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High-accuracy adaptive vibrational control for uncertain systems

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High-accuracy adaptive vibrational control for uncertain systems

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Content HIGH-ACCURACY ADAPTIVE VIBRATIONAL CONTROL FOR UNCERTAIN SYSTEMS by Saeid Jafari A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2015 Copyright 2015 Saeid Jafari Acknowledgments I would like to express my gratitude to my family for their love and endless support. A special appreciation goes to my Ph.D. advisor, Professor Petros Ioannou, for his invaluable support, encouragement, and generous advice. He is a dedicated professor who knows how to motivate his students; it has been a great honor for me to work with him. Without his guidance and constructive suggestions, I would not be able to pursue this research. I am also grateful to my dissertation and qualifying exam committee members: Prof. Edmond Jonckheere, Dr. Perez-Arancibia, Dr. Paul Bogdan, and Dr. Ketan Savla, for their valuable comments and advice on my research and presentation. I would also like to express my sincere thanks to all my friends and colleagues for their help and kindness during my Master and Ph.D. studies, including: Sadegh Bolouki, Amir Ajorlou, Alireza Louni, Aboutaleb Haddadi, Yihang Zhang, Yuzhuo Ren, Ben Fitzpatrick, Lael Rudd, Amin Rahimian, Omid Abadi, Delara Rashidi, Seyed Reza Mirnezami, Reza Jalayer, Afshin Abadi, Tooraj Rajabioun, Ardeshir Hakhamian, Alireza Javadi, Hamid Mahboubi, Ali Naghashpour, and Mohammad Ali Aghasafari. Finally, I would like to acknowledge all the sta members of the department, in particular, Diane Demetras, Shane Goodo, and Tim Boston for their energy and support. i Table of Contents List of Figures iv Abstract vii 1 Introduction 1 1.1 Motivation and Related Work . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . 8 2 SISO Discrete-Time Uncertain Systems 10 2.1 Preliminaries and Notations . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Control System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Known Disturbance Case . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Unknown Disturbance Case . . . . . . . . . . . . . . . . . . . 18 3 SISO Continuous-Time Uncertain Systems 29 3.1 Notations and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Controller Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.1 Design of the LTI lter F (s) . . . . . . . . . . . . . . . . . . . 40 3.4.2 Design of the adaptive lter K(s;(t)) . . . . . . . . . . . . . 44 3.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 MIMO Discrete-Time Uncertain Systems 55 4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Existence of a Periodic Disturbance Rejecting Controller . . . . . . . 59 ii 4.3 Adaptive Suppression of Unknown Periodic Components . . . . . . . 64 4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 MIMO Continuous-Time Uncertain Systems 87 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Robust Adaptive Rejection of Periodic Disturbances . . . . . . . . . . 91 5.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6 Plants with Unknown Parameters 105 6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2 Controller Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2.1 Non-adaptive Scenario: Known-parameter case . . . . . . . . . 107 6.2.2 Adaptive Scenario: Unknown-parameter case . . . . . . . . . . 113 6.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7 Conclusion and Directions for Future Work 124 References 126 iii List of Figures Figure 1.1: A stable known plant aected by unmeasurable disturbances . 3 Figure 2.1: Structure of the discrete-time closed-loop system . . . . . . . . 12 Figure 2.2: Control performance for the case of known disturbance fre- quencies. The control input is applied at t = 20 sec. (a) xed dis- turbance frequency ! = 25 rad/sec and dierent lter orders N; increasing N improves the performance. (b) xed lter order N = 7 and dierent disturbance frequencies. . . . . . . . . . . . . . . . . . . 19 Figure 2.3: Structure of adaptive control system . . . . . . . . . . . . . . 20 Figure 2.4: The performance of the adaptive scheme. The control input is applied at t = 10 sec. At t = 30 sec new sinusoidal terms are abruptly added to the disturbance; the adaptive controller updates its parameters at this time to suppress the eect of new terms. . . . . 28 Figure 3.1: The acoustic model with one control loadspeaker and one mi- crophone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 3.2: The block diagram of the acoustic system with modeling un- certainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 3.3: Structure of the controller . . . . . . . . . . . . . . . . . . . . 33 Figure 3.4: The magnitude plot of (a) G 0 (s) and (b) G 0 0 (s) = F (s)G 0 (s) with 0 = 1 and = 300. The plant nominal model G 0 (s) has zeros on the j! axis at s = 0 and s =2j. F (s) attens the magnitude plot of the plant model over the frequency range of interest. . . . . . 43 Figure 3.5: The magnitude plots of G 0 (s) and G 0 0 (s) = F (s)G 0 (s) for 0 = 0:5 =6 dB and = 500. . . . . . . . . . . . . . . . . . . . . . 52 iv Figure 3.6: The performance of the proposed scheme forN = 20 (a) with- out lter F (s), the speed of adaptation is small. (b) with lter F (s) much better performance is achieved. In both cases, the control sig- nal u(t) is applied at t = 4 sec. The adaptive controller with lter F (s) quickly identies and rejects the periodic components of the disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 3.7: The performance of the proposed scheme when the disturbance characteristics changed. The control input is switched on att = 4 sec. From t = 6 to t = 16 sec, ! 1 in (3.29) varies linearly from 70 to 100 rad/sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 4.1: The structure of the open-loop plant model with multiplicative output uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 4.2: The structure of the controller . . . . . . . . . . . . . . . . . . 57 Figure 4.3: The structure of the adaptive closed-loop system . . . . . . . . 65 Figure 4.4: The structure of the adaptive controller with a compensator . 70 Figure 4.5: The singular value plot of the unshaped nominal G 0 (z) . . . . 83 Figure 4.6: The performance of the adaptive control system with N = 20 without pre-ltering modication. The control is applied att = 10 sec and att = 25 sec, new frequencies are added to the existing disturbance. 84 Figure 4.7: The singular value plot of the shaped plantG 0 0 (z) =G 0 (z)F 0 (z). The singular values are almost aligned. . . . . . . . . . . . . . . . . . 85 Figure 4.8: The performance of the adaptive control system with N = 20 and pre-ltering modication. . . . . . . . . . . . . . . . . . . . . . . 85 Figure 4.9: The performance of the adaptive control system with larger adaptive lter order N = 60 together with pre-ltering modication. . 86 Figure 5.1: The structure of the controller and the closed-loop system . . 89 v Figure 5.2: The singular value plots of (a) G 0 (s) and the shaped plant G 0 (s)F (s), and (b) m (s)G 0 (s) and m (s)G 0 (s)F (s). The singular values of G 0 (s)F (s) and m (s)G 0 (s)F (s) are aligned. . . . . . . . . . 101 Figure 5.3: The performance of the proposed scheme with (a) N = 16, (b) N = 40. The control input is switched on at t = 10 sec. . . . . . . 103 Figure 5.4: (a) The plant output signals forN = 40. The control input is switched on att = 10 sec. Fromt = 20 tot = 30 sec,! 1 varies linearly with respect to time from 70 to 100 rad/sec and ! 4 changes from 60 to 80 rad/sec and att = 30 sec,! 3 abruptly changes from 275 rad/sec to 350 rad/sec and ! 6 changes from 310 rad/sec to 200 rad/sec. (b) The estimate of the controller parameters (t). . . . . . . . . . . . . . 104 Figure 6.1: The bode plots of the plant model G 0 (z) and its forth-order approximate G 0 (z) (Example 1). The approximate model G 0 (z) does not have any (nontrivial) zero. The approximation can be arbitrarily improved by increasing the lter order n. . . . . . . . . . . . . . . . . 112 Figure 6.2: The structure of the adaptive closed-loop system . . . . . . . . 113 Figure 6.3: The performance of the adaptive control system. The con- trol is applied at t = 10 sec and at t = 20 sec a new disturbance term 0:7 sin(100t) + 0:8 sin(300t) is abruptly added to the existing disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Figure 6.4: The performance of the adaptive control system. The control is applied at t = 10 sec and at t = 20 sec the unknown plant gain k 0 abruptly changes from 0:5 to 0:05. . . . . . . . . . . . . . . . . . . . . 123 vi Abstract Attenuation of unwanted sound and vibrations is a key enabling technology in a vast array of aerospace and industrial control applications. In many defence, indus- trial, and medical applications, unwanted sound and vibrations which are primarily caused by rotating or reciprocating components are typically dominated by a num- ber of harmonics. Eective suppression of these types of disturbances is essential to achieve accurate control. Examples include optical/laser jitter suppression, vi- bration reduction in helicopters, precise pointing/tracking of spacecraft, acoustic noise reduction in powertrain and aerial vehicles. In these and other applications, disturbance attenuation requires adaptation both in the estimation and rejection of the jitter source as well as the determination of the system itself. Vibrational dis- turbances often have multiple narrow-band components combined with broadband random noise which can be modeled as sum of noise and some sinusoidal terms with unknown and possibly time-varying frequencies, phases, and amplitudes. Moreover, there always exists discrepancy between the actual dynamical system and its iden- tied model. Therefore, a controller designed for attenuation of disturbances must be able to handle types of uncertainties. The proposed research focuses on output- feedback adaptive control of uncertain systems. The main objective is to design and vii analyze a robust adaptive control scheme which ensures globally asymptotic sup- pression of unknown unmeasurable disturbances with time-varying characteristics for some classes of uncertain plants. In almost all past work in this area, the prob- lem of robustness with respect to plant unmodeled dynamics and bounded noise disturbances has not been adequately addressed and it is not clear that how the performance can be improved. In fact, most adaptive laws proposed in literature for the problem of disturbance rejection were shown not to be robust back in the early 80's. These issues indicate the importance of establishing the robustness of these schemes with respect to inevitable modeling errors. In some applications, the plant model is almost linear and time-invariant (LTI) and can be identied o-line with relatively high accuracy over some frequency range. In the rst step, we consider known single-input single-output (SISO) LTI plants in the presence of unknown unmodeled dynamics. We examine robust stabil- ity and performance of a robust adaptive control scheme and discuss the trade-o between robust stability and performance improvement and practical design consid- erations. We show that over-parameterization of the internal model of the distur- bances provides structural exibility to reduce the sensitivity of the plant output with respect to noise and provides further improvements. We demonstrate that pre-ltering of the plant input together with over-parameterization signicantly improves the performance especially when the zeros of the internal model of the disturbance are close to those of the plant. We generalize the idea to multi-input multi-output (MIMO) LTI plants for both discrete-time and continuous-time sys- tems and show that how the proposed control scheme designed for SISO plants can be modied to be applicable to MIMO plants with signicant cross-couplings. In some other