible values and the notch 55 filter parameters computed. These computed values are stored and the correct parame- ters selected online based on the estimated µ i and linearly interpolating between stored values. This will now be explained in further detail. Referring to Fig. 3.1, the feedback signals used for the rigid-body controller are F(s;£ ¤ )G(s;£ ¤ )u. The notch filter bank can be expressed as a transfer function matrix F(s;£ ¤ ), comprised of only diagonal elementsF i (s;µ ¤ i ) and all zero off diagonal entries. Now the optimization problem is solved for each F i (s;µ ¤ i ) separately, assuming all the parameters in each filter path are known. Therefore G i (s;µ ¤ i ), which takes the form of (3.4), is treated as a known transfer function matrix. We wish to design the notch filter in this path,F i (s;µ ¤ i ), so that the slowest flexible mode is completely suppressed. Using a superscript of m to denote the modal part needing suppression and the subscript ij corresponding the element ofG(s;£ ¤ ), the flexible mode is expressed as G m i1 (s;µ ¤ i )= µ ¤ i1 s 2 +2µ ¤ i2 s+µ ¤ i6 s 2 +2µ ¤ i5 s+µ ¤ i6 (3.14) and G m i2 (s;µ ¤ i )= µ ¤ i3 s 2 +2µ ¤ i4 s+µ ¤ i6 s 2 +2µ ¤ i5 s+µ ¤ i6 : (3.15) Which can also be expressed, in a more meaningful way, as G m i1 (s)= µ ! 2 D ! 2 i1 ¶ s 2 +2³ i1 ! i1 s+! 2 i1 s 2 +2³ D ! D s+! 2 D (3.16) 56 and G m i2 (s)= µ ! 2 D ! 2 i2 ¶ s 2 +2³ i2 ! i2 s+! 2 i2 s 2 +2³ D ! D s+! 2 D : (3.17) Theith adaptive notch filter takes the form F i (s;µ ¤ i )= b 2 (µ ¤ i )s 2 +b 1 (µ ¤ i )s+b 0 (µ ¤ i ) s 2 +a 1 (µ ¤ i )s+a 0 (µ ¤ i ) ; (3.18) which can also be expressed in a clearer form F i (s)= µ ! iR ! iZ ¶ s 2 +2³ iZ ! iZ s+! 2 iZ s 2 +2³ iR ! iR s+! 2 iR ; (3.19) where the ³ iZ < ³ iR and ! iZ ¸ ! iR will determine the shape of the filter. These will now be generalized since each ith filter path, where i = 1;:::;5, is identical in it’s construction. The generalized flexible modesM 1 (s);M 2 (s) and filterN(s) used for the optimization problem become M 1 (s)= µ ! 2 ! 2 1 ¶ s 2 +2³ 1 ! 1 s+! 2 1 s 2 +2³!s+! 2 (3.20) M 2 (s)= µ ! 2 ! 2 2 ¶ s 2 +2³ 2 ! 2 s+! 2 2 s 2 +2³!s+! 2 (3.21) N(s)= µ ! 2 R ! 2 Z ¶ s 2 +2³ Z ! Z s+! 2 Z s 2 +2³ R ! R s+! 2 R : (3.22) 57 An optimization problem is now posed where we wish to minimize the phase lag of the filter at some desired frequency. This becomes min ³ Z ;³ R ;! Z ;! R · arctan µ 2³ R ! R ! c ! 2 R ¡! 2 c ¶ ¡arctan µ 2³ Z ! Z ! c ! 2 Z ¡! 2 c ¶¸ (3.23) where ! c is a design parameter that specifies the point at which the phase lag is mini- mized. For our simulations we choose! c = 1 2 !, which is half of the frequency at which the peak occurs in the frequency responses ofM 1 (s) andM 2 (s). Now there are several constraints for this minimization problem. The first couple retain the notch shape of the filter and are 1. ³ Z <³ R 2. ! Z ¸! R . since the peak of the flexible dynamics will occur at! we place the maximum magnitude suppression at the same frequency, leading to the constraint 3. ! Z =!. Next we add in the constraints to ensure that the flexible dynamics that occur in each filter path are suppressed below some level, 4. jN(s)M 1 (s)j 1 ·a 5. jN(s)M 2 (s)j 1 ·a, 58 where a is the maximum desired magnitude of the flexible mode and notch filter com- bination. Since the flexible dynamics and notch filter are normalized so as to have a dc-gain of unity, it may be desired to completely suppress the flexible mode and there- fore seta = 1. However this can pose problems for the optimization routine, so a value slightly larger than one may be a better choice. Another constraint is added which keeps the maximum magnitude of the notch filter below unity 6. jN(s)j 1 ·1. At this point this design methodology works to suppress a known flexible mode in two transfer functions, however in practice the modal parameters are unknown. Even with the addition of an online estimator, the notch filters should be designed to accept some uncertainty in (3.20) and (3.21). This is accomplished by adding four constraints based on some user specified acceptable uncertainty. These uncertainties are expressed as !§ ¹ ! ! 1 § ¹ ! 1 ! 2 § ¹ ! 2 (3.24) ³ § ¹ ³ ³ 1 § ¹ ³ 1 ³ 2 § ¹ ³ 2 (3.25) where the overbar variables are constants that specify the deviation from the nominal quantity. It may be helpful to express these overbar parameters as a percentage of the nominal value, ie. ¹ ! =0:1!, thereby signifying a 10% variation in complex pole natural frequency. The worst case modal shapes, in terms of largest magnitude, are extracted 59 from the uncertainties and then expressed as two transfer functions for each original generalized mode in Eqs. (3.20), (3.21). From these the constraints are formed 7. jN(s)V 1 1 (s)j 1 ·a 8. jN(s)V 2 1 (s)j 1 ·a 9. jN(s)V 1 2 (s)j 1 ·a 10. jN(s)V 2 2 (s)j 1 ·a. This method of employing several constraints to design the notch filters is similar to the method in [11], where an envelope is created that the notch filter must suppress. Once the minimization problem is posed and the constraints are set, the optimization toolbox in MATLAB is used to solve the problem and find an optimal notch filterN(s). If the overbar uncertainty parameters in Eqs. (3.24) - (3.25) are constant, and constraint 3 is upheld, then the notch filter solution can be viewed as a function of ! 1 ! , ! 2 ! , ³, ³ 1 , and³ 2 . Therefore a solution of different notch filters can be found by varying these five parameters through some known space where they may occur, which is equivalent to varying aµ i vector. The parameters of the notch filter solution, ! R , ! Z , ³ Z , and ³ R , are used to com- pute the coefficients, a i ;b i in (3.18) as a function of a particular µ ¤ i . The notch filter 60 coefficients and µ ¤ i vector are then stored after each optimization is completed, form- ing a lookup table with N j points for each µ ¤ ij element. The indices to the table can be represented as µ ¤1 ij <µ ¤2 ij <¢¢¢<µ ¤N j ij ; (3.26) and aµ ¤ ij value lying between 2 consecutive indices µ ¤l j ij <µ ¤ ij <µ ¤l j +1 ij : (3.27) The parameters for the notch filter in (3.18) are computed, all in a similar fashion b 2 (µ ¤ i )= 6 X j=1 µ ¤ ij b l j 2m +b l j 2b (3.28) whereb l j 2m andb l j 2b are the slope and intercept associated with theµ ¤l j ij index. 3.3.2 Robust Online Estimator The robust online estimator presented here is for the case of a single unknown flexible mode. However it can be easily expanded to estimate more unknown flexible modes. In the case of many aircraft, the flexible mode with the slowest natural frequency, the first bending mode, will have the greatest impact on the rigid-body controller bandwidth. Therefore the slowest mode has been modeled in simulations as the uncertain and/or changing mode and the robust online estimator should be able to return an estimate of this 61 -25 -20 -15 -10 -5 0 5 Magnitude (dB) 10 0 10 1 10 2 10 3 -135 -90 -45 0 45 Phase (deg) Figure 3.2: Bode plot of two notch filters designed with the optimization scheme pre- sented. Both filters are designed with constraints 4, 7, and 8 neglected, ¹ ³ = 0, and ¹ ³ 2 =0. Blue line: A narrow notch filter design with ¹ ! =0:01! and ¹ ! 2 =0:01! 2 . Green line: A wider notch filter design with ¹ ! =0:05! and ¹ ! 2 =0:05! 2 . mode. The parameters estimated, and used in the adaptive notch filter, are coefficients of the second order transfer function for the slowest flexible mode in the linearized model. These transfer functions are the complex conjugate poles and zeros that create the peak in the bode plot which must be suppressed. Each row of G(s;£) will create a separate online estimator, leading to five estimators which run in parallel. Now (3.4) is placed in the form of a continuous time parametric equation to be used in theith estimator, where i=1;:::;5 z i (t)=µ ¤T i Á i (t) (3.29) z(t)= s 2 D k (s) ¤(s) y i (t) (3.30) 62 Á i (t)= · Á i1 (t) Á i2 (t) Á i3 (t) Á i4 (t) Á i5 (t) Á i6 (t) ¸ T (3.31) Á i1 (t)= s 2 N k i1 (s) ¤(s) ± T (t) Á i2 (t)= 2sN k i1 (s) ¤(s) ± T (t) (3.32) Á i3 (t)= s 2 N k i2 (s) ¤(s) ± e (t) Á i4 (t)= 2sN k i2 (s) ¤(s) ± e (t) (3.33) Á i5 (t)=¡ 2sD k (s) ¤(s) y i (t) (3.34) Á i6 (t)=¡ D k (s) ¤(s) y i (t)+ N k i1 (s) ¤(s) ± T (t)+ N k i2 (s) ¤(s) ± e (t) (3.35) µ ¤ i = · ! ¤2 D ! ¤2 i1 ! ¤2 D ³ ¤ i1 ! ¤ i1 ! ¤2 D ! ¤2 i2 ! ¤2 D ³ ¤ i2 ! ¤ i2 ³ ¤ D ! ¤ D ! ¤2 D ¸ T : (3.36) Here ¤(s) is a polynomial added to make proper transfer functions, and takes the form ¤(s) = (s+¸) 11 , where¸ is a design parameters which will determine the speed of the filter 1 ¤(s) . The unknown parameters for theith estimator,µ ¤ i =[µ ¤ i1 :::µ ¤ i6 ] T , are estimated online as µ i = [µ i1 :::µ i6 ] T . This µ i vector is then used in the adaptive notch filter in (3.18) of the previous section. The parameters are updated using some a priori known bounds on 63 the damping ratios and natural frequencies which can be calculated from the bounds i and will give the following 1¸³ u i1 ¸³ ¤ i1 ¸³ l i1 >0 1¸³ u i2 ¸³ ¤ i2 ¸³ l i2 >0 1¸³ u D ¸³ ¤ D ¸³ l D >0 ! u i1 ¸! ¤ i1 ¸! l i1 >0 ! u i2 ¸! ¤ i2 ¸! l i2 >0 ! u D ¸! ¤ D ¸! l D >0 (3.37) which leads to the following update algorithm _ µ i1 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i1 " i Á i1 if ³ (! u D ) 2 (! l i1 ) 2 >µ i1 > (! l D ) 2 (! u i1 ) 2 ´ orµ i1 = (! l D ) 2 (! u i1 ) 2 and" i Á i1 ¸0 orµ i1 = (! u D ) 2 (! l i1 ) 2 and" i Á i1 ·0; 0 otherwise (3.38) _ µ i2 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i2 " i Á i2 if ³ (! u D ) 2 ³ u i1 ! l i1 >µ i2 > (! l D ) 2 ³ l i1 ! u i1 ´ orµ i2 = (! l D ) 2 ³ l i1 ! u i1 and" i Á i2 ¸0 orµ i2 = (! u D ) 2 ³ u i1 ! l i1 and" i Á i2 ·0; 0 otherwise (3.39) 64 _ µ i3 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i3 " i Á i3 if ³ (! u D ) 2 (! l i2 ) 2 >µ i3 > (! l D ) 2 (! u i2 ) 2 ´ orµ i3 = (! l D ) 2 (! u i2 ) 2 and" i Á i3 ¸0 orµ i3 = (! u D ) 2 (! l i2 ) 2 and" i Á i3 ·0; 0 otherwise (3.40) _ µ i4 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i4 " i Á i4 if ³ (! u D ) 2 ³ u i2 ! l i2 >µ i4 > (! l D ) 2 ³ l i2 ! u i2 ´ orµ i4 = (! l D ) 2 ³ l i2 ! u i2 and" i Á i4 ¸0 orµ i4 = (! u D ) 2 ³ u i2 ! l i2 and" i Á i4 ·0; 0 otherwise (3.41) _ µ i5 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i5 " i Á i5 if ¡ ³ u D ! u D >µ i5 >³ l D ! l D ¢ orµ i5 =2³ l D ! l D and" i Á i5 ¸0 orµ i5 =2³ u D ! u D and" i Á i5 ·0; 0 otherwise (3.42) _ µ i6 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i6 " i Á i6 if( ¡ ! u D ) 2 >µ i6 >(! l D ) 2 ¢ orµ i6 =(! l D ) 2 and" i Á i6 ¸0 orµ i6 =(! u D ) 2 and" i Á i6 ·0; 0 otherwise (3.43) m 2 i =1+n is +m is (3.44) n is (t)=C is Á T i Á i (3.45) 65 _ n id =¡± i0 n id +± i1 (y 2 i +± 2 e +± 2 T ) (3.46) " i = z i ¡µ T i Á i m 2 i (3.47) The online estimator uses a gradient algorithm with robustness modifications [36]. For robustness the update term is normalized using a dynamic term which is calculated in (3.46). Parameter projection is used to take advantage of the known region of the parameters, expressed in (3.37), based on a priori knowledge of the vehicle structure. The upper and lower bounds are denoted as ³ u and ³ l , respectively. The adaptation gains are design parameter that satisfy, ° ij > 0. The C is ;± i0 ;± i1 terms and are design parameters that must be greater than zero. 3.3.3 Pointwise Stability We now justify our approach of wrapping feedback aroundG(s;£) with the rigid-body controller, now defined asK 0 (s), and notch filterF(s;£), which is developed by linearly interpolating the coefficients of the optimal notch filters, by establishing that the closed- loop system is internally stable for any constant£ ¤ 2. Verifying this stability property also provides an intermediate result used to establish closed-loop stability of the adaptive system, where an online estimate of£ ¤ is tuned by an adaptive law. The approach taken 66 to establish stability is to express the closed-loop dynamics in the M¢-structure form, shown in Fig. 3.3, and then check the condition kMk ¢ <1; M internally stable (3.48) to establish stability. By the above we mean kMk ¢ =sup !2< ¹ ¢ (M(j!)) (3.49) where ¢ has some specified structure. In Fig. 3.3, the M block is the closed-loop dynamics for some nominal value of £ ¤ , and the ¢ block captures the effect of the £ ¤ variations away from nominal. Towards expressing the closed-loop system in the M¢- structure, we rewrite the closed-loop system y =G(s;£ ¤ )u; u=K(s;£ ¤ )y (3.50) 67 ' ' e M ' d Figure 3.3: M¢-structure used for the stability analysis. in a state-space realization convenient for ”pulling out” the ¢ block. Consider the con- trollable canonical realization of the transfer function fromu toy i _ x P i = 2 6 6 6 6 6 6 6 6 6 6 4 ¡® i r+1 I 2£2 ¡® i r I 2£2 ¢¢¢ ¡® i 1 I 2£2 ¡® i 0 I 2£2 I 2£2 0 ¢¢¢ 0 0 . . . . . . . . . . . . . . . 0 0 ¢¢¢ I 2£2 0 3 7 7 7 7 7 7 7 7 7 7 5 x P i + 2 6 6 6 6 6 6 6 6 6 6 4 I 2£2 0 . . . 0 3 7 7 7 7 7 7 7 7 7 7 5 u (3.51) y i = · ¯ i 1;r+1 ¯ i 2;r+1 ¢¢¢ ¯ i 1;0 ¯ i 2;0 ¸ (3.52) or _ x P i =A P i (µ ¤ i )x P i +B P i u (3.53) y i =C P i (µ ¤ i )x P i : (3.54) 68 Observe that each element ofA P i (µ ¤ i ) andC P i (µ ¤ i ) is affine inµ ¤ i because each coef- ficient® i j ,¯ i jk is affine inµ ¤ i . Thus, the complete plant can be expressed as _ x P =A P (£ ¤ )x P +B P u (3.55) y =C P (£ ¤ )x P (3.56) where A P (£ ¤ )= diag(A P 1 (µ ¤ 1 );:::;A P 5 (µ ¤ 5 )) (3.57) B P = £ B T P 1 ;:::;B T P 5 ¤ T (3.58) C P (£ ¤ )=[C P 1 (µ ¤ 1 );:::;C P 5 (µ ¤ 5 )] (3.59) x P = £ x T P 1 ;:::;x T P 5 ¤ T : (3.60) Once again, the elements of A P (£ ¤ ) and C P (£ ¤ ) are affine in £ ¤ . Now we look at the realization of the composite controller which is K(s;£ ¤ )=K 0 (s)F(s;£ ¤ ) (3.61) or K(s;£ ¤ )= 2 6 6 4 K 11 (s)F 1 (s;µ ¤ 1 ) ¢¢¢ K 15 (s)F 5 (s;µ ¤ 5 ) K 21 (s)F 1 (s;µ ¤ 1 ) ¢¢¢ K 25 (s)F 5 (s;µ ¤ 5 ) 3 7 7 5 : (3.62) 69 Although the coefficients of F(s;£ ¤ ) are not necessarily affine in £ ¤ on the do- main, the coefficients ofF(s;£ ¤ ) are, by construction, affine between any consecutive breakpoint in the lookup table. So for now we restrict £ ¤ to S i , the space between any set of breakpoints. Later, we check stability for the complete space by checking the conditionkMk ¢ < 1 on eachS i . Because the coefficients ofK(s;£ ¤ ) are affine in£ ¤ , the controllable canonical realization is _ x C =A C (£ ¤ )x C +B C y (3.63) u=C C (£ ¤ )x C (3.64) where the elements ofA C (£ ¤ ) andC C (£ ¤ ) are affine in£ ¤ . Then, from (3.55), (3.56), (3.63), and (3.64) the closed-loop system is given by 2 6 6 4 _ x P _ x C 3 7 7 5 = 2 6 6 4 A P (£ ¤ ) B P C C (£ ¤ ) B C C P (£ ¤ ) A C (£ ¤ ) 3 7 7 5 2 6 6 4 x P x C 3 7 7 5 : (3.65) Now we will exploit the fact that A P , A C , C P , and C C are affine in £ ¤ and rewrite each as A P (£ ¤ )=A P 0 + 5 X i=1 6 X j=1 µ ¤ ij A P ij (3.66) A C (£ ¤ )=A C 0 + 5 X i=1 6 X j=1 µ ¤ ij A C ij (3.67) 70 C P (£ ¤ )=C P 0 + 5 X i=1 6 X j=1 µ ¤ ij C P ij (3.68) C C (£ ¤ )=C C 0 + 5 X i=1 6 X j=1 µ ¤ ij C C ij (3.69) for some constant matricesA P 0 ,A P ij ,A C 0 ,A C ij ,C P 0 ,C P ij ,C C 0 , andC C ij of appropriate dimensions. Furthermore, if we rewrite eachµ ¤ ij as µ ¤ ij = ¹ µ ij +w ij ± ij (3.70) ¹ µ ij = µ ¤l j +1 ij +µ ¤l j ij 2 ; w ij = µ ¤l j +1 ij ¡µ ¤l j ij 2 (3.71) where± ij is a constant satisfyingj± ij j<1 andµ ¤ ij lies between two breakpoints such that µ ¤l j ij <µ ¤ ij <µ ¤l j +1 ij . Combining the above expressions ofµ ¤ ij with (3.66) - (3.69) and then substituting the result into (3.65) yields theM block 2 6 6 4 _ x P _ x C 3 7 7 5 = · A 0 ¸ 2 6 6 4 x P x C 3 7 7 5 +B ¢ d ¢ (3.72) e ¢ = £ x T P x T C ¤ T (3.73) where A 0 = 2 6 6 4 A P 0 B P C C 0 B C C P 0 A C 0 3 7 7 5 + 5 X i=1 6 X j=1 ¹ µ ij 2 6 6 4 A P ij B P C C ij B C C P ij A C ij 3 7 7 5 (3.74) 71 B ¢ = 5 X i=1 6 X j=1 w ij 2 6 6 4 A P ij B P C C ij B C C P ij A C ij 3 7 7 5 (3.75) as well as the¢ block d ¢ =¢e ¢ (3.76) where ¢= diag(± 11 I 7£7 ;:::;± 16 I 7£7 ;± 21 I 7£7 ;:::;± 56 I 7£7 ): (3.77) We now write M(S i ) to make explicit M’s depending on the subset S i 2 between adjacent breakpoints in the lookup table. Since there areN j elements for eachµ ¤ ij there will be N = 6 Y j=1 N j (3.78) total subsets. By checking the conditionkM(S i )k ¢ < 1 we establish stability for any £ ¤ 2 S i . By straightforward extension, stability of the closed-loop system for any£ ¤ 2 is established if and only if max i kM(S i )k ¢ <1; i=1;:::;N (3.79) 3.4 Discussion An adaptive mode suppression scheme has been presented for use on MIMO systems. The intended application is aircraft where the rigid-body controller and structural filters 72 are generally designed separately, however the integration of the two must be perfect. With the scheme presented here the notch filters are made adaptive and the pointwise stability result given above, ensures that the system is internally stable for any fixed£ ¤ . This is a requirement which is levied on the rigid-body controller, as well as the design of the adaptive notch filters. Since the notch filters have been optimized in an offline algorithm, the correct parameters are chosen online based on estimates of the modal parameters. In the future it is hoped that the stability proof can be completed, however it will take a very different approach than that of the SISO case given in the previous chapter. Also it may be possible to explore the design of the notch filters and interaction with the rigid-body controller. A systematic design procedure would be beneficial and the stability result may prove to aid in this research. 73 Chapter 4 Disturbance Rejection Scheme 4.1 Introduction Tracking control of various mechanical systems can be formulated as disturbance rejec- tion problems, where the goal is to reduce the effect of disturbances at the output. Such systems include hard disk drives (HDD)[33], fast steering mirrors (FSM) for optical communication systems[70], missile seekers [55], and aircraft autopilots. These distur- bances may come from imperfections in the mechanical system, vibrations, atmospheric effects, or electrical noise. In mechanical systems dealing with rotating machinery, such as the HDD servo system or DC motor control systems, the disturbance can be separated into two parts: a part which has known single frequency components and a part which may be unknown. In a HDD system this equates to repeatable runout (RRO) and non- repeatable runout (NRRO). The RRO is produced by imperfections and eccentricities on 74 the tracks, while NRRO is produced by aggregated effects of disk drive vibrations, imper- fections in the ball-bearings, and electrical noise. Research has been conducted over the years to cancel the effects of these disturbances and acquire better tracking performance [93, 79, 72, 71, 33, 42, 94, 52, 95, 39, 62]. It has been shown that the RRO can be suppressed with adaptive feedforward meth- ods [79, 80], and by repetitive control [71]. Both of these methods use the existing knowledge of the frequencies at which the RRO disturbance occurs to suppress its effect and obtain better tracking. The feedforward method employs the injection of the negative of an estimated sinusoidal disturbance model. The amplitude and phase of the sinusoidal disturbance are estimated online through a gradient update algorithm. An internal model is used to synthesize a linear time invariant controller that rejects sinusoidal disturbances of known frequencies in the repetitive control method. The contribution of this chapter is an adaptive neural disturbance rejection scheme which may be added to a system already incorporating other disturbance rejection tech- niques and control designs. In particular the neural scheme will be added to an adaptive feedforward disturbance rejection scheme which eliminates RRO. There has been work done in using neural networks for feedforward disturbance rejection [55, 74, 66]. Re- jection of the disturbance torque for missile seekers using neural networks is presented in [55]. A multilayer neural network uses the measurable load disturbance to cancel to cancel the degrading effects through a feedforward controller. Many systems, such as the conventional HDD, do not have the luxury of extra sensors for the disturbance, so 75 the only possible input to a neural network is the position signal. In [74], a general ap- proach with simulations show the benefit of adapting a dynamic neural network to cancel an unknown disturbance. Here the idea of passing the disturbance estimate through an estimated plant inverse is used. The methods proposed in [66] use multilayer neutral networks to model disturbances as outputs of dynamical systems and then expands the plant model to try to reduce the adverse effects. Radial basis functions (RBF) have been used to model sea-clutter noise in radar ap- plications [31]. The approach uses RBFs to remove the noise from radar signal data, since the clutter noise has been shown to be chaotic, thus providing the ability to detect small targets in the clutter. Training data is used to adapt the neural parameters before being implemented on actual test data. A similar approach was taken here except we add an extra term in the neural model to account for extra delays and there is no training set of data; the adaptive neural disturbance rejector is both adapted and implemented in real-time. Also the adaptation of the neural parameters uses a deadzone modification not present in [31], which allows adaptation to cease once the performance begins to de- grade. The neural modeled disturbance rejection is added to either RRO rejection scheme to obtain better performance than the RRO scheme alone. This increase in performance is due to the ability of the adaptive neural network to provide a model for a dynamic nonlinear disturbance which is the residual from a RRO rejection schemes. 76 ) (z P ) ( 0 k d ) (k y m ) (k y p ) (z C Figure 4.1: Closed loop system used for disturbance rejection. y m (k) is the output to be tracked,y p (k) is the measurable output,d 0 (k) is the disturbance,C(z) is a LTI controller, andP(z) is a LTI plant. 4.2 Adaptive Disturbance Rejector The goal of the disturbance rejection scheme is to reduce the tracking errore=y m ¡y p in Fig. 4.1 where C and P have been designed to give a stable closed loop system. To reduce the tracking error we will inject a negative of an estimate of the disturbance. To do this we need a closed loop model of the system G(z) = C(z)P(z) 1+C(z)P(z) , which is stable, and then the system can be viewed as in Fig. 4.2. This new output disturbance is d(k)= 1 1+C(z)P(z) d 0 (k) (4.1) and the input signaly c (k)=u c (k)+y m (k) whereu c (k) is a disturbance rejection signal that will be generated. The disturbance rejection problem is then divided into two parts: repeatable runout disturbance rejection and neural modeled disturbance rejection. The former has proven experimental results as shown in [79]. We make no modification to the algorithm other than applying it to a large number of frequencies. There are other RRO compensators 77 ) (z G ) (k d ) (k y c ) (k y p Figure 4.2: System used for disturbance rejection design. y c (k) is the output to be tracked plus disturbance rejection signal, y p (k) is the measurable output, d(k) is the output dis- turbance, andG(z) is the closed-loop system. available in the literature, such as repetitive control, which has been done experimentally in [71] and also compared in [41]. Other RRO compensators could be substituted for the RRO scheme adopted in this paper, which is used as one example to show integration of the adaptive notch filter and disturbance rejection scheme. Once the repeatable runout disturbance rejection is applied there is still disturbance that creates non-perfect tracking. This remaining disturbance is modeled using neural techniques and adaptively updated online, which has been experimentally verified in [51, 50]. This neural scheme could also be substituted for other nonlinear compensators that have been developed such as the ones in [32, 33, 72]. 4.2.1 Repeatable Runout Disturbance Rejection The repeatable runout (RRO) disturbance occurs at evenly spaced frequencies Âm Hz where m = 1;2;:::;n and Â is some constant, this is due to the fixed revolution speed 78 of the machinery. A control input is designed such that it will adaptively cancel this disturbance. The disturbance,d(k)=d RRO (k) can be modeled as d RRO (k)= n X i=1 a i (k)sin µ 2¼ik N rev ¶ +b i (k)cos µ 2¼ik N rev ¶ ; (4.2) wherei is the index for the harmonic andN rev is the number of samples per revolution. If the system is modeled as in Fig. 4.2 then the output is y p (k)=G(z)[y c (k)]+d RRO (k): (4.3) To cancel the disturbance the control signal should beu c (k) =¡ ^ G ¡1 (z)[ ^ d RRO (k)]. The identified inverse, ^ G a ¡1 (z), will have an effect on the magnitude and phase of the distur- bance estimate, ^ d RRO (k). Since the magnitude and phase of the sinusoidal disturbance are being estimated, the system inverse can be ignored and the new control signal be- comesu c (k)=¡ ^ d RRO (k). The disturbance estimate is ^ d RRO (k)= n X i=1 ^ a i (k)sin µ 2¼ik N rev ¶ + ^ b i (k)cos µ 2¼ik N rev ¶ : (4.4) The update equations for the estimated parameters are ^ a i (k)= ^ a i (k¡1)+° i y(k¡1)sin µ 2¼ki N rev +Á i ¶ (4.5) 79 ^ b i (k)= ^ b i (k¡1)+° i y(k¡1)cos µ 2¼ki N rev +Á i ¶ : (4.6) Where the ° i are adaptation gains, chosen differently for each harmonic. A phase ad- vance modification is added to reduce the sensitivity and allow for more harmonics to be canceled as was done previously in [79]. The Á i =\ ^ G a (j! i ) and ! i is the angular frequency of theith harmonic. 4.2.2 Neural Modeled Disturbance Rejection Since the disturbance does not consist entirely of harmonics from the rotation of the machinery, another disturbance rejection algorithm is added. The new disturbance is modeled as d(k)=d RRO (k)+d NN (k): (4.7) So the system output now becomes y p (k)=G(z)[y c (k)]+d RRO (k)+d NN (k): (4.8) To cancel the disturbance the control signal should be u c (k)=¡ ^ d RRO (k)¡ ^ G ¡1 (z)[ ^ d NN (k)]: (4.9) 80 Since the computed model ^ G(z) of may be non-minimum phase, the inverse is unstable. Also the computed model may be strictly proper which makes a state space inverse not realizable. The unstable zeros of ^ G(z) are reflected across the unit circle, and then the system is augmented with very fast zeros making the degree of the numerator equal to the degree of the denominator. Then the inverse is taken. This new inverse, ¹ G ¡1 (z), is used in the computation of the control signal, making itu c (k)=¡ ^ d RRO (k)¡ ¹ G ¡1 (z)[ ^ d NN (k)], and therefore causes an extra delay that will be dealt with. The following disturbance rejection scheme uses gaussian radial basis functions (RBF) from neural networks to attempt to model the disturbance. The disturbance estimate takes the form ^ d NN (k)= L X q=1 M X i=1 µ q;i (k)R q;i ^ d(k¡±¢(q¡1)¡1) (4.10) R q;i =ª i ³ ^ d(k¡±¢(q¡1)¡1) ´ : (4.11) TheR q;i is computed using an RBF and theith gaussian RBF is ª i (x)=exp " ¡ µ x¡c i º ¶ 2 # : (4.12) The parameters that specify the shape of the ith gaussian RBF are the center c i and the width º. There are a total of M gaussian RBFs, and their centers are linearly spaced across the range of input. The current disturbance estimate, ^ d NN (k), is a function of L previous disturbances that are spaced± samples apart. 81 The reason for the spacing ± is the delay associated with passing the disturbance estimate through the system inverse. One method of coping with the delay would be to estimate the disturbance at the next sample, and then use this estimate to create another future estimate, and continue iterating to find some ^ d NN (k+¢) in the future [74]. This method may not always work as the estimation error can grow very large with each future estimate. Instead the disturbance is thought of as a function of previous evenly spaced disturbances. The ^ d NN (k) can be viewed as a future disturbance estimate when compared to the sample rate. It should be noted that the algorithm still creates a new disturbance estimate at every sample of the outputy p (k). The model of the disturbance is motivated by the assumption that the disturbance is a nonlinear function of previous disturbance values. The ^ d(k¡±¢(q¡1)¡1) term that is multiplied by the output of the RBF in (4.10) is the added term that allows the model to work for the disturbance in this HDD. This added term makes the model different from previously used RBF neural predictors [31, 74]. Now the disturbance can be thought of as an autoregressive filter with spacing± and nonlinear coefficients that are modeled with the RBF’s. A simple example with ± = 2 and L = 3 is shown in Fig. 4.3 to help view the modeling of the disturbance. The ^ d(k¡±¢(q¡1)¡1) used in the estimate is not produced by the estimator but rather is a measured disturbance estimate which can be calculated from an identified model of the system, ^ G(z), using ^ d(k¡1)=y p (k¡1)¡ ^ G(z)[y c (k¡1)]: (4.13) 82 1 < 2 < M < ) 1 ( ˆ k d ¦ 1 , 1 T 2 , 1 T M , 1 T ¦ ) 1 ( ˆ k d 1 < 2 < M < ) 3 ( ˆ k d ¦ 1 , 2 T 2 , 2 T M , 2 T 1 < 2 < M < ) 5 ( ˆ k d ¦ 1 , 3 T 2 , 3 T M , 3 T ) 3 ( ˆ k d ) 5 ( ˆ k d ) ( ˆ k d NN Figure 4.3: Simple example of how ^ d NN (k) is computed from previous values of the disturbance. Here± =2 andL=3. The unknown parameters should be updated with the current modeling error, which is " NN (k)=y p (k¡1)¡ ^ G(z)[ ^ d RRO (k¡1)]; (4.14) and the parameters that caused that error, ^ d(k¡±q¡1). This leads to the update equations µ q;i (k)= 8 > > < > > : µ q;i (k¡1)+® q;i (k) " new ·½¢" old µ 0 q;i otherwise (4.15) ® q;i (k)= l NN m 2 (k) " NN ¹ R q;i ^ d(k¡±q¡1) (4.16) ¹ R q;i =ª i ³ ^ d(k¡±q¡1) ´ (4.17) 83 m 2 (k)=1+m s (k) (4.18) m s (k)=± 0 m s (k¡1)+ ^ d 2 (k¡1): (4.19) EveryN " samples the following are calculated " new = N " ¡1 X n=0 " 2 NN (k¡n) (4.20) " old = 8 > > < > > : " new " new <" old " old otherwise (4.21) µ 0 q;i = 8 > > < > > : µ q;i (k) " new <" old µ 0 q;i otherwise (4.22) The update is an instantaneous gradient algorithm with a couple robustness modifica- tions. The adaptation, or learning, rate isl NN and is greater than zero. The update term is normalized with a dynamic term to add robustness, this term is calculated in (4.19). The parameter± 0 is chosen between 0 and 1. The other robustness modification is one that is added to stop adaptation when the performance starts to degrade. In practice the estimation error will never become zero and so the parameters will continue to update. There will be a point at which the esti- mation error is small and on the same level as the noise and modeling error. Usually a simple deadzone is added to stop adaptation when the current error is below some thresh- old. In some applications the estimation error, which is the system output, may be noisy 84 and must be averaged over N " samples. Instead of averaging, the sum squared error is easier to calculate online via (4.20). The adaptation will continue as long as this new sum squared error," new , is less than the old sum squared error," old , to within some small range. The ½ term is selected to be greater than 1 to allow adaptation even if the sum squared error did not decrease. This gives some room for noisy measurements and lets the algorithm continue. If the algorithm is doing a good job and the new sum squared error is strictly less than the old, the current µ q;i (k)’s are saved and the " old is updated. Theµ q;i (k)’s are saved so that when the sum squared error is too large, the algorithm can revert back to the best known parameters. The neural modeled disturbance rejector has a large number of parameters that must be updated online. Although this scheme has been implemented in real-time and the experimental results using a Matlab xPC Target presented in [51, 50], it will require more computational power than may be suitable for a production embedded real-time control system. 4.2.3 Stability Analysis of Neural Model The adaptive neural modeled disturbance rejection scheme can be placed in a form that follows the framework of adaptive estimators in [36]. Starting with the modeled distur- bance d NN (k)= L X q=1 M X i=1 µ ¤ q;i (k)R q;i ^ d(k¡±¢(q¡1)¡1) (4.23) 85 R q;i =ª i ³ ^ d(k¡±¢(q¡1)¡1) ´ : (4.24) And placing it in the form of z(k)=µ ¤T Á(k)+´(k); (4.25) where z(k)=d(k) (4.26) µ ¤ =[µ ¤ 1;1 ;:::;µ ¤ L;M ] T (4.27) Á(k)=[f 1;1 ;:::;f L;M ] T (4.28) f q;i =R q;i ^ d(k¡±¢(q¡1)¡1) (4.29) and´(k) is the modeling error. The estimation model and estimation error are given as ^ z =µ T (k)Á(k) (4.30) ²(k)= z(k)¡ ^ z m 2 (k) = z(k)¡µ T (k)Á(k) m 2 (k) ; (4.31) where m(k) is the normalizing signal designed to boundjÁ(k)j andj´(k)j from above. Only the ^ d(k¡±¢(q¡1)¡1) part of theÁ(k) needs to be bounded bym(k) since the RBF’s,R q;i , are bounded by definition. Using the gradient law in (4.15) the parameters are adaptively updated. As shown in [36] the adaptive laws defined above will guarantee 86 thatµ q;i (k);²(k)2` 1 and²(k);jµ q;i (k)¡µ q;i (k¡1)j2S(g 0 +´ 2 0 ), where´ 0 is an upper bound of j´(k)j m(k) ·´ 0 , andg 0 is bounded by½¢" 0 ¸g 0 ¸½¢" old . Hereg 0 is a value which is bounded by the choice of the design parameters ½ and " 0 , which is the initialization value of design parameter" old , and the computed values of" old . This will define the set S(g 0 +´ 2 0 ). 4.3 Discussion The neural network based adaptive disturbance rejector presented has the ability to re- duce the tracking error in a system possessing a nonlinear dynamic disturbance. Ex- perimental results of this scheme are presented in an upcoming chapter and have been presented in [51, 50], although this was done for a single HDD unit. There are many practical considerations when dealing with this scheme. There are a very large number of parameters that must be updated online, which may take a large amount of computing power. Depending on the processing power of the real-time control system, the num- ber of parameters may need to be reduced which may reduce the effectiveness of the disturbance rejector. Also there are a variety of design variables such as l NN ;L;±,etc. that must be chosen for the particular application. These parameters may be very hard to chose in practice, however the empirical results point to the potential of the adaptive disturbance rejection scheme to reduce the tracking error for a HDD application. The stability result presented above guarantees stability and boundedness of the neu- ral disturbance rejector, together with Fig. 4.2 and the fact that G(z) is stable, implies 87 thaty p (k) will remain bounded. However this does not prove that the tracking error will always be reduced, although the system may be stable, the adaptive disturbance rejector is not guaranteed to improve performance. In fact it may even decrease performance, however with proper tuning, simulation, and testing, the neural modeled adaptive distur- bance rejection scheme is shown to be beneficial. 88 Chapter 5 Adaptive Time-Delay Compensation 5.1 Introduction Time delays are a significant problem in the field of process control as they limit the bandwidth and hence performance of control systems. These time delays arise from the presence of travel times of fluids, recycle loops in chemical processes, and also long computation times [84, 8, 75]. Robustness and stability will be affected by the addition of known time delays, but unknown or changing time delays present an even greater problem. The time delay may vary from the congestion of pipes, changing of filters, or variation in the properties of the chemicals in the process being controlled. With a variation in the time delays a decrease in performance is usually realized and even instabilities can occur. For this reason unknown and varying time delays in process control systems is a problem that requires ongoing research. 89 Research has been conducted trying to solve the problem of time delays in process control systems. One of the oldest solutions is the so called Smith predictor [85] which uses a model of the plant and time delay to allow for an increase in overall system gain and increase in performance. The Smith predictor has since been modified and improved to include steady-state tracking [91] and then further modified for ease of tuning and increase in robustness [5, 58]. However these methods require a well known model of the plant and time delay, so work has been done to improve control systems while estimating parameters online [68]. A simple adaptive smith predictor with some analysis was completed in [8], but there have been other efforts to create algorithms to estimate the time delay during a realtime process [44, 28]. One method of estimation incorporates an overparameterization of the plant in the discrete domain and then an RLS algorithm to estimate the plant parameters. An algorithm is performed to extract the plant parameters and the time delay, which are then used in an extended minimal variance controller and a deadbeat controller, however this method has the disadvantage of poor convergence and growing number of parameters with an increase in sampling rate. An adaptive Smith predictor is compared to a non-adaptive robust predictive controller in [4] where the benefit of the adaptive controller are evident under an unknown time delay, however no analysis of the adaptive scheme is given. The idea of using a rational representation of a time delay was used for estimation in [24] and then furthered in [3, 90]. This polynomial representation and estimation was used for control in [77, 76] however no stability results are presented. fully analyzed and simulated adaptive controller for time delay systems 90 was published in [86], where a polynomial representation of the time delay is used in an RLS algorithm to estimate the plant parameters, which are then incorporated into a pole placement controller. More advanced neural network approaches have been used for systems with unknown time delays, however the controllers are more complex and not necessarily intuitive for a classical control designer [25, 14]. In this work we will attempt to estimate the time delay which may be unknown or change with time. The estimated time delay will be used to update an adaptive Smith predictor and a non-adaptive PI controller will be combined to the control scheme. This method is different from that of [86] in that an adaptive Smith predictor is added to a fixed controller, while in [86] the controller used is basically an adaptive pole placement scheme using the polynomial approximation of the time delay. 5.2 Plant Model The plants considered for this scheme are those with a time delay, thereby taking the form y p =G p (s)e ¡Ts u p (5.1) 91 whereu p andy p are the input and output respectively. HereT is the time delay in seconds andG p (s) is a rational transfer function of the form G p (s)= Z p (s) R p (s) ; (5.2) where it is assumed thatR p (s) is a monic Hurtwitz polynomial whose degreen is known and the degree of Z p (s) < n. In the field of process control most plants can be ap- proximated by first or second order systems, which will simplify the control design and analysis. However, we will stick to the general case for now and assume that the plant without the time delayG p (s) is stabilizable. 5.3 Adaptive Smith Predictor The adaptive Smith predictor is broken into several components which will be further explained in the upcoming sections. There is an online estimator which will identify the time delay in a realtime system, and then the adaptive Smith predictor will use this estimate. There is also a non-adaptive controller which is designed to meet certain per- formance specifications. This controller can be designed robust to uncertainties in the plant, however since the Smith predictor is model-based it is less robust to plant uncer- tainties. For this reason the time delay in the Smith predictor will be updated online, and in future work the plant parameters will be updated online as well. First the Smith predictor control scheme, assuming knowledge of all the parameters, will be explained. 92 5.3.1 Smith Predictor Control Scheme A Smith predictor is an added controller which enables the time delay to be removed from the characteristic equation, thereby making control design simpler and more intuitive. The control scheme is depicted in Fig. 5.1. In the figure C(s) is a controller designed to achieve the performance and stability required for the application. A model of the systemG p (s) is denoted as ~ G p (s) and a model of the time delay is ~ T . The controller then effectively has a transfer function C S (s)= C(s) 1+C(s) ~ G p (s) ¡ 1¡e ¡ ~ Ts ¢; (5.3) leading to the control lawu p =C S (s)(y m ¡y p ). The closed loop dynamics of the system assuming perfect matching, that is ~ G p (s)=G p (s) and ~ T =T , become y p = C(s)G p (s)e ¡Ts 1+C(s)G p (s) y m : (5.4) Notice that the time delay now only appears in the numerator and not in the characteristic equation of the closed loop transfer function. This assumes perfect matching between the model and the actual system, although the robustness of mismatched parameters has been researched and published in the literature. 93 ) (s G p Ts e ) (s C s T p e s G ~ 1 ) ( ~ p u m y p y ) (s C S Figure 5.1: Overall diagram of the Smith predictor control scheme. Herey p is the plant output,y m is the reference signal, andu p is the control signal. For the control here we will assume that the controller C(s) = Z c (s) R c (s) stabilizes the model of the plant without the time delay ^ G p (s). This leads to R p (s)R c (s)+Z p (s)Z c (s)= ¹ A(s) (5.5) where ¹ A(s) is a Hurwitz polynomial. We have the degree ofR c = m and the degree of Z c < m is Since we are mainly interested in the Smith predictor and compensating for the time delay, we are not setting the roots of ¹ A(s), but rather assuming a controllerC(s) is given which leads to the Hurwitz polynomial ¹ A(s), and then focus our attention on the Smith predictor. For our design we will assume they m is a constant and since we would like to have perfect tracking we will assume that R c (s) contains an internal model of a constant, which is simply a factor ofs. 94 5.3.2 Estimation of Unknown Parameters Now we will consider that the time delay is uncertain or changing. The goal is to estimate this parameter by placing the system into a linear parametric model. First the time delay is approximated by e ¡Ts ¼ 2=T ¡s 2=T +s = Z D R D : (5.6) The system then used for the estimator becomes y p = Z p (s) R p (s) (2=T ¡s) (2=T +s) (1+¢ m (s))u p : (5.7) ¢ m (s)= 2+Ts 2¡Ts e ¡Ts ¡1 (5.8) Next we can place the system in (5.7) into the form of a linear static parametric model z =µ ¤ Á+´ (5.9) z = sR p (s) ¤ p (s) y p + sZ p (s) ¤ p (s) u p (5.10) µ ¤ = 1 T (5.11) 95 Á=¡ 2R p (s) ¤ p (s) y p + 2Z p (s) ¤ p (s) u p (5.12) where ¤ p (s) is a monic Hurwitz polynomial of degree n + 1. The modeling error is expressed as ´ = Z p (2=T ¡s) ¤ p (s) ¢ m u p (5.13) Placing the system in this form allows for the use of various estimation laws. For this presentation a gradient update algorithm will be used to compute the estimates of the unknown parameter, which will be denoted µ = 1 ^ T . Since we know that T > 0 and we can use parameter projection to ensure that ^ T ¸ T 0 , where T 0 > 0 is some known lower bound on the time delay. The adaptive law is _ µ = 8 > > > > > > < > > > > > > : °"Á if( 1 µ > 1 T 0 ) or(µ =T 0 and"Á¸0) 0 otherwise (5.14) m 2 s =1+n d (5.15) _ n d =¡± 0 n d +u 2 p +y 2 p (5.16) n d (0)=0: (5.17) 96 In the above, the parameter° > 0 is an adaptive gain which is chosen by the control designer. The adaptive law guarantees (i) ";"m s ;µ; _ µ2L 1 (ii) ";"m s ; _ µ2S( ´ 2 m 2 s ). 5.3.3 Adaptive Control Law The adaptive control law is obtained by substituting the estimated parameters into the original control law. The controller C(s) remains the same as before however we now get ^ C S (s)= C(s) 1+C(s) ~ G p (s) ³ 1¡e ¡ ^ Ts ´; (5.18) where we now have ^ T which come from the estimator in the previous section and we will assume that ~ G p (s) = G p (s), that is the model of the plant used in the Smith predictor is the actual plant. As seen in Fig. 5.2, the adaptive Smith predictor control scheme adds an adaptive component to compensate for the time delay in the plant. 5.3.4 Stability analysis The adaptive Smith predictor scheme has stability properties which are defined in the following theorem: 97 ) (s G p Ts e ) (s C s T p e s G ˆ 1 ) ( ~ p u m y p y ) ( ˆ s C S Estimator Figure 5.2: Overall diagram of the adaptive Smith predictor control scheme. Here y p is the plant output,y m is the reference signal, andu p is the control signal. Theorem 5.3.1 There exists a± ¤ >0 such that if ¢ 2 2 <± ¤ ; where ¢ 2 , ° ° ° ° Z p (2=T ¡s) ¤ p (s) ¢ m (s) ° ° ° ° 2± 0 ; (5.19) the the system described by (5.4), (5.14) - (5.17), (5.18) guarantees that all signals are bounded and the tracking errore 1 =y p ¡y m satisfies Z t 0 e 2 1 d¿ ·c¢ 2 2 +c; 8t¸0 (5.20) Proof: The proof is carried out using the properties of the adaptive law and the following four steps: Step 1. Write the input and output in terms of the estimation error The objective of this step is to get the inputu p and outputy p in terms of the estimation error". The control law in (5.18) and normalized estimation error can be written as Z c R p y f + ³ R c R p +Z c Z p (1¡e ¡ ^ Ts ) ´ u f =y m1 (5.21) 98 ^ R D R p ¤ q y f ¡ ^ Z D Z p ¤ q u f ="m 2 s (5.22) where¤=¤ p ¤ q is a monic Hutwitz polynomial and 2µ¡s 2µ+s = ^ Z D ^ R D : (5.23) The degree of the arbitrary monic Hurwitz polynomial¤ q =m¡1 leading to the degree of¤=n+m. The filtered input and output are u f , 1 ¤ u p ; y f , 1 ¤ y p (5.24) and lastly the filtered reference signal isy m1 , ^ Z c R p 1 ¤ y m 2L 1 The polynomials may be expressed as ^ R D R p ¤ q =s n+m + ¹ µ T 1 ® n+m¡1 (s) (5.25) ^ Z D Z p ¤ q = ¹ µ T 2 ® n+m¡1 (s) (5.26) Z c R p =¯ T 1 ® n+m¡1 (s) (5.27) ¹ A=s n+m +¯ T 2 ® n+m¡1 (s) (5.28) Z c Z p =¯ T 3 ® n+m¡1 (s) (5.29) ® i (s),[s i ;s i¡1 ;:::;s;1] T (5.30) 99 where order of some of the polynomials are increased by adding parameters with zero coefficients. Now defining the state x(t), h y (n+m) f (t);:::; _ y f (t);y f (t);u (n+m) f (t);:::; _ u f (t);u f (t) i T (5.31) the system _ x(t)=A(t)x(t)+A D x(t¡ ^ T)+b 1 "m 2 s +b 2 y m1 (5.32) is formed, where A(t)= 2 6 6 6 6 6 6 6 6 6 6 4 ¡ ¹ µ T 1 j ¡ ¹ µ T 2 I n+m ¹ O j O (n+m)£(n+m+1) ¡¯ T 1 j ¡¯ T 2 O (n+m)£(n+m+1) j I n+m ¹ O 3 7 7 7 7 7 7 7 7 7 7 5 (5.33) A D = 2 6 6 6 6 6 6 6 6 6 6 4 O (n+m+1)£(1) j O (n+m+1)£(1) O (n+m)£(n+m+1) j O (n+m)£(n+m+1) O (n+m+1)£(1) j ¯ T 3 O (n+m)£(n+m+1) j O (n+m)£(n+m+1) 3 7 7 7 7 7 7 7 7 7 7 5 (5.34) b 1 =[1;0;:::;0; | {z } n+m+1 0;0;:::;0 | {z } n+m+1 ] T (5.35) 100 b 2 =[0;:::;0; | {z } n+m+1 1;0;:::;0 | {z } n+m+1 ] T : (5.36) andO (n+m)£(n+m+1) is an(n+m) by(n+m+1) matrix, ¹ O is a1 by(n+m) matrix, both with all elements equal to zero. Because u p = ¤u f = u (n+m+1) f +¸ T ® n+m (s)u f andy p =¤y f =y (n+m+1) f +¸ T ® n+m (s)y f where¤=s n+m+1 +¸ T ® n+m (s), we have u p (t)=[0;:::;0 | {z } n+m+1 1;0;:::;0 | {z } n+m+1 ]_ x(t)+[0;:::;0 | {z } n+q+1 ¸ T ]x(t) (5.37) y p (t)=[1;0;:::;0 | {z } n+m+1 0;:::;0 | {z } n+m+1 ]_ x(t)+[¸ T 0;:::;0 | {z } n+m+1 ]x(t): (5.38) Step 2. Establish exponential stability (e.s.) of the homogeneous part At each frozen timet the state space in (5.32) gives the output as y f (t)= 1 K 1 ³ ^ R s "m 2 s + ^ Z D Z p ¤ q y m1 ´ (5.39) K 1 =¤ q ³ ^ Z D ¢Z s Z p + ^ R s ¢ ^ R D R p ´ (5.40) where the adaptive control law gives ^ C S = Z s ^ R s = R p Z c R c R p +Z c Z p ¢ ³ 1¡e ¡ ^ Ts ´ = R p Z c ¹ A¡e ¡ ^ Ts Z c Z p : (5.41) The characteristic equation of (5.32) isK 1 which is stable at each timet. This can be verified through the use of the pseudo-delay method from [27] in which the time-delay is replaced with a transformation, namely the one used in the parametric model, and the 101 new characteristic equation is calculated K 2 = ¤ q ^ R D R p ¹ A. First we check the roots of K 1 at ^ T = 0 which gives2¤ q R p ¹ A, which has Hurwitz roots. Checking the roots ofK 2 we see that it is stable for all ^ T and has no complex roots by definition. Therefore we can be sure thatK 1 is a stable characteristic equation, which meansA(t);A D are stable at each timet. The adaptive law guarantees thatµ2L 1 and _ µ2S(¢ 2 2 ). The elements of the matrix A(t) are linear combinations of the parameterµ, thenkA(t)k2L 1 . Also the elements of _ A(t) are linear combinations of _ µ2 S(¢ 2 2 ). Now applying Theorem A.8.8(b) in [36] we have for ¢ 2 < ¢ ¤ for some ¢ ¤ > 0, the homogeneous part of (5.32) is uniformly asymptotically stable which is equivalent to exponentially stable. Step 3. Establish boundedness The exponentially weightedL 2± norm is defined as kx t k 2± , µZ t 0 e ¡±(t¡¿) x T (¿)x(¿)d¿ ¶ 1 2 ; (5.42) where±¸0 is a constant. We say thatx2L 2± ifkx t k 2± exists. For clarity of presentation we will denote theL 2± norm ask¢k. From (5.32) and using Lemma A.5.10 from [36] combined withA(t);A D (t);b 1 (t)2L 1 we have kxk·ck"m 2 s k+c; (5.43) 102 for any0<± <± 0 . From (5.32), (5.37), and (5.38) we have ku p k;ky p k·ck"m 2 s k+c: (5.44) Therefore the fictitious normalizing signal m 2 f ,1+ky p k 2 +ku p k 2 (5.45) satisfies m 2 f ·ck"m 2 s k 2 +c (5.46) and the signalm f boundsm s ;x;u p ;y p from above. This is established by using Lemma A.5.9 from [36] to show thatÁ=m f and thereforex=m f 2L 1 . From the fact that± <± 0 we have thatm s =m f 2L 1 . Since the elements ofA(t) are bounded and"m s 2L 1 we have that _ x=m f 2L 1 and thereforeu p =m f ;y p =m f 2L 1 . Now we express m 2 f ·c+ck"m s m f k 2 (5.47) which is m 2 f ·c+c Z t 0 e ¡±(t¡¿) ~ g 2 (¿)m 2 f (¿)d¿ (5.48) 103 where ~ g = "m s 2 S(¢ 2 2 ). Applying the Bellman-Gronwell Lemma (Lemma A.6.3 in [36]) we get m 2 f ·ce ¡±t e c R t 0 ~ g 2 (¿)d¿ +c± Z t 0 e ¡±(t¡s) e c R t s ~ g 2 (¿)d¿ ds: (5.49) Sincec R t s ~ g 2 (¿)d¿ ·c¢ 2 2 (t¡s)+c it follows that forc¢ 2 2 <±, we then havem f inL 1 . The boundedness of the rest of the signals follows from this fact. So the condition on¢ 2 for stability is ¢ 2 2 <min[±;¢ ¤2 ] (5.50) where the bound on ¢ 2 < ¢ ¤ for some ¢ ¤ > 0 is required to make the homogeneous part of the state space realization u.a.s. Step 4. Establish tracking error bounds Starting with a variation on (5.22) above we have "m 2 s = ^ R D R p 1 ¤ p y p ¡ ^ Z D Z p 1 ¤ p u p (5.51) and then we will filter each side with ^ R S 1 ¤ q to get ^ R S 1 ¤ q ¡ "m 2 s ¢ = ^ R S 1 ¤ q µ ^ R D R p 1 ¤ p y p ¡ ^ Z D Z p 1 ¤ p u p ¶ : (5.52) 104 Now using the fact that¤=¤ q ¤ p and the using the definition of ^ R S from (5.41) we get ^ R S 1 ¤ q ¡ "m 2 s ¢ = ¹ A ^ R D R p ¤ y p ¡ e ¡ ^ TS Z c Z p ^ R D R p ¤ y p ¡ ¹ A ^ Z D Z p ¤ u p + e ¡ ^ Ts Z c Z p ^ Z D Z p ¤ u p : (5.53) Using Swapping Lemma 1 (Lemma A.11.1 in [36]) on the third term of the RHS ¹ A ^ Z D Z p ¤ u p = ^ Z D Z p ¹ A ¤ u p +r 1 (5.54) where r 1 ,W c1 (s) ³ ¡ W b1 (s)® T n (s)u p ¢ _ ¹ µ 3 ´ (5.55) ^ Z D Z p = ¹ µ T 3 ® n (s) (5.56) andW c1 (s);W b1 (s) are defined as in Swapping Lemma 1 withW = ¹ A ¤ in (5.55). Since _ ¹ µ 3 is a linear combination of _ µ it follows that _ ¹ µ 3 2 S(¢ 2 2 ), and using the fact thatu p 2L 1 we have thatr 1 2S(¢ 2 2 ). Doing a similar operation on the last term of the RHS of (5.53) with the fact thate ¡ ^ Ts Z c =e ¡ ^ Ts ¢Z c =Z c ¢e ¡ ^ Ts we have e ¡ ^ Ts Z c Z p ^ Z D Z p ¤ u p = ^ Z D Z p ¢e ¡ ^ Ts Z c Z p ¤ u p +e ¡ ^ Ts r 2 (5.57) where r 2 ,W c1 (s) ³ ¡ W b1 (s)® T n (s)u p ¢ _ ¹ µ 3 ´ (5.58) 105 and W c1 (s);W b1 (s) are defined as in Swapping Lemma 1 with W = Z c Z p ¤ in (5.58). Similar logic applies to allowr 2 2S(¢ 2 2 ). We can now rewrite (5.53) as ^ R S 1 ¤ q ¡ "m 2 s ¢ = ¹ A ^ R D R p ¤ y p ¡ e ¡ ^ TS Z c Z p ^ R D R p ¤ y p ¡ ^ Z D Z p ¢ ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ ¤ u p ¡r 1 +e ¡ ^ Ts r 2 : (5.59) Using the fact that the adaptive control law can be written as ¹ A¡e ¡ ^ Ts Z c Z p ¤ u p =¡ Z c R p ¤ e 1 (5.60) we substitute into (5.59) to get ^ R S 1 ¤ q ¡ "m 2 s ¢ = ¹ A ^ R D R p ¤ y p ¡ e ¡ ^ TS Z c Z p ^ R D R p ¤ y p + ^ Z D Z p Z c R p ¤ e 1 ¡r 1 +e ¡ ^ Ts r 2 : (5.61) We will now turn our attention to the y p terms on the RHS of (5.61). Starting with the first term and applying Swapping Lemma 1 ¹ A ^ R D R p ¤ y p = ^ R D R p ¹ A ¤ y p +r 3 (5.62) where r 3 ,W c1 (s) ³ ¡ W b1 (s)® T n+1 (s)y p ¢ _ ¹ µ 4 ´ (5.63) 106 ^ R D R p = ¹ µ T 4 ® n+1 (s) (5.64) andW c1 (s);W b1 (s) are defined as in Swapping Lemma 1 withW = ¹ A ¤ in (5.63). Since we have that _ ¹ µ 4 is a linear combination of _ µ 2 S(¢ 2 2 ) and y p 2 L 1 , it follows that r 3 2 S(¢ 2 2 ). Taking the other y p term on the RHS of (5.61) and applying Swapping Lemma 1 e ¡ ^ Ts Z c Z p ^ R D R p ¤ y p = ^ R D R p e ¡ ^ Ts Z c Z p ¤ y p +e ¡ ^ Ts r 4 (5.65) where r 4 ,W c1 (s) ³ ¡ W b1 (s)® T n+1 (s)y p ¢ _ ¹ µ 4 ´ (5.66) andW c1 (s);W b1 (s) are defined as in Swapping Lemma 1 withW = Z c Z p ¤ in (5.66). Also it follows, using similar analysis used above, thatr 4 2S(¢ 2 2 ). Rewriting (5.61) we get ^ R S 1 ¤ q ¡ "m 2 s ¢ = ^ R D R p ¢ ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ ¤ y p + ^ Z D Z p Z c R p ¤ e 1 ¡r 1 +e ¡ ^ Ts r 2 +r 3 ¡e ¡ ^ Ts r 4 : (5.67) Sincey p =e 1 +y m we write ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ y p = ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ (e 1 +y m ) (5.68) 107 which becomes ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ y p = ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ e 1 + ³ R c R p +Z c Z p ¡Z c Z p ¢e ¡ ^ Ts ´ y m (5.69) and sincey m is a constant andR c containss as a factor, to serve as the internal model of y m , we have ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ y p = ³ ¹ A¡e ¡ ^ Ts Z c Z p ´ e 1 : (5.70) Plugging into (5.67), using ¹ A¡e ¡ ^ Ts Z c Z p = ^ R S andZ c R p =Z S and ^ R D R p = ^ R D ¢R p ^ R S 1 ¤ q ¡ "m 2 s ¢ = ^ R D ¢R p ¢ ^ R S ¤ e 1 + ^ Z D ¢Z p Z c R p ¤ e 1 ¡r 1 +e ¡ ^ Ts r 2 +r 3 ¡e ¡ ^ Ts r 4 (5.71) and now we can collect terms and solve fore 1 to get e 1 = ¤ B Ã ^ R S ¤ q "m 2 s +r 1 ¡e ¡ ^ Ts r 2 ¡r 3 +e ¡ ^ Ts r 4 ! (5.72) B = ^ R D ¢R p ¢ ^ R S + ^ Z D ¢Z p Z S : (5.73) 108 The roots of (5.73) are stable, which was shown in Step 2 above. Now all of the transfer functions are proper and stable, however they are irrational, with inputs that are inS(¢ 2 2 ), we use Lemma A.5.8 in [36] to show that e 1 2S(¢ 2 2 ) (5.74) which is the same as saying Z t 0 e 2 1 d¿ ·c¢ 2 2 t+c (5.75) which implies that the mean value ofe 2 1 is of the order of the modeling error characterized by¢ 2 . ¥ 5.4 Numerical Simulations A numerical simulation will be conducted to show the performance of the adaptive Smith predictor scheme presented here. In the field of process control, many systems can be represented by first order systems with time delays, so here we take the same plant used in [77] which is y p = 1:4e ¡8s 1:1s+1 u p : (5.76) This model comes from a flow control problem where the object is to regulate the flow into a tank which could be used in the food, drink, or chemical industries. The flow rate is measured by a flow meter and the control variable a pump voltage. The model given 109 above is constructed based on an experimental setup given in [77], however with the time delay enlarged to create a time delay dominant system. In [77] a PI style controller is given, which will also be used here asC(s) and that is C(s)=K p µ 1+ K i s ¶ = 0:714(s+0:9091) s : (5.77) The gains K p ;K i are set so that K p = 1 1:4 and K i = 1 1:1 . For comparison purposes the same reference signal will be used which is a square-wave with an amplitude of 0:15, DC offset of0:75, and a period of80s. Now that the fixed controller C(s) is set, the adaptive Smith predictor needs to be designed. First, the parametric model for the online estimator is z =µ ¤ Á (5.78) z = 2(1:1s+1) (s+1) 2 y p ¡ 2(1:4) (s+1) 2 u p (5.79) µ ¤ =T (5.80) Á=¡ s(1:1s+1) (s+1) 2 y p ¡ 1:4s (s+1) 2 u p (5.81) 110 The adaptive law is then _ µ =¡"Á (5.82) _ µ = 8 > > > > > > < > > > > > > : °"Á if(µ >1) or(µ =1 and"Á¸0) 0 otherwise (5.83) m 2 s =1+ÁÁ (5.84) since we know that the time delay is bounded asT ¸ 1. Using the same starting values for plant parameters as in [77], the initial estimates are µ(0) = 6. The simulation is conducted and the set point and output flow from the plant are plotted in Fig. 5.3 and the estimate of the time delay is seen in Fig. 5.4. With each step command the estimate changes slightly, however this could be compensated with more logic in the estimator or possibly a different adaptive law, such as a gradient with integral cost or least squares. When a fixed non-adaptive Smith predictor is used the system goes unstable, although as seen in the figures, the use of the adaptive Smith predictor ensured performance is retained even when the plant is unknown. 5.5 Discussion We have shown an adaptive Smith predictor control scheme, which incorporates a fixed controller as well as an adaptive Smith predictor. The adaptive Smith predictor uses a time delay estimate which come from an online estimator. Based on a linear parametric 111 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (s) Flow dm 3 /min Figure 5.3: Time series of the reference signaly m in red and the outputy p in blue. 0 50 100 150 200 5.5 6 6.5 7 7.5 8 8.5 9 Time (s) Time-delay Estimate (ρ) Figure 5.4: Time series of the estimate of the time delay, which is ½ = ^ T . The correct value of the time delay is 8 seconds. 