applications, the plant model may have large parametric uncer- tainties and/or the parameters may vary with time. For instance, change in ight viii condition in a ight vehicles, partial failure in some components of the system, or change in environment may lead to signicant change in the plant model. Therefore, the controller must be able to adjust its parameters to counteract these variations and at the same time try to minimize the eect of vibrational disturbances. In such cases, both disturbance and plant model are unknown, ensuring the stability and performance of such a nonlinear and time-varying closed-loop system is very complicated as there is coupling and a nonlinear relation between the unknown pa- rameters of the internal model of the disturbance and the unknown parameters of the plant. This makes separate estimations of the plant model and the internal model of the disturbance problematic. In spite of eorts done on this problem, the case of unknown plant model and unknown disturbance remains an open problem as no solution with guaranteed global stability has been yet proposed for practical implementation. We propose a practical solution to the problem of attenuation of unknown narrow-band disturbances acting on unknown linear systems in the pres- ence of unstructured unmodeled dynamics. ix 1 Introduction 1.1 Motivation and Related Work The problem of sound and vibrations suppression is very important in many sys- tems in precision engineering as the system's performance is signicantly aected by various sources of vibrations [1]. This problem arises in many applications such as: (i) Jitter Suppression in Laser Beam Pointing: The applications of laser sys- tems have grown in recent decades, including areas such as communications [2, 3], detection and ranging [4], imaging [5], medicine [6], and defense [3,7,8]. Of partic- ular interest in many laser applications are problems of beam control, ranging from pointing and tracking to wavefront control. Performance of beam control systems is often adversely aected by dicult-to-characterize disturbances that arise from the medium of propagation, structural vibrations in the platform, or other external factors [9]. In particular, much of the jitter in laser beam control is due to periodic disturbances whose frequencies and amplitudes are unknown and could vary with time. (ii) Structural Vibration Control: One of the applications of vibrational con- 1 trol is isolation of sensitive equipment from the vibrations of a base structure. In some cases, sensitive equipments may be supported by a structure which vibrates due to unknown oscillatory forces. A set of actuators and sensors connected by a feedback loop can be employed to minimize the eects of vibrations and to prevent the propagation of vibrational disturbances to sensitive parts of the systems [10,11]. Examples of such applications include: Vibration Reduction in Helicopters: The main purposes of suppression of vibrations in helicopters are to improve the comfort of the pilot, to reduce fatigue in the rotor and structure of the helicopter, and to protect onboard equipment from damage. The goal of vibration control is to reduce vibrations in the rotor as the main source of periodic vibrations before they prop- agate into the fuselage [12{15]. Spacecraft Vibration Control: In high-performance spacecraft, some components generate periodic disturbances which are detrimen- tal to performance. For example disturbances caused by gyroscopes and cryogenic coolers [16]. Identifying and attenuating the eect of disturbances can improve the performance of the instruments. (iii) Active Noise Control: The control of unwanted acoustic noises in a wide class of systems has been the subject of much engineering research over the re- cent decades. In many industrial applications, undesired sound can be classied as periodic or quasi-periodic disturbances which are mainly caused by components such as electric motors, compressors, engines, cooling systems, fans, propellers, air- conditioning systems, or can be generated by resonance coils in magnetic resonance imaging system [17{22]. In active noise control, the main objective is to minimize the noise level in an environment by producing anti-noise at sensor position, i.e., generating noise from speakers (control actuators) such that noise level at micro- phones (error sensors) position is made as small as possible. The active noise con- trol primarily deals with low-frequency acoustical noise (typically less than 500 Hz); 2 high frequency components can be eectively suppressed by using passive sound absorbers [23, 24]. Some successful applications including reduction of propeller- induced cabin noise in aircraft, engine and road noise in cars, and noise in acoustic ducts can be found in [25{27]. In most research eorts of the past two decades, the problem of attenuation/ rejection of unknown periodic disturbances is formulated as follows. Figure 1.1 shows the plant that is assumed to be known, linear time-invariant (LTI) and stable whose output is corrupted by the disturbance term d. It is inherent in the assumption that if the plant was unstable but known, a robust LTI controller could be designed to end up with the conguration of Figure 1.1 involving a stable LTI plant with a known (nominal) mathematical model. In many control applications, the plant has already been stabilized with a xed-gain controller, and the plant in Figure 1.1 is an augmentation of the baseline original control design (see [28{30]). The disturbance Figure 1.1: A stable known plant aected by unmeasurable disturbances term d can be modeled as sum of some sinusoidal terms with unknown amplitude, frequency, and phase. Another formulation of the disturbance term is to view it as the output of a lter with poles on the unit circle (in discrete-time formulations), or on the imaginary axis (in continuous-time formulations) and with impulse function as the input of the lter. It should be noted that an input disturbance with similar characteristics can be added in Figure 1.1. The control problem is to nd the control input u which will reject the periodic 3 component of the disturbance from the output and force y to go to zero as close as possible. This control problem has been extensively studied in the control literature [29{42]. It is assumed that the plant has already been stabilized using known robust control techniques and the control objective is simply to attenuate the eect of the output disturbance using an adaptive feedback control law. In [33], properties and limitations of dierent approaches for rejection of unknown periodic disturbances have been discussed. In many applications, a nominal linear model for the dynamical system can be obtained through oine experimentations. However, for plants with large parametric uncertainties and/or time variations in the dynamics, online plant identication is necessary, that is the controller must be able to identify the plant as well as disturbance parameters. In [37, 38, 43, 44], some internal model-based adaptive control approaches have been proposed. These approaches are divided into two classes of schemes: the indi- rect and direct schemes. In the indirect adaptive case an on-line estimator estimates the parameters of the internal model of the disturbance at each time instant and these estimated parameters are used to calculate the controller parameters at each instant of time as done in indirect adaptive control schemes. In the direct adap- tive case, the Youla-Kucera parametrization is used to replace the unknown internal model parameters with the controller parameters enabling a suitable parametriza- tion that allows the direct adaptation of the controller parameters without inter- mediate calculations. State-space approaches involving the design of adaptive ob- servers together with on-line frequency identiers are also used to reject periodic disturbances [45]. In [46, 47], an approach based on the concept of a phase-locked loop is proposed where the disturbance frequency estimation and the disturbance cancelation are performed simultaneously using a single error signal. The prob- lem of vibration attenuation is also studied using harmonic steady-state methods 4 and averaging in which the plant is approximated by its steady-state sinusoidal response [15,48]. Adaptive pole placement control has been used in [49, 50] for rejection of un- known disturbances aecting unknown LTI systems. It is assumed that the plant order and the number of distinct frequencies of the disturbance are known, but the plant and disturbance parameters are unknown. The global stability and con- vergence of the algorithm has been established without requirement of persistently exciting signals. The method however has some major drawbacks impairing its us- ability in practical situations. In addition to diculties in extraction of the estimate of the internal model of the disturbance from a composite polynomial especially for high-order cases, the procedure is susceptible to numerical problems due to possible division by zero. No eective procedure has been proposed for construction of some design parameters; moreover some unrealistic assumptions are made which limit its practical use. In [51], an adaptive harmonic steady-state algorithm for rejection of sinusoidal disturbances acting on unknown linear systems has been proposed, but the disturbance frequencies are assumed to be known. In [52], the idea has been ex- tended for unknown disturbances. In both cases, the local stability of the closed-loop system has been established, but no analysis for the size of the region of attraction has been provided. In fact, the case of unknown plant model and unknown dis- turbance remains an open problem as no practical solution with guaranteed global stability has been yet proposed. A state-derivative feedback adaptive controller has been proposed in [53] for cancelation of unknown sinusoidal disturbances acting on a class of continuous-time LTI systems with unknown parameters. Recently, in [54], an LTI plant model in the controllable canonical form with unknown parameters and full-state measurable is considered. A state-feedback adaptive control scheme is proposed for rejection of an unknown sinusoidal disturbance term. It is how- 5 ever not clear that this approach can be practically implemented as the robustness properties of the scheme in the presence of unmodeled dynamics has not been stud- ied; moreover, it has not been addressed how the proposed controller may perform for rejection of disturbances with multiple frequencies and possibly time-varying characteristics. Despite the considerable number of publications in the area the following prob- lems of high practical importance remain open. 1. In practice the plant transfer function in Figure 1.1 is never known exactly let alone be LTI. The eect of the inevitable plant uncertainties on any scheme proposed based on Figure 1.1 conguration is of great practical importance. In almost all publications the problem of robustness with respect to inevitable plant uncertainties and noise has been taken for granted under the argument that persistence of excitation (PE) of the regressor in the adaptive law is guaranteed due to sucient excitation by the periodic disturbance terms. The PE property guarantees exponential convergence of the estimated disturbance model parameters close to their real values, which in turn guarantees a level of robustness. This assumption however is based on a parameterization that uses the exact number of the unknown frequencies in the disturbance. Assuming an upper bound for the number of frequencies which is the practical thing to do given that the number of frequencies may change with time leads to an over parametrization in which case the regressor cannot be PE. In the absence of PE most if not all of the adaptive laws proposed in literature and even experimented with are non robust as small disturbances can lead to parameter drift and instability as pointed out using simple examples in [1]. This serious practical problem has been completely overlooked in the literature on adaptive rejection of periodic vibrations. 6 2. In most publications and applications the focus is to reject the periodic dis- turbances. In the absence of noise this objective can be achieved exactly provided the LTI plant is exactly known and stable. In the presence of noise however a scheme that rejects the periodic disturbances exactly in the absence of noise may amplify the noise and lead to a worse performance than with- out the feedback. A practical feedback control scheme should have enough structural exibility to reject the periodic disturbances without amplifying the noise. This structural exibility should be supported analytically as part of the overall design. The amplication of noise usually occurs when the zeros of the internal model of the disturbance model are close to the zeros of the plant in addition to other cases. Addressing and understanding these issues and nding appropriate solutions is critical to applications. 3. The assumption that the plant is exactly known, stable and LTI is fundamental to all approaches proposed for adaptive vibration control. The justication behind this assumption is that o line identication is used to estimate the parameters of a dominant plant model and a xed robust controller is designed to stabilize it. While this assumption may be valid under normal operations, changes in the plant parameters over time due to tear and wear or due to some component failures may easily lead to failure of the xed controller. Designing a robust adaptive control scheme which can simultaneously stabilize a plant with unknown or changing parameters in addition to rejecting periodic disturbances is recognized to be an open problem and of practical importance. 4. In most past work, it is assumed that the frequencies of the periodic distur- bance are unknown and could be changing with time. The analysis however how time variations in the values of these frequencies will aect adaptation and stability has not been carried out. 7 5. The exact cancellation of periodic disturbances often leads to a worse per- formance than that of the open loop in the case when some of the zeros of the plant are very close to some of the zeros of the internal model of the dis- turbance. In such situations it may be better to suppress rather than reject the periodic disturbances leading to a dierent adaptive control scheme with important trade os that are yet to be analyzed and understood. Since the internal model of the disturbance is unknown the problem is a challenging one and becomes even more challenging when the zeroes of the plant are also unknown. This dissertation aims to address all the above points and propose practical solutions for the problem and to provide guidelines for selection od design parameter for implementation. 1.2 Organization of the Dissertation In the following section we present some preliminary results for the cases that the nominal model of the dynamical system deviates from the actual one. We developed an adaptive control scheme that is robust with respect to the plant modeling error both for SISO and MIMO plants (in both discrete-time and continuous-time formu- lations) and motivate the approach to be taken in addressing the above problems. Then, we study the case of plants with unknown parameters. The dissertation is organized as follows. Chapter 2 proposes a robust adaptive scheme for rejection of unknown periodic components of the disturbance and analyze its stability and per- formance properties for discrete-time SISO plant. In Chapter 3, the same problem is studied for continuous-time SISO plants with application in acoustic system in which the time variations of disturbance parameters are taken into consideration and an input ltering is proposed to improve performance and robust stability. In 8 Chapter 4 and Chapter 5 we study the problem of unknown disturbance attenuation in multi-channel systems in both discrete-time and continuous-time formulations. In Chapter 6, an output-feedback adaptive controller is designed and analyzed to reject the eect of unknown additive output disturbances in linear systems with unknown parameters. Finally, future research directions are presented in Chapter 7. 9 2 SISO Discrete-Time Uncertain Systems The purpose of this chapter is to propose a robust adaptive scheme for rejection of unknown periodic components of the disturbance and analyze its stability and performance properties for discrete-time single-input single-output (SISO) systems. First, we consider the ideal case (non-adaptive) when complete information about the characteristics of the disturbance is available. We show that the rejection of periodic terms may lead to amplication of output noise and in some cases lead to a worse output performance. The way to avoid such undesirable noise amplication is to increase the size of the feedback control lter in order to have the exibility to achieve rejection of the periodic disturbance terms while minimizing the eect of the noise on the output. The increased lter order leads to an over-parameterized scheme where persistence of excitation is no longer possible, and this shortcoming makes the use of robust adaptation essential. With this important insight in mind, the coecients of the feedback lter whose size is over parameterized are adapted using a robust adaptive law. We show analytically that the proposed robust adaptive control scheme guarantees stability, performance and robustness with respect to unmodeled dynamics and bounded broadband random noise disturbances. We use numerical simulations to demonstrate the results. The results of this chapter have 10 been accepted for published in IEEE Transactions on Automatic Control [55]. 2.1 Preliminaries and Notations An m-input, q-output nite-dimensional LTI discrete-time system with transfer functionH(z), inputu(k), and outputy(k) may be expressed asy(k) =H(z)[u(k)]. For a signal x, the truncated function x k is dened as x k (i) = x(i), for 0 i k and x k (i) = 0, for i > k. TheH 1 andL 1 norms of a system with transfer matrix H(z) are dened askH(z)k 1 = max 2[0;] max (H(e j )), where max denotes the maximum singular value, andkH(z)k 1 = max q i=1 P m j=1 kh ij k 1 , where h ij is the im- pulse response corresponding to theij-element ofH(z), andkh ij k 1 = P 1 k=0 jh ij (k)j. We say signal x is -small in the mean square sense and write x2S(), when 1 N k+N1 X i=k x(i) T x(i)c 0 + c 1 N ;8k;N; and some constant c 0 ;c 1 0 that are independent of the constant [1,56]. 2.2 Control System Model We consider stable plants of the form y(k) =G(z)[u(k)] +d(k) =G 0 (z)(1 + m (z))[u(k)] +d(k) (2.1) where z =e j!Ts is the operator of the z-transformation, T s is the sampling period, y(k) =y(kT s ) is the observed output,u(k) is the control input,d(k) is an unknown (and unobserved) bounded disturbance. The plant discrete-time model G(z) (a representation of the sampled continuous-time model combined with the zero-order hold) is typically not perfectly known. We model G 0 (z) = Z 0 (z)=R 0 (z), where 11 R 0 (z) is a monic Hurwitz polynomial of degree n, as a known estimate of G(z) (possibly non-minimum phase) with m (z) representing an unknown multiplicative uncertainty such that m (z)G 0 (z) is proper with stable poles. A wide class of plant modeling errors can be expressed in the form of (2.1) [1]. In many, if not most, control applications, the plant has already been stabilized with a xed-gain controller, and the plant (2.1) is an augmentation of the baseline original control design (see [28{30]). We assume that the disturbance term d(k) is modeled as d(k) = n! X i=1 i sin(! i T s k +' i ) +v(k) =d s (k) +v(k) (2.2) where the parameters i , ! i , ' i , and the number of distinct frequencies n ! are all unknown andv is a bounded random noise disturbance with amplitude much smaller than that ofd s . It should be noted that the selection of sampling periodT s depends on the characteristics of the disturbance such as the upper bound on the highest signicant frequency present in d s in (2.2). The control objective is to minimize the eect of d on the plant output y and make the norm of y as small as possible. Figure 2.1 illustrates the structure of the closed-loop system in which the stable strictly proper lter F (z) must be designed to achieve the objective. Figure 2.1: Structure of the discrete-time closed-loop system 12 From Figure 2.1 and the small-gain theorem [57], a sucient condition for sta- bility of the closed-loop system is that k m (z)G 0 (z)F (z)k< 1; (2.3) where z = e j , = wT s 2 [0;], andkk can be any norm satisfying the multi- plicative property such asH 1 norm orL 1 norm [57]. It should be noted that in general condition (2.3) may not be necessary for stability of the closed-loop system; it is a sucient condition and states that nonzero but small size modeling errors can be tolerated in the feedback control system. Clearly, with F (z) = G 1 0 (z) for any modeling error satisfyingk m k < 1, the plant output will be zero for any distur- bance with any characteristics; however, such a lter is not realizable. In the next section, we present the design and analysis of a realizable Finite Impulse Response (FIR) lter structure for disturbance suppression in the presence of unmodeled plant dynamics and bounded noise. 2.3 Controller Design In this section we begin with the ideal case where the frequencies of the disturbance d s (k) in (2.2) are assumed to be known exactly. The known frequency case allows us to identify what is the best performance that can be achieved and set the reference performance and robustness levels to be compared with in the more realistic case where the frequencies are unknown and changing with time. 2.3.1 Known Disturbance Case This subsection considers the case when the frequencies of the disturbance are known. The analysis of the closed-loop system in this case provides insights that can 13 be used to deal with the unknown disturbance case using robust adaptive control techniques. Let F be an N th -order FIR lter of the form F (z;) = (z)=z N , where (z) = 0 + 1 z +::: + N1 z N1 , which can be written as F (z;) = N1 X i=0 i z iN = T (z) (2.4) where = [ 0 ; 1 ;:::; N1 ] T 2 R N and (z) = [z N ;z 1N ;:::;z 1 ] T . The control objective is to nd the parameter vector so that the norm of y is minimized. If the frequencies of d s are known, then its internal model D s (z) = Q n! i=1 (z 2 2 cos(! i T s )z + 1) is also known. It is obvious that if (2.3) is satised and the zeros of the sensitivity transfer function H from d =d s +v to y given by y =H(z)[d], 1G 0 F 1 + m G 0 F [d s +v] (2.5) contain the roots of D s , then the eect of d s on y will be completely eliminated. Lemma 2.1 Consider the lter (2.4) and the transfer functionH in (2.5). For any G 0 =Z 0 =R 0 , there exist a vector so that the zeros of H contain the roots of D s , if and only if Z 0 and D s are coprime and Nn d , where n d = deg(D s ). Furthermore, the vector is unique if N =n d . Proof: To prove this lemma, we need to establish that the existence of polyno- mials (z) andL(z) satisfyingZ 0 +D s L =z N R 0 is equivalent toZ 0 andD s being coprime with N n d . The proof follows by writing the above polynomial equa- tion as a Sylvester-type matrix equation Sx =, where x = [ T ;l T ] T and matrix S contains the coecients of Z 0 and D s , vector contains the coecients of z N R 0 , and vectors,l are unknown coecients of andL, respectively (see [58], Ch. 2). 14 If N = n d , the unique satisfying the conditions of Lemma 2.1 may lead to a large peak magnitude of G 0 F . This situation is not desirable, because it may lead to noise amplication, and the performance may be much worse than the case with no control input. Moreover, from (2.3), the size of the modeling error which can be tolerated may be too small leading to smaller stability margins. In order to completely reject the eect of the periodic components of the distur- bance on y, improve the robustness with respect to unmodeled dynamics without amplifying the output noise, we choose a higher order lter, N > n d . In this case, we have an innite number of vectors that achieve the rejection of the disturbance periodic terms and therefore have more exibility in selecting the best parameter vector that ts all the criteria, i.e. rejecting the periodic terms of the disturbance without amplifying the noise eect and reducing the impact on the robust stability margins. At this point, we may look for vectors from this set that minimize the objective function J() =kG 0 Fk 1 = Z 0 z N R 0 1 ; s.t. Z 0 +D s L =z N R 0 ; (2.6) for some polynomial L(z). In other words, we nd a such that at z i = e j! i Ts , F (z i ;) = G 1 0 (z i ) and J() is minimized. The existence of a minimizer is the subject of the following lemma. Lemma 2.2 Consider the closed-loop system shown in Figure 2.1. Assume the lter is of the form (2.4) and the output disturbance can be represented as in (2.2). If Z 0 and D s have no common factors and N > n d , then there exists a withjj r, for some r > 0, for which the eect of d s on y is completely rejected and the cost function (2.6) is minimized. 15 Proof: The existence of a for which we can achieve complete rejection of the sinusoidal part of the disturbance is guaranteed from Lemma 2.1. We dene Q =f :9lj Sx = g, where S, x, and are dened in the proof of Lemma 2.1. The setQ is nonempty by Lemma 2.1 and ane by inspection, and hence closed and convex. The function J is continuous and convex. As such it attains its minimum on any compact set. The problem of minimizing (2.6) can be written as an LMI optimization problem and can be solved by using the Matlab YALMIP Toolbox. Lemma 2.2 shows the existence of a constant parameter vector for complete rejection of the periodic components of disturbance signals of the class dened by (2.2). It also provides a smaller maximum peak of the magnitude of G 0 F which is equivalent to minimizing the eect of noise on the output. The following theorem gives the properties of the closed-loop system. Theorem 2.1 Consider the closed-loop system shown in Figure 2.1 with the lter (2.4). Assume the frequencies of d s are known, Nn d , and is a solution of the optimization problem (2.6). If the inequality k m (z)G 0 (z)F (z; )k 1 < 1 (2.7) holds, then lim T!1 sup kT jy(k)j 1G 0 (z)F (z; ) 1 + m (z)G 0 (z)F (z; ) 1 v 0 c(1 +kG 0 (z)F (z; )k 1 )v 0 ; (2.8) wherev 0 = sup k jv(k)j andc is a nite positive constant. In addition, in the absence of the noise (i.e., whenv 0 = 0), the plant outputy(k) goes to zero exponentially fast with time. 16 Proof: The existence of a minimizer for the constrained optimization problem (2.6) is guaranteed by Lemma 2.1, 2.2. Then, from (2.5) and using the small gain theorem [57], if for all frequencieskG 0 (z)F (z; ) m (z)k 1 < 1, then the stability of the closed-loop system is guaranteed. Since with the zeros of 1G 0 F contain the roots ofD s , theny(k) =H(z)[v(k)]+ 0 (k), where 0 (k) is an exponentially decaying term. Thus, lim T!1 sup kT jy(k)jkH(z)k 1 v 0 . The second upper bound follows by noting thatkHk 1 k1=(1 + m G 0 F )k 1 k1G 0 Fk 1 c(1 +kG 0 Fk 1 ), wherec is a generic symbol for a positive constant. This completes the proof. Theorem 2.1 shows that we can design a lter which completely rejects the eect of d s on y. However, in the presence of noise, a large magnitude of the sensitivity function H(z; ) given by (2.5) may lead to noise amplication, especially at high frequencies. The situation may be worse if the plant has very small gain at the frequency range of the disturbance and larger gain at high frequencies. In such case, the design of a pre-compensator to shape the frequency response of the plant to achieve a good compromise between performance and robustness will be a possible remedy. We illustrate these results using the following example. Example 1: Consider the following 3 rd -order model: G 0 (z) = 0:00146(z 0:1438)(z 1) (z 0:7096)(z 2 0:04369z + 0:01392) (2.9) Figure 2.2 shows the performance of the o-line designed controllers obtained from (2.6) for complete rejection of the periodic term of a disturbance, where d s is a single-frequency sinusoidal signal with amplitude one, v is a zero-mean Gaussian noise with standard deviation 0:02 andjvj 0:1. In all simulations, the sampling period is T s = 1=480 sec. In Figure 2.2(a), the frequency of d s is ! = 25 rad/sec. The performance of the controllers with dierent orders N are shown with dierent 17 colors. After closing the loop at t = 20 sec, the sinusoidal part is completely rejected but the noise part is amplied for low order lters. As the order of the lter increases it provides the exibility to reject the sinusoidal disturbance terms without amplifying the noise. Figure 2.2(b) shows the performance for dierent disturbance frequencies ! = 25, 45, and 85 rad/sec, where the lter order N = 7 is xed. The plant has a zero atz = 1 and the low plant gain (and low coprimeness of Z 0 (z) and D s (z)) at the low frequency of 25 rad/sec makes it dicult to reject the disturbance. In fact, the performance is worse than that of the open-loop system due to the amplication of the output noise. For higher frequencies however the gain of the plant corresponding to these frequencies is higher leading to a better performance as shown in Figure 2.2(b). The control design and analysis of the known frequency case is very useful as it establishes analytically that for better performance the order of the lter F has to be much larger than n d in order to provide the structural exibility to design the coecients of the lter to simultaneously reject the periodic components of the disturbance and minimize the eect of the output noise on the plant output. This analysis provides the form and dimension of the controller lter F in the adaptive case where the frequencies of the disturbances are completely unknown. We treat this case in the following subsection. 2.3.2 Unknown Disturbance Case In practice, the characteristics of the disturbance signal are unknown and may be time-varying, so an adaptive control scheme is required to meet the control objective. In this subsection, our aim is to design and analyze an adaptive FIR lter of the form F (z; ^ (k)) = N1 X i=0 ^ i (k)z iN = ^ T (k)(z) (2.10) 18 Figure 2.2: Control performance for the case of known disturbance frequencies. The control input is applied att = 20 sec. (a) xed disturbance frequency! = 25 rad/sec and dierent lter ordersN; increasingN improves the performance. (b) xed lter order N = 7 and dierent disturbance frequencies. in the feedback loop, modifying Figure 2.1 to include adaptation, as shown in Fig- ure 2.3, where 0 is a nonzero scalar. The adaptively tuned lter (2.10) is designed to minimize the output variance, suppressing to the extent possible the disturbance d. From Figure 2.3, we haveu = 0 F (z; ^ )[], where =yG 0 (z)[u], and ^ is the esti- mate of the unknown optimum parameter vector which at timet =kT s is ^ (k1). Then, we write the control signal as u(k) = 0 ^ T (k 1)w(k), where w =(z)[]. The plant output can be written as y = G 0 (z)[u] + = 0 G 0 (z)[ ^ T w] +. From 19 Figure 2.3: Structure of adaptive control system the Swapping Lemma [1] we have G 0 (z)[ ^ T w] k = ^ T G 0 (z)[w] k +G 0c (z)[(G 0b (z)[w T ]) ^ ] k ; where ^ (k) = ^ (k+1) ^ (k),G 0c (z) =C T 0 (zIA 0 ) 1 ,G 0b (z) =z(zIA 0 ) 1 B 0 , and the quadruple (A 0 ;B 0 ;C 0 ;D 0 ) represents a minimal realization of G 0 . Then, we rewrite y as y = ^ T 0 G 0 (z)[w] 0 + = ~ T 0 G 0 (z)[w] 0 +; where 0 = 0 G 0c (z)[(G 0b (z)[w T ]) ^ ], where = (1 0 G 0 (z)F (z; ))[v +u ], u = m (z)G 0 (z)[u], and ~ = ^ , where is such that 1 0 G 0 (z)F (z; ) has zero gain at the frequencies of the disturbance. The existence of such a desired vector has already been established in the previous section. From the above presentation, we have the following parametric model; (k) = T (k) +(k); (2.11) 20 where = yG 0 (z)[u] and = 0 G 0 (z)[(z)[]] are available for measurement and the unknown parameter appears linearly with respect to the known regressor signal vector. The error term is driven by the modeling error and noise and may also depend on exponentially decaying to zero terms due to initial conditions that for the sake of clarity are considered to be zero as they do not aect the analysis. From the design and analysis of the known frequency case we require N > n d in order to have the structural exibility for the adaptive control law to choose the parameters that reject the periodic disturbance terms and minimize the eect of the noise. Since the number of frequencies of d s is unknown, a conservative upper bound for this number is used. This choice leads to an over-parametrization, and the regressor vector in (2.11) cannot be persistently exciting (PE) even when the periodic disturbance terms provide adequate excitation [1]. For this reason the adaptive law used to update the estimate of the parameter vector needs to be robust, otherwise the presence of noise and modeling error terms may cause parameter drift as pointed out in [1]. This is an interesting case as even though PE can be guaranteed by the design due to the richness of the disturbance, which in turn can guarantee PE of a minimal dimensional regressor, our analysis of the known frequency case dictates that we have to over-parameterize and therefore reject the powerful property of PE in order to achieve better disturbance rejection properties. The parametric model (2.11) can be used to develop a wide class of robust parameter estimators using the approach in [56] as follows: If ^ (k1) is the estimate of at timek1, the predicted value of the signal(k) can be generated as ^ (k) = ^ T (k1)(k). We dene a normalized estimation error as"(k) = ((k) ^ (k))=m 2 (k), wherem 2 (k) = 1 +c 0 T (k)(k), withc 0 > 0, is a normalizing signal. To develop an 21 adaptation scheme for ^ we consider the robust pure least-squares algorithm [56] P (k) =P (k 1) P (k 1)(k) T (k)P (k 1) m 2 (k) + T (k)P (k 1)(k) ^ (k) = proj( ^ (k 1) +P (k)"(k)(k)) (2.12) where P (0) =P T (0)> 0 and projection operator proj() is used to guarantee that ^ (k)2S,8k, where S is a compact set dened as S( max ) =fx2R N j x T x 2 max g; where max > 0 is chosen so that the coecient vector of the optimum lter, , belongs toS. Since we do not have a priori knowledge on the norm of , max must be chosen suciently large. The projection may be implemented as (k) = ^ (k 1) +P (k)"(k)(k) (k) =P 1=2 (k) (k) (k) = 8 > > < > > : (k) if (k)2 S 0 (k) =? proj of (k) on S if (k)62 S ^ (k) =P 1=2 (k)(k); (2.13) where P 1 is decomposed into (P 1=2 ) T (P 1=2 ) at each instant k to guarantee all properties of the corresponding recursive least-squares algorithm without projection. The setS( max ) is transformed to S such that if (k)2S( max ), then (k)2 S. Since S is chosen to be convex and (k) =P 1=2 (k) (k) is a linear transformation, then S is also a convex set [59]. An alternative to projection is a xed -modication where no bounds forj j are required [1]. An additional constraint that is often used in practice to ensure 22 that the control signalu is within certain limitsu max is a saturation block inserted right afterF (z; ^ ) in Figure 2.3. The above adaptive control law in contrast to those proposed in literature for similar applications has all the robustness modications to guarantee stability in the presence of modeling errors something that needs to be analytically established as we will show below. In implementing (2.12) we also need to use modications such as covariance resetting [56] which monitors the covariance matrix P (k) to make sure it does not become small in some directions, i.e. its minimum eigenvalue is always greater than a small positive constant. Such modications are presented and analyzed in [1, 56] and other references on adaptive control. The performance of this adaptive system can be guaranteed as is detailed in the following theorem in which it is assumed that the characteristics of the distur- bance are unknown, but there exists a (unknown to designer) to achieve the best performance. Theorem 2.2 Consider the adaptive controlller (2.10) and (2.12), and chooseN > 2 n ! , where n ! is an upper bound for the number of distinct frequencies in the range of interest in the sinusoidal part of the disturbance. If the modeling error satises M k 0 m (z)G 0 (z)k 1 < 1, where M = max 2S kk 1 , then all signals in the closed- loop system are uniformly bounded and the plant output satises 1 T k+T1 X i=k jy(i)j 2 c( 2 +v 2 0 ) + c T ;8k 0;8T > 1; (2.14) and some nite constant c independent of T;k, where is a constant proportional to the size of the unmodeled dynamics, and v 0 = sup k jv(k)j. In addition, in the absence of modeling error and noise (i.e., when m = 0 and v 0 = 0), the adaptive control law guarantees the convergence of y(k) to zero. 23 Proof: From Figure 2.3, the signal can be written as =yG 0 (z)[u] =u +d = m (z)G 0 (z)[ 0 ^ T w] +d; k k k 1 k 0 m (z)G 0 (z)k 1 k( ^ T w) k k 1 +d 0 ; where d 0 = sup k jd(k)j. The projection operator ensures the boundedness of ^ and we can write k( ^ T w) k k 1 k max i=0 k ^ (i 1)k 1 kw k k 1 (max ^ 2S k ^ k 1 )kw k k 1 M k(z)k 1 k k k 1 = M k k k 1 : Thus,k k k 1 M kk 0 m (z)G 0 (z)k 1 k k k 1 +d 0 . Then, if the modeling uncertainty term is such that M k 0 m (z)G 0 (z)k 1 < 1, then(k) and therefore all signals in the closed-loop system are uniformly bounded. The normalized estimation error can be expressed as "(k) = T (k) ~ (k 1) +(k) m 2 (k) = y(k) + 0 (k) m 2 (k) : From (2.12), P (k) is bounded and the covariance resetting ensures that P (k) is positive denite. It can be shown (see [56], Ch. 4) that the parameter error ~ can be written as ~ (k) =P (k)P 1 (k 1) ~ (k 1) +P (k)(k) (k) m 2 (k) : 24 Consider the Lyapunov-like function V (k) = (k) T P 1 (k) (k) = ( (k) ) T ( (k) ); whose boundedness is guaranteed due to the projection and covariance resetting, where =P 1=2 (k) 2 S. If (k)2 S, V (k) = ~ T (k)P 1 (k) ~ (k), then we have V (k) =V (k)V (k 1) = 0 (k)" 2 (k)m 2 (k) + 1 (k)"(k)m(k) (k) m(k) + 2 (k) 2 (k) m 2 (k) ; where 0 (k) = 1=(1 + T (k)P (k 1) (k)), 1 (k) = 2( 0 (k) + T (k)P (k) (k) 1), 2 (k) = 2 0 (k) T (k)P (k) (k), and (k) = (k)=m(k). Then, from the boundedness of (k) and P (k), it follows that V 0 " 2 m 2 + 1 j"mjj=mj + 2j=mj 2 ; where 0 = 1=(1 + ' 0 max (P 0 )), ' 0 = sup k j (k)j 2 , 1 = 2= 0 are positive constants. Using the inequalityax 2 +bx(a=2)x 2 +b 2 =(2a) we have V( 0 =2)" 2 m 2 + (2 + 2 1 =(2 0 )) 2 =m 2 c" 2 m 2 +c 2 =m 2 ; where c is a generic symbol for a positive constant. The error term in (2.11) is = (1 0 G 0 (z)F (z; ))[v +u ]; then j=mj 2 c(v 2 0 + 2 ), 2 ; for some nitec> 0. Thus, V (k)c" 2 (k)m 2 (k)+c 2 and by taking summation 25 on both sides we obtain 1 T T X k=1 " 2 (k)m 2 (k)c 2 + c T (V (0)V (T ))c 2 + c T ; that is "m2S( 2 ). Since m 2 1, we have " 2 " 2 m 2 , therefore "2S( 2 ). From (2.12), ^ =P" =P"m , then j ^ j 2 2 max (P )j"mj 2 j j 2 2 max (P 0 ) ' 0 j"mj 2 ; which implies thatj ^ j2S( 2 ). If (k)62 S, then we have V (k) = ( 0 (k) ) T ( 0 (k) ), where 0 (k) is an orthogonal projection of (k) on S. Then, since 2 S, ( 0 (k) ) T ( 0 (k) ) ( (k) ) T ( (k) ) = ~ T (k)P 1 (k) ~ (k): This enables us to establish all properties of the adaptive law without projection [59]. Since w is bounded and ^ 2S( 2 ), then 0 2S( 2 ). Also, from y ="m 2 0 , we have jyj 2 2 m 2 j"mj 2 + 2j 0 j 2 ; where m 2 is an upper bound for m 2 , which implies y2S( 2 ). This completes the proof of Theorem 2.2. Theorem 2.2 states that the average energy of the plant output is of the order of the size of the modeling error and the noise level. It follows from (2.14) that as T goes to innity the term c=T that is due to initial conditions also goes to zero. In the absence of the modeling error and noise, the output converges to zero despite the presence of completely unknown periodic components. We demonstrate 26 the performance of the proposed robust adaptive control scheme using the follow- ing example where we assume unknown frequencies, amplitudes, and phases of the periodic terms in the disturbance. Remark 2.1 The use of positive scalar 0 is to increase the level of the regressor signal vector which is benecial to convergence. A large value of 0 , however, reduces the stability margin given by Theorem 2.2. This trade o can be improved by replacing the constant 0 with a strictly proper stable lter 0 (z) which can be designed to improve convergence and have a positive rather than negative impact on robustness. In addition, 0 (z) can be chosen to improve the gain of G 0 (z) when the zeros of G 0 (z) are very close to the zeros of the generating polynomial of the disturbance. Example 2: Consider the plant model (2.9). For simulation purposes, we consider the multiplicative uncertainty m (z) = 0:0002 (z + 0:99) 2 ; whose magnitude (for z = e j!Ts ) is of the order of80 dB for ! < 900 rad/sec and increases to 6 dB as !! =T s = 1508 rad/sec. The sampling period is T s = 1=480 sec, N = 50, c 0 = 1, P (0) = 20I, 0 = 100, v is a zero-mean Gaussian noise with standard deviation 0:02, andjvj 0:1. The periodic term of the disturbance that we like to reject is d s (t) = 0:7 sin(25t + 3 ) + 0:5 sin(225t + 4 ): Figure 2.4 shows the performance of the adaptive scheme, where at t = 10 sec the feedback is switched on and at t = 30 sec new sinusoidal terms 0:6 sin(85t 6 ) and 27 0:4 sin(125t + 2 ) are abruptly added to the disturbance. It is clear from simulations that the feedback control rejects the periodic terms in the plant output and adjusts itself to reject new ones at t = 30 sec automatically with a very small transient despite the presence of noise and unmodeled dynamics. Figure 2.4: The performance of the adaptive scheme. The control input is applied at t = 10 sec. At t = 30 sec new sinusoidal terms are abruptly added to the disturbance; the adaptive controller updates its parameters at this time to suppress the eect of new terms. 28 3 SISO Continuous-Time Uncertain Systems Eective attenuation of the noise level is an important problem in acoustic systems. In this chapter, we propose a robust adaptive output feedback control scheme that can considerably attenuate narrow-band noises made up of periodic signals mixed with random noise in the presence of modeling uncertainties for continuous-time single-input single-output (SISO) systems. The amplitude, phase and frequencies as well as the number of periodic terms are unknown and could vary with time. The performance and robustness of the proposed scheme with respect to unstructured modeling uncertainties are analyzed for continuous-time SISO systems; the results, however, are extendable to multi-channel systems. The successful attenuation of the unknown periodic components of the disturbance despite the time variations, modeling errors, and random noise is demonstrated using simulations. In addition, guidelines how to choose certain design parameters for performance improvement have been presented. The results of this chapter have been accepted for published in Journal of Vibration and Control [60]. 29 3.1 Notations and Preliminaries For a signalx(t), the truncated signalx t for anyt2 [0;1) is dened asx t () =x(), for 0t, andx t () = 0, for >t [1]. A nite-dimensional linear time-invariant (LTI) system with transfer functionG(s), inputu(t) and outputy(t) may be written as y(t) = G(s)[u(t)] =L 1 fG(s)U(s)g, where U(s) is the Laplace transform of u(t),L 1 () denotes the inverse Laplace transform, and s is the Laplace variable. TheH 1 -norm of a system is dened askG(s)k 1 = max ! jG(j!)j and theL 1 - norm iskG(s)k 1 = R 1 0 jg()jd, where g(t) is the impulse response of the system. For a system with minimal realization of degree n, the two norms are related as kG(s)k 1 kG(s)k 1 (2n + 1)kG(s)k 1 [1,57,61]. We say functionx : [0;1)!R n is -small in the mean square sense and write x2S(), if for a given constant 0, we have Z t+T t x T ()x()dc 0 T +c 1 ; for any t;T 0 and some nite constants c 0 ;c 1 0 independent of [1]. Swapping Lemma [1]: Let x(t);y(t) : R + ! R n and x(t) be dierentiable. Let W (s) be a proper stable rational transfer function with a minimal realization (A;B;C;D), i.e., W (s) =C T (sIA) 1 B +D. Then, W (s)[x T (t)y(t)] =x T (t)W (s)[y(t)] +W c (s)[(W b (s)[y(t)]) T _ x(t)]; where W c (s) =C T (sIA) 1 and W b (s) = (sIA) 1 B. 3.2 Problem Formulation We consider an acoustic feedback conguration with a single loadspeaker (actuator) and a single microphone (error sensor) as shown in Figure 3.1, where y(t) is the 30 output signal from the microphone andu(t) is the input signal to the loadspeaker. Figure 3.1: The acoustic model with one control loadspeaker and one microphone The objective is to produce a control signalu(t) to minimize the noise level at the microphone position. The block diagram of the system is shown in Figure 3.2, where d(t) is an unknown acoustical disturbance which is not directly observable whose characteristics may be time-varying, and G(s) is the transfer function from u(t) to y(t) which is typically not perfectly known consisting of speaker, acoustic path, and microphone dynamics. We consider G 0 (s) as the known modeled part of the plant with the multiplicative uncertainty term m (s) representing modeling errors, time delays, etc. [1]. Since in real physical systems a very accurate linear model over a wide range of frequencies is usually unavailable especially at high frequency ranges, then any model must include such sources of uncertainties and its eect on stability and performance must be carefully investigated. Figure 3.2: The block diagram of the acoustic system with modeling uncertainty Based on Figure 3.2, the plant dynamics are expressed as G(s) =G 0 (s)(1 + m (s)); (3.1) 31 where G 0 (s) is assumed to have stable poles, i.e. poles in the open left half plane with no restriction on the location of its zeros. The modeling uncertainty term m (s) accounts for neglected dynamics and is assumed to be small in some norm sense in the low frequency range which is the range of interest, and may be large at high frequencies. It may represent unmodeled input delay, fast actuator dynamics and other modeling errors. An example of such uncertainty is m (s) =s, where 0 < 1 is a small constant, which may represent the approximation of a time delay in the input, i.e., u(t) whose Laplace is e s [u(t)] (1s)[u(t)]. The parameters and structure of m (s) are unknown and it is just assumed that G 0 (s) m (s) is proper and BIBO stable. In many applications, the unknown acoustical disturbance d(t) is dominated by periodic unknown signals and can be modeled as a sum of some sinusoidal terms corrupted with low-amplitude broadband random noise as d(t) = n f (t) X i=1 a i (t) sin(! i (t)t +' i (t)) +v(t); (3.2) where the parameters a i ;! i , ' i , the number of distinct frequencies n f are unknown and may vary with time and v is a broadband random noise with amplitude much smaller than that of the periodic terms. We assume that an upper bound n max for the maximum number of distinct frequencies of the periodic components of the disturbance and an upper bound ! max for its largest frequency are known, i.e., n f (t) n max and ! i (t) ! max , for any t. If no knowledge about these parameters is available, one may choose conservative upper bounds. In active acoustic noise control problems, we often deal with disturbances at low frequency ranges, because passive sound absorbers can eciently suppress distur- bances with high frequency components [24]. The proposed control scheme in this chapter however can be used for any range of frequencies over which the size of 32 modeling uncertainties is suciently small. The control law must produce the control signal u(t) by using the knowledge of the output signal y(t) and nominal plant model G 0 (s) to counteract the eect of the unknown disturbance d(t) on y(t) as much as possible. It is obvious from Figure 3.2 that if G(s)[u(t)] =d(t), then y(t) = 0. This choice of u(t) however is not acceptable as this control law cannot be implemented in practice as G(s) even if known exactly may have unstable zeros and its inverse is not causal. In the following section, we present a feedback adaptive control that can meet the objective of attenuating the unknown output disturbance despite the presence of random noise and modeling uncertainties. 3.3 Controller Architecture Since the characteristics of the disturbance are unknown and may vary with time, an adaptive control law is designed to achieve the control objective. We consider the controller structure shown in Figure 3.3, whereF (s) is a xed-parameter BIBO stable lter and K(s;) is an adaptive lter whose parameter vector (t) is to be generated at each time t. Figure 3.3: Structure of the controller We assume the adaptive lter K has the following form K(s;(t)) = N1 X k=0 k (t) Nk (s +) Nk ; (3.3) 33 where(t) = [ 0 (t); 1 (t);:::; N1 (t)] T is the controller parameter vector, the scalar > 0, and integer N are design parameters. The parameter vector (t) is to be adjusted at each time t by an adaptive law to be designed. Before designing the adaptive law we examine whether there is exists a desired vector (which (t) is trying to nd), which meets the control objective. From Figure 3.2 and 3.3, the transfer function from d to y is given by y(t) =S(s;)[d(t)] = 1G 0 0 (s)K(s;) 1 + m (s)G 0 0 (s)K(s;) [d(t)]; (3.4) where G 0 0 (s) =G 0 (s)F (s) and the lter F (s) (to be designed) is such that G 0 0 (s) is BIBO stable and strictly proper. The eect of the periodic components of d on y is rejected if the zeros of S(s) include s =j! i , for i = 1;:::;n f . In other words, the periodic components ofd(t) can be rejected exactly ifS(s) contains the internal model of those components. This can be achieved if the ltersF (s) andK(s;) are chosen so that S(s) contains the internal model of all the periodic terms in d(t). Such choice however should be such that the poles of S(s) remain in the stable region. Assuming that we have perfect knowledge of the frequencies ind(t), is the struc- ture of the controller given in Figure 3.3 able to meet the control objective for some choice of the lters F (s) and K(s;)? In particular, before we proceed with the case of the unknown disturbance we address the following questions: Does there exist lters F (s) and K(s;) that reject the periodic terms in d(t)? If such lters exist how do they aect stability and performance? How does the plant uncertainty m (s) aects stability and performance? The following theorem provides answers to the above questions and becomes the reference of the best that can be achieved in the case unknown disturbance frequencies. 34 Theorem 3.1 Consider the closed-loop system shown in Figure 3.2 and 3.3 with an LTI lter of the form (3.3). Let ! 1 ;:::;! n f be the distinct frequencies of the disturbance. There exists a such that K(s; ) completely rejects the periodic components of the disturbances if and only if G 0 0 (s) = G 0 (s)F (s) has no zero at s = j! i , for i = 1;:::;n f , and N 2n f , provided the stability condition k m (s)G 0 0 (s)K(s; )k 1 < 1 is satised. The choice of is unique if N = 2n f . In addition, all signals in the closed-loop system are guaranteed to be uniformly bounded and the plant output satises lim t!1 sup t jy()jkS(s; )k 1 v 0 c(1 +kG 0 0 (s)K(s; )k 1 )v 0 ; (3.5) where v 0 = sup t jv(t)j and c = 1 1+m(s)G 0 0 (s)K(s; ) 1 is a nite positive constant. Moreover, in the absence of broadband noise (v(t) = 0), y(t) converges to zero exponentially fast. Proof:: From (3.4), sinceG 0 0 (s) andK(s;) for any parameter vector are BIBO stable transfer functions, from the small gain theorem [1,57,62], the transfer function from d(t) to y(t) is BIBO stable ifk m (s)G 0 0 (s) K(s; )k 1 < 1. This condition guarantees that all signals in the closed-loop system are uniformly bounded. From (3.4), the eect of the periodic components of d(t) on y(t) is completely rejected for any m (s) satisfying the stability condition, if and only if 1G 0 0 (s)K(s;) has zero gain at the frequencies of the disturbance! i 's,i = 1;:::;n f , that is ats =j! i , K(s;) must be equal to the inverse of G 0 0 (s). So,G 0 0 (s) must not have any zero on 35 the j!-axis at the frequencies of the disturbance. From (3.3), we can write K(s;) = 0 N + 1 N1 (s +) +::: + N1 (s +) N1 (s +) N = 0 + 1 s +::: + N1 s N1 (s +) N , X(s) (s +) N ; (3.6) where i 's can be uniquely obtained from i 's. LetG 0 0 (s) =Z 0 (s)=R 0 (s), whereR 0 (s) is a nomic Hurwitz polynomial of degreen, then complete rejection of periodic terms ind(t) is possible if and only if the numerator of 1G 0 0 (s)K(s;) has the generating polynomial of the period disturbances as a factor, i.e., when there exist polynomials X(s) of degree at most N 1 and Y (s) of degree at most n +N 2n f such that Z 0 (s)X(s) +D s (s)Y (s) = (s +) N R 0 (s) (3.7) where D s (s) = Q n f i=1 (s 2 +! 2 i ) is the generating polynomial of the period part of the disturbance. The polynomial Diophantine equation (3.7) can be written as an algebraic system of linear equations withN +n + 1 equations and 2N +n 2n f + 1 unknowns, then the number of independent equations does not exceed the number of unknowns if N 2n f . The Diophantine equation (3.7) with N 2n f is solvable if and only if every common divisor ofZ 0 (s) andD s (s) divides (s+) N R 0 (s). Since all roots ofD s (s) are on thej!-axis and (s +) N R 0 (s) is Hurwitz, then we always can nd polynomials X(s) and Y (s) when Z 0 (s) and D s (s) are coprime and N 2n f . From (3.4) and the input-output stability results [1, 57, 62], for any periodic disturbance rejecting parameter vector we have lim t!1 sup t jy()jkS(s; )k 1 v 0 : The second upper bound in (3.5) follows from the multiplicative property of the 36 L 1 -norm and triangle inequality, i.e., kS(s; )k 1 = 1G 0 0 (s)K(s; ) 1 + m (s)G 0 0 (s)K(s; ) 1 1 1 + m (s)G 0 0 (s)K(s; ) 1 (1 +kG 0 0 (s)K(s; )k 1 ): This completes the proof of Theorem 3.1. Theorem 3.1 gives conditions under which we can completely reject the periodic components of the disturbance when the frequencies of the disturbance are known and xed with time. It shows that if the plant uncertainty satises an upper bound, the output is of the order of the broadband random noise level. When this noise is absent the output converges to zero exponentially fast demonstrating that the periodic terms of the disturbance are completely rejected. It should be noted that with N = 2n f , there exists a unique for which complete rejection of the period part of the disturbance is possible. Such a unique value however does not provide any exibility to improve performance and/or robust stability margins. As shown in the proof of Theorem 3.1, forN > 2n f there is an innite number of vectors that guarantee the results of Theorem 3.1. In such case, one may choose the that minimizeskG 0 0 (s)K(s; )k 1 . The minimization ofkG 0 0 (s)K(s; )k 1 , while maintaining rejection of the periodic terms and stability, guarantees that the norm sensitivity function from the output to the random noisev(t) is also minimized over the set of all possible . If the frequencies of the disturbance are varying with time in some sense then (t) is a function of time. It can be easily shown that if the time derivatives of the unknown frequencies of the disturbance are -small in the mean square sense then _ (t) is also -small in the mean square sense i.e. for some constants c 0 ; 0 0, R t+T t k _ ()k 2 d 0 T +c 0 [1,63]. 37 3.4 Adaptive Control Design From Figure 3.3, the control input signal is given by u(t) =F (s)[K(s;(t))[z(t)]] z(t) =y(t)G 0 (s)[u(t)]; (3.8) where(t) = [ 0 (t);:::; N1 (t)] T is the estimate of a desired parameter vector (t) at each time, and K(s;(t)) = 0 (t) N + 1 (t) N1 (s +) +::: + N1 (t)(s +) N1 = (s +) N : (3.9) In order to estimate online, we express it in the form of a parametric model as follows [1,64]. We dene w(t),G 0 0 (s)[K(s; (t))[z(t)]] =G 0 0 (s)[ T (t)(s)[z(t)]]; (3.10) where (s) = [ N (s+) N ;:::; s+ ] T is a BIBO stable transfer function matrix and G 0 0 (s), G 0 (s)F (s). Since G 0 0 (s)K(s; (t)) at each frozen time t is stable, then from Theorem 3.4.11 in [1], there exists a 0 > 0 such that ifk _ (t)k2S( 0 ) for any 0 2 [0; 0 ), the LTV system from z(t) to w(t) is BIBO stable. The vector (t) is such that at each time t, S(s; (t)), 1G 0 0 (s)K(s; (t)) 38 has zero gain at the frequencies of the disturbance ! i , i = 1;:::;n f , then e(t),z(t)w(t) =S(s; (t))[z(t)] =S(s; (t))[d(t) + m (s)G 0 (s)[u(t)]] =S(s; (t))[v(t) + m (s)G 0 (s)[u(t)]]; (3.11) where the possible exponentially decaying terms are neglected as they do not aect the analysis. From the Swapping Lemma (see Section 3.1), w(t) =G 0 0 (s)[ T (t)(s)[z(t)]] = T (t)G 0 0 (s)(s)[z(t)] +f( _ (t)); (3.12) where f( _ (t)) =G 0 0c (s) h (G 0 0b (s)[((s)[z(t)]) T ]) _ (t) i ; G 0 0b (s) = (sIA g ) 1 B g ; G 0 0c (s) =C T g (sIA g ) 1 ; (3.13) and the triple (A g ;B g ;C g ) is a minimal state-space realization of G 0 0 (s). The last term in the right hand side of (3.12) is due to the time variations in the parameters of the disturbance, that acts as a disturbance term. By adding z(t) to both sides of (3.12) we obtain z(t) = T (t)G 0 0 (s)(s)[z(t)] + (z(t)w(t)) +f( _ (t)); 39 Then, a parametric model is obtained as z(t) = T (t)(t) +(t); (t) =G 0 0 (s)(s)[z(t)] =F (s)G 0 (s)(s)[z(t)]; (t) =e(t) +f( _ (t)); (3.14) where z(t) = y(t)G 0 (s)[u(t)] and (t) are available for measurement, the error term(t) is unknown and goes to zero ifv(t) = 0, m (s) = 0, and _ (t) = 0 (i.e., in the absence of broadband noise, modeling error, and if disturbance characteristics do not change with time). Using parametric model (3.14), a robust adaptive law can be designed to estimate (t) and achieve the control objective. The amplitude of the regressor vector(t) at the frequencies of the disturbance is an important factor contributing to the rate of adaptation. In (3.14), the regressor(t) is obtained by passingz(t) throughW (s) = F (s)G 0 (s)(s). If W (s) severely suppresses some of the frequency components of z(t), there will not be adequate information about those frequencies in the regressor and this will lead to a very small or zero learning rate of the unknown frequencies. As a result, the feedback controller will have diculty attenuating the respective disturbance periodic terms associated with these frequencies. In order to remedy or improve this situation we design the LTI lter F (s) such that F (s)G 0 (s)(s) has a large enough gain over the frequency range of 0 ! ! max . It is to be noted that is chosen such that (s) = [ N (s+) N ;:::; s+ ] T has a large enough gain over the frequency range of interest. 3.4.1 Design of the LTI lter F (s) The objective of introducing the lterF (s) is to atten the spectrum of all elements of W (s) = F (s)G 0 (s)(s) over the frequency range ! 2 [0; ! max ] to the extent 40 possible. Since all elements of (s) are chosen to be low-pass lters with DC gain one and bandwidth greater than ! max , we just need to choose F (s) such that G 0 0 (s) = F (s)G 0 (s) has a suciently large gain at almost all frequencies in the frequency range of ! ! max . Let ~ G 0 (s) be a rational transfer function obtained from G 0 (s) by modifying the numerator of G 0 (s) as follows. 1. Let ~ G 0 (s) =G 0 (s). 2. If ~ G 0 (s) has a zero on the j!-axis at s = j! 0 , change it to s = 0 +j! 0 , where 0 > 0 is a small positive constant. 3. If ~ G 0 (s) has an unstable zero at s = 0 j! 0 with 0 > 0, change it to s = 0 j! 0 , i.e., re ect it across the j!-axis and make it stable. Now, we let F (s) = 0 m (s +) m ~ G 1 0 (s), 0 F (s); (3.15) where m > 0 is an integer greater than the relative degree of G 0 (s) and > 0 is a design constant chosen such that the low-pass lter m (s+) m has a large enough gain over the frequency range of interest. Then G 0 0 (s) = F (s)G 0 (s) is a BIBO stable strictly proper transfer function with gain about 0 at most frequencies over ! ! max except at and around the frequencies whereG 0 (s) has zeros on thej!-axis. Example: Assume the nominal plant model is given by G 0 (s) = s(s 2 + 4)(s 0:8)(s + 1:4) (s + 0:5) 3 (s + 2) 2 (s + 3) : 41 Then the lter F(s) can be designed as: F (s) = 0 2 (s + 0:5) 3 (s + 2) 2 (s + 3) (s +) 2 (s + 0 )((s + 0 ) 2 + 4)(s + 0:8)(s + 1:4) ; where 0 is a positive small constant, e.g., 0 = 0:01. Let 0 = 1 and assume ! max = 300 rad/sec, then we choose = 300. Figure 3.4 shows the magnitude plot of G 0 (s) and G 0 0 (s) =F (s)G 0 (s) which has gain of 0 = 1 over the frequency range of interest except at and around ! = 0 and 2 rad/sec at which G 0 (s) has zeros on the j! axis. It should be noted that from Theorem 3.1, for this plant model, rejection of disturbances with frequencies ! = 0 and 2 rad/sec is impossible as the plant has zero gain at these frequencies. The above procedure increases the gain of the nominal plant by introducing a lterF (s) and attening the magnitude of its frequency response over the frequency range of interest. However, some other considerations must be taken into account in order to ensure the stability of the closed-loop system in the presence of unmodeled dynamics. We will show below that the stability of the adaptive closed-loop system is guaranteed if the norm of F (s)G 0 (s) m (s) is suciently small. With lter F (s) obtained from the above procedure, the gain of F (s)G 0 (s) at low frequency range ! < is less than or equal to 0 and at very high frequencies ! is very small. Since the size of the modeling error term m (s) is usually small at low frequencies and is possibly large at high frequencies, then if the magnitude of m (s) at low frequencies is small enough and the roll-o rate of F (s)G 0 (s) for ! is high enough, the size of F (s)G 0 (s) m (s) is small in the low and high frequency range; however, in the moderate frequency range, especially when ! max is large, F (s)G 0 (s) m (s) may have a large peak magnitude which may lead to instability of the closed-loop system. Since the frequency range where the unmodeled dynamics term m (s) is dominant is unknown, we may need to choose conservative parameters 42 Figure 3.4: The magnitude plot of (a)G 0 (s) and (b)G 0 0 (s) =F (s)G 0 (s) with 0 = 1 and = 300. The plant nominal modelG 0 (s) has zeros on thej! axis ats = 0 and s =2j. F (s) attens the magnitude plot of the plant model over the frequency range of interest. for the lter F (s). In (3.15), the parameters 0 and must be chosen carefully to have a good stability margin and a satisfactory performance. For example, increasing the value of 0 can speed up adaptation and improve performance, but it may reduce the margin of stability in the presence of modeling error. Also, decreasing reduces the bandwidth of F (s)G 0 (s) and compensates the high-frequency unmodeled dy- namics; hence improves the stability margin, but adversely aects the performance if the disturbance has dominant terms with frequencies beyond the bandwidth of F (s)G 0 (s). These trade os are well known in robust control and robust adaptive control and need to be kept in mind in choosing the various design parameters [1]. 43 In [65] a conceptually similar approach has been proposed to improve the gain of G 0 (s) by using the Youla-Ku cera parametrization and an FIR lter in the feedback loop. 3.4.2 Design of the adaptive lter K(s;(t)) Using the parametric model (3.14) and employing the techniques discussed in [1,56] we design a robust adaptive law to estimate the unknown parameter vector (t). The estimated model of (3.14) is obtained by replacing (t) by its estimate(t) at time t and ignoring the unknown error term (t) as ^ z(t) = T (t)(t) (3.16) where ^ z(t) is the predicted value ofz(t) based on(t) at timet. In order to measure the dierence between(t) and (t) at each time, we dene the estimation error as "(t) = z(t) ^ z(t) m 2 s (t) m 2 s (t) = 1 + 0 T (t)(t) (3.17) where m 2 s (t) is a normalizing signal and 0 > 0 is a design constant. Following the design procedure of [1,56] we consider the robust least squares algorithm to generate the estimates of (t): _ P (t) =P (t) (t) T (t) m 2 s (t) P (t) (3.18a) _ (t) = proj (P (t)"(t)(t)); (3.18b) where (0) = 0, P (0) =P 0 =P T 0 > 0 is a design constant matrix. To prevent P (t) in (3.18a) from becoming singular, we can use covariance resetting modication [1], 44 i.e., set P (t + r ) = 0 I, where t r is the time at each min (P ) 1 , and 0 > 1 > 0 are some design constants (where min denotes the minimum eigenvalue). This modication guarantees thatP (t) for all time is positive denite and never becomes close to singularity. The projection operator proj() in (3.18b) is used to ensure that the estimated vector (t) belongs to a known compact set dened as S =f2 R N jg(), T 2 max 0g, where max > 0 is chosen such that (t)2S,8t. Since no a priori knowledge on the norm of is available, max must be chosen suciently large. Some alternatives to projection and other robust modications can be found in [1]. The projection operator in (3.18b) may be implemented as _ (t) = 8 > > > > > > > > > > < > > > > > > > > > > : P" if 2S 0 or if 2S and (P") T rg 0; P"P rgrg T rg T Prg P" otherwise (3.19) whereS 0 =f2R N jg() = T 2 max < 0g andS =f2R N jg() = T 2 max = 0g denote the interior and the boundary of S. The following theorem summarizes properties of the adaptive closed-loop system shown in Figure 3.2 and 3.3. Theorem 3.2 Consider the closed-loop system shown in Figures 3.2 and 3.3 with an adaptive lter of the form (3.3), and choose N 2 n max . If the unmodeled dynamics satisfy 0 M k F (s)G 0 (s) m (s)k 1 < 1; (3.20) where M = max 2S kk 1 , 0 > 0 is a design constant, and F (s) is dened in (3.15), 45 then all signals in the closed-loop system are uniformly bounded and the plant output satises lim T!1 sup 1 T Z t+T t jy()j 2 dc(v 2 0 + 2 m + 2 _ ); (3.21) for any t and some nite constant c > 0 independent of T;t, where v 0 = sup t v(t), m > 0 is a constant proportional to the size of the unmodeled dynamics, and _ > 0 is a constant proportional to the size ofk _ k. In addition, in the absence of broadband noise and modeling error and when the disturbance frequencies do not vary with time (i.e., when v(t) = 0, m (s) = 0, and _ = 0), the plant output y(t) converges to zero. Proof: From (3.8) and Figure 3.2 we have z(t) =y(t)G 0 (s)[u(t)] =G 0 (s) m (s)[u(t)] +d(t) = 0 G 0 (s) m (s) F (s)[ T (t)(s)[z(t)]] +d(t) where F (s) = m (s+) m ~ G 1 0 (s) is dened in Section 3.4.1. Then, kz t k 1 0 k F (s)G 0 (s) m (s)k 1 k( T (s)[z]) t k 1 +d 0 ; (3.22) where d 0 > 0 is an upper bound for the magnitude of d(t). Since the boundedness 46 of (t) is guaranteed by the projection, then we have k( T (s)[z]) t k 1 = sup 0t j T ()(s)[z()]j sup 0t j T ()j 1 j(s)[z()]j 1 M k((s)[z]) t k 1 M k((s)[z]) t k 1 M k(s)k 1 kz t k 1 = M kz t k 1 (3.23) where M = max 2S kk 1 . From (3.22) and (3.23) we obtain kz t k 1 0 M k F (s)G 0 (s) m (s)k 1 kz t k 1 +d 0 ; (3.24) then, if 0 M k F (s)G 0 (s) m (s)k 1 < 1, then z(t) and hence all signals in the closed- loop system are uniformly bounded. From (3.8), (3.16), and using the Swapping Lemma [1] we have y(t) =z(t)G 0 0 (s)[ T (t)(s)[z(t)]] =z(t) T (t)G 0 0 (s)(s)[z(t)]f( _ (t)) =z(t) ^ z(t)f( _ (t)); where f( _ (t)) = G 0 0c (s) h (G 0 0b (s)[((s)[z(t)]) T ]) _ (t) i , and G 0 0c (s) and G 0 0b (s) are de- ned in (3.13). Then, the estimation error (3.17) can be written as "(t) = z(t) ^ z(t) m 2 s (t) = ~ T (t)(t) +(t) m 2 s (t) = y(t) +f( _ (t)) m 2 s (t) ; (3.25) where ~ (t) =(t) (t). From (3.18a), _ P (t) 0,8t, thenP (t) is nonincreasing and 47 covariance resetting modication guarantees that P (t) is always positive denite. Then, P 1 (t) is a positive denite bounded matrix for any t. Now, consider the following Lyapunov-like function V (t) = 1 2 ~ T (t)P 1 (t) ~ (t) which is ensured to be bounded and its time-derivative is given by _ V (t) = 1 2 ~ T (t) d dt (P 1 (t)) ~ (t) + ~ T (t)P 1 (t) _ ~ (t) where d dt (P 1 (t)) =P 1 (t) _ P (t)P 1 (t) = (t) T (t) m 2 s (t) _ ~ (t) = _ (t) _ (t) =P (t)"(t)(t) (t) _ (t) where = 8 > > > > > > > > > > < > > > > > > > > > > : 0 if 2S 0 or if 2S and (P") T rg 0; P rgrg T rg T Prg P" otherwise: Also, from (3.25), ~ T ="m 2 s +, then _ V = ( ~ T ) 2 2m 2 s +" ~ T ~ T P 1 (t) ~ T P 1 _ = 1 2 " 2 m 2 s + 1 2 2 m 2 s ~ T P 1 (t) ~ T P 1 _ : (3.26) 48 The third term in the right-hand side of (3.26) which is due to projection is zero if 2S 0 or if2S with (P") T rg 0. SinceS is convex and that belongs toS, then if 2S, ~ T rg 0. Therefore, for 2S with (P") T rg> 0 we have ~ T P 1 (t) = ( ~ T rg)((P") T rg) rg T Prg > 0: Since _ T rg 0 the projection never allows to leaveS, then this term only makes _ V more negative, then _ V 1 2 " 2 m 2 s + 1 2 2 m 2 s ~ T P 1 _ : (3.27) From (3.11) and (3.14), the error term (t) can be written as (t) =S(s; (t))[v(t) + m (s)G 0 (s)[u(t)]] +f( _ (t)); then j(t)j 2 m 2 s (t) c(v 2 0 + 2 m + 2 _ ); (3.28) where v 0 = sup t v(t), m > 0 is a constant proportional to the size of unmodeled dynamics, _ > 0 is a constant proportional to the size of _ , and c is a positive constant independent of v(t), m (s), and _ . Integrating both sides of (3.27) we obtain Z t+T t " 2 m 2 s d = 2(V (t)V (t +T )) + Z t+T t 2 m 2 s 2 ~ T P 1 _ d: 49 Since ~ and P 1 are bounded, it follows that "m s 2S(v 2 0 + 2 m + 2 _ ). Then, _ =P" =P ("m s )( m s )2S(v 2 0 + 2 m + 2 _ ) and y =m s ("m s )f( _ )2S(v 2 0 + 2 m + 2 _ ); which implies (3.21). This completes the proof of Theorem 3.2. Theorem 3.2 states that with the proposed controller if the size of modeling uncertainty is small enough, the energy of the output signal will be of the order of broadband noise level and the size of unmodeled dynamics. Even though the bounds in (3.20) and (3.21) cannot be explicitly calculated as they depend on the unknown parameter and m (s), they oer some indication how to choose the lter F (s) and other design parameters. The performance bound (3.21) is typical in robust adaptive control. It does not prevent the output y(t) to assume large values over short interval of times a phenomenon known as \bursting" [1]. This phenomenon can be prevented by further modifying the adaptive law using a dead-zone in addition to projection. 3.5 Numerical Simulation Consider the system shown in Figure 3.2. The actual transfer function from u to y is given by G(s) =G 0 (s)(1 + m (s)), where G 0 (s) = 0:5(s 0:2) s 2 +s + 1:25 50 is the known modeled part of the plant and m (s) =s, with = 0:001, is the unknown unmodeled dynamics. The magnitude of m (s) is very small at low fre- quencies and is large at high frequencies. Assume the number of distinct frequencies of the periodic terms of the disturbance is at most n max = 5 and the largest fre- quency of interest of the disturbance is less than ! max = 600 rad/sec. The unknown disturbance signal is given by d(t) = 0:6 sin(! 1 t + 4 ) + 0:7 sin(! 2 t + 2 ) +v(t); (3.29) where! 1 = 70 rad/sec and! 2 = 187 rad/sec,v(t) is a zero mean Gaussian noise with standard deviation 0:02 andjv(t)j 0:1. The objective is to design the feedback control signal u to minimize the eect of d on y. As shown in Figure 3.5, the magnitude bode plot of G 0 (s) is very small at the expected frequency range of the disturbance. This small plant gain may drastically slow down the adaptation and adversely aect the performance of the proposed control scheme. In order to increase the plant gain over the frequency range of interest of the disturbance, we use the procedure proposed in Section 3.4.1 and design a stable lter F (s) such that the compensated plant model G 0 0 (s) =F (s)G 0 (s) has a large enough gain over the frequency range of interest. The lter F (s) designed for this plant is given by F (s) = 0 2 2 (s 2 +s + 1:25) (s +) 2 (s + 0:2) : Figure 3.5 shows the magnitude bode plot of G 0 0 (s) = F (s)G 0 (s) for = 500 and 0 = 0:5 =6 dB. For a xed value of, increasing 0 increases the gain of the plant as well as the level of the regressor signal vector(t) over the frequency range of interest; however, 51 Figure 3.5: The magnitude plots of G 0 (s) and G 0 0 (s) = F (s)G 0 (s) for 0 = 0:5 = 6 dB and = 500. from the bound in (3.20), in the presence of unmodeled dynamics, increasing 0 reduces the margin of stability and that may lead to instability. By choosing within the expected frequency range of disturbance 0 < ! max , the gain or magnitude of G 0 0 (s) = F (s)G 0 (s) for s = j! with ! < is about 0 . Then, by choosing reasonable values of 0 , the gain of G 0 0 (s) is high enough in the frequency range of interest and low in the range where the unmodeled dynamics become important which is outside the frequency range of interest. If however the frequency range of interest is expanded to come close to the range where the unmodeled dynamics are dominant we may run into instability problems. This phenomenon suggests the classical trade o between robustness and performance. Therefore, a proper choice of lter F (s) can provide an appropriate trade-o between performance and robust stability. Consider the adaptive control scheme proposed in Section 3.4.2 with P (0) = 500I, (0) = 0, N = 20, 0 = 0:5, = 500, and = 500. We assume the sampling period for simulations isT s = 10 4 sec. Figure 3.6 shows the plant outputy(t), where the control input u(t) is applied at t = 4 sec. Figure 3.6(a) is the performance of the proposed scheme without lter F (s) in the loop (i.e., F (s) = 1). The rate of adaptation in this case is very small as the plant modelG 0 (s) has a very small gain 52 at the frequencies of the disturbance! 1 = 70 and! 2 = 187 rad/sec (see Figure 3.5). It is clear that the feedback adaptive controller with F (s) = 1 andN = 20 failed to improve performance by rejecting the periodic disturbances at least during the rst 20 seconds. We should note that by increasing the size of the adaptive lter, i.e. N, the performance shown in Figure 3.6(a) can be improved but not as eectively as using the lter F (s) above with , 0 = 0:5, = 500 for the same N = 20. As shown in Figure 3.6(b) the periodic terms have been rejected very fast right after the control input is switched on at t = 4 sec. The output is driven by mostly the output broadband random noise. Figure 3.6: The performance of the proposed scheme for N = 20 (a) without lter F (s), the speed of adaptation is small. (b) with lterF (s) much better performance is achieved. In both cases, the control signalu(t) is applied att = 4 sec. The adap- tive controller with lterF (s) quickly identies and rejects the periodic components of the disturbance. To show the performance of the proposed scheme in case of change in the char- 53 acteristics of the disturbance, we assume from t = 6 tot = 16 sec, the frequency ! 1 in (3.29) changes from 70 to 100 rad/sec linearly with time, i.e., ! 1 (t) = 8 > > > > > > < > > > > > > : 70 if t 6 3(t 6) + 70 if 6<t 16 100 if t> 16: As shown in Figure 3.7, the adaptive controller readjusts its parameters and rejects the new disturbance periodic terms. As predicted by analysis, during the time the frequency varies with time the output error is proportional to the speed of variation i.e., the derivative of the frequency which for this example is rather high i.e., 3 for 6<t 16. Despite the time variation the adaptive scheme is still able to maintain small regulation error which reduces further when the frequency of the disturbance becomes constant. Figure 3.7: The performance of the proposed scheme when the disturbance char- acteristics changed. The control input is switched on at t = 4 sec. From t = 6 to t = 16 sec, ! 1 in (3.29) varies linearly from 70 to 100 rad/sec. 54 4 MIMO Discrete-Time Uncertain Systems This chapter studies unknown time-varying disturbance suppression for multi-input multi-output (MIMO) linear time-invariant discrete-time systems in the presence of unstructured modeling uncertainties and output random noise. The main contri- butions of this chapter are: i) design and analysis of an output feedback adaptive controller which is robust with respect to unmodeled dynamics, output noise, and time variations in the values of the unknown disturbance characteristics as well as over-parameterizations in the estimated parameters, ii) use of a pre-compensator to increase the gain of the plant over some frequency range which is expected to con- tain the unknown disturbance frequencies. This modication is benecial to both performance and robustness with respect to noise and high-frequency unmodeled dynamics. 4.1 Problem Formulation Consider an uncertain multi-input multi-output plant y(k) =G(z)[u(k)] +d(k) = (I + m (z))G 0 (z)[u(k)] +d(k) (4.1) 55 wherey2R ny is the measurable output,u2R nu is the control input,d2R ny is an unknown and unmeasurable bounded disturbance, G(z) = (I + m (z))G 0 (z) is the n y n u transfer matrix of the plant, G 0 (z) = [G 0ij (z)] nynu is the known modeled part of the plant which is bounded-input bounded-output (BIBO) stable and is possibly non-minimum phase, and m (z) is the multiplicative output uncertainty which is unknown but small in some norm sense in the low frequency range and may be relatively large at high frequencies (a qualitative assumption we later make more precise). We assume (z) = m (z)G 0 (z) is proper and analytic injzj 1 which is a standard assumption in robust control theory for permissible perturbations. No knowledge about the structure and parameters of the unmodeled dynamics m (z) is assumed to be known. The structure of the open-loop plant model is shown in Figure 4.1. Figure 4.1: The structure of the open-loop plant model with multiplicative output uncertainty In many applications, the dominant part of narrow-band vibrational disturbances are often modeled as a sum of some sinusoidal terms with unknown frequencies, magnitude, and phases. For each output channel we assume the disturbance can be 56 expressed as d j (k) =d s j (k) +v j (k) = n fj (k) X i=1 a ij (k) sin(! ij (k)k +' ij (k)) +v j (k) (4.2) for j = 1;:::;n y , where the sum of the periodic terms d s j (k) is the dominant part of the disturbance, n fj (k) is the number of distinct frequencies at sample time k, v j (k) is a zero-mean bounded noise whose amplitude is much smaller than that of d s j (k), and the parameters a ij , ! ij , and ' ij are all unknown and may vary with time. The number of distinct frequencies in the disturbance vector is unknown, but an upper bound may be available. Then we write the disturbance vector as d(k) =d s (k) +v(k). The objective is to design a control signal u to suppress the eect of d on y as much as possible. In this chapter we use the structure shown in Figure 4.2 for the controller where the lterK(z) must be designed such that all signals in the closed- loop system are uniformly bounded and the norm of output signal y is minimized. Figure 4.2: The structure of the controller From the plant and controller structure shown in Figure 4.1 and 4.2, the transfer matrix from d to y is given by y(k) = (IG 0 K) (I + m G 0 K) 1 [d(k)] = (IG 0 K) [ d(k)]; (4.3) 57 where d(k) = (I + m G 0 K) 1 [d(k)]: (4.4) From the small gain theorem [1,62] if k m (z)G 0 (z)K(z)k 1 < 1; (4.5) then d dened in (4.4) is ensured to be a bounded disturbance which contains all frequencies ofd and can be modeled as a sum of some unknown noise-corrupted sinu- soidal terms as d = d s + v. The transfer matrix from d toy isS(z),IG 0 (z)K(z). IfK(z) is such that at the frequencies of the disturbance, S(z) has zero gain in the direction of d s , then the controller completely rejects the periodic parts of the dis- turbance; however, this controller may have the undesirable action of amplifying the noise part of the disturbance. Moreover, the characteristics of the disturbance are unknown and time-varying, so that a xed-parameter robust controller cannot meet the control objective. Hence, a robust adaptive control scheme that has the ability to learn about the unknown parameters online must be employed to achieve the control objective. That is, in the controller structure shown in Figure 4.2, K must be an adaptive lter which adjusts its parameters to produce an appropriate control signal u to counteract the eect of a noisy, time-varying d on y. We consider a transfer matrix for lter K whose elements are Finite Impulse Response (FIR) lters of order N of the form K(z;) = [K ij (z; ij )] nuny ; K ij (z; ij ) = ij T (z); (z), [z N ;z 1N ;:::;z 1 ] T ; (4.6) 58 where ij 2 R N is the parameter vector of the ij-th element of the lter and = [ T 11 ; T 12 ;:::; T nuny ] T 2R Nnuny is the concatenation of ij 's. Before designing and analyzing an adaptive controller, we rst examine condi- tions under which there exists a desired parameter vector for which the periodic terms of the disturbance can be completely rejected and that how the eect of noise on the output can be minimized. We also investigate factors contributing to the performance of the closed-loop system. 4.2 Existence of a Periodic Disturbance Rejecting Controller In this section we examine the existence of a lter K(z;) of the form (4.6) which rejects the periodic components of the disturbance. We show under what condi- tions we can nd a controller parameter vector at each time for which the periodic components of the disturbance are completely rejected. Let be an optimal parameter vector at a time t, for which complete rejection of the periodic parts of the disturbance is possible. For a given disturbance vector d(k) with frequencies! i 's, one may solve the system of equationsG 0 (z i )K(z i ) d(! i ) = d(! i ) for , where z i = e j! i and d(! i ) is the phasor of the sinusoidal disturbance at frequency ! i , e.g., if d(k) = [a i sin(w i k +' i );b i sin(w i k + i )] T , then d(! i ) = [a i e j' i ;b i e j i ] T ; however, this equation is not always solvable. The existence of a disturbance rejecting parameter vector at each time for any disturbance vector is the subject of the following theorem. It should be noted that with such a parameter vector the stability condition (4.5) must be satised, otherwise, the closed-loop system may be unstable. This point will be discussed later. Theorem 4.1 Let ! 1 ;:::;! n f be all distinct frequencies in the disturbance vector d at timet. There exists a such thatK(z; ) completely rejects periodic disturbances 59 in any direction if and only if n y n u , rank(G 0 (e j! i )) =n y , for i = 1;:::;n f , and N 2n f . Proof : Rejection of sinusoidal disturbances in any direction is possible if and only if the transfer matrix S(z) = IG 0 (z)K(z) has zero gain in all input direc- tions at z i = e j! i . Then, the numerator of all elements of S must be zero at the frequencies of the disturbance. The condition S(z i ) =IG 0 (z i )K(z i ) = 0 implies K is equal to the right inverse of G 0 at z i 's which exists if and only if G 0 (z i ) is full row rank, i.e., rank(G 0 (z i )) = n y u u . Moreover, from S(z i ) = 0, the numer- ator of the elements of S(z) must contain the disturbance generating polynomial D s (z) = Q n f i=1 (z 2 2 cos(! i )z + 1) as a factor. So, we haven y n u polynomial Dio- phantine equations which can be expressed in a matrix form. It is straightforward to show that such equations can be written as n y Sylvester matrix equations with full-row rank matrices and if N 2n f , the number of independent equations does not exceed the number of unknowns and this guarantees that the system is always solvable. For plants with fewer inputs than outputs, for degenerate plants rank(G 0 (z))< minfn y ;n u g,8z, and for plant with transmission zeros at the frequencies of the disturbance, complete rejection of disturbances in all directions is impossible. It should be noted that Theorem 4.1 provides necessary and sucient conditions for rejection of periodic disturbance signals in any direction; however, for a given dis- turbance vector, these conditions are sucient as for a specic direction we just need S =IG 0 K to have zero gain in the direction of the disturbance vector not in all directions. In a MIMO system, a signal vector u with frequency ! 0 can pass through G 0 (z) even if G 0 (z) has a transmission zero at this frequency; which is the case when the