112 model of the plant, treating the time delay as a polynomial approximation, the estimator is able to accurately estimate unknown or changing parameter. Since the Smith predictor is model based, it incorporates a model of the actual plant, it may less robust to plant variations. For this reason the estimator will be upgraded to be able to estimate the plant parameters as well as the time delay. Some stability results have been published in [86], however the adaptive time delay compensation controller is not a Smith predictor. 113 Chapter 6 Applications 6.1 Adaptive Mode Suppression and Disturbance Rejec- tion Scheme with application to Disk Drives 6.1.1 Introduction Hard disk drives (HDD) are a form of data storage that are present in just about every computer system. In a HDD, rotating disks, sputtered with a thin magnetic layer or recording medium, are written with data in concentric circles, called tracks [13]. A head positioning servomechanism is a control system which positions the head (mounted on the actuator) over a desired track and repositions the head from one track to another. The time needed to reposition the head as well as the position accuracy of the head over the center of the track are the most important performance characteristics of any HDD control system [2, 63]. In order to control the position of the head, the controller needs 114 to have a measure as to how far the head is from the desired position which is ideally the center of the desired track. This measure of deviation is known as the position error signal (PES). In most of today’s disk drives, the PES is generated using prewritten position data on each track. The position data are written on a number of chosen sectors, referred to as servo burst sectors that are symmetrically located on each track, thus placing a constraint on the sampling frequency of the PES. There has been a large amount of research activity into two types of control problems dealing with the HDD: track-seeking and track-following [13]. The former deals with motion control of the head between tracks, and the latter with maintaining the head on the center of the HDD track. This paper deals with track-following. A track-following con- troller must be able to maintain stability and meet performance requirements, whereas running strictly off the PES signal which is only available at a preset sampling frequency, f s . The control objective is to position the center of the head over the center of a data track. Thus, the typical measure of HDD tracking performance is the deviation of the center of the head from the center of a given track, which is often called track misreg- istration (TMR) [63]. There exist many indexes used to quantify TMR. Here we adopt TMR =3¾: Where¾ is the empirical standard deviation (STD) of the control error sig- nal. It is common to express 3¾ as a percentage of the track pitch [42, 63], which must be less than10% in order to be considered acceptable. TMR values larger than this fig- ure will produce excessive errors during the reading and recording processes. In Section 6.1.2 the problem statement will be presented and then in Section 6.1.3 the integrated 115 adaptive mode suppression and disturbance rejection scheme is explained. Section 6.1.4 presents simulation results and conclusions are drawn in section 6.1.5. 6.1.2 Problem statement The HDD has many high frequency resonant modes which must be suppressed by the control system. This is usually done through the use of notch filters. However these modes are often uncertain and vary between units, creating a need for wider notch filters or adaptive notch filters. The adaptive notch filter has been studied in research [87, 78, 20] as well as various applications, such as the HDD [69], launch vehicles [21], aircraft [61], and space structures [54]. Since the sampling frequency of the HDD is fixed, and many of these resonant modes may lie near or beyond the nyquist frequency, the use of multirate notch filters is necessary [92]. Since the controller must be designed to work in conjunction with the multirate adaptive notch filter, an adaptive bandwidth controller is incorporated. This design uses a multiple model approach, where numerous controllers are designed offline and the correct controller is selected online. The use of multiple model controllers has been researched before [67, 65, 6], however the scheme presented here uses the parameter estimates from a robust online estimator as the selection criteria for the controller. The track-following controller must also reject disturbances. The disturbance can be separated into repeatable runout (RRO) and non-repeatable runout (NRRO). The RRO is produced by imperfections and eccentricities on the tracks, whereas NRRO is produced 116 by aggregated effects of disk drive vibrations, imperfections in the ball-bearings, and electrical noise. Research has been conducted over the years to cancel the effects of these disturbances and acquire better track following capabilities [93, 79, 72, 71, 33, 42, 94, 52, 95, 39, 62]. It has been shown that the RRO can be suppressed with adaptive feedforward methods [79], and by adaptive repetitive control [71]. In this paper we use the adaptive feedfor- ward disturbance rejection scheme to eliminate the RRO and then focus on reducing the disturbance even further. There has also been work done in using neural networks for feedforward disturbance rejection [55, 74, 66]. Radial basis functions (RBF) have been used to model sea-clutter noise in radar applications [31], a similar approach was taken here. Simulations verify the neural model of the disturbance that is adapted online and used for disturbance rejection to obtain more precise track-following. This paper will present a multirate adaptive notch filter that is able to track the reso- nant modes of the plant, even if they exist near or above the nyquist frequency, without the addition of a probe signal. This is done through the use of plant parametrization and a novel deadzone modification. Also an adaptive bandwidth controller will be added to maintain stability and performance requirements as the multirate adaptive notch filter changes online. This is the same scheme presented in [47, 50], but added is an adaptive disturbance rejection scheme that uses a neural model of the disturbance [51]. These two adaptive schemes work together to provide one integrated adaptive mode suppression 117 and disturbance rejection scheme for track-following of a hard disk drive [45]. The con- trol scheme is designed with application to the HDD however components of the scheme could be changed based on the specific application. There are also possible applications to other digital control systems with flexible modes and unknown disturbances such as flight control systems, space structures, and manufacturing processes. The integrated adaptive control scheme can be appropriately modified to enhance performance on such applications. 6.1.3 Integrated adaptive mode suppression and disturbance rejec- tion scheme The integrated adaptive scheme presented consists of an adaptive multirate notch filter and an adaptive bandwidth controller for mode suppression with the addition of an adap- tive feedforward disturbance rejector. A diagram of the entire scheme is presented in Fig. 6.1. Each of the units will be presented in the following subsections. The integration of separate adaptive schemes to work together to achieve good track-following is done by only having one scheme at a time updating online. The following sections will explain how this is accomplished. 6.1.3.1 Adaptive Mode Suppression Scheme In this subsection the control scheme with the multirate adaptive notch filter and adaptive bandwidth controller will be explained. The closed loop system for mode suppression 118 HDD Dynamics r(t) Multirate Adaptive Notch Filter y(t) Adaptive Bandwidth Controller Robust Online Estimator Adaptive RRO Disturbance Rejector Adaptive Neural Disturbance Rejector Adaptive Mode Suppression Adaptive Disturbance Rejection Figure 6.1: Overall diagram of the integrated adaptive mode and disturbance suppression scheme. Herey(t) is the sample-and-hold version of the HDD PES signal andr(t) is the reference track input. can be seen in Fig. 6.2, whereG(s) is the HDD plant,F(z) is the multirate adaptive notch filter, C(z) is the adaptive bandwidth controller, and Est is the robust online estimator. Zero-order holds and sample-and-holds are represented by ZOH and S/H respectively. Also the frequency at which each element updates is written within each block. The robust online estimator will estimate the resonant mode frequency of the plant, which will be used as the center frequency of the multirate adaptive notch filter and also to select the adaptive bandwidth controller. Robust Online Estimator The robust online estimator that is presented is for the case of a single unknown resonant mode of the plant, however it can be easily expanded to estimate more unknown resonant mode frequencies. The continuous time model of the HDD can be represented as G(s)= N(s) D(s) : (6.1) 119 Which can be rewritten to be used in the online estimation scheme as follows: G(s)= N(s) D k (s)(s 2 +2³! n s+! n 2 ) ; (6.2) where D k (s) is the known part of the denominator of G(s). The goal is to estimate the unknown parameters³ and! n with a robust online estimator. To do this (6.2) is placed in the form of the continuous time parametric equation z(t)=µ ¤T Á(t) (6.3) z(t)=z y (t)¡z u (t) (6.4) z y (t)= s 2 D k (s) ¤(s) y(t) (6.5) z u (t)= N(s) ¤(s) u(t) (6.6) Á(t)= · ¡2sD k (s) ¤(s) y(t) ¡D k (s) ¤(s) y(t) ¸ T (6.7) µ ¤ = · ³! n ! n 2 ¸ T : (6.8) Here ¤(s) is a polynomial added to make proper transfer functions, and takes the form ¤(s)=(s+¸) n . Here¸ andn are design parameters which will determine the speed of the filter 1 ¤(s) . 120 The above equations are then discretized using the Tustin approximation at the fre- quency of the online estimator, which in this case will be2f s . This means the estimator will be run at two times the sampling frequency of the HDD. Since we only want to esti- mate the parameters when their is a sufficient level of persistent excitation (PE), the best time for estimation is only during the very beginning of the track-following routine. The track-following controller will be turned on at the end of the track-seeking routine, when the HDD head is close to the track center. It will then be the job of the track-following controller to bring the head over the center of the track and maintain this position for the desired length of the following command. This appears as a step function input to the track-following controller and therefore will result in a good level of PE until the head settles. The estimator therefore only has until the settling time to perform its best estimation. After this time, there will be a low level of PE and the inputs to the estimator will be highly corrupted with noise and disturbance effects making accurate estimation difficult. It is for this reason that the estimator is run at the frequency of 2f s so as to estimate as fast as possible before the settling time. The online estimator has two inputs: a signal based on the HDD PESy E (k) and the one from the control signal u E (k). Since the estimator is running at a frequency of 2f s and the HDD PES signal is only available for measurement at a frequency off s , the PES signal must be passed through a zero-order hold as seen in Fig. 6.2. It is for this reason that the estimator is not run at a higher frequency. As the frequency of estimation is increased, the estimation error will grow larger, causing large overshoots and incorrect 121 G(s) C(z) (f s ) ZOH (4f s ) r(k) Est. (2f s ) F(z) (4f s ) u(t) y(t) y F (k) ZOH (cont) y(k) ZOH (2f s ) u F (k) u C (k) u C * (k) y E (k) u E (k) S/H (2f s ) S/H (f s ) S/H (4f s ) Figure 6.2: Closed loop system used for mode suppression. The sampling frequency of each block is shown in parenthesis. Here y(k) is the real sampled HDD output, u(t) is the continuous input to the HDD, andr(k) is the reference input. parameter estimates. This is because the PES signal input to the estimator is held constant with the zero-order hold, so the estimates are updated based on incorrect information. By estimating at only two times the sampling frequency, it allows the estimator to update faster, but not fast enough to cause large errors and overshoots. This is emphasized in Fig. 6.3, where the estimated mode frequency is seen for different estimation speeds, and the actual plant frequency is 5.00 kHz. The other input to the estimator, based on control signal, is available at a frequency of 4f s because of the multirate adaptive notch filters. Therefore the control signal does not require a zero-order hold before the estimator. The unknown parameters are updated online using the following algorithm ¹ µ(k)=µ(k¡1)+¡Á(k)("(k)+g) (6.9) µ(k)= ¹ µ(k)min µ 1; M j ¹ µ(k)j ¶ (6.10) 122 g = 8 > > < > > : 0 if¯ E (k)>b 0 ¡"(k) if¯ E (k)·b 0 (6.11) ¯ E (k)=min(a 11 ;a 22 ;:::;a nn ) (6.12) A= 2 6 6 6 6 6 6 6 6 6 6 4 a 11 a 12 ¢¢¢ a 1n a 21 a 22 ¢¢¢ a 2n . . . . . . . . . . . . a n1 a n2 ¢¢¢ a nn 3 7 7 7 7 7 7 7 7 7 7 5 = k X j=k¡l Á(j)Á T (j) m 2 (j) (6.13) "(k)= z(k)¡µ T (k¡1)Á(k) m 2 (k) (6.14) m 2 (k)=1+n s (k)+m s (k) (6.15) n s (k)=C s Á T (k)Á(k) (6.16) m s (k)=± 0 m s (k¡1)+u 2 (k¡1)+y 2 (k¡1): (6.17) The online estimator uses a discrete gradient algorithm with a couple of robustness modifications [36]. The update term is normalized with a dynamic term to add robust- ness, this term is calculated in (6.17) where the parameter± 0 is chosen between 0 and 1. Also parameter projection is used in (6.10), since there is a known region of the param- eters,jµ ¤ j· M for some knownM > 0, just based on a priori knowledge of the HDD. The adaptation gain¡ is a design parameter that satisfies,¡ = ¡ T > 0, and theC s term in (6.16) is another design parameter that must be larger than zero. In (6.11), the term 123 0 0.5 1 1.5 2 2.5 3 4.6 4.8 5 5.2 5.4 5.6 5.8 Time (ms) ω n (kHz) Figure 6.3: Simulation results of estimated mode frequencies for different estimation speeds. The actual mode of the plant is 5.00 kHz. Solid line: Estimator running at f s . Dashed line: Estimator running at2f s . Dotted line: Estimator running at4f s . b 0 is used to set the level of the deadzone. This parameter is chosen based on a priori knowledge of the HDD. For online implementation theA matrix in (6.13) does not need to be directly computed since we are only interested in the elements along the diagonal. These diagonal entries can be computed directly, which in this example only requires 2 calculations. Also in (6.13) the terml is used to set how many values are summed. There is also a novel deadzone based on the level of energy in the plant output. This deadzone is added to stop adaptation when the level of energy becomes low, meaning there is not sufficient information to update the parameters. As described before, this is to allow estimation to only occur before the settling time of the track-following routine. This is a derivation of the initial plan to estimate based on the level of PE. The reason to 124 use PE is that in the gradient adaptive law of (6.9), it is established that if Á(k) m(k) is PE, it satisfies k+l¡1 X j=k Á(j)Á T (j) m 2 (j) ¸® 0 lI; (6.18) where ® 0 > 0 is the level of PE and l > 1 is some fixed integer, then µ(k) ! µ ¤ exponentially fast, where µ ¤ is the actual system parameters. To use this condition in a deadzone modification, the following could be computed online ¯ =¸ min ( k+l¡1 X j=k Á(j)Á T (j) m 2 (j) ) : (6.19) This value¯ is not the level of PE, but instead a value which has similar significance and can be compared to some design parameter, similar to b 0 in (6.11), to determine when adaptation occurs. Simulations were performed with this deadzone technique using¯ as the requirement for adaptation. However finding the eigenvalues of large matrices can be computationally intensive and difficult to perform online in a real system. So a variation of this deadzone technique is used. The¯ that is computed in (6.19) is similar to finding a matrix A= 2 6 6 6 6 6 6 6 6 6 6 4 a 11 a 12 ¢¢¢ a 1n a 21 a 22 ¢¢¢ a 2n . . . . . . . . . . . . a n1 a n2 ¢¢¢ a nn 3 7 7 7 7 7 7 7 7 7 7 5 = k+l¡1 X j=k Á(j)Á T (j) m 2 (j) (6.20) 125 and then finding the minimum value along the diagonal ¯ diag (k)=min(a 11 ;a 22 ;:::;a nn ): (6.21) A variation of this is seen in (6.12) and (6.13), but with a change of the summation limits to allow for realistic online processing. Therefore the level of PE, or energy, in the system can also be thought of as the minimum of the squared summation of the previous l values of the individual components in the Á(k) vector. This can be seen through a HDD simulation where ¯ and ¯ E are both computed online and their values compared in Fig. 6.4. The online computed¯ E is compared to some design parameterb 0 to determine whether adaptation should occur. One method of selectingb 0 is to find the level of ¯ E (k) that corresponds to the settling time. In Fig. 6.4 the settling time of the HDD is around1:4 ms, therefore a value of0:002 is chosen forb 0 . As the energy in the system decreases, adaptation will stop and the parameters will become frozen until the level of energy increases again. Multirate Adaptive Notch Filter The multirate adaptive notch filter is implemented to suppress the uncertain or chang- ing high frequency modes of the HDD. The notch filter’s center frequency will use the estimate, ^ ! n from the previously described robust online estimator to track the frequency of the mode. Since the notch filter will accurately follow the mode it can therefore be 126 0 0.5 1 1.5 2 2.5 3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Time (ms) Magnitude Figure 6.4: HDD simulation to compare the deadzone parameters. The simulation is run using the parameters that are described later in this paper, with a step reference input. Solid line: ¯ E , the minimum sum squared diagonal element. Dashed line: ¯, the minimum eigenvalue. designed narrower, providing less phase lag at lower frequencies. This will enable the design of higher bandwidth controllers. To be able to suppress modes near or above the nyquist frequency, the notch filter is run using a multirate scheme. The estimates of the modal frequency are updated at a frequency of 2f s and the notch filter itself is run at a frequency of 4f s . The system in Fig. 6.2 can be analyzed by the method in [92]. The discrete time transfer function of the HDD plant,G(z), sampled at4f s and used for the notch filter design, is the transfer function from the notch output,u F (k), to the output of the first sample and hold,y F (k). This valuey F (k) is purely fictitious and only used for analysis. 127 The multirate adaptive notch filter takes the form F(z)= z 2 ¡2® N cos( ^ !n 4fs )z+® 2 N z 2 ¡2® D cos( ^ ! n 4f s )z+® 2 D ; (6.22) ® N >® D , and define the width and depth of the filter, and ^ ! n is the estimate of the mode frequency of the plant. It should be noted that in this paper only the natural frequency of the mode is assumed to be uncertain or changing, however the damping of the mode may also vary. The damping ^ ³ is also estimated and could be used to change the shape of the notch filter. The® N and® D parameters could be dynamic terms based on the estimated damping. (6.22) can also be represented in the companion state space form, for ease of implementation, as u F (k)=C F x F (k¡1)+D F u ¤ C (k) (6.23) x F (k)=A F x F (k¡1)+B F u ¤ C (k) (6.24) A F = 2 6 6 4 2® D cos( ^ ! n 4f s ) ¡® 2 D 1 0 3 7 7 5 (6.25) B F = 2 6 6 4 1 0 3 7 7 5 (6.26) C F = · ¡2(® N ¡® D )cos( ^ !n 4f s ) ® 2 N ¡® 2 D ¸ (6.27) D F =1: (6.28) 128 Where u ¤ C (k) and u F (k) are the input and output of the notch filter, respectively, and x F (k) is the state vector. The adaptive notch filter scheme presented differs from schemes previously reported in the literature for various reasons. The adaptive notch filter presented in [21] is used on the model of a booster from the Advanced Launch System (ALS) program. The least squares estimator in the publication uses a simple undamped resonator as the model and functions well since the resonant mode is very pronounced. However in the HDD, and other applications, full plant parameterizations is necessary as the flexible mode may not be as significant. Another strategy for the estimation of the center frequency can be found in [69], where frequency weighting functions are used. The downside is there are several failure modes that are known and avoidance requires some modal information a priori. A stochastic state space algorithm for mode frequency estimation is presented in [61]; however it relies on the injection of a probe signal which is not needed in the scheme presented here. The indirect adaptive compensation (IAC) scheme in [94] also requires a probe signal to complete the estimation. The adaptive mode suppression scheme in [54] uses a LMS algorithm to update filter coefficients and then the modal parameters are extracted from the filter. This is opposite as to what is being presented in this paper, where the modal parameters are first estimated and then used in the adaptive notch filter. The estimation scheme also allows for the identification of multiple modes simultaneously if the model in (6.2) is expanded to include more unknown harmonics. Adaptive Bandwidth Controller 129 The controller must be designed to work in conjunction with the multirate adaptive notch filter. Since the notch filter is designed narrower, providing less phase lag at low frequencies, the controller can attain a higher bandwidth. However if the mode of the plant is at a lower frequency, or changes to a lower frequency, than the notch filter may interfere with the controller. Since the notch filter is going to track the mode of the plant, the center frequency will decrease, and the phase lag from the filter will effect the stability margins of the control scheme. To prevent this from happening the controller has an adaptive bandwidth design based on a multiple model technique. The adaptive bandwidth is accomplished by designing multiple controllers offline which meet stability and performance margins. Separate controllers are designed for various mode frequencies, and therefore various notch filter center frequencies. A single controller is then selected online, based on the online parameter estimate of the modal frequency, ^ ! n . In this work, all of the controllers designed offline are of the same or- der, thereby simplifying the selection scheme. The state space matrices of the individual controllers are stored in a database, and the correct matrices are found online by inter- polation. For example, the correct A matrix can be found online when the estimated modal frequency, ^ ! n lies between the two frequencies where controllers were designed. If the frequencies are ! n1 and ! n2 , where ! n1 < ! n2 , and designed controller have the following matricesA 1 andA 2 , respectively, then A=A 1 + ^ ! n ¡! n1 ! n2 ¡! n1 (A 2 ¡A 1 ): (6.29) 130 This is then repeated for the rest of the state space matrices. Interpolation is a fairly quick computational procedure, and for the HDD only a small number of designed controllers are needed. The controllers themselves are designed using the previously designed multirate notch filters and discretized plant. Again referring to Fig. 6.2, the transfer function used for controller design, sampled atf s is found by computing the transfer function fromu C (k) toy(k). The complete open loop transfer function, sampled atf s is computed by taking the combination ofF(z) andG(z), dividing the sampling rate by four, and combining it withC(z). 6.1.3.2 Adaptive Disturbance Rejector The goal of the disturbance rejection problem is to reduce the PES, y(k) in Fig. 6.2, by injecting the negative of an estimate of the disturbance. To do this, a closed loop model from r(k) to y(k), called G a (z), is needed. This then allows us to view the sys- tem as Fig. 6.5, an appropriate way for the disturbance rejection scheme. However the system G a (z) is not LTI, since it possesses adaptive controllers. To overcome this and allow the adaptive disturbance rejection scheme to work with the previously mentioned adaptive mode suppression scheme, the adaptive disturbance rejection scheme is turned on when the adaptive mode suppression scheme is turned off. The online estimator for the adaptive mode suppression is only updated when there is a certain level of energy in the system, to avoid being influenced by noise and disturbances. Since the disturbance 131 ) (z G a ) (k d ) (k r ) (k y Figure 6.5: Block diagram of the system used for disturbance rejection. G a (z)= closed- loop system with adaptive controllers;d(k) = disturbance;r(k) = disturbance rejection signal and reference input. rejection scheme only needs to be effective once the head is near the center of the track, it can turn on when the level of energy in the system is low. Therefore once the online estimator for the adaptive mode suppression scheme stops updating, the adaptive distur- bance scheme can begin updating. Since the estimated parameters are frozen in the mode suppression scheme, the multirate adaptive notch filter and adaptive bandwidth controller can be treated as LTI. This allowsG a (z) to also be treated as LTI, and an estimate ^ G a (z) can be computed online. An estimate of the HDD dynamics, ^ G(z), is known as well as the controllerC(z) and notch filtersF(z), allowing us to compute ^ G a (z). The disturbance rejection problem is divided into two parts: repeatable runout distur- bance rejection and neural modeled disturbance rejection. The former has proven experi- mental results as shown in [79]. This paper makes no modification to the algorithm other than applying it to a large number of frequencies. There are other RRO compensators available in the literature, such as repetitive control, which has been done experimentally in [71] and also compared in [41]. Other RRO compensators could be substituted for the RRO scheme adopted in this paper, which is used as one example to show integration of the adaptive notch filter and disturbance rejection scheme. 132 Once the repeatable runout disturbance rejection is applied there is still disturbance that creates non-perfect tracking. This remaining disturbance is modeled using neural techniques and adaptively updated online, which has been experimentally verified in [51]. This neural scheme could also be substituted for other nonlinear compensators that have been developed such as the ones in [32, 33, 72]. The complete disturbance rejection scheme is updated and run at the HDD sampling frequency,f s . Repeatable Runout Disturbance Rejection The repeatable runout (RRO) disturbance occurs at frequencies120m Hz wherem= 1;2;:::;n due to the 7200 rpm speed of the disk. A control input is designed such that it will adaptively cancel this disturbance. The disturbance, d(k) = d RRO (k) can be modeled as d RRO (k)= n X i=1 a i (k)sin µ 2¼ik N rev ¶ +b i (k)cos µ 2¼ik N rev ¶ ; (6.30) wherei is the index for the harmonic andN rev is the number of samples per revolution. If the system is modeled as in Fig. 6.5 then the output is y(k)=G a (z)[r(k)]+d RRO (k): (6.31) To cancel the disturbance the control signal should ber(k) =¡ ^ G a ¡1 (z)[ ^ d RRO (k)]. The identified inverse, ^ G a ¡1 (z), will have an effect on the magnitude and phase of the distur- bance estimate, ^ d RRO (k). Since the magnitude and phase of the sinusoidal disturbance 133 are being estimated, the system inverse can be ignored and the new control signal be- comesr(k)=¡ ^ d RRO (k). The disturbance estimate is ^ d RRO (k)= n X i=1 ^ a i (k)sin µ 2¼ik N rev ¶ + ^ b i (k)cos µ 2¼ik N rev ¶ : (6.32) The update equations for the estimated parameters are ^ a i (k)= ^ a i (k¡1)+° i y(k¡1)sin µ 2¼ki N rev +Á i ¶ (6.33) ^ b i (k)= ^ b i (k¡1)+° i y(k¡1)cos µ 2¼ki N rev +Á i ¶ : (6.34) Where the ° i are adaptation gains, chosen differently for each harmonic. A phase ad- vance modification is added to reduce the sensitivity and allow for more harmonics to be canceled as was done previously in [79]. The Á i =\ ^ G a (j! i ) and ! i is the angular frequency of theith harmonic. Neural Modeled Disturbance Rejection Since the disturbance does not consist entirely of harmonics from the rotation of the disk, another disturbance rejection algorithm is added. The new disturbance is modeled as d(k)=d RRO (k)+d NN (k): (6.35) 134 So the system output now becomes y(k)=G a (z)[r(k)]+d RRO (k)+d NN (k): (6.36) To cancel the disturbance the control signal should be r(k)=¡ ^ d RRO (k)¡ ^ G a ¡1 (z)[ ^ d NN (k)]: (6.37) Since the computed model ^ G a (z) of the HDD may be, and the case presented in simu- lations is, non-minimum phase, the inverse is unstable. Also the computed model may be strictly proper which makes a state space inverse not realizable. The unstable zeros of ^ G a (z) are reflected across the unit circle, and then the system is augmented with very fast zeros making the degree of the numerator equal to the degree of the denominator. Then the inverse is taken. This new inverse, ¹ G a ¡1 (z), is used in the computation of the control signal, making itr(k)=¡ ^ d RRO (k)¡ ¹ G a ¡1 (z)[ ^ d NN (k)], and therefore causes an extra delay that will be dealt with. The following disturbance rejection scheme uses gaussian radial basis functions (RBF) from neural networks to attempt to model the disturbance. The disturbance estimate takes the form ^ d NN (k)= L X q=1 M X i=1 µ q;i (k)R q;i ^ d(k¡±¢(q¡1)¡1) (6.38) R q;i =ª i ³ ^ d(k¡±¢(q¡1)¡1) ´ : (6.39) 135 TheR q;i is computed using an RBF and theith gaussian RBF is ª i (x)=exp " ¡ µ x¡c i º ¶ 2 # : (6.40) The parameters that specify the shape of the ith gaussian RBF are the center c i and the width º. There are a total of M gaussian RBFs, and their centers are linearly spaced across the range of input. The current disturbance estimate, ^ d NN (k), is a function of L previous disturbances that are spaced± samples apart. The reason for the spacing ± is the delay associated with passing the disturbance estimate through the system inverse. One method of coping with the delay would be to estimate the disturbance at the next sample, and then use this estimate to create another future estimate, and continue iterating to find some ^ d NN (k+¢) in the future [74]. This method did not work as the estimation error grew with each future estimate. Instead the disturbance is thought of as a function of previous evenly spaced disturbances. The ^ d NN (k) can be viewed as a future disturbance estimate when compared to the HDD sample rate. It should be noted that the algorithm still creates a new disturbance estimate at every sample of the HDD output. The model of the disturbance is motivated by the assumption that the disturbance is a nonlinear function of previous disturbance values. The ^ d(k¡±¢(q¡1)¡1) term that is multiplied by the output of the RBF in (6.38) is the added term that allows the model to work for the disturbance in this HDD. This added term makes the model different from 136 1 < 2 < M < ) 1 ( ˆ k d ¦ 1 , 1 T 2 , 1 T M , 1 T ¦ ) 1 ( ˆ k d 1 < 2 < M < ) 3 ( ˆ k d ¦ 1 , 2 T 2 , 2 T M , 2 T 1 < 2 < M < ) 5 ( ˆ k d ¦ 1 , 3 T 2 , 3 T M , 3 T ) 3 ( ˆ k d ) 5 ( ˆ k d ) ( ˆ k d NN Figure 6.6: Simple example of how ^ d NN (k) is computed from previous values of the disturbance. Here± =2 andL=3. previously used RBF neural predictors [31, 74]. Now the disturbance can be thought of as an autoregressive filter with spacing± and nonlinear coefficients that are modeled with the RBF’s. A simple example with ± = 2 and L = 3 is shown in Fig. 6.6 to help view the modeling of the disturbance. The ^ d(k¡±¢(q¡1)¡1) used in the estimate is not produced by the estimator but rather is a measured disturbance estimate which can be calculated from an identified model of the system, ^ G a (z), using ^ d(k¡1)=y(k¡1)¡ ^ G a (z)[u(k¡1)]: (6.41) 137 The unknown parameters should be updated with the current modeling error, which is " NN (k)=y(k¡1)¡ ^ G a (z)[ ^ d RRO (k¡1)]; (6.42) and the parameters that caused that error, ^ d(k¡±q¡1). This leads to the update equations µ q;i (k)= 8 > > < > > : µ q;i (k¡1)+® q;i (k) " new ·½¢" old µ 0 q;i otherwise (6.43) ® q;i (k)= l NN m 2 (k) " NN ¹ R q;i ^ d(k¡±q¡1) (6.44) ¹ R q;i =ª i ³ ^ d(k¡±q¡1) ´ (6.45) m 2 (k)=1+m s (k) (6.46) m s (k)=± 0 m s (k¡1)+ ^ d 2 (k¡1): (6.47) EveryN " samples the following are calculated " new = N " ¡1 X n=0 " 2 NN (k¡n) (6.48) " old = 8 > > < > > : " new " new <" old " old otherwise (6.49) 138 µ 0 q;i = 8 > > < > > : µ q;i (k) " new <" old µ 0 q;i otherwise (6.50) The update is an instantaneous gradient algorithm with a couple robustness modifica- tions. The adaptation, or learning, rate isl NN and is greater than zero. The update term is normalized with a dynamic term to add robustness, this term is calculated in (6.47). The parameter± 0 is chosen between 0 and 1. The other robustness modification is one that is added to stop adaptation when the performance starts to degrade. In practice the estimation error will never become zero and so the parameters will continue to update. There will be a point at which the estima- tion error is small and on the same level as the noise and modeling error. Usually a simple deadzone is added to stop adaptation when the current error is below some threshold. In the HDD application the estimation error, which is the HDD output, is noisy and must be averaged over N " samples. Instead of averaging, the sum squared error is easier to calculate online via (6.48). The adaptation will continue as long as this new sum squared error," new , is less than the old sum squared error," old , to within some small range. The ½ term is selected to be greater than 1 to allow adaptation even if the sum squared error did not decrease. This gives some room for noisy measurements and lets the algorithm continue. If the algorithm is doing a good job and the new sum squared error is strictly less than the old, the currentµ q;i (k)’s are saved and the" old is updated. Theµ q;i (k)’s are saved so that when the sum squared error is too large, the algorithm can revert back to the best known parameters. 139 -100 -50 0 50 Magnitude (dB) 10 2 10 3 10 4 10 5 -360 0 360 720 1080 1440 1800 Phase (deg) Bode Diagram Frequency (Hz) Figure 6.7: Bode plot of the HDD plant,G(s). The neural modeled disturbance rejector has a large number of parameters that must be updated online. Although this scheme has been implemented in real-time and the experimental results using a Matlab xPC Target presented in [51], it will require more computational power than may be suitable for a production HDD. 6.1.4 Simulations Simulations were performed with the described adaptive mode suppression and adaptive disturbance rejection scheme. First the adaptive mode suppression scheme is compared to a non-adaptive mode suppression scheme, with fixed wider notch filters and a single fixed controller. The plant used for demonstration models that of a commercial HDD 140 with several high frequency modes, which must be suppressed. It is similar to the iden- tified plant in [38]. A bode plot of the plant, G(s), is shown in Fig. 6.7. The mode occurring at 5.6 kHz is modeled as uncertain, and allowed to change in simulation. This is chosen because it is the slowest mode and will have the greatest impact on the band- width of the control design. The sampling frequency of this HDD is f s = 12.78 kHz, making the multirate notch filter design necessary. There are high frequency modes that exist near and beyond the nyquist frequency of 6.39 kHz that need to be attenuated. Performance and stability requirements for the controller are taken from [37]: ² The controller order should be less than 12. ² The maximum magnitudes of the sensitivity and complementary sensitivity func- tions should not exceed 7.5 and 6 dB, respectively. ² The magnitude of the sensitivity function at 120 Hz should be less than -24 dB. ² The magnitude of the complementary sensitivity function at 10 kHz should be less than -8 dB. ² There should be 6 dB gain margin and 40 degree phase margin. ² The output should settle as fast as possible with zero tracking at steady state. First, the non-adaptive controller is designed. It consists of three fixed, non-adaptive, multirate notch filters which are run at the frequency of 4f s , allowing them to suppress the modes at 5.60 kHz, 7.76 kHz, and 9.98 kHz. These notch filters follow the same form 141 -100 -80 -60 -40 -20 0 20 Magnitude (dB) 10 2 10 3 10 4 10 5 -180 -90 0 90 180 Phase (deg) Bode Diagram Frequency (Hz) Figure 6.8: Bode plot of the multirate notch filters. Solid line: Adaptive scheme. Dashed line: Non-adaptive scheme. as (6.22), except ^ ! n is replaced by known quantities. The notch filter at 5.6 kHz is wider than the other two, so it can deal with the small variations in the HDD mode frequency. The controller, C(s), is designed to run at the HDD sampling frequency, f s . It consists of an integrator and three phase lead controllers C(s)= 695787(s+269:9)(s+2253) 2 s(s+4491)(s+69990) 2 (6.51) and provides a closed loop bandwidth of 1.91 kHz. This is then discretized at the fre- quency of f s using the Tustin approximation, which gives the resulting controller C(z) in Fig. 6.2. 142 Next, the adaptive control scheme is designed. It consists of the same three multirate notch filters, however the filter at 5.60 kHz is adaptive. It is designed narrower than the non-adaptive case, since the center frequency will be updated online with the estimate of the mode frequency from the robust online estimator. The filters at 7.76 kHz and 9.98 kHz are the same ones from the non-adaptive design. A comparison of the adaptive and non-adaptive notch filters is seen in Fig. 6.8. The adaptive bandwidth controller consists of three controllers designed offline for varying mode frequencies. A controller was de- signed to meet the stability and performance specifications with the mode frequency, and therefore multirate adaptive notch filter center frequency, at 4.50 kHz. This is done by modeling the plant as having a mode occur at 4.50 kHz, instead of the original 5.60 kHz, and placing the notch filter at this frequency. The open loop combination of the notch filters and plant is then used to design a controller which meets the requirements. This design procedure was repeated for 5.60 kHz and 6.50 kHz, resulting in three controllers designed offline. The controller designed with the mode at 4.50 kHz is C 1 (s)= 584147(s+269:9)(s+2061) 2 s(s+3661)(s+69990) 2 : (6.52) The controller designed with the mode at 5.60 kHz is C 2 (s)= 618320(s+269:9)(s+2061) 2 s(s+3679)(s+69990) 2 : (6.53) 143 The controller designed with the mode at 6.50 kHz is C 3 (s)= 645212(s+269:9)(s+2061) 2 s(s+3714)(s+69990) 2 : (6.54) The above controllers provide a closed loop bandwidth of 2.07 kHz, 2.22 kHz, and 2.28 kHz respectively. They are discretized at the sampling frequencyf s using the Tustin approximation, and their state space matrices stored in a database. A single controller, C(z) is then chosen online based on the estimated mode frequency, ^ ! n , and using inter- polation as described previously. Simulations are run using a step reference input, which is equal to about one third the track width. This is to simulate the time when the track-following controller becomes active and is required to place the head over the center of the track, and then maintain good tracking. To increase the fidelity of the simulation, a real measured HDD distur- bance signal is added to the measured outputy(k), and then quantized. The disturbance used is similar to the one from the commercial HDD used in [72, 51]. The total time of the simulation is 300 ms, and the3¾ values are computed for different frequencies of the 5.60 kHz mode, and results are shown in Fig. 6.9. The simulations show the benefit of the adaptive scheme. As the mode frequency is decreased, the non-adaptive system becomes unstable. This is why no non-adaptive results are seen below 5.3 kHz . However, as the mode frequency is increased, the system remains stable in both the adaptive and non-adaptive case. The adaptive case outperforms 144 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 5.5 5.55 5.6 5.65 5.7 5.75 5.8 5.85 5.9 5.95 Plant mode frequency (kHz) 3σ (% track width) Figure 6.9: Simulation results for various mode frequencies. x: Adaptive scheme. o: Non-adaptive scheme. the non-adaptive case because as the mode frequency increases, so does the bandwidth of the adaptive bandwidth controller. This results in a smaller3¾. A simulation where the mode frequency is decreased from the original 5.60 kHz to 5.00 kHz is now displayed in more detail. The estimated mode frequency during the simulation is shown in Fig. 6.10, where it is evident that adaptation occurs rather quickly due to the estimator running at2f s . Also it can be seen that adaptation stops around 1.5 ms, this is due to the deadzone. The plot of the computed value ¯ E used for stopping adaptation was displayed in Fig. 6.4. The open loop bode plots for the adaptive scheme at the start and end of the simulation are shown in Fig. 6.11. At the beginning of the simulation the system is actually unstable, but as the notch filter tracks the mode of the plant the system becomes stable and meets the stability and performance requirements. 145 0 0.5 1 1.5 2 2.5 3 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Time (ms) ω n (kHz) Figure 6.10: Estimated mode frequency during simulation Solid line: Estimated fre- quency. Dashed line: Actual frequency . -80 -60 -40 -20 0 20 Magnitude (dB) 10 2 10 3 10 4 -720 -540 -360 -180 0 180 360 Phase (deg) Bode Diagram Frequency (Hz) Figure 6.11: Bode plot of open loop system with the adaptive notch filter and adaptive bandwidth controller. Solid line: Start of simulation, before adaptation occurs. Dashed line: End of simulation, after adaptation occurs. 146 -80 -60 -40 -20 0 20 Magnitude (dB) 10 2 10 3 10 4 -720 -540 -360 -180 0 180 360 Phase (deg) Bode Diagram Frequency (Hz) Figure 6.12: Bode plot of open loop system with the non-adaptive notch filter and fixed bandwidth controller. Fig. 6.12 shows the open loop bode plot of the non-adaptive system, which is unstable throughout the entire simulation since the notch filter does not track the mode frequency. The time series data for the simulation can be seen in Fig. 6.13, which shows the non- adaptive scheme going unstable, whereas the adaptive case is able to retain stability and performance. Now, the simulation where the mode frequency is increased to 6.20 kHz is examined to show the benefit of the adaptive bandwidth controller. Since the mode frequency is higher, the multirate adaptive notch filter will track the mode frequency, thereby sup- plying less phase lag at lower frequencies. This means the bandwidth of the controller can be increased and provide better disturbance rejection capabilities. The PSD of the 147 0 2 4 6 8 10 12 -40 -20 0 20 PES (% of track) Time (ms) 0 2 4 6 8 10 12 -40 -20 0 20 PES (% of track) Time (ms) Figure 6.13: Time series simulation data with the mode decreased to 5.00 kHz. Top plot: Non-adaptive scheme. Bottom plot: Adaptive scheme. 10 2 10 3 -55 -50 -45 -40 -35 -30 -25 -20 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.14: PSD simulation data with the mode increased to 6.20 kHz. Solid line: Adaptive scheme. Dashed line: Non-adaptive scheme. 148 adaptive and non-adaptive schemes are shown in Fig. 6.14 to display the advantage of the adaptive bandwidth controller. Now the adaptive disturbance rejection scheme is added. For these simulations, the same HDD model, multirate adaptive notch filter, and adaptive bandwidth controller that was simulated above is used. The same step function is used as reference input for track-following. The adaptive mode suppression scheme will adapt its parameters until the energy is below the threshold, ¯ E (k) · g 0 , as stated earlier. The adaptive disturbance rejector is then turned on when the modal estimator is turned off, also at this moment ^ G a (z) and ¹ G ¡1 a (z) are computed for use in the disturbance rejection scheme. The adaptive feedforward disturbance rejection scheme, denoted as K RRO , is used to canceln = 33 harmonics. For this scheme there are a total of 66 gains, all of which are greater than zero, for the online updating that must be chosen accordingly. These values are tuned using an LTI equivalent representation of the feedforward disturbance rejection scheme, shown in [80]. From this LTI representation the gains can be tuned to achieve the best suppression of the RRO disturbance while retaining stability. Next Monte-Carlo simulations were conducted while slightly varying the adaptive gains to achieve even greater performance. Once these values are fixed the parameters of the neural model disturbance rejection scheme could be tuned. The neural disturbance rejection scheme is denoted as K NN , and has parameters M = 15;L = 9;N " = 500;½ = 1:5, and ± = 2. This value of N " is the same that 149 was used in [51] which presents experimental results for the adaptive disturbance rejec- tion scheme. These parameters were also tuned through Monte-Carlo simulations while varying the number of basis functionsM, adaptive gainl NN , and the number of past pa- rametersL. A range of values for each parameter appropriate to the residual disturbance that is to be rejected with the neural model is tested in simulation. The number of basis functions is increased, starting from 5, until a diminishing return is noticed. Increasing the number of basis functions over 15 will increase the computation time required in exchange for only slightly increased performance. The adaptive gain was varied over a range:1<l NN < 10 and the past parameters valueL was chosen in a similar fashion to the choice ofM. The learning rate, l NN , is scheduled with the estimated frequency of the 5.60 kHz mode of the HDD. This is done to achieve the best performance at varying estimated frequency, and therefore varying bandwidth controller. Thea i (k) andb i (k) of the RRO disturbance rejector, and the µ q;i (k) of the neural scheme are all initialized to zero and adapted online. The overall performance of the adaptive disturbance rejection schemes when combined with the adaptive mode suppression scheme can be seen in Fig. 6.15, where the 5.60 kHz mode of the plant is changed and the 3¾ values are computed for a one second simulation. A maximum improvement of 14:3% in 3¾ is seen with the integrated adaptive mode suppression and disturbance rejection scheme over the non- adaptive counterpart with no disturbance rejection. 150 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 Plant Mode Frequency (kHz) PES (% of track width) Figure 6.15: Simulation results for various mode frequencies and adaptive disturbance rejection schemes when combined with the adaptive mode suppression scheme. x: No adaptive disturbance rejection. o: K RRO . +: BothK RRO andK NN . The direct impact of the adaptive disturbance rejection scheme can be seen for a sin- gle case when the 5.60 kHz mode of the plant is changed to 5.00 kHz. In Fig. 6.16, the time series data of the HDD output is displayed . The case where the adaptive distur- bance rejection schemes are implemented has better tracking performance. The PSDs are shown in Fig. 6.17, where attenuation of the disturbance can be seen. A closeup view on a linear frequency scale in Fig. 6.18 displays the elimination of the RRO which occurs at frequencies of120m Hz wherem=1;2;:::;33. 151 0 0.2 0.4 0.6 0.8 1 -5 0 5 Time (s) PES (% of track width) 0 0.2 0.4 0.6 0.8 1 -5 0 5 Time (s) PES (% of track width) Figure 6.16: PSD simulation data with the mode decreased to 5.00 kHz. Top Plot: PSD with no adaptive disturbance rejection. Bottom Plot: PSD with adaptive disturbance rejection. (K RRO andK NN ) 10 0 10 1 10 2 10 3 -55 -50 -45 -40 -35 -30 -25 -20 -15 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.17: PSD simulation data with the mode decreased to 5.00 kHz. Solid line: Adaptive disturbance rejection on. Dashed line: Adaptive disturbance rejection off. 152 1000 1500 2000 2500 3000 -45 -40 -35 -30 -25 -20 Frequency (Hz) Power / Frequency (dB / Hz) 1000 1500 2000 2500 3000 -45 -40 -35 -30 -25 -20 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.18: Closeup of PSDs in Fig. 6.17. Top Plot: PSD with no adaptive disturbance rejection. Bottom Plot: PSD with adaptive disturbance rejection. (C, U, K RRO , and K NN ). 6.1.5 Discussion This paper presented a multirate adaptive notch filter combined with an adaptive band- width controller and adaptive disturbance rejector. The robust online estimator added a deadzone modification based on the available energy in the system, which stopped adap- tation when the HDD was near the center of the track. This prevented incorrect parameter estimates and allowed for a fast adaptation rate, and can possibly be used with other on- line estimators as a means to stop adaptation. The adaptive bandwidth controller was able to use the resonant mode frequency estimate to change the bandwidth of the con- troller online. The control scheme was verified through simulation on a HDD to show the 153 ability to maintain stability, as well as improve in performance, when a resonant mode of a HDD is uncertain. This paper also presented an adaptive feedforward disturbance rejection scheme for a HDD. The RRO of the disturbance is attenuated through the use of an adaptive feed- forward model and the remaining disturbance is modeled with neural techniques using radial basis functions.The combination of the adaptive mode suppression scheme and adaptive disturbance rejection scheme was made possible with the deadzone modifica- tion based on the energy in the system, and the fact that the closed loop system can be treated as LTI once the parameter estimates are frozen. The limitation of this method is that both adaptive systems cannot be updating simultaneously. Different adaptive or non- adaptive disturbance rejection schemes could be substituted or added into the integrated scheme to possibly attain better performance depending on the specific application. For the HDD application chosen here the simulation results provided show the benefit of the combination of the two adaptive schemes in reducing the TMR of a disk drive. 154 6.2 Adaptive Notch Filter Applied to a Flexible Laser Pointing System 6.2.1 Introduction Flexible dynamics occur in numerous mechanical systems where control systems are de- sired to maintain stability or increase tracking performance. Unless accounted for by the control scheme, these dynamics can cause instabilities or degradation in performance. A notch filter is sometimes used to suppress the flexible modes, however in many applica- tions the modal frequencies are often uncertain and can even vary over time creating the need for a wide notch filter. One such system is a pointing system, where the flexible actuators may cause the need for a notch filter. As the notch filter becomes wider it also induces greater magnitude and phase lag at lower frequencies resulting in a lower band- width system. One solution to such a problem is the adaptive notch filter, a notch filter whose center frequency varies online to track the modal frequencies of the system. The adaptive notch filter has been studied in signal processing research [87, 78, 20] as well as various applications, such as the hard disk drive (HDD) [69], launch vehicles [21], aircraft [61], and space structures [54]. The adaptive notch filter presented in [21] is used on the model of a booster from the Advanced Launch System (ALS) program. The least squares estimator in the publica- tion uses a simple undamped resonator as the model for estimation and functions well since the resonant mode is very pronounced. However in other applications, full plant 155 parameterizations is necessary as the flexible mode may not be as significant. Another strategy for the estimation of the center frequency can be found in [69], where frequency weighting functions are used. The downside is there are several failure modes that are known and avoidance requires some modal information a priori. A stochastic state space algorithm for mode frequency estimation is presented in [61]; however it relies on the in- jection of a probe signal which is not needed in the scheme presented here. The indirect adaptive compensation (IAC) scheme in [94] also requires a probe signal to complete the estimation. The adaptive mode suppression scheme in [54] uses a LMS algorithm to update filter coefficients and then the modal parameters are extracted from the filter. This is opposite as to what is being presented in this paper, where the modal parameters are first estimated and then used in the adaptive notch filter. This paper will present a control scheme which makes use of an adaptive notch filter. The adaptive notch filter is designed to suppress the modal dynamics of the system while working in harmony with another controller designed for the rigid system, that is the sys- tem without the flexible modes. This second controller, called the rigid-body controller, can be designed using a variety of methods since the elastic dynamics can be, for the most part, neglected in the design. In this paper we will present a design requirement, on the interaction of the rigid-body control and adaptive notch filter, which ensures stability of the closed loop adaptive system. Since the design of each component can be done in a more separated method it may be useful for systems where the modes are not precisely known early in the control design process or for systems where the modal parameters 156 vary between production units. In this paper, unlike the pole placement design in [50], the rigid-body controller will be designed using classical loop shaping techniques. How- ever the complete stability analysis in [50] can be used for the designs within this paper. Other methods of rigid-body control with an adaptive notch filter of this type are a LQ controller for aircraft control in [48] and a classical phase lead design for a HDD in [47, 45]. The design procedure for the adaptive notch filter scheme is given to serve as an example for other future designs. To showcase the ability of the adaptive notch filter to work in a real-time control system the laser-beam pointing experiment of [70] will be used. In this experiment, the plant contains a single lightly damped complex pole. This type of system is prevalent in mechanical systems which, by design, have a flat re- sponse below the bandwidth of the elastic components. The adaptive notch filter enables a higher bandwidth control system with better performance, in terms of disturbance re- jection capabilities. The general adaptive notch filter control scheme is given in Section 6.2.2. The control design for the single complex pole is presented in Section 6.2.3 and then simulations and experiments of the design are discussed in Section 6.2.4. Finally conclusions are drawn in Section 6.2.5. 6.2.2 General Adaptive Notch Filter The setup for this problem is a rigid plant with a single unknown lightly damped flexible mode which only contains a single pair of complex poles. A controller is designed to 157 achieve good performance in the presence of disturbances, which means shaping the sensitivity and complementary sensitivity functions appropriately. The control objective may include tracking a certain class of reference signal y m 2 L 1 by using the internal model principle. So, we design the controller to include Q m (s), which is an internal model ofy m and is a known monic polynomial of degreeq with all roots in<[s]·0 and with no repeated roots on thej!-axis. The plant takes the form y p =G p (s)M(s)u p +d= Z p (s) R p (s) Z m (s) R m (s) u p +d; (6.55) where Z m (s) R m (s) is a mode of the plant, Zp(s) R p (s) is the non-modal part of the plant, and d is a bounded output disturbance. The flexible part of the plant takes the form Z m (s) R m (s) = ! 2 d s 2 +2³! d s+! 2 d (6.56) where ³ > 0 is the damping and ! d > 0 is the natural frequency of the mode. It is assumed that the order of R p (s) is n, and since we are concerned with suppressing the flexible modes, the non-modal part of the plantG p (s) = Z p (s) Rp(s) must be stabilizable so a rigid-body controller, later denoted asC(s), can be designed for the rigid system. 6.2.2.1 Known parameter case The control scheme includes a narrow adaptive notch filter centered at the natural fre- quency of the flexible pole in (6.55), and a compensator designed for G p (s) using any 158 ) (s G p ) (s M ) (s C ) (s F p u m y p y d Figure 6.19: Feedback system diagram for the mode suppression schemes. design technique while, for the most part, completely neglecting the flexible dynamics. Here we design a controller which includes an internal modelQ m (s), however this is not necessary for the adaptive notch filter to function properly. We pose the control problem in this format for clarity of presentation, the rigid-body control design is not the main topic of importance in this paper. The control loop is seen in Fig. 6.19 and input u p =¡F(s)C(s)(y p ¡y m ) (6.57) C(s)= P(s) Q m (s)L(s) (6.58) F(s)= Z f (s) R f (s) : (6.59) We assume that the rigid-body controllerC(s) is proper and realizable, and is designed such that the polynomial equation L(s)Q m (s)R p (s)+P(s)Z p (s)=A(s); (6.60) 159 gives a HurwitzA(s), which are the desired closed loop poles when the flexible dynamics and notch filter are neglected. The filterF(s) in (6.57) is Z f (s) R f (s) = s 2 +2³ z ! d s+! 2 d s 2 +2³ r ! d s+! 2 d (6.61) where ! d is the same as in (6.56), ³ z ;³ r > 0 and ³ z < ³ r . We would like to design the filter to fully suppress the flexible mode at the resonant frequency, this gives us a condition that must be met: ¯ ¯ ¯ ¯ ! 2 d ! 2 n Z m (j!)Z f (j!) R m (j!)R f (j!) ¯ ¯ ¯ ¯ ·k m (6.62) wherek m is the desired margin. Now treating the modes and notch filter as uncertainty, and ignoring the disturbance, the system can be put into the form of Fig. 6.20, whose characteristic equation is 1+C(s)G p (s)(1+¢(s))=0 (6.63) which, due to the stable roots of1+C(s)G p (s)=0, implies 1+ C(s)G p (s) 1+C(s)G p (s) ¢(s)=0: (6.64) 160 For stability of the system the following must be satisfied ° ° ° ° P(s)Z p (s) A(s) ¢(s) ° ° ° ° 1 <1 (6.65) ¢(s)=F(s)M(s)¡1: (6.66) The above comes from neglecting the flexible dynamics and notch filter and treating them as an uncertainty ¢(s) and then applying the small gain theorem. We use this form of the uncertainty in (6.66) for ease of the stability proof which is presented in [50]. It can be shown that the tracking errore 1 =y p ¡y m is e 1 = ¡ L(s)R p (s) Qm(s)L(s)Rp(s)+P(s)Zp(s)(1+¢(s)) Q m y m + L(s)Rp(s) Q m (s)L(s)R p (s)+P(s)Z p (s)(1+¢(s)) Q m d (6.67) which is a proper stable transfer function since (6.65) is satisfied. Now we have e 1 = ¡ L(s)Rp(s) Q m (s)L(s)R p (s)+P(s)Z p (s)(1+¢(s)) Q m d +" t (6.68) where" t is a term exponentially decaying to zero. Therefore the control law will cause e 1 to converge exponentially to the set D e =fe 1 jke 1 k·cd 0 g; (6.69) 161 ) (s G p ) (s C m y p y d ) (s ' Figure 6.20: Feedback system diagram with the notch filter and mode expressed as an uncertainty¢(s). whered 0 is an upper bound forjdj andc > 0 is a constant. It should be noted this result is for the system when all the parameters are known and the requirement in (6.65) is met. 6.2.2.2 Estimation of plant parameters The adaptive mode suppression scheme that is used when the flexible dynamics are un- certain or changing will now be designed. Starting with the system in (6.55) we have y p = Z p (s) R p (s) Z ¤ m (s) R ¤ m (s) u p +d; (6.70) where Z ¤ m (s) R ¤ m (s) is the unknown mode of the plant Z ¤ m (s) R ¤ m (s) = ! ¤2 d s 2 +2³ ¤ ! ¤ d s+! ¤2 d (6.71) and Z p (s) R p (s) is the known part of the plant. Z p (s);R p (s);Z ¤ m (s); andR ¤ m (s) follow the all the same assumptions made in the known parameter case. The polynomials denoted with the star are polynomials whose coefficients are the actual values of the real system, which 162 are treated as unknown. Similarly, the parameters with a star are the actual parameters of the system. The parametric model to estimate the unknown modal frequency is as follows: z =µ ¤T Á+´; (6.72) where´ is the used to represent the disturbance where ´ = R p (s)R ¤ m (s) ¤ p (s) d2L 1 (6.73) z = s 2 R p (s) ¤ p (s) y p (6.74) Á= · ¡sRp(s) ¤ p (s) y p Zp(s) ¤ p (s) u p ¡ Rp(s) ¤ p (s) y p ¸ T (6.75) µ ¤ = · 2³ ¤ ! ¤ d ! ¤2 d ¸ T (6.76) and ¤ p (s) is a monic Hurwitz polynomial of degree n + 2. The parametric model in (6.72) is achieved by taking (6.70) and (6.71), multiplying by a common denominator, collecting unknown terms, and then making proper transfer functions by dividing by a Hurwitz polynomial ¤ p (s). The creation of this type of parametric model is well docu- mented in [36] and the exact representation of the flexible modes in this form allows for estimation of the unknown modal parameters. 163 Our goal is to estimate the modal frequency and damping. A wide class of adaptive laws can be used to estimate the unknown parameters, but we adopt the gradient algo- rithm with parameter projection and a deadzone. Allow Á = [Á 1 ;Á 2 ] T ;µ = [µ 1 ;µ 2 ] T , and also some a priori known bounds on the damping and natural frequency such that 1 ¸ ³ u ¸ ³ ¤ ¸ ³ l > 0 and ! u d ¸ ! ¤ d ¸ ! l d > 0 are satisfied. These bounds are used for projection and a deadzone modification is added to ensure robust adaptation in the presence of the bounded disturbance. The update equations are _ µ 1 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 1 ("+g)Á 1 if(2³ u ! u d >µ 1 >2³ l ! l d ) or(µ 1 =2³ l ! l and"Á 1 ¸0) or(µ 1 =2³ u ! u and"Á 1 ·0); 0 otherwise (6.77) _ µ 2 = 8 > > > > > > > > > > < > > > > > > > > > > : ° 2 ("+g)Á 2 if((! u d ) 2 >µ 2 >(! l d ) 2 ) or(µ 2 =(! l ) 2 and"Á 2 ¸0) or(µ 2 =(! u ) 2 and"Á 2 ·0); 0 otherwise (6.78) "= z¡µ T Á m 2 s (6.79) m 2 s =1+Á T Á: (6.80) 164 g = 8 > > < > > : 0 ifj"m s j>g 0 ; ¡" ifj"m s j·g 0 (6.81) In the update equations, the bounds ³ l ;³ u ;! l d ;! l d are constants determined a priori, and ° 1 ;° 2 ;g 0 > 0 are also design parameters chosen a priori. The deadzone will ensure that adaptation stops when the estimation error is below the level of the disturbance, so that only good information is used to update the parameters. The above estimation law guarantees that (i) µ2L 1 (ii) ";"m s ; _ µ2S(g 0 + ´ 2 m 2 s ) (iii) _ µ2L 1 T L 2 (iv) lim t!1 µ(t)= ¹ µ; where ¹ µ is a constant vector. 6.2.2.3 Adaptive control law The adaptive control law is formed by replacing the notch filter in (6.57), which has the form of (6.61), with an adaptive notch filter. The online estimates used in the adaptive notch filter come from the online estimator and are µ = · 2 ^ ³ d ^ ! d ^ ! 2 d : ¸ T : (6.82) 165 The adaptive control law becomes u p =¡ ^ Z f (s) ^ R f (s) P(s) L(s)Q m (s) (y p ¡y m ): (6.83) In the above control law, the notch filter ^ Z f (s) ^ R f (s) is designed to cancel the unknown mode of the plant Z ¤ m (s) R ¤ m (s) . By denoting the polynomials with a hat we are saying the coefficients are time-varying estimates which come from the online estimator. This is done by using the estimate of the modal frequency as the center frequency thereby making it an adaptive notch filter. The filter becomes ^ Z f (s) ^ R f (s) = s 2 +2³ z ^ ! d s+ ^ ! 2 d s 2 +2³ r ^ ! d s+ ^ ! 2 d (6.84) where ^ ! d is the estimate of the modal frequency and the damping ratios are set a priori using (6.62) as a reference. We also must make sure that (6.65) is satisfied at every frozen timet, which leads to ° ° ° ° P(s)Z p (s) A ¤ (s) ¢¢(s;t) ° ° ° ° 1 <1 (6.85) ¢(s;t)= ^ F(s;t)¢ ^ M(s;t)¡1: (6.86) By frozen time we mean that the time-varying coefficients of the polynomials are treated as constants when two polynomials are multiplied. Therefore the controller C(s) must be designed such that (6.85) is always satisfied. This implies a priori knowledge of the 166 bounds on the unknown parameters which leads to a convex setµ2S, that the estimator must use for projection. These bounds, through the updating of the parameters in the adaptive notch filter, create a convex set of possible ¢(s;t) which we use to obtain a weight W(s) used for control design. Now, denote ¹ F(s;µ); ¹ M(s;µ) as the frozen time versions of the transfer functions ^ F(s;t); ^ M(s;t). That is to say, the overbar versions have estimated parameters that come from the setS but are frozen in time, and therefore LTI systems. We have l(!)=max µ2S ¯ ¯¹ F(j!;µ) ¹ M(j!;µ)¡1 ¯ ¯ (6.87) and a rational transfer function weight jW(j!)j¸l(!); 8!: (6.88) This weight can be substituted in (6.85) to acquire the LTI stability requirement as ° ° ° ° C(s)G p (s) 1+C(s)G p (s) W(s) ° ° ° ° = ° ° ° ° P(s)Z p (s) A ¤ (s) W(s) ° ° ° ° 1 <1: (6.89) This requirement for stability can be achieved offline from knowledge of the parameter bounds and the adaptive notch filter design. This stability requirement is used in the analytical proof presented in [50] which proves boundedness of the parameters as well as convergence of the error signal. 167 -60 -40 -20 0 20 40 60 Magnitude (dB) 10 1 10 2 10 3 0 90 180 270 360 Phase (deg) Bode Diagram Frequency (Hz) Figure 6.21: Bode plot of the open loop plant. A single decoupled axis of the FSM experimental setup. 6.2.3 Control design The adaptive notch filter scheme will now be designed for a system with a single complex pole, which is representative of mechanical systems with a flat frequency response up to the frequency of a lightly damped elastic mode. This type of system is similar to that of the MEMS fast steering mirror (FSM) which will be used as the actuator in the simulations and experiments described later. For this system the rigid part of the plant G p (s) is unity and the modal part is as in (6.71) leading to a system with a bode plot seen in Fig. 6.21. For this section we will allow the control design to be done in the Laplace domain and in the next section the controllers will be discretized for implementation on a digital computer. 168 Plant Dynamics r(t) Adaptive Notch Filter y(t) Rigid-body Controller Robust Online Estimator Figure 6.22: Closed loop diagram for the experimental setup. For the non-adaptive scheme the adaptive notch filter is replaced by a fixed notch filter and the online esti- mator is removed. Herer(t) is the reference signal which is equal to zero andy(t) is the measured output. Some fictitious stability and performance requirements are created to show the benefit of the adaptive mode suppression scheme. Since we will be desiring perfect tracking and elimination of disturbances, our performance metric will be the standard deviation of the tracking error. To accomplish this, requirements are levied on the closed loop sensitivity function which are a magnitude of at most -55 dB at 1 Hz and a maximum magnitude of 12 dB. Now a requirement will be added to limit the bandwidth of the closed loop system, this may be necessary for a variety of reasons. In a real implemented system used for commercial use, the sampling rate of the feedback error signal may be only slightly faster than the flexible dynamics and high frequency noise may be present, so a limited bandwidth would be desired. These factors contribute to the requirement of limiting the closed loop complementary sensitivity to at most -60 dB at 500 Hz with an overall maximum value of 12 dB. With these requirements in place, two controllers will be designed and the closed loop diagram is in Fig. 6.22. A control scheme utilizing an adaptive notch filter (ANF) 169 and a scheme with a fixed non-adaptive notch filter (NA) will be created. The non- adaptive scheme will utilize a wide notch filter to account for variations in the flexible mode frequency of up to 5% and variations of the damping of up to 5%. This wider notch filter will add phase lag at the lower frequencies and limit the performance of the system. However the adaptive scheme will have a much narrower notch filter, adding less phase lag and thereby allowing for better disturbance rejection. Both of the notch filters are displayed in Fig. 6.23. The frozen time adaptive notch filter is ¹ F A (s;µ))= s 2 +2(9:6£10 ¡4 ) p µ 2 s+µ 2 s 2 +2(0:38) p µ 2 s+µ 2 (6.90) and the non-adaptive notch filter is F NA (s)= s 2 +2(1:0£10 ¡3 )(799:64)s+(799:64) 2 s 2 +2(1:0)(799:64)s+(799:64) 2 : (6.91) To graphically visualize the benefit of the narrower adaptive notch filter we will go back to the stability requirement where the notch filter and flexible mode are treated as uncertainties. If we allow the the unknown parameters to come from the setS, which is created from the a priori bounds on the unknown parameters, we can devise a covering function for both the adaptive and non-adaptive cases. First for the adaptive case we will create the filterW A (j!) by using (6.87) and (6.88) and the bounds specified by the 5% variation in natural frequency and damping. For this we use the adaptive notch filter 170 -70 -60 -50 -40 -30 -20 -10 0 Magnitude (dB) 10 1 10 2 10 3 -90 -45 0 45 90 Phase (deg) Bode Diagram Frequency (Hz) ANF NA Figure 6.23: Bode plot of the notch filters. The narrower adaptive notch filter (ANF) adds less phase lag than the non-adaptive notch filter (NA). The ANF is one where the center frequency is frozen at the same value as the non-adaptive notch filter, therefore it can be treated as LTI. given in (6.90), which is treated as a frozen time LTI system whenµ is constant, and the mode given by ¹ M(s;µ)= µ 2 s 2 +µ 1 s+µ 2 : (6.92) For the non-adaptive notch filter case we will construct the filterW NA (j!) by using l NA (!)=max µ2S ¯ ¯ F NA (j!;µ) ¹ M(j!;µ)¡1 ¯ ¯ (6.93) and then jW NA (j!)j¸l NA (!); 8!: (6.94) 171 The stability criteria in (6.89) can now be used in the form of jT(j!)j< ¯ ¯ ¯ ¯ 1 W(j!) ¯ ¯ ¯ ¯ ; 8! (6.95) where T(s) is the complementary sensitivity function. Plotting the bode plots of the inverse of the filters W A (s) and W NA in Fig. 6.24 shows how the adaptive system can allow an increase in bandwidth as well as disturbance rejection. It should also be noted that the adaptive notch filter can deal with a much larger variation in damping since the center frequency will track the flexible mode and suppress the mode. However the suppression capabilities of the fixed non-adaptive notch filter decrease exponentially as the actual modal frequency of the plant is displaced from the center frequency of the notch filter. The rigid body controller for the two cases are slightly different, the adaptive scheme has a slightly higher gain and faster zeros allowing for an increase in performance while still meeting the requirements, which is permitted due to the narrower adaptive notch filter. Both rigid body controllers are designed to maximize performance for the given notch filters in their respective schemes. The rigid controllers are C ANF (s)= 129609:0(s+75:66) 2 s(s+635:8)(s+10:87)(s+9:86) (6.96) C NA (s)= 107090:6(s+61:44) 2 s(s+635:8)(s+10:87)(s+9:86) ; (6.97) 172 -20 0 20 40 60 From: Output To: Out(1) Magnitude (dB) 10 0 10 1 10 2 10 3 45 90 135 180 225 Phase (deg) Bode Diagram Frequency (Hz) ANF NA Figure 6.24: Bode plot of inverse of the filters W A (s) and W NA . The narrower adap- tive notch filter (ANF) has a looser constraint on the rigid body control design, when compared to the non-adaptive notch filter (NA). whereC ANF (s) is the rigid controller for the adaptive notch filter scheme andC NA (s) is the rigid controller for the non-adaptive scheme. The sensitivity functions of each scheme is seen in Fig. 6.25 where the disturbance rejection capability of the adaptive scheme is clear. Since the adaptive notch filter is narrower than the non-adaptive coun- terpart, the rigid controller can be designed more aggressively causing the lower mag- nitude of the sensitivity function. The closed loop bandwidth of the adaptive scheme is also slightly higher at 65 Hz as opposed to that of the non-adaptive scheme which has a bandwidth of 51 Hz, although both schemes meet the sensitivity and complementary 173 -100 -80 -60 -40 -20 0 20 Magnitude (dB) 10 -1 10 0 10 1 10 2 -45 0 45 90 135 180 225 Phase (deg) Bode Diagram Frequency (Hz) ANF NA Figure 6.25: Bode plot of the closed loop sensitivity functions. The narrower adaptive notch filter (ANF) allows for better disturbance rejection than the non-adaptive notch filter (NA) due to the increase in bandwidth of the rigid body controller. The ANF can be treated as LTI since we fix the value for the center frequency to be the same value used in the non-adaptive notch filter, which is the nominal plant modal frequency. sensitivity function requirements that were imposed. With the controllers and notch fil- ters designed, using the bounds for the unknown parameters, we can check the stability condition of (6.89) using theW A (j!) andW NA (j!) covering filters created earlier. The online estimator is designed for the adaptive scheme in the same way presented in the previous section. The¤ p (s) filter is designed to maximize the signal to noise content of the estimator, this can be done by designing bandpass filters in regions where the flexible modal frequency is thought to occur. The deadzone modification is the estimator will turn off estimation when the estimation error becomes less than some designated 174 1 2 3 4 Figure 6.26: Photograph of the laser beam system preset design value. This is necessary due to the disturbance signal, which will cause the estimates to drift based on incorrect estimation information. 6.2.4 Simulations and Experiments The simulations presented in this section use the dynamical models of the laser-beam system shown in Fig. 6.26, where the plant displays a lightly damped flexible mode. The details of the experimental setup in Fig. 6.26 are described in [70], however, a brief overview is given here. As shown in the photograph in Fig. 6.26, a laser beam leaves the source at position 1 °, reflects off the fast steering mirror FSM-C at position 2 °, then reflects off the fast steering mirror FSM-D at position 3 ° and finally reaches the optical position sensor at position 4 °. Two lenses in the optical path focus the beam on FSM-D and the sensor. The mirrors FSM-C and FSM-D are identical Texas Instruments (TI) MEMS mirrors used in laser communications for commercial and defense applications. FSM-C is the control actuator, and FSM-D is used to add disturbance. 175 The open-loop discrete-time plant of the system is the transfer function that maps the two-channel digital control command to the sampled two-channel output of the optical position sensor. Thus open-loop plant of the system is the two-input/two-output digital transfer function for the lightly damped fast steering mirror FSM-C with a gain deter- mined by the optical position sensor and the laser path length. Output channels 1 and 2 represent horizontal and vertical displacements, respectively, of the beam; input chan- nels 1 and 2 represent commands that drive FSM-C about its vertical and horizontal axes, respectively. As shown in [70], the two channels of the system can be decoupled, creating two separate SISO systems. Only Channel 1 will be used in the simulations considered here, because the results of this paper pertain to a single SISO system. A model of Channel 1, identified with a sample-and-hold rate of 5 kHz, is shown in Fig. 6.21, where the lightly damped flexible mode at 127 Hz can easily be observed. Notice, that the magnitude of the frequency response of Channel 1 at 0 Hz is 0 dB. This is due to the fact that the transfer function of Channel 1 has been scaled, so that, changes in the distances between the components in optical path in Fig. 6.26 do not change the models of the system. A slightly different configuration to the one in Fig. 6.26, but with the same mirrors FSM-C and FSC-D (i.e., essentially the same dynamics), is used for the experiments to be described in Subsection 6.2.4.2. There, the effectiveness of the proposed adaptive mode suppression scheme is demonstrated. In this case, the closed-loop system is run with a sampling-and-hold rate of 5 kHz. Although the sampling frequency of the system 176 is 5 kHz, a commercial application of the FSM may have a sampling frequency that is significantly smaller, due to time associated with the response of the detector, processing of the detector signal, or calculation of the position error. Since the controllers, notch filters, and estimator are all designed in the continuous- time domain, they must be discretized for implementation on the real-time system. This is done using the bilinear transformation on both rigid controllers as well as the estimator filters, however the non-adaptive notch filter is discretized using the matched pole-zero technique [23]. The adaptive notch filter is converted to a digital notch filter for use in the real-time system by the following method. Allow ! f =tan µ ^ ! d t s 2 ¶ ; (6.98) wheret s is the sampling time. Then calculate the following values a 0 =1+2³ r ! f +! 2 f (6.99) a 1 =2! 2 f ¡2 (6.100) a 2 =1¡2³ r ! f +! 2 f (6.101) b 0 =1+2³ z ! f +! 2 f (6.102) b 1 =2! 2 f ¡2 (6.103) b 2 =1¡2³ z ! f +! 2 f (6.104) 177 and the digital notch filter becomes ^ F(z)= µ a 0 +a 1 +a 2 b 0 +b 1 +b 2 ¶ b 0 z 2 +b 1 z+b 2 a 0 z 2 +a 1 z+a 2 : (6.105) This filter can be implemented in a real-time system in a number of ways. Here the filter is placed in a canonical state space form and the states are updated using the standard discrete state space equations. 6.2.4.1 Simulations A series of simulations are first completed using Matlab and Simulink, where the goal is to test the adaptive notch filter scheme before running real-time experiments. The distur- bance that will be generated with the second FSM-D in the experiment is incorporated into the simulation. For the simulations the reference signal is a slowly varying 2 Hz sinusoid with an amplitude of 10 V . The adaptive mode suppression scheme is compared to the non-adaptive control scheme that is described previously, for both simulations the flexible modal frequency is thought to be at 93% of the nominal value, therefore the notch filters are centered at this incorrect frequency. At this amount of variation in modal frequency, the non-adaptive scheme will be unstable, as the notch filter is only meant to suppress a variation of up to 5%. The time series plots of the output and tracking error are displayed in Fig. 6.27, where the system begins to grow unstable as the flexible mode is not adequately suppressed. However, the adaptive scheme’s output and tracking error are seen in Fig. 6.28. Here the system displays a large tracking error initially as the notch 178 0 1 2 3 4 5 -10 -5 0 5 10 Time (s) Position from center (V) 0 1 2 3 4 5 -1 0 1 Tracking error (V) Time (s) Figure 6.27: Simulation results for the non-adaptive scheme when the notch filter is placed at 93% of the plant’s modal frequency. Top plot: Position outputy p . Bottom plot: Tracking error. (ANF). center frequency is incorrect, but as adaptation occurs the notch filter is able to suppress the mode thereby retaining performance and stability. The estimated modal frequency and estimation error are seen in Fig. 6.29. 6.2.4.2 Experiments The experiments are conducted on the same day and run several times for further veri- fication of the results. In Fig. 6.30 the time series of the error signal when the adaptive scheme is run is displayed. For the first 5 seconds the system is open loop, so only the disturbance created by FSM-D is seen in the error signal. The initial estimate of the plant’s modal frequency is assumed to be at 95% of the actual value, thereby placing 179 0 1 2 3 4 5 -10 -5 0 5 10 Time (s) Position from center (V) 0 1 2 3 4 5 -2 -1 0 1 2 Tracking error (V) Time (s) Figure 6.28: Simulation results for the adaptive scheme when the initial value of the estimated modal frequency ^ ! d is placed at 93% of the plant’s modal frequency. Top plot: Position outputy p . Bottom plot: Tracking error. (ANF). 0 1 2 3 4 5 -50 0 50 Time (s) Estimation error 0 1 2 3 4 5 115 120 125 130 Estimated frequency (Hz) Time (s) Figure 6.29: Simulation results for the adaptive scheme. the actual modal frequency is 127.26 Hz. Top plot: Estimation error. Bottom plot: Estimated modal frequency. 180 0 1 2 3 4 5 6 7 8 9 10 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Time (s) Error (V) Figure 6.30: Time series from the FSM experiment, the adaptive notch filter is initially placed at 95% of the actual modal frequency of the plant, however the adaptive notch filter will update the center frequency online. The control loop is closed at 5 seconds. the adaptive notch filter in the incorrect location. This variation could be due to envi- ronmental effects, variations between FSMs, or degradation over time. Since the notch filter center frequency is incorrect, the flexible mode is excited and creates a large error shortly after the loop is closed. Adaptation then occurs, the notch filter tracks the modal frequency, the flexible mode is suppressed, and the error signal attenuates. As compared to a non-adaptive scheme, when the non-adaptive notch filter is placed perfectly (ie. the plant model is correct and exactly known), the adaptive scheme decreases the standard deviation of the error signal by 14%. This increase in performance is due to the more aggressive rigid-body controller associated with the narrow adaptive notch filter. 181 10 0 10 1 10 2 -80 -60 -40 -20 Frequency (Hz) Power/Frequency (dB/Hz) 10 0 10 1 10 2 -80 -60 -40 -20 Frequency (Hz) Power/Frequency (dB/Hz) Non-adaptive NF ANF Figure 6.31: PSDs computed from the error signal of the FSM experiment. The data used is collected from the 10 second mark until the 25 second mark. Top plot: Open loop system, only disturbance. Bottom plot: Non-adaptive scheme and adaptive notch filter (ANF). The power spectral densities (PSDs) of error signal from the open loop system, adap- tive scheme after adaptation, and non-adaptive scheme are presented in Fig. 6.31. The bottom plot displays the rejection of the disturbance with the adaptive scheme. Fig. 6.32 shows the estimation error and estimated modal frequency for the adaptive scheme. The plots are time series which begin at 5 seconds into the experiment, which is the time at which the loop is closed and the adaptive control scheme is turned on. Initially the esti- mated modal frequency, and therefore notch center frequency, is incorrect. This causes the system to be unstable and the tracking error to grow large, which in turn causes the estimation error to grow larger than the disturbance level. This level of excitation is suffi- cient enough to cause the online estimator to begin adapting the parameters online. Once 182 5 5.2 5.4 5.6 5.8 6 -100 0 100 200 Time (s) Estimation error 5 5.2 5.4 5.6 5.8 6 120 125 130 135 Estimated frequency (Hz) Time (s) Figure 6.32: Data from the adaptive scheme when the loop is closed at 5 seconds, the actual modal frequency is 127.26 Hz. Top plot: Estimation error. Bottom plot: Estimated modal frequency. the level of the error decreases below the deadzone threshold the estimation is halted and the parameters remain constant. It should also be noted that the estimated frequency does not exactly converge to the real plant modal frequency, but this is acceptable since the system becomes stable and the performance is improved. The adaptive mode suppression scheme does not guarantee that ¹ µ =µ ¤ . Another case of the non-adaptive scheme is run, but this time the non-adaptive notch filter’s center frequency is displaced. The center frequency is set at 97% of the plant mode frequency and the system remains stable, however the standard deviation of the tracking error is increased by 10%. This is due to the lightly damped mode being excited by the control system which can be seen in Fig. 6.33. The plot shows the PSDs of the 183 10 2 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 Frequency (Hz) Power/Frequency (dB/Hz) Non-adaptive incorrect Non-adaptive Figure 6.33: PSDs computed from experimental data comparing a non-adaptive notch filter with an incorrect center frequency (97% of nominal) and one which is correct. two cases, where the plant mode at 127 Hz can be seen as a spike in the plot due to the incorrectly placed non-adaptive notch filter. These experimental results show that the adaptive scheme is able to provide better performance than the non-adaptive version even when the non-adaptive notch filter is centered exactly on the flexible mode. With the center frequency slightly perturbed the non-adaptive scheme remains stable, however there is a degradation in tracking error. The adaptive scheme can track and adjust for such an incorrect plant model, only after estimation error has sufficient information for adaptation to occur. 184 6.2.5 Discussion This paper presented an adaptive mode suppression scheme which incorporates an adap- tive notch filter. An estimator using plant parameterization is used to track the modal frequency of the flexible dynamics of the plant. This frequency estimate is then used to update the center frequency of the adaptive notch filter. Since the adaptive notch filter will track the flexible mode, it can be designed narrower, which will allow for an in- crease in bandwidth of the closed loop system. The adaptive scheme is compared to a non-adaptive scheme empirically through the use of a laser beam pointing system. The experimental setup displays a plant with a lightly damped flexible mode near the desired closed loop bandwidth. Starting the incorrect parameters, the estimator of the adaptive scheme is able to track the modal frequency of the plant in real-time and cause the adap- tive notch filter to suppress the flexible mode. The benefit of the narrow adaptive notch filter is seen in through the improved tracking performance of the laser beam system, however the actual control design for a different system will vary. This experiment is meant to be a single example of how the adaptive mode suppression scheme may be designed an implemented as a real-time control system. 185 6.3 Experimental Results of a Neural-Network based Adap- tive Disturbance Rejection Scheme for Disk Drives 6.3.1 Introduction Hard disk drives (HDD) are a form of data storage that are present in just about every computer system. As the storage capacity grows so does the track density which puts tighter constraints on the servo control system. With tracks placed closer together in the radial direction there is a need to increase the positioning accuracy. There has been a large amount of research activity into two types of control problems: track-seeking and track-following [13]. The former deals with motion control of the head between tracks, and the latter with maintaining the head on the center of the HDD track. This paper deals with track-following which can be formulated as a disturbance rejection problem [33, 72, 42]. The disturbance can be separated into repeatable runout (RRO) and non-repeatable runout (NRRO). The RRO is produced by imperfections and eccentricities on the tracks, while NRRO is produced by aggregated effects of disk drive vibrations, imperfections in the ball-bearings, and electrical noise. Research has been conducted over the years to cancel the effects of these disturbances and acquire better track following capabilities [93, 79, 72, 71, 33, 42, 94, 52, 95, 39, 62]. It has been shown that the RRO can be suppressed with adaptive feedforward meth- ods [79, 80], and by repetitive control [71]. Both of these methods use the existing knowledge of the frequencies at which the RRO disturbance occurs to suppress its effect 186 and obtain better tracking. The feedforward method employs the injection of the negative of an estimated sinusoidal disturbance model. The amplitude and phase of the sinusoidal disturbance are estimated online through a gradient update algorithm. An internal model is used to synthesize a linear time invariant controller that rejects sinusoidal disturbances of known frequencies in the repetitive control method. The contribution of this paper is the experimental verification of a scheme that makes use of either the adaptive feedforward disturbance rejection scheme or the repetitive con- troller to eliminate the RRO in combination with another adaptive neural disturbance rejection scheme that focuses on reducing the tracking error even further. There has been work done in using neural networks for feedforward disturbance rejection [55, 74, 66]. Rejection of the disturbance torque for missile seekers using neural networks is presented in [55]. A multilayer neural network uses the measurable load disturbance to cancel to cancel the degrading effects through a feedforward controller. The conventional HDD does not have the luxury of extra sensors for the disturbance, so the only possible input to a neural network is the position error signal. In [74], a general approach with sim- ulations show the benefit of adapting a dynamic neural network to cancel an unknown disturbance. Here the idea of passing the disturbance estimate through an estimated plant inverse is used. The methods proposed in [66] use multilayer neutral networks to model disturbances as outputs of dynamical systems and then expands the plant model to try to reduce the adverse effects. 187 Radial basis functions (RBF) have been used to model sea-clutter noise in radar ap- plications [31]. The approach uses RBFs to remove the noise from radar signal data, since the clutter noise has been shown to be chaotic, thus providing the ability to detect small targets in the clutter. Training data is used to adapt the neural parameters before being implemented on actual test data. A similar approach was taken here except we add an extra term in the neural model to account for extra delays and there is no training set of data; the adaptive neural disturbance rejector is both adapted and implemented in real-time. Also the adaptation of the neural parameters uses a deadzone modification not present in [31], which allows adaptation to cease once the performance begins to de- grade. The neural modeled disturbance rejection is added to either RRO rejection scheme to obtain better performance than the RRO scheme alone. This increase in performance is due to the ability of the adaptive neural network to provide a model for a dynamic nonlinear disturbance which is the residual from the RRO rejection schemes. The paper is organized as follows. In section 6.3.2 the experiment and real-time im- plementation issues are explained. Section 6.3.3 describes the baseline control to which the disturbance rejection schemes are added. These adaptive feedfoward disturbance re- jectors are explained in section 6.3.4. Section 6.3.5 presents experimental results and conclusions are drawn in section 6.3.6. 188 6.3.2 Description of the Experiment A HDD is a mechatronic device that uses rotating platters to store data. Information is recorded on, and read from concentric cylinders or tracks by read-write magnetic trans- ducers called heads, that fly over the magnetic surfaces of the HDD platters. The position of the heads over the platters is changed by an actuator that consists of a coil attached to a link, which pivots about a ball bearing. This actuator connects to the head by a steel leaf called a suspension [2], [63]. This description of the HDD is shown in Fig. 6.34. The control objective is to position the center of the head over the center of a data track. Thus, the typical measure of HDD tracking performance is the deviation of the center of the head from the center of a given track, which is often called track misregis- tration (TMR) [63]. There exist many indexes used to quantify TMR. Here we adopt TMR =3¾: (6.106) Where ¾ is the empirical standard deviation (STD) of the control error signal. It is common to express 3¾ as a percentage of the track pitch [42, 63], which must be less than 10% in order to be considered acceptable. TMR values larger than this figure will produce excessive errors during the reading and recording processes. The experiment was performed with a 2-platter (10 GB/platter), 4-head, 7200 rpm, commercial HDD, and aMathworks r xPC Target system for control. The sample-hold rate of9:36 KHz, used for communication, control and filtering, is internally determined 189 7200 rpm Tracks or Cylinders Sector Head Suspension Actuator Figure 6.34: Schematic idealization of the hard disk drive (HDD) system. by the HDD and transmitted through a clock signal to the target PC used for control. Both systems must operate in a synchronized manner, as shown in the diagram of the experiment (Fig. 6.35). The position of a given HDD head is digitally transmitted by the use of two signals. The first conveys the track number (TN) over where the head is positioned. The second is the position error signal (PES), which conveys the position of the head on the track pitch. Thus, the measured positiony is a function of both the TN and PES signals. The loop is closed when the digital controller outputs the sequence x which is con- verted into an analog signal to command the HDD actuator. At this stage, we pose the control problem in the discrete-time domain, defining the mapping from x to y as the open-loop plantP . 190 y =f(TN;PES) ¾ Digital Controller ? D/A - Digital Reader Digital Reader ¾ ¾ HDD - - - Clock Reader x y DSP (xPC Target System) clock TN PES Figure 6.35: Diagram of the experiment. 6.3.3 Baseline Control The work in this paper utilizes controllers that were previously developed and imple- mented in [72] as baseline controllers for adding the disturbance rejection schemes. This includes a simple LTI controller and a controller which is tuned using the inverse QR- RLS algorithm, both will be described briefly. 6.3.3.1 Controller Design An open-loop model of the HDD, ^ P , is first found by the method described in [72]. A simple LTI feedback controller C shown in Fig. 6.36 was designed using discrete-time domain classical techniques. It consists of a digital integrator and a digital notch filter. The integrator gain and notch parameters were tuned to maximize the output-disturbance rejection bandwidth. 191 P(z) -h - ? ? C(z) ¾ ? h - U(z) 6 ¾ h ¾ ¾ - u y y ref ~ y d o G(z) G 1 (z) ¡ ¡ Figure 6.36: Block diagram of the control system. P(z) = open-loop plant; C(z) = simple LTI controller; U(z) = converged inverse QR-RLS controller; y = position of the head; d 0 = aggregate disturbance; y ref = position reference; ~ y = PES;u = control signal; G 1 (z) = closed-loop plant with C(z); G(z) = closed-loop plant with C(z) and U(z). The controller tuned with the inverse QR-RLS is developed using a model of the closed-loop plantG 1 shown in Fig. 6.36. An identified model of this closed-loop plant, ^ G 1 , is found using the n4sid algorithm and truncated to a 4 th order model. Now the control objective is to minimize the RMS value of the position error. The control problem is posed as a least squares problem and solved using the inverse QR-RLS algorithm in [81]. The algorithm is allowed to converge to steady-state and the controller is denoted asU(z) in Fig. 6.36. 6.3.3.2 Closed-loop Model The baseline LTI and inverse QR-RLS controllers are placed in the loop with the HDD dynamics and a closed-loop model fromu to ~ y is formed, denoted asG. The new system diagram is shown in Fig. 6.37, the outputy of this system is the PES, which is the same 192 as ~ y in Fig. 6.36 since y ref is constant. An identified model of this system, ^ G, is again found by the n4sid algorithm and truncated to a 10 th order model. The bode plot of the identified ^ G is seen in Fig. 6.38. This is the system that will be used for disturbance rejection throughout the rest of this paper. 6.3.4 Disturbance Rejection The goal of the disturbance rejection schemes that will be added to the baseline control, explained above, is to reduce the PES which isy in Fig. 6.37. This will be done in two parts. First the RRO will be suppressed with either the adaptive feedforward scheme demonstrated in [79] or a repetitive controller similar to the one demonstrated in [71]. Once the repeatable runout disturbance rejection is applied there is still disturbance that creates non-perfect tracking. The second part of the disturbance rejection scheme models the remaining disturbance using neural techniques and is adaptively updated online. This technique was demonstrated in [51]. 6.3.4.1 Adaptive Feed Forward RRO Disturbance Rejection This RRO disturbance rejection scheme will be injecting a negative of the estimated dis- turbance into the system and has proven experimental results as shown in [79, 80]. This paper makes no modification to the algorithm other than applying it to a large number of frequencies. The repeatable runout (RRO) disturbance occurs at frequencies 120m Hz wherem=1;2;:::;n due to the 7200 rpm speed of the disk. A control input is designed 193 such that it will adaptively cancel this disturbance. The disturbance, d(k) = d RRO (k) can be modeled as d RRO (k)= n X i=1 a i (k)sin µ 2¼ik N rev ¶ +b i (k)cos µ 2¼ik N rev ¶ (6.107) wherei is the index for the harmonic andN rev is the number of samples per revolution. If the system is modeled as in Fig. 6.37 then the output is y(k)=G[u(k)]+d RRO (k) (6.108) To cancel the disturbance the control signal should be u(k) = ¡ ^ G ¡1 [ ^ d RRO (k)]. The identified inverse, ^ G ¡1 , will have an effect on the magnitude and phase of the disturbance estimate, ^ d RRO (k). Since the magnitude and phase of the sinusoidal disturbance is being estimated, the system inverse can be ignored and the new control signal becomesu(k)= ¡ ^ d RRO (k). The disturbance estimate is ^ d RRO (k)= n X i=1 ^ a i (k)sin µ 2¼ik N rev ¶ + ^ b i (k)cos µ 2¼ik N rev ¶ (6.109) The update equations for the estimated parameters are ^ a i (k)= ^ a i (k¡1)+° i y(k¡1)sin µ 2¼ki N rev +Á i ¶ (6.110) 194 ^ b i (k)= ^ b i (k¡1)+° i y(k¡1)cos µ 2¼ki N rev +Á i ¶ (6.111) Where the ° i are adaptation gains, chosen differently for each harmonic. A phase ad- vance modification is added to reduce the sensitivity and allow for more harmonics to be canceled as was done previously in [79]. The Á i = \G(j! i ) and ! i is the angular frequency of theith harmonic. It was shown previously in [80] that the adaptive feed forward disturbance rejection scheme has an LTI equivalent representation. By treating this RRO rejection scheme as LTI, the sensitivity functionS C¡U¡U RRO =[1+P (C¡U¡U RRO )] ¡1 fromd o toy can be computed. Also the shape of N C¡U¡U RRO =P(C¡U¡U RRO )[1+P(C¡U¡U RRO )] ¡1 (6.112) fromy ref toy is computed, whereU RRO is the LTI representation for the complete adap- tive feedforward scheme withn = 33 harmonics. Estimates for both transfer functions, denoted with^ ², are shown in red in Fig. 6.39 and Fig. 6.40 respectively. The attenuation at the RRO harmonic frequencies is very pronounced as is some amplification in the low frequency region. However the experimental results will show that this amplification is in fact an attenuation, contrary to what the sensitivity plot shows. 195 6.3.4.2 Repetitive Control Two prominent methods for dealing with RRO disturbances have been described in the literature. One is the adaptive feedforward rejection method described in the previous subsection. Another one is repetitive control, which uses the concept of internal model in [7] for synthesizing linear time invariant (LTI) controllers. In this paper we employ a repetitive control scheme presented in [71], which has been demonstrated to be suitable for integrating repetitive and adaptive elements simultaneously. The design is carried out as follows. First, we considerG to be stable and that ^ G=G, and we represent the aggregated effects of all the disturbances acting on the system by the output disturbanced, i.e.,d = [1+P (C¡U)] ¡1 d 0 . This is illustrated in Fig. 6.37. Then we choose the internal model D = 1¡ q(z;z ¡1 )z ¡N , where q is a zero–phase low–pass filter andN is the period of the periodical disturbance to be attenuated. Notice that the operatorq allows us some flexibility over the frequency range of disturbances to be canceled while maintaining stability. The filter D has a combed shape with notches matching the frequencies of the periodic disturbance signals. Thus, ideally we would like to search for a filterK that makes the frequency response of the LTI system1¡KG close to zero at the same periodic frequencies. This is achievable by solving the B´ ezout identity RD+KG=1; (6.113) 196 whereR andK are the unknowns. For (6.113) the existence of stable solutions forR andK will be assured if the numer- ators and denominators of G and D can be arranged to have a polynomial Diophantine equation satisfying the coprimeness condition in [1]. Furthermore, (6.113) character- izes a whole family of stabilizing internal model type repetitive controllers, because the system 1¡GKo is stable as long as G and Ko remain stable. Following the general guidelines in [89] and [88] a particular solution is presented here. The method starts by separating G into its minimum and non–minimum phase parts G o and G i respectively. Thus, G= B A = B + B ¡ A =G i G o ; G o = B + A ; G i =B ¡ : (6.114) WhereB + andB ¡ are the cancelable and uncancelable parts of the numerator B ofG. Now, substituting (6.114) into (6.113) we can write RD+K 0 G i =1; K 0 =KG o : (6.115) 197 Among the infinity many solutions to (6.115) it is verifiable by simple algebraic manip- ulations that one of the solutions is given by R o = 1 1¡(1¡°G ¤ i G i )qz ¡N ; K 0 o =q°G ¤ i z ¡N R o ; K o =K 0 o G ¡1 o : (6.116) Here,G ¤ i is defined asG ¤ i (z ¡1 )=G i (z), and° as a positive real number. At this point, questions on the causality and the stability of the controller K o arise. The zero–phase filter q is noncausal and the plants G i and G ¤ i might not be causal as well. Nonetheless, the causality ofK o is guaranteed for a sufficiently largeN, sincez ¡N is a factor of both(1¡°G ¤ i G i )qz ¡N and°qG ¤ i z ¡N . Also, it is verifiable, by the use of the small gain theorem [19] that the stability ofK o and the stability ofR o are ensured by the sufficient condition j1¡°G ¤ i (e jµ )G i (e jµ )j< 1 jq(e jµ )j ; 8 µ2[0;¼]: (6.117) In (6.117) the real number ° can be thought of as a stability and performance tuning parameter. Fig. 6.41 shows the fulfilment of the condition (6.117) and the achievable performances for three different values of °: 4:5£ 10 ¡7 , 1:5£ 10 ¡6 and 5:0£ 10 ¡6 . Clearly, when considering the three cases in Fig. 6.41, there exists a trade–off between performance and stability robustness. For example, for the case ° = 5:0£ 10 ¡6 the 198 condition (6.117) is amply satisfied, however, there exists a noticeable internotch am- plification in the sensitivity function 1¡ GKo, from d to y, shown on the bottom of Fig. 6.41. On the other hand, the converse is true for the case° =4:5£10 ¡7 , i.e., the in- ternotch amplification is almost unnoticeable and the condition (6.117) is satisfied with a smaller amplitude. It is important to remark that there is no a linear relation between the magnitude of° and performance or stability robustness and that the purpose of Fig. 6.41 is simply exemplify that ° can be thought of as a tuning parameter. Here, we choose ° =4:5£10 ¡7 because this value is a good compromise. The bottom plot in Fig. 6.41 shows the feedforward1¡GK o , which is the sensitivity function from d to y. However, what is more interesting at this point is the shape of the overall sensitivity functionS C¡U¡U REP = [1+P (C¡U¡U REP )] ¡1 fromd o toy, and also the shape of N C¡U¡U REP = P(C ¡ U ¡ U REP )[1+P(C¡U¡U REP )] ¡1 fromy ref toy, withU REP =¡K o (1¡K o G) ¡1 (recall thatd o is the original open–loop output disturbance andy ref is the reference signal). Estimates for both transfer functions, denoted with a hat^, are shown in red in Fig. 6.42 and Fig. 6.43 respectively. Notice that this internal–model–based controller is not only able to create deep notches but also to improve the rejection over low frequencies at the expense of some amplification on inter–notch regions that can be attenuated using the neural–networks method. 199 6.3.4.3 Neural Modeled Disturbance Rejection In the above sections there have been several controllers developed. Initially there is a simple LTI controller designed to stabilize the HDD, then a controller tuned with the inverse QR-RLS is added to achieve better tracking performance. This control scheme can be yet improved using available information of the frequency of the RRO distur- bance. Using this a priori knowledge the adaptive feedforward RRO disturbance rejec- tion scheme and the repetitive controller are designed. With the RRO eliminated there still exists a disturbance that is nonlinear and dynamic. A neural model of this dynamic nonlinear disturbance is then created to reduce the tracking error even further. The new disturbance is modeled as d(k)=d RRO (k)+d NN (k): (6.118) So the system output now becomes y(k)=G[u(k)]+d RRO (k)+d NN (k) (6.119) The neural modeled disturbance rejector assumes that the disturbance from the RRO, d RRO (k) is suppressed with either the adaptive feedfoward rejection of repetitive con- troller. This means the only disturbance of concern at this point is thed NN (k). To cancel the disturbance the control signal should be u(k) = u RRO (k)¡ ^ G ¡1 [ ^ d NN (k)], where 200 u RRO (k) is the input generated by either RRO suppression scheme. Since the identified model of the HDD is non-minimum phase the inverse is unstable. The unstable zero ofG is reflected across the unit circle and the inverse is taken. This new inverse, ¹ G ¡1 , is used in the computation of the control signal, making itu(k)=u RRO (k)¡ ¹ G ¡1 [ ^ d NN (k)], and therefore causes an extra delay that will be dealt with. The following disturbance rejection scheme uses gaussian radial basis functions (RBF) from neural networks to attempt to model the disturbance. The disturbance estimate takes the form ^ d NN (k)= L X q=1 M X i=1 µ q;i (k)R q;i ^ d(k¡±¢(q¡1)¡1) (6.120) R q;i =ª i ³ ^ d(k¡±¢(q¡1)¡1) ´ (6.121) WhereR q;i is computed using an RBF and theith gaussian RBF is ª i (x)=exp " ¡ µ x¡c i ¯ ¶ 2 # (6.122) The parameters that specify the shape of the ith gaussian RBF are the center c i and the width ¯. There are a total of M gaussian RBFs, and their centers are linearly spaced across the range of input. The current disturbance estimate, ^ d NN (k), is a function of L previous disturbances that are spaced± samples apart. The reason for the spacing ± is the delay associated with passing the disturbance estimate through the system inverse. One method of coping with the delay would be to 201 estimate the disturbance at the next sample, and then use this estimate to create another future estimate, and continue iterating to find some ^ d NN (k+¢) in the future [74]. This method did not work as the estimation error grew with each future estimate. Instead the disturbance is thought of as a function of previous evenly spaced disturbances. The ^ d NN (k) can be viewed as a future disturbance estimate when compared to the HDD sample rate. It should be noted that the algorithm still creates a new disturbance estimate at every sample of the HDD output. The model of the disturbance is motivated by the assumption that the disturbance is a nonlinear function of previous disturbance values. The ^ d(k¡±¢(q¡1)¡1) term that is multiplied by the output of the RBF in (6.120) is the added term that allows the model to work for the disturbance in this HDD. This added term makes the model different from previously used RBF neural predictors [31, 74]. Now the disturbance can be thought of as an autoregressive filter with spacing± and nonlinear coefficients that are modeled with the RBF’s. A simple example with± = 2 andL = 3 is shown in Fig. 6.44 to help view the modeling of the disturbance. The ^ d(k¡±¢(q¡1)¡1) used in the estimate is not produced by the estimator but rather is a measured disturbance estimate which can be calculate from an identified model of the system, ^ G(z), as follows ^ d(k¡1)=y(k¡1)¡ ^ G[u(k¡1)] (6.123) 202 The unknown parameters should be updated with the current modeling error, which is the plant outputy(k¡1), and the parameters that caused that error, ^ d(k¡±q¡1). This leads to the update equations µ q;i (k)= 8 > > < > > : µ q;i (k¡1)+® q;i (k) " new ·½¢" old µ 0 q;i otherwise (6.124) ® q;i (k)= l NN m 2 (k) y(k¡1) ¹ R q;i ^ d(k¡±q¡1) (6.125) ¹ R q;i =ª i ³ ^ d(k¡±q¡1) ´ (6.126) m 2 (k)=1+m s (k) (6.127) m s (k)=± 0 m s (k¡1)+ ^ d 2 (k¡1) (6.128) EveryN " samples the following are computed " new = N" X n=1 y 2 (k¡n) (6.129) " old = 8 > > < > > : " new " new <" old " old otherwise (6.130) µ 0 q;i = 8 > > < > > : µ q;i (k) " new <" old µ 0 q;i otherwise (6.131) 203 The update is an instantaneous gradient algorithm with a couple robustness modifica- tions. The adaptation, or learning, rate isl NN and is greater than zero. The update term is normalized with a dynamic term to add robustness, this term is calculated in (6.128). The parameter± 0 is chosen between 0 and 1. The other robustness modification is one that is added to stop adaptation when the performance starts to degrade. In practice the estimation error will never become zero and so the parameters will continue to update. There will be a point at which the estima- tion error is small and on the same level as the noise and modeling error. Usually a simple deadzone is added to stop adaptation when the current error is below some threshold. In the HDD application the estimation error, which is the HDD output, is noisy and must be averaged over N " samples. Instead of averaging, the sum squared error is easier to cal- culate online via (6.129). The adaptation will continue as long as this new sum squared error," new , is less than the old sum squared error," old , to within some small range. The ½ term is selected to be greater than 1 to allow adaptation even if the sum squared error did not decrease. This gives some room for noisy measurements and lets the algorithm continue. If the algorithm is doing a good job and the new sum squared error is strictly less than the old, the currentµ q;i (k)’s are saved and the" old is updated. Theµ q;i (k)’s are saved so that when the sum squared error is too large, the algorithm can revert back to the best known parameters. 204 6.3.4.4 Parameter Tuning for Neural Model The parameters were first tuned in an offline simulation. Experimental disturbance data was collected by allowing the LTI and converged inverse QR-RLS controllers to run and measuring the PES. This PES data is the disturbance for both of the adaptive disturbance rejection schemes. The number of harmonics and adaptive gain for each harmonic of the RRO rejection scheme were tuned first. This was done through Monte-Carlo simulations while varying the adaptive gain. The number of harmonics was simply increased until no performance benefit was seen. Once these values were fixed the parameters of the neural model disturbance rejection could be tuned. Monte-Carlo simulations were run while varying the number of basis functions,M, adaptive gain,l NN , and the number of past parameters,L. TheN " and½ needed to be tuned online once at a single head / track combination to ensure adaptation stopped at the appropriate time to maximize performance. It should be noted that each time the algorithm begins " old is initialized by " old = " 0 , where " 0 is a design parameter that should be chosen large at first. This way after the firstN " samples " old can be updated via (6.130). 6.3.5 Experimental Results In the experiments described here, the sample-and-hold rate for control was 9.36 KHz, externally determined by the HDD clock. The controllers used in this section include 205 the LTI controller, denoted by C. The U that is used is a 36 th order tuned inverse QR- RLS, tuned using the head 0 over track 15,000. The U RRO is the adaptive feedforward disturbance rejection scheme used to canceln=33 harmonics and the gains were tuned offline using a previously acquired PES signal. U REP is the repetitive controller which is designed entirely offline as well. Lastly theK NN , which is the adaptive neural distur- bance rejection scheme withM = 15;L = 9;l NN = 1;N " = 500;½ = 1:5, and± = 2. Thea i (k) andb i (k) of the RRO disturbance rejector, and theµ q;i (k) of the neural scheme are all initialized to zero and adapted online. The adaptive feedforward RRO disturbance rejection scheme and repetitive controller are first tested in the experiment. Each RRO rejection scheme was tested with theC and U controllers but independently of one another. A 10 second period of HDD PES was collected using the head0 over track 15,000 and the PSDs are calculated. The logarithmic and linear PSDs are shown in Fig. 6.45 and Fig. 6.46, respectively. The top plot in each shows the baseline control with no RRO disturbance rejection, and the bottom plot is the case when the RRO is eliminated. There are a couple differences between the two experimental results that should be pointed out The U REP appears to have better low frequency attenuation and this is because the zero frequency is eliminated in the repetitive controller but not in theU RRO scheme. Although the zero frequency can also be eliminated with the U RRO scheme this will not help our ultimate goal of improving our TMR metric which is3¾. Also both figures show that theU REP has deeper notches at lower frequency than theU RRO , which is true in this design but may not be in general. 206 TheU RRO does attenuate the overall disturbance in the frequency range from 5 – 400 Hz, and not just RRO harmonics like theU REP . Fig. 6.47 clearly shows the rejection of the RRO harmonics at multiples of 120 Hz. Both RRO rejection schemes do a similar job in this frequency region of canceling the repeatable-runout but at the same time amplifying the disturbance in between the harmonics. Both RRO rejection schemes perform well and some tabulated results are in Fig. 6.50 and Table 6.1 where the 3¾ value of the PES as a percentage of the track width is calculated for different heads and tracks of the HDD. In just about every scenario the adaptive feed forward RRO disturbance rejection scheme outperforms the repetitive controller, which could be due to the tuning in this case. However, the neural modeled disturbance rejection scheme improves the TMR in every case tested and is able to work in conjunction with either RRO rejection scheme to make a mean improvement in of 2:3%. The PSD of the neural scheme in combination with the RRO rejection schemes is displayed in Fig. 6.48, where the low frequency disturbance rejection from 0 – 400 Hz is improved in both cases. The impact of the adaptive schemes is most noticeable in Fig. 6.49, where the time series data from the HDD at head 0 and track 15,000 has been plotted. Initially only the baseline controllers are active and then at 5 seconds the adaptive disturbance rejectors,U RRO andU NN , are switched on. The results compiled in Table 6.1 come from experiments performed on the same day around the same time. This was done for comparison purposes, however the different schemes were run hundreds of times on various days to assure the ability to work under 207 various conditions. The disturbance was quantitatively different during various tests and the improvement from the RRO schemes varied. However, in all of the tests completed the neural disturbance rejector was able to improve the performance by similar amounts seen in this paper. In all the cases tested the RRO rejection schemes performed quite eventhough the schemes were tuned on a single track and head combination. Differ- ent tracks and heads result in different RRO disturbance spectrums, but the measurable tracking performance of the schemes presented points to the robustness of the design. Each of the disturbance rejection schemes possesses implementation issues. As dis- cussed in [80] the adaptive feedforward disturbance rejector will in general require more computation than the repetitive controller. However in this experiment there are 66 pa- rameters that are updated online using sine lookup tables, while the repetitive controller is a filter with192 internal states. The repetitive controller requires a mathematical model of the plant, while the adaptive feedforward method can be implemented with just exper- imental frequency response data. The most computation intensive part of this scheme is the neural modeled disturbance rejector. For this experiment it requires a model of the plant, a model of stable inverse, and the online updating of270 neural model parameters. The identified plant and identified stable inverse both use10 states and their outputs must also be computed online. This large amount of computation required may be too large for production disk drives, where computation and memory budgets are of importance. However, this is one application of the scheme where the benefit in tracking to high per- formance systems can be experimentally verified. The disturbance rejection scheme can 208 possibly be applied to other systems where computational power is not of paramount concern. Table 6.1: 3¾ Value of the Position Error Signal (PES) as a Percentage of the Track Width Head 0 y ref =10 4 y ref =1:5¢10 4 y ref =2¢10 4 C andU 5.0641 5.0371 4.6399 C,U, andU REP 4.3918 4.1531 4.1311 C,U, andU RRO 4.2809 4.0642 3.8694 C,U,U REP , andK NN 4.3452 4.0574 4.0770 C,U,U RRO , andK NN 4.1926 3.9714 3.8634 Head 1 y ref =10 4 y ref =1:5¢10 4 y ref =2¢10 4 C andU 5.0905 5.1992 4.5877 C,U, andU REP 4.2409 4.1162 3.9287 C,U, andU RRO 4.2666 3.9954 3.7213 C,U,U REP , andK NN 4.1402 3.9981 3.8746 C,U,U RRO , andK NN 3.9930 3.8760 3.6445 6.3.6 Discussion This paper presented an adaptive disturbance rejection scheme for a HDD. The RRO of the disturbance is attenuated through the use of an adaptive feedforward model or a repet- itive controller and the remaining disturbance is modeled with neural techniques using radial basis functions. The complete control scheme was experimentally tested without retuning at various head and track positions on a HDD to show the improvement in TMR 209 by as much as25:4%. However, the addition of the neural disturbance rejector improved the tracking by as much as 6:4%, but the performance increase was measurable on all track and head combinations tested. It is difficult to quantitatively compare the results presented here to other published results since variations in HDDs and disturbances will have a large impact, but another adaptive repetitive control scheme was done on the same HDD and published in [71]. The NRRO suppression ability of the neural scheme is su- perior to the adaptive scheme tested in [71], while both add some complexity in terms of computation. The increase in computation for the neural scheme results in better perfor- mance while remaining implementable on the experimental setup. Unlike the QR-RLS scheme in [72], where the order can be increased by 100 resulting in simulated perfor- mance benefits but the computation time required is too great for the computing power of the experimental setup. The performance benefit of the neural scheme is clear from the experimental results provided, but the added complexity may make implementation on commercial disk drives difficult. As technology and research improves, this scheme presented may be simplified and the computations accelerated with new algorithms. Examples from the past are RLS filters, Kalman filters, and matrix operations, which have all progressed in terms of their ability to utilize modern processing power. Future research should focus on improving the speed of the neural algorithm while also expanding the range of applications. The HDD is one application of the disturbance rejection scheme that demonstrates the ability 210 to reduce tracking error through innovative techniques. This scheme can also be applied to other systems where computational power is not the driving constraint. 211 G(z) - -h ? - u y d Figure 6.37: Block diagram of the system used for disturbance rejection. G(z)= closed- loop system with baseline controllers;d= disturbance;u= control signal. 10 1 10 2 10 3 −40 −20 0 20 40 Magnitude (dB) 10 1 10 2 10 3 −400 −300 −200 −100 0 Phase (deg) Frequency (Hz) Figure 6.38: Bode plot of identified close-loop system ^ G 212 -120 -80 -40 0 40 Magnitude (dB) 10 -1 10 0 10 1 10 2 10 3 -180 -90 0 90 180 Phase (deg) Sensitivity Functions Frequency (Hz) Estimate for S C - U - U RRO Estimate for S C - U Estimate for S C Figure 6.39: Bode plots for the computed output sensitivity transfer functions. -60 -50 -40 -30 -20 -10 0 10 Magnitude (dB) 10 -1 10 0 10 1 10 2 10 3 -180 -90 0 90 180 Phase (deg) Reference Sensitivity Functions Frequency (Hz) Estimate for N C - U - U RRO Estimate for N C - U Estimate for N C Figure 6.40: Bode plots for the computed mappings ^ N C , ^ N C¡U , and ^ N C¡U¡U RRO from y ref toy, for controllersC,C¡U, andC¡U¡U RRO respectively. 213 10 0 10 1 10 2 10 3 -15 -10 -5 0 5 10 15 20 Frequency (Hz) Magnitude (dB) Stability Condition: |1-γG i * G i |<|q| -1 |q| -1 |1-γG i * G i |, γ=4.5e-7 |1-γG i * G i |, γ=1.5e-6 |1-γG i * G i |, γ=5.0e-6 10 0 10 1 10 2 10 3 -100 -50 0 Frequency (Hz) Magnitude (dB) 1-GK o γ=4.5e-7 γ=1.5e-6 γ=5.0e-6 Figure 6.41: Top plot: Sufficient stability condition for different values of °. Bottom plot: Estimated sensitivity function fromd toy,1¡GK o , for different values of°. -120 -80 -40 0 40 Magnitude (dB) 10 -1 10 0 10 1 10 2 10 3 -180 -90 0 90 180 Phase (deg) Sensitivity Functions Frequency (Hz) Estimate for S C - U - U REP Estimate for S C - U Estimate for S C Figure 6.42: Bode plots for the computed output sensitivity transfer functions. 214 -60 -50 -40 -30 -20 -10 0 10 Magnitude (dB) 10 -1 10 0 10 1 10 2 10 3 -180 -90 0 90 180 Phase (deg) Reference Sensitivity Functions Frequency (Hz) Estimate for N C - U - U REP Estimate for N C - U Estimate for N C Figure 6.43: Bode plots for the computed mappings ^ N C , ^ N C¡U and ^ N C¡U¡U REP from y ref toy, for controllersC,C¡U, andC¡U¡U REP respectively. 1 < 2 < M < ) 1 ( ˆ k d ¦ 1 , 1 T 2 , 1 T M , 1 T ¦ ) 1 ( ˆ k d 1 < 2 < M < ) 3 ( ˆ k d ¦ 1 , 2 T 2 , 2 T M , 2 T 1 < 2 < M < ) 5 ( ˆ k d ¦ 1 , 3 T 2 , 3 T M , 3 T ) 3 ( ˆ k d ) 5 ( ˆ k d ) ( ˆ k d NN Figure 6.44: Simple example of how ^ d NN (k) is computed from previous values of the disturbance. Here± =2 andL=3. 215 10 0 10 1 10 2 10 3 -100 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) 10 0 10 1 10 2 10 3 -100 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.45: Experiment performed on head 0 and track 15;000. Top Plot: PSD with no adaptive disturbance rejection (C and U). Bottom Plot: Red Line: PSD with adap- tive feedforward RRO disturbance rejection (C, U, and U RRO ). Blue Line: PSD with repetitive control (C,U, andU REP ). 216 0 500 1000 1500 2000 2500 3000 3500 4000 4500 -100 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 -100 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.46: Experiment performed on head 0 and track 15;000. Top Plot: PSD with no adaptive disturbance rejection (C and U). Bottom Plot: Red Line: PSD with adap- tive feedforward RRO disturbance rejection (C, U, and U RRO ). Blue Line: PSD with repetitive control (C,U, andU REP ). 217 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.47: Closeup of PSD in Fig.6.46. Top Plot: PSD with no adaptive disturbance rejection (C and U). Bottom Plot: Red Line: PSD with adaptive feedforward RRO disturbance rejection (C, U, andU RRO ). Blue Line: PSD with repetitive control (C, U, andU REP ). 218 10 0 10 1 10 2 10 3 -100 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) 10 0 10 1 10 2 10 3 -100 -90 -80 -70 -60 -50 -40 Frequency (Hz) Power / Frequency (dB / Hz) Figure 6.48: Experiment performed on head 0 and track 15;000. Top Plot: Red Line: PSD with repetitive control and neural modeled disturbance rejection (C,U,U REP , and K NN ). Blue Line: PSD with repetitive control. (C, U, and U REP ). Bottom Plot: Red Line: PSD with adaptive feedforward RRO and neural modeled disturbance rejection (C, U, U RRO , and K NN ). Blue Line: PSD with adaptive feedforward RRO disturbance rejection (C,U, andU RRO ). 219 0 2 4 6 8 10 12 14 -6 -4 -2 0 2 4 6 Time (s) PES (% of track) Figure 6.49: Time series data from experiment performed on head 0 and track 15;000. At 5 seconds the adaptive disturbance rejection is switched on (C,U,U RRO , andK NN ). 0 1 2 3 4 5 6 10,000 15,000 20,000 10,000 15,000 20,000 RLS REP REP+NN AFC AFC+NN 3 V RLS REP REP+NN AFC AFC+NN C, U C, U, U REP C, U, U REP , K NN C, U, U RRO C, U, U RRO , K NN Head 0 10,000 Head 0 15,000 Head 0 20,000 Head 1 10,000 Head 1 15,000 Head 1 20,000 Track 0 1 2 3 4 5 6 Figure 6.50: 3¾ value of the position error signal as a percentage of the track width for various head and track locations. 220 6.4 Adaptive Mode Suppression Scheme for a Hypersonic Cruise Vehicle 6.4.1 Introduction In recent years there has been considerable interest and a flurry of research efforts de- voted to system-level modeling and control of hypersonic flight vehicles that use air- breathing propulsion systems. These aircraft have application in a variety of military missions including reconnaissance, tactical, and time sensitive targets, as well as low cost space access. In addition many researchers now believe the dream of high-speed civil transport using a scramjet powered aircraft - for example flying from Brussels to Sydney in two to four hours - is closer to reality. Several countries have ongoing pro- grams that identify challenges and evaluate design ideas. An accepted configuration for this class of vehicles dictated by the characteristics of the flight and the requirement of the scramjet propulsion system is an X-43 like rectangular integrated engine-airframe configuration which uses the forebody as part of the engine inlet. This configuration although optimal for the propulsion system it poses significant challenges for control en- gineers necessitating their involvement early in the conceptual design phase [82, 83]. The challenge comes from the coupling between the aerodynamics and the propulsion sys- tem; each affecting the other during even subtle maneuvers especially in the longitudinal mode. 221 Studies also indicate that there could be coupling between the structural and rigid- body dynamics [9, 12, 18]. In addition to the coupling between the aerodynamics and propulsion system, an aerodynamically optimized full scale vehicle should have a long slender forebody configuration with slow flexible modes not separated enough from the aircraft rigid-body modes which pose additional control challenges. These low frequency flexible modes can get excited during certain maneuvers thereby causing adverse aeroser- voelastic effects, degrading performance, and causing potential structural damage and instabilities. What makes the control problem even more challenging is that these modes and their associated frequencies may involve significant uncertainties, especially dur- ing the early phase of design where structural details of the aircraft have not been fully decided. The modal frequencies and damping ratios may change based on aircraft con- figuration, cg location, payload for a required mission, or due to atmospheric heating [57, 56, 59, 60, 30]. The flexible modal frequencies may drop by as much as 5-30% due to the aerothermoelastic effects. A robust control scheme for a hypersonic aircraft must adequately deal with the flex- ible structural modes. Mode suppression in hypersonic vehicles has been studied and reported in the literature [15, 64, 29]. A linear parameter-varying (LPV) synthesis ap- proach to account for the changing flexible mode due to atmospheric heating has been studied in [57, 56]. Mode suppression schemes for conventional aircraft usually incor- porate notch filters to suppress the flexible modes [73, 22, 16, 11]. The notch filters are added on top of the rigid-body controller in a complete aircraft control scheme. However, 222 researchers have also applied integrated techniques, which do not explicitly use structural filters. This is accomplished through robust controller design using¹-synthesis[40], dy- namics inversion [26], or adaptive dynamic inversion [10]. A conventional notch filter is ineffective when the frequencies and damping ratios of the structural modes involve large uncertainties and/or may change during flight as is the case with airbreathing hyper- sonic vehicles. In these cases an adaptive notch filter may be used to deal with changing structural dynamics of the vehicle. The adaptive notch filter has been studied in signal- processing research [87, 78, 20] as well as other applications, such as the HDD [69, 47], launch vehicles [21], aircraft [61], and space structures [54]. In this paper we present application of an adaptive notch filter for control of an air- breathing hypersonic vehicle. The adaptive notch filter is applied to the longitudinal model of a full scale flexible vehicle where structural modes change during the flight. The notch filter is based on plant parametrization and hence does not require a probe sig- nal. The hypersonic vehicle model has been used previously for control design [43, 35], but suppression of changing or uncertain structural modes have not been taken into con- sideration. The control scheme presented here will incorporate a robust online estimator that tracks the flexible dynamics of the elastic modes which are then used to adjust the parameters of the notch filter. The adaptive notch filter is combined with a robust con- troller for the rigid-body modes to present a complete adaptive control scheme which suppresses vehicle flexible modes if they are excited. We have applied the scheme to the 223 longitudinal model of a generic hypersonic vehicle, CSULA-GHV , which we have de- veloped. The CFD-based model described below includes aero-propulsion couplings and vehicle flexible dynamics. As these vehicles are inherently statically unstable, the GHV represents an unstable nonlinear multi-input/multi-out (MIMO) vehicle model. We have conducted a number of simulations in vehicle cruise mode demonstrating the scheme’s ability to adapt online and fully suppress changing structural modes. 6.4.2 CSULA-GHV Mathematical Model A longitudinal model of a full-scale generic airbreathing hypersonic vehicle developed at the Multidisciplinary Flight Dynamics and Control Laboratory (MFDCLab) at the Cali- fornia State University, Los Angeles (CSULA), is used as the main simulation platform for the adaptive notch filter demonstration[18, 17]. The GHV model shown in Fig. 6.51 is a CFD/FEA-based two-dimensional air-breathing generic hypersonic flight vehicle. The aerodynamic and propulsion data are generated using a combination of CFD and theoretical analyses. The flow is assumed to be an inviscid ideal gas (thermally perfect gas) with changes in specific heat calculated using kinetic theory. The model includes a complete set of aero-propulsion data at Mach numbers 8, 10 and 12; eleven angles of attack (® = -5 deg to 5 deg); nine elevon settings (± e = -20 deg to 20 deg); and six different fuel settings based on equivalence ratio (± T = 0 to 0.3). The elastic modes are extracted from a NASTRAN FEA-based model of the vehicle. The GHV model contains a variable geometry engine cowl, to account for the changes in oblique shockwave at the 224 underside of the forebody, which changes angle with altitude, Mach number, and angle of attack. The engine cowl moves to maximize flow rate of air through the engine and prevent shock train inside the scramjet. The equations of motion for the elastic vehicle starts with the rigid body dynamics which then are augmented with the elastic effects. By noting that the first two bending modes manifest themselves mainly in tip and the tail of the GHV , their effect can be accounted for as changes in the angle of attack and elevon deflection respectively. Terms are added to the rigid body equations to account for these changes as the vehicle structure deforms when these modes are excited. The resulting elastic equations of motion of the GHV are _ V = Tcos(® r )¡D m ¡ ¹sin(°) r 2 (6.132) _ ° = L+Tsin(® r ) mV ¡ (¹¡V 2 r)cos(°) Vr 2 (6.133) _ h=Vsin(°) (6.134) _ ® r =q¡ _ ° (6.135) _ q = M yy I yy (6.136) Ä ´ i =¡2³ si ! si _ ´ i ¡! 2 si +k N ± e (6.137) ± e =± e;r +± e;e (6.138) ® =® r +® e (6.139) 225 L= 1 2 ½V 2 SC L (M;®;± T ;± e ) (6.140) D = 1 2 ½V 2 SC D (M;®;± T ;± e ) (6.141) T = 1 2 ½V 2 SC T (M;®;± T ;± e ) (6.142) M yy = 1 2 ½V 2 S¹ cC M (M;®;± T ;± e ): (6.143) In the above equations the coefficients C L , C D , C T , and C M are computed using lookup tables withM;®;± T , and± e as the index parameters. The± e and® are the effec- tive angle of attack and elevon deflection which accounts for rigid-body motion as well as vehicle structural deformations.. These values are the sum of the rigid body deflection and the elastic deflection. The³ si and! si are the damping and natural frequency of the ith mode, fori = 1;2;3, andk N is the effective normal force at the elevon. The natural frequencies and mode shapes are computed using finite element analysis. The first three elastic modes are included which have natural frequencies 20.33 rad/s, 58.62 rad/s, and 116.18 rad/s. The damping ratio for all three is set to³ si =0:02. This damping is smaller than the damping used in [18, 43, 35] but more in agreement with the damping used in [15, 73, 9, 12, 57]. The smaller damping makes the flexible modes more pronounced in simulation studies better demonstrating the effectiveness of the proposed control design. The elastic deflections are calculated ± e;e = 3 X i=1 ¿ t;i ´ i (6.144) 226 Figure 6.51: Overall geometry of the CSULA-GHV used for simulation. ® e = 3 X i=1 ¿ n;i ´ i (6.145) where ¿ t;i and ¿ n;i are the tail and nose scaling factor, respectively, which is based on the mode shape. As these equations show, an elastic deflection at the nose will add to the rigid angle of attack to result in an effective angle of attack. Similarly, a deflection at the tail will result in an effective elevon deflection. These effective values are then used in the lookup tables for the aerodynamic and propulsion data. Therefore, in this model, the engine cowl location will remain optimized even when an elastic deformation is present (this may not be possible in a real situation and is a drawback of the GHV model). Accordingly, the GHV model has two control inputs: elevon deflection± e;r and throttle ± T , and five outputs: velocity V , flight-path angle °, altitude h, angle of attack ® r , and pitch rateq. The limits of the control inputs in simulations are determined by the range of inputs for which data has been generated in the GHV model. The flexible equations of motion for the GHV , Eqs. (6.132 - 6.136), are nonlinear, but can be linearized around one point in the flight envelope. This will lead to the the linearized model y =G(s)u (6.146) 227 where y = [ V ° h ® r q ] T is the output vector, u = [ ± T ± e;r ] T is the input vector, and G(s) is the transfer function matrix. An element in the ith row and jth column of G(s) can be written as G ij . Bode plots of the transfer functions G 31 (s) and G 32 (s) can be seen in Fig. 6.52. The CSULA-GHV model flexible modes, which are driven by elevon deflections, do not influence the transfer function from ± T to altitude appreciably as seen in the bode plot. We will assume some of the parameters in the model are unknown and will be denoted as £ ¤ and in the following sections these parameters will be estimated online and replaced with the estimates£, so the transfer function matrix becomes G(s;£ ¤ )= 2 6 6 6 6 6 6 4 G 1 (s;µ ¤ 1 ) . . . G 5 (s;µ ¤ 5 ) 3 7 7 7 7 7 7 5 (6.147) where µ ¤ i = [µ ¤ i1 :::µ ¤ i6 ] T 2 i ½< 6 are the actual parameters and = 1 £¢¢¢£ 5 and £=[µ ¤ 1 :::µ ¤ 5 ]2½< 5£6 : (6.148) Now consider the transfer functions matrix G i (s;µ ¤ i )= 1 d i (s;µ ¤ i ) £ n i 1 (s;µ ¤ i ) n i 2 (s;µ ¤ i ) ¤ (6.149) where d i (s;µ ¤ i )=D k (s)(s 2 +2µ ¤ i5 s+µ ¤ i6 ) (6.150) 228 n i 1 (s;µ ¤ i )=N k i1 (s)(µ ¤ i1 s 2 +2µ ¤ i2 s+µ ¤ i6 ) (6.151) n i 2 (s;µ ¤ i )=N k i2 (s)(µ ¤ i3 s 2 +2µ ¤ i4 s+µ ¤ i6 ) (6.152) and D k (s)=s r + ¹ ® r¡1 s r¡1 +¢¢¢+ ¹ ® 1 s+ ¹ ® 0 (6.153) N k i1 (s)= ¹ ¯ i 1;r¡1 s r¡1 +¢¢¢+ ¹ ¯ i 1;1 s+ ¹ ¯ i 1;0 (6.154) N k i2 (s)= ¹ ¯ i 2;r¡1 s r¡1 +¢¢¢+ ¹ ¯ i 2;1 s+ ¹ ¯ i 2;0 (6.155) where r is the order of the polynomial D k (s). In the above equations the polynomials D k (s), N k i1 (s), and N k i2 (s) are considered known and encompass both the rigid-body dynamics and any assumed known flexible dynamics. We then have d i (s;µ ¤ i )=s r+2 +® i r+1 (µ ¤ i )s r+1 +¢¢¢+® i 1 (µ ¤ i )s+® i 0 (µ ¤ i ) (6.156) n i 1 (s;µ ¤ i )=¯ i 1;r+1 (µ ¤ i )s r+1 +¢¢¢+¯ i 1;1 (µ ¤ i )s+¯ i 1;0 (µ ¤ i ) (6.157) n i 2 (s;µ ¤ i )=¯ i 2;r+1 (µ ¤ i )s r+1 +¢¢¢+¯ i 2;1 (µ ¤ i )s+¯ i 2;0 (µ ¤ i ) (6.158) 6.4.3 Adaptive Mode Suppression Scheme Hypersonic aircraft, such as the one modeled in the previous section, will travel at very high velocities through the atmosphere on a variety of missions. These missions will 229 -100 -50 0 50 100 150 200 Magnitude (dB) 10 -4 10 -2 10 0 10 2 -270 -180 -90 0 90 180 270 Phase (deg) Bode Diagram Frequency (rad/sec) Figure 6.52: Bode plot of the complete aeroelastic linearized GHV transfer functions. Blue line: Transfer function from the throttle to altitude, G 31 (s). Green line: Transfer function from the elevon to altitude,G 32 (s). require payloads that vary in terms of mass and location on the aircraft, which will affect the center of gravity and the flexible modes. Variations in the flexible dynamics of this kind are common in military aircraft and flight control systems generally incorporate notch filters to suppress the structural modes [73, 22, 16, 11]. However, as stated earlier, hypersonic vehicles will also experience extreme heating of the fuselage during flight. The thermal properties of the aircraft will cause the flexible modes to change in terms of mode shapes, damping ratios, and natural frequencies [57, 56, 59, 60, 30]. One possible solution to the changing dynamics is an adaptive notch filter, a notch filter that is able to track and suppress the flexible modes online. The adaptive notch filter presented in [21] is used on the model of a booster from the Advanced Launch System (ALS) program. The least squares estimator in the publication uses a simple undamped resonator as the model for estimation and functions well since the resonant mode is very pronounced. However in the GHV , and other applications, full plant parameterizations is 230 necessary as the flexible mode may not be as significant. Another strategy for the estima- tion of the center frequency can be found in [69], where frequency weighting functions are used. The downside is there are several failure modes that are known and avoidance requires some modal information a priori. A stochastic state space algorithm for mode frequency estimation is presented in [61]; however it relies on the injection of a probe signal which is not needed in the scheme presented here. The indirect adaptive compen- sation (IAC) scheme in [94] also requires a probe signal to complete the estimation. The adaptive mode suppression scheme in [54] uses a LMS algorithm to update filter coeffi- cients and then the modal parameters are extracted from the filter. This is opposite as to what is being presented in this paper, where the modal parameters are first estimated and then used in the adaptive notch filter. The adaptive mode suppression scheme to control the CSULA-GHV will be pre- sented in the following subsections. An overview of the control scheme is shown in Fig. 6.53. Here the GHV has measurable outputs y 2 < 5 and control inputs u 2 < 2 which will be used by a robust online estimator to estimate the flexible dynamics of the vehi- cle while in flight without the use of a probing signal. These estimated parameters of the modal dynamics will be used to tune to the notch filters, thereby creating adaptive notch filters. There are five separate adaptive notch filters, one for each measured output. The estimated flexible dynamics of the hypersonic vehicle are used as inputs to functions which will return parameters that define the shape of each filter. These functions are found offline based on a priori knowledge of possible modal parameters of the vehicle. 231 CSULA-GHV G(s) r Robust Online Estimator Rigid-Body Controller u y Adaptive Notch Filter F(s) Figure 6.53: Overall diagram of the adaptive mode suppression scheme for the generic hypersonic vehicle. Herey2< 5 is all the measurable outputs from the GHV ,u2< 2 is the control inputs, andr2< 5 is the reference input shown for demonstration purposes. The rigid-body controller is designed to control the rigid-body dynamics while ignoring the flexible modes and adaptive notch filters. The adaptive notch filters and flexible dy- namics are treated as an uncertainty for the design. Because our objective in this paper is to demonstrate the performance of the adaptive notch filter in suppressing flexible modes we have used a simple LQ design with an integral action for the rigid-body control. This controller has been demonstrated to produce good altitude and velocity tracking in cruise condition. Each of these systems are further described below. 6.4.3.1 Robust Online Estimator The robust online estimator presented here is for the case of a single unknown flexible mode. However it can be easily expanded to estimate more unknown flexible modes. In the case of the hypersonic vehicle model, the flexible mode with the slowest natural frequency, the first bending mode, will have the greatest impact on the rigid-body con- troller bandwidth. Therefore this mode which has a natural frequency of 20.33 rad/s has been modeled in simulations as the uncertain and/or changing mode. The robust online 232 estimator should be able to return an estimate of this mode in the linearized model of the aircraft. The modal frequency that is varied in the mathematical model is! si in (6.137). This frequency is not however one of the parameters that the online estimator is designed to estimate. The parameters estimated, and used in the adaptive notch filter, are coeffi- cients of the second order transfer function for the slowest flexible mode in the linearized model. These transfer functions are the complex conjugate poles and zeros that create the peak in the bode plot which must be suppressed. Due to the interaction of the flex- ible and the rigid-body dynamics in the linearized model these frequencies may not be necessarily the same as! si . Each row ofG(s;£) will create a separate online estimator, leading to five estimators which run in parallel. Now (6.149) is placed in the form of a continuous time parametric equation to be used in theith estimator, wherei=1;:::;5 z i (t)=µ ¤T i Á i (t) (6.159) z(t)= s 2 D k (s) ¤(s) y i (t) (6.160) Á i (t)= · Á i1 (t) Á i2 (t) Á i3 (t) Á i4 (t) Á i5 (t) Á i6 (t) ¸ T (6.161) Á i1 (t)= s 2 N k i1 (s) ¤(s) ± T (t) Á i2 (t)= 2sN k i1 (s) ¤(s) ± T (t) (6.162) Á i3 (t)= s 2 N k i2 (s) ¤(s) ± e (t) Á i4 (t)= 2sN k i2 (s) ¤(s) ± e (t) (6.163) 233 Á i5 (t)=¡ 2sD k (s) ¤(s) y i (t) (6.164) Á i6 (t)=¡ D k (s) ¤(s) y i (t)+ N k i1 (s) ¤(s) ± T (t)+ N k i2 (s) ¤(s) ± e (t) (6.165) µ ¤ i = · ! ¤2 D ! ¤2 i1 ! ¤2 D ³ ¤ i1 ! ¤ i1 ! ¤2 D ! ¤2 i2 ! ¤2 D ³ ¤ i2 ! ¤ i2 ³ ¤ D ! ¤ D ! ¤2 D ¸ T : (6.166) Here ¤(s) is a polynomial added to make proper transfer functions, and takes the form ¤(s) = (s+¸) 11 , where¸ is a design parameters which will determine the speed of the filter 1 ¤(s) . The unknown parameters for theith estimator,µ ¤ i =[µ ¤ i1 :::µ ¤ i6 ] T , are estimated online asµ i = [µ i1 :::µ i6 ] T . Thisµ i vector is then used in the adaptive notch filter in (6.182) of the previous section. The parameters are updated using some a priori known bounds on the damping ratios and natural frequencies which can be calculated from the bounds i and will give the following 1¸³ u i1 ¸³ ¤ i1 ¸³ l i1 >0 1¸³ u i2 ¸³ ¤ i2 ¸³ l i2 >0 1¸³ u D ¸³ ¤ D ¸³ l D >0 ! u i1 ¸! ¤ i1 ¸! l i1 >0 ! u i2 ¸! ¤ i2 ¸! l i2 >0 ! u D ¸! ¤ D ¸! l D >0 (6.167) 234 which leads to the following update algorithm _ µ i1 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i1 " i Á i1 if ³ (! u D ) 2 (! l i1 ) 2 >µ i1 > (! l D ) 2 (! u i1 ) 2 ´ orµ i1 = (! l D ) 2 (! u i1 ) 2 and" i Á i1 ¸0 orµ i1 = (! u D ) 2 (! l i1 ) 2 and" i Á i1 ·0; 0 otherwise (6.168) _ µ i2 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i2 " i Á i2 if ³ (! u D ) 2 ³ u i1 ! l i1 >µ i2 > (! l D ) 2 ³ l i1 ! u i1 ´ orµ i2 = (! l D ) 2 ³ l i1 ! u i1 and" i Á i2 ¸0 orµ i2 = (! u D ) 2 ³ u i1 ! l i1 and" i Á i2 ·0; 0 otherwise (6.169) _ µ i3 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i3 " i Á i3 if ³ (! u D ) 2 (! l i2 ) 2 >µ i3 > (! l D ) 2 (! u i2 ) 2 ´ orµ i3 = (! l D ) 2 (! u i2 ) 2 and" i Á i3 ¸0 orµ i3 = (! u D ) 2 (! l i2 ) 2 and" i Á i3 ·0; 0 otherwise (6.170) _ µ i4 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i4 " i Á i4 if ³ (! u D ) 2 ³ u i2 ! l i2 >µ i4 > (! l D ) 2 ³ l i2 ! u i2 ´ orµ i4 = (! l D ) 2 ³ l i2 ! u i2 and" i Á i4 ¸0 orµ i4 = (! u D ) 2 ³ u i2 ! l i2 and" i Á i4 ·0; 0 otherwise (6.171) 235 _ µ i5 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i5 " i Á i5 if ¡ ³ u D ! u D >µ i5 >³ l D ! l D ¢ orµ i5 =2³ l D ! l D and" i Á i5 ¸0 orµ i5 =2³ u D ! u D and" i Á i5 ·0; 0 otherwise (6.172) _ µ i6 = 8 > > > > > > > > > > < > > > > > > > > > > : ° i6 " i Á i6 if( ¡ ! u D ) 2 >µ i6 >(! l D ) 2 ¢ orµ i6 =(! l D ) 2 and" i Á i6 ¸0 orµ i6 =(! u D ) 2 and" i Á i6 ·0; 0 otherwise (6.173) m 2 i =1+n is +m is (6.174) n is (t)=C is Á T i Á i (6.175) _ n id =¡± i0 n id +± i1 (y 2 i +± 2 e +± 2 T ) (6.176) " i = z i ¡µ T i Á i m 2 i (6.177) The online estimator uses a gradient algorithm with robustness modifications [36]. For robustness the update term is normalized using a dynamic term which is calculated in (6.176). Parameter projection is used to take advantage of the known region of the parameters, expressed in (6.167), based on a priori knowledge of the vehicle structure. The upper and lower bounds are denoted as ³ u and ³ l , respectively. The adaptation 236 gains are design parameter that satisfy, ° ij > 0. The C is ;± i0 ;± i1 terms and are design parameters that must be greater than zero. This online estimator must also be gain-scheduled for the full two-dimensional flight envelope since the vehicle dynamics are nonlinear. The model is linearized around var- ious points in the space of V;h and the known transfer functions N k ij (s), N k ij (s), and D k (s) are computed. State space representations of the transfer functions are then stored in a database. Online, the correct transfer functions are be chosen using two-dimensional linear interpolation. 6.4.3.2 Adaptive Notch Filter The adaptive notch filter is implemented to suppress the slowest flexible mode of the GHV by filtering each of the outputs from the plant, which may also be considered the outputs from measurable sensors. The frequency of the flexible mode may, for various reasons given earlier, be uncertain or change. The desired shape of the notch filter in each sensor path is determined by solving an optimization problem offline. This approach to designing the notch filters by optimization has been done in a MIMO setting in [11] and a single loop problem in [16]. In both of these references the authors were concerned with meeting the Military Specification for gain margin as it pertains to flexible modes. The design is posed similarly here, except instead of using gain margin we will be attempting to suppress the flexible dynamics in each sensor path individually and add some new con- straints for the amount of variation tolerated. The parameters for the offline optimization 237 problem will be varied over a range of possible values and the notch filter parameters computed. These computed values are stored and the correct parameters selected online based on the estimatedµ i and linearly interpolating between stored values. This will now be explained in further detail. Referring to Fig. 6.53, the feedback signals used for the rigid-body controller are F(s;£ ¤ )G(s;£ ¤ )u. The notch filter bank can be expressed as a transfer function matrix F(s;£ ¤ ), comprised of only diagonal elementsF i (s;µ ¤ i ) and all zero off diagonal entries. Now the optimization problem is solved for each F i (s;µ ¤ i ) separately, assuming all the parameters in each filter path are known. Therefore G i (s;µ ¤ i ), which takes the form of (6.149), is treated as a known transfer function matrix. We wish to design the notch filter in this path,F i (s;µ ¤ i ), so that the slowest flexible mode is completely suppressed. Using a superscript of m to denote the modal part needing suppression and the subscript ij corresponding the element ofG(s;£ ¤ ), the flexible mode is expressed as G m i1 (s;µ ¤ i )= µ ¤ i1 s 2 +2µ ¤ i2 s+µ ¤ i6 s 2 +2µ ¤ i5 s+µ ¤ i6 (6.178) and G m i2 (s;µ ¤ i )= µ ¤ i3 s 2 +2µ ¤ i4 s+µ ¤ i6 s 2 +2µ ¤ i5 s+µ ¤ i6 : (6.179) Which can also be expressed, in a more meaningful way, as G m i1 (s)= µ ! 2 D ! 2 i1 ¶ s 2 +2³ i1 ! i1 s+! 2 i1 s 2 +2³ D ! D s+! 2 D (6.180) 238 and G m i2 (s)= µ ! 2 D ! 2 i2 ¶ s 2 +2³ i2 ! i2 s+! 2 i2 s 2 +2³ D ! D s+! 2 D : (6.181) Theith adaptive notch filter takes the form F i (s;µ ¤ i )= b 2 (µ ¤ i )s 2 +b 1 (µ ¤ i )s+b 0 (µ ¤ i ) s 2 +a 1 (µ ¤ i )s+a 0 (µ ¤ i ) ; (6.182) which can also be expressed in a clearer form F i (s)= µ ! iR ! iZ ¶ s 2 +2³ iZ ! iZ s+! 2 iZ s 2 +2³ iR ! iR s+! 2 iR ; (6.183) where the ³ iZ < ³ iR and ! iZ ¸ ! iR will determine the shape of the filter. These will now be generalized since each ith filter path, where i = 1;:::;5, is identical in it’s construction. The generalized flexible modesM 1 (s);M 2 (s) and filterN(s) used for the optimization problem become M 1 (s)= µ ! 2 ! 2 1 ¶ s 2 +2³ 1 ! 1 s+! 2 1 s 2 +2³!s+! 2 (6.184) M 2 (s)= µ ! 2 ! 2 2 ¶ s 2 +2³ 2 ! 2 s+! 2 2 s 2 +2³!s+! 2 (6.185) N(s)= µ ! 2 R ! 2 Z ¶ s 2 +2³ Z ! Z s+! 2 Z s 2 +2³ R ! R s+! 2 R : (6.186) 239 An optimization problem is now posed where we wish to minimize the phase lag of the filter at some desired frequency. This becomes min ³ Z ;³ R ;! Z ;! R · arctan µ 2³ R ! R ! c ! 2 R ¡! 2 c ¶ ¡arctan µ 2³ Z ! Z ! c ! 2 Z ¡! 2 c ¶¸ (6.187) where ! c is a design parameter that specifies the point at which the phase lag is mini- mized. For our simulations we choose! c = 1 2 !, which is half of the frequency at which the peak occurs in the frequency responses ofM 1 (s) andM 2 (s). Now there are several constraints for this minimization problem. The first couple retain the notch shape of the filter and are 1. ³ Z <³ R 2. ! Z ¸! R . since the peak of the flexible dynamics will occur at! we place the maximum magnitude suppression at the same frequency, leading to the constraint 3. ! Z =!. Next we add in the constraints to ensure that the flexible dynamics that occur in each filter path are suppressed below some level, 4. jN(s)M 1 (s)j 1 ·a 5. jN(s)M 2 (s)j 1 ·a, 240 where a is the maximum desired magnitude of the flexible mode and notch filter com- bination. Since the flexible dynamics and notch filter are normalized so as to have a dc-gain of unity, it may be desired to completely suppress the flexible mode and there- fore seta = 1. However this can pose problems for the optimization routine, so a value slightly larger than one may be a better choice. Another constraint is added which keeps the maximum magnitude of the notch filter below unity 6. jN(s)j 1 ·1. At this point this design methodology works to suppress a known flexible mode in two transfer functions, however in practice the modal parameters are unknown. Even with the addition of an online estimator, the notch filters should be designed to accept some uncertainty in (6.184) and (6.185). This is accomplished by adding four constraints based on some user specified acceptable uncertainty. These uncertainties are expressed as !§ ¹ ! ! 1 § ¹ ! 1 ! 2 § ¹ ! 2 (6.188) ³ § ¹ ³ ³ 1 § ¹ ³ 1 ³ 2 § ¹ ³ 2 (6.189) where the overbar variables are constants that specify the deviation from the nominal quantity. It may be helpful to express these overbar parameters as a percentage of the nominal value, ie. ¹ ! =0:1!, thereby signifying a 10% variation in complex pole natural frequency. The worst case modal shapes, in terms of largest magnitude, are extracted 241 from the uncertainties and then expressed as two transfer functions for each original generalized mode in (6.184), (6.185). From these the constraints are formed 7. jN(s)V 1 1 (s)j 1 ·a 8. jN(s)V 2 1 (s)j 1 ·a 9. jN(s)V 1 2 (s)j 1 ·a 10. jN(s)V 2 2 (s)j 1 ·a. This method of employing several constraints to design the notch filters is similar to the method in [11], where an envelope is created that the notch filter must suppress. Once the minimization problem is posed and the constraints are set, the optimization toolbox in MATLAB is used to solve the problem and find an optimal notch filterN(s). If the overbar uncertainty parameters in (6.188) - (6.189) are constant, and constraint 3 is upheld, then the notch filter solution can be viewed as a function of ! 1 ! , ! 2 ! , ³, ³ 1 , and³ 2 . Therefore a solution of different notch filters can be found by varying these five parameters through some known space where they may occur, which is equivalent to varying aµ i vector. The parameters of the notch filter solution, ! R , ! Z , ³ Z , and ³ R , are used to com- pute the coefficients, a i ;b i in (6.182) as a function of a particular µ ¤ i . The notch filter 242 coefficients and µ ¤ i vector are then stored after each optimization is completed, form- ing a lookup table with N j points for each µ ¤ ij element. The indices to the table can be represented as µ ¤1 ij <µ ¤2 ij <¢¢¢<µ ¤N j ij ; (6.190) and aµ ¤ ij value lying between 2 consecutive indices µ ¤l j ij <µ ¤ ij <µ ¤l j +1 ij : (6.191) The parameters for the notch filter in (6.182) are computed, all in a similar fashion b 2 (µ ¤ i )= 6 X j=1 µ ¤ ij b l j 2m +b l j 2b (6.192) whereb l j 2m andb l j 2b are the slope and intercept associated with theµ ¤l j ij index. In the GHV model used for simulation in this paper we have the luxury of only requiring the suppression of a flexible mode in the second column of the G(s) transfer function matrix. This is because the structural dynamics are excited by the elevon input only, allowing for constraints 4, 7, and 8 to be discarded from the optimization problem. Fig. 6.54 shows two notch filters, each designed with the optimization scheme presented. One of the filters is designed with ¹ ! = 0:01!, ¹ ! 2 = 0:01! 2 , ¹ ³ = 0, and ¹ ³ 2 = 0, representing the adaptive case where the damping ratios are known exactly and a 1% variation in natural frequencies is tolerated. The other filter is designed to represent a 243 -25 -20 -15 -10 -5 0 5 Magnitude (dB) 10 0 10 1 10 2 10 3 -135 -90 -45 0 45 Phase (deg) Figure 6.54: Bode plot of two notch filters designed with the optimization scheme pre- sented. Both filters are designed with constraints 4, 7, and 8 neglected, ¹ ³ = 0, and ¹ ³ 2 =0. Blue line: A narrow notch filter design with ¹ ! =0:01! and ¹ ! 2 =0:01! 2 . Green line: A wider notch filter design with ¹ ! =0:05! and ¹ ! 2 =0:05! 2 . non-adaptive case, so the variation is increased to 5%, although larger may be necessary for a more practical design. The phase lag of the adaptive filter at 1 2 ! is 24.42 degrees, while the wider non-adaptive filter has a phase lag of 46.11 degrees. This difference in notch filters shows the advantage of narrow notch filters designed to accept only a small variation on flexible mode frequencies. The simplified optimization scheme leads to a smaller database due to reducing the number of parameters to be varies by two. For the GHV used for simulation in this paper, a database of notch filter parameters was created by varying ! 2 ! ,³, and³ 2 through some values which may occur during flight. To have a clearer understanding of the optimization results, the µ ¤ i solutions are not plotted, but instead the parameters ! R ! Z , ³ Z , and ³ R for various ! 2 ! and ³ are shown in Figs. 6.55, 6.56, and 6.57, respectively. A 244 0.015 0.02 0.025 0.8 1 1.2 1.4 0.7 0.8 0.9 1 1.1 ζ ω 2 / ω ω R / ω Z Figure 6.55: Plot showing an optimized notch filter parameter ! R ! Z as a function of ! 2 ! and ³. plot that displays the phase lag for each filter designed is given in Fig. 6.58. This was done using a 1% variation in all parameters and setting ³ 2 = 0:02. These plots show that a variation in³ has a negligible impact on ! R ! Z and³ R as expected, since the complex zero of the notch filter causes the suppression so desired by the filter. The ³ parameter will change the magnitude of the peak in the flexible mode and therefore cause a need to change the suppression capabilities of a filter, thereby changing the optimized³ Z which is the damping of the complex zero of the notch filter. The minimum phase lag around ! 2 ! =1 is not uprising either, since around this point a notch filter is barely required since the complex pole and zero of the flexible mode cancel. 245 0.015 0.02 0.025 0.03 0.8 1 1.2 1.4 0.02 0.025 0.03 0.035 0.04 0.045 0.05 ζ ω 2 / ω ζ Z Figure 6.56: Plot showing an optimized notch filter parameter³ R as a function of ! 2 ! and ³. 0.015 0.02 0.025 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 ζ ω 2 / ω ζ R Figure 6.57: Plot showing an optimized notch filter parameter³ Z as a function of ! 2 ! and ³. 6.4.3.3 Rigid body controller The rigid-body controller is independent of the proposed mode suppression scheme and can be any controller which robustly stabilizes the rigid-body dynamics of the GHV , 246 0.015 0.02 0.025 0.8 1 1.2 1.4 0 10 20 30 40 50 ζ ω 2 / ω Phase lag (degrees) Figure 6.58: Plot showing an optimized notch filter phase lag as a function of ! 2 ! and³. meets performance requirements, and is robust to treating the adaptive notch filters and flexible dynamics as uncertainties. For ease of design and implementation in simulation, a LQ tracking controller used previously for the control of a GHV [35] and a F-16 [34] is used for this study. A summary of the rigid-body LQ design is presented for com- pleteness and use for simulation. The nonlinear equations of motion for the GHV is linearized at a specific velocity and altitude. Only the rigid-body dynamics are used for the rigid-body control design, since the adaptive notch filters will suppress the slowest structural mode and the higher order modes are treated as uncertainties. The complete rigid and flexible linearized GHV system becomes y = Gu wherey 2< 5 andu2< 2 . The dynamics can be separated intoG = G r +G e whereG r andG e represent the rigid and elastic dynamics of the vehicle, respectively. This approach for separation of the 247 structural modes was taken in [57, 56]. The rigid body controller will be designed forG r only, which can be expressed as _ x=Ax+Bu (6.193) y =Cx (6.194) z =Hx (6.195) where x = [ V ° h ® r q ] T , u = [ ± T ± e;r ] T , and C is the identity matrix. We will be performing feedback using all of the states, which are measurable in this model. The output vector used for tracking is z = [ V h ]. This z is chosen since a velocity and altitude tracking problem is considered in the simulations presented in this paper. The objective is to track a commanded velocity and altitude,V com andh com respectively, which makes a command vector z c = · V com h com ¸ T . First Eq. (6.193) and Eq. (6.194) are augmented with new states,w, representing the integral of the tracking errors: _ w = 2 6 6 4 V com ¡V h com ¡h 3 7 7 5 : (6.196) So that the augmented system of equations becomes _ x aug =A aug x aug +B aug u+ 2 6 6 4 z c 0 3 7 7 5 (6.197) 248 y = · 0 H ¸ x aug (6.198) x aug = 2 6 6 4 w x 3 7 7 5 (6.199) A aug = 2 6 6 4 0 ¡H 0 A 3 7 7 5 (6.200) B aug = 2 6 6 4 0 B 3 7 7 5 : (6.201) Using the model described by Eqs. (6.197 - 6.201) the tracking problem is reduced to the standard regulator problem to which the LQ solution can be applied. Denote the deviations of the states x aug and control inputs u from their steady-state values as ~ x aug and~ u, respectively. The LQ cost function, assuming the augmented system is stabilizable and detectible, is J = Z 1 0 (~ x T aug Q~ x aug + ~ u T R~ u)dt (6.202) where Q > 0 and R > 0 are the chosen weights. The weights are chosen to limit the control bandwidth so as not to excite the flexible modes or other unmodeled dynamics in the high frequency range. However the bandwidth is close to the natural frequency of the slowest flexible mode therefore creating the need for a notch filter. The control input becomes ~ u=¡K~ x aug =¡ K 1 s (z c ¡z)¡K 2 ~ x (6.203) 249 where K = R ¡1 B T aug P , and P is the solution of the algebraic Ricatti equation. In the simulations it is assumed that all the original states, x = [ V ° h ® r q ] T , can be measured andw can be computed by simply integrating the tracking error. As shown in [34] this control scheme can either be gain-scheduled for the flight envelope or converted to an adaptive LQ design where the original system matrices A and B in Eq. (6.193) are estimated online. In this paper the nonlinear vehicle model is linearized at various points in the two-dimensional flight envelope, based on velocity and altitude, and an optimal gain matrix K is calculated and stored in a database where the best controller gain is chosen by using two-dimensional linear interpolation. 6.4.4 Simulations The simulations presented are performed on the full aero-elastic nonlinear CSULA-GHV model presented earlier. The initial set of simulations shows how the adaptive notch filter is necessary to fully suppress the effect of the uncertain or changing flexible mode. The slowest flexible mode, which occurs at 20.33 rad/s is modeled in this simulation as changing. The change in modal frequency, which could be due to thermal heating of the fuselage, could lower the frequencies of the elastic modes by as much as 30%[56]. A non-adaptive notch filter cannot deal with such a change in frequency. Whereas the adaptive notch filter presented here is able to track the parameters of the flexible mode as shown in the simulations. For the simulations the model is linearized around a nominal cruise point ofM =10 andh= 100,000 ft, the same conditions used in [43]. 250 The vehicle is commanded to track a 500 ft/s velocity and 5,000 ft altitude step change. This command is passed through a prefilter P(s)= (0:05) 2 s 2 +2(0:05)s+(0:05) 2 : (6.204) The modal frequency of 20.33 rad/s is subjected to a filtered step change to 15.33 rad/s beginning at 5 seconds. This assumed abrupt change clearly is far more severe than an actual mode gradual change due to thermal heating of the fuselage in a real flight situation and selected to demonstrate the robustness of the proposed control scheme. The example demonstrates the effectiveness of the online estimator to track fast changing modal dynamics. This modal frequency change is input into the model as a change in! 1 , which appears in the mathematical model in Eq. (6.137). As seen in Fig. 6.59, simulations show that the commanded outputs are tracked closely, even with the assumed changing flexible mode frequency. This change can be easily observed as it appears in the control inputs shown in Fig. 6.60. The modal fre- quency begins to appear after about 50 seconds, but then fades as it is suppressed once the adaptive notch filter starts tracking the modal dynamics. The center frequency for the adaptive notch filter F 1 (s) is plotted in Fig. 6.61. It is seen that the frequency estimate does not immediately pickup the change which we have introduced at the start of the simulation; the adaptation begins around 50 seconds and converges in approximately 15 seconds. The change starts taking place only after the incorrect notch filter results in 251 0 10 20 30 40 50 60 70 80 90 -200 0 200 400 600 Time (s) Velocity change (ft/s) 0 10 20 30 40 50 60 70 80 90 0 2000 4000 6000 Time (s) Altitude change (ft) Figure 6.59: Velocity and altitude for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. Blue line: Command response. Green line: Actual response. the control inputs exciting the flexible mode creating a large enough system excitation required for the online estimation to work. The estimates of the frequencies for the com- plex zeros in the first row of the transfer function matrix G(s) are shown in Fig. 6.62. These estimates are used in the selection of the notch filter parameters, although the es- timates are incorrect for the first half of the simulation, the flexible mode does not get excited immediately to create a large enough excitation to require adaptation. However, to more clearly see the effect of the flexible mode on the GHV , the normal acceleration at the nose is plotted in Fig. 6.63. As the control inputs excite the flexible modes, the nose of vehicle begins to vibrate at this flexible frequency. This vibration, in a real system, will affect the bow shock placement on the lip of the engine cowl since a variable geometry cowl may not be able to compensate for the elastic deformations of the 252 0 10 20 30 40 50 60 70 80 90 0.05 0.1 0.15 0.2 0.25 Time (s) δ T 0 10 20 30 40 50 60 70 80 90 -10 -5 0 5 Time (s) δ e,r (degree) Figure 6.60: Control inputs for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. 0 10 20 30 40 50 60 70 80 90 14 16 18 20 22 Time (s) ω s1 (rad/s) 0 10 20 30 40 50 60 70 80 90 16 17 18 19 20 Time (s) ω D (rad/s) Figure 6.61: Vehicle structural modal frequency ! s1 and estimated complex pole fre- quency! D , which isµ 16 , for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. 253 0 10 20 30 40 50 60 70 80 90 16 17 18 19 20 Time (s) ω 11 (rad/s) 0 10 20 30 40 50 60 70 80 90 18 19 20 21 22 Time (s) ω 12 (rad/s) Figure 6.62: Estimate frequencies for the complex zeros inG 11 (s) andG 12 (s), for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. nose. In turn, this shock placement will greatly affect the thrust and aerodynamics forces on the vehicle. However, in the GHV model, as mentioned earlier , the engine cowl is as- sumed to move and captures the bow shock when the angle of attack changes and hence leave the aerodynamic and propulsion performance of the vehicle unaffected. However, the vibration of the nose could, in addition to affecting the propulsion system perfor- mance, cause deterioration of the structure over time and therefore should be suppressed quickly. This is accomplished in simulations using the adaptive notch filter. The vibra- tions damp out, as it is seen to a negligible level around 70 seconds. The control inputs and normal acceleration at the nose for the case when adaptation is turned off are seen in Fig. 6.64 and Fig. 6.65. The flexible mode appears as large fluctuations in the control 254 0 10 20 30 40 50 60 70 80 90 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Time (s) Normal accleration at nose (g) Figure 6.63: Normal acceleration at the nose for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. inputs and the normal acceleration at the nose. The system appears stable, however the very large fluctuations would surely exceed the structural limits of the vehicle. 6.4.5 Discussion The design of an adaptive notch filter for suppression of structural modes in an aeroelas- tic airbreathing hypersonic cruise vehicle model is presented. A robust online estimator tracks the dynamics of the slowest flexible mode, which may include significant uncer- tainty, especially during the early phases of design, where structural detail are not known or it can change during an actual flight due to heating of the fuselage. Estimation of the flexible modes is accomplished by parameterizing the linearized version of the nonlinear aircraft model. The parameters of the notch filters are calculated using the estimated 255 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 Time (s) δ T 0 10 20 30 40 50 60 70 80 90 -20 -10 0 10 20 Time (s) δ e,r (degree) Figure 6.64: Control inputs for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is off. 0 10 20 30 40 50 60 70 80 90 -15 -10 -5 0 5 10 15 20 Time (s) Normal accleration at nose (g) Figure 6.65: Normal acceleration at the nose for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is off. 256 modal parameters to suppress flexible adverse effects on performance and structural in- tegrity. An integrated aero-propulsion elastic GHV model, CSULA-GHV is used for simulaions. Simulations for the adaptive notch filter and non-adaptive notch filter are shown and discussed. The adaptive scheme is shown to suppress the excited flexible mode before the oscillations become too large to cause structural damage. Simulations using a non-adaptive filter result in large oscillations in the control inputs and accelera- tion at the nose. 257 Chapter 7 Future Work This report has given a presentation of the work completed on the topic of flexible modes, disturbances, and time delays and the adaptive schemes which have been designed to combat the problems they cause. The adaptive schemes presented are all a type of add- on controller which can be appended to a fixed non-adaptive controller to enhance the performance or robustness of the system. There is much work to be done on the topics presented: ² The completion of the stability results for the adaptive mode suppression scheme for MIMO systems. The adaptive scheme and a pointwise stability result, some- times referred to as certainty equivalence, have been presented. The analysis of the system may also give insight into the design and interaction of the rigid-body controller and adaptive notch filters. ² The improvement in the adaptive Smith predictor control scheme which was pre- sented. The scheme should be updated to be able to estimate the plant parameters 258 as well as the time-delay. There are several difficulties in doing such a task and using the Smith predictor control structure, as overparameterization and poor con- vergence can occur. ² Explore the design of an adaptive bandwidth controller in conjunction with an adaptive notch filter. The stability result of a SISO adaptive bandwidth controller will follow from the work done in Chapter 2, however the design of such a system is more complex. For the cases presented, the controllers were design empirically and the pointwise stability guaranteed a priori. A design procedure which may be somewhat automated and more intuitive should be created. 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Abstract (if available)

Abstract Control systems are used in a variety of applications where tracking and regulation are required, although there exist numerous problems which can cause detrimental effects for standard controllers. The field of adaptive control has been researched and applied to solve some of these difficulties, usually through the use of complete adaptive control strategies. This research proposes adaptive controllers which may be added to fixed non-adaptive controllers to form a single unified scheme. This allows the traditional control designer some intuition in the development while reaping the benefits of a controller which can compensate for unknown or changing dynamics. There are three specific problems tackled in this research: flexible modes, disturbances, and time delays. Since each of these may be uncertain or may change over time, the addition of an adaptive controller may allow the system to meet performance requirements or retain stability.

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Asset Metadata

Creator Levin, Jason M. (author)

Core Title Practical adaptive control for systems with flexible modes, disturbances, and time delays

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/06/2009

Defense Date 05/18/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag adaptive control,adaptive notch filter,adaptive smith predictor,fast steering mirror,flexible mode suppression,hard disk drive,hypersonic aircraft,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ioannou, Petros A. (committee chair), Flashner, Henryk (committee member), Safonov, Michael G. (committee member)

Creator Email levinj@usc.edu,levinj27@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2326

Unique identifier UC1458446

Identifier etd-Levin-3009 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-574976 (legacy record id),usctheses-m2326 (legacy record id)

Legacy Identifier etd-Levin-3009.pdf

Dmrecord 574976

Document Type Dissertation

Rights Levin, Jason M.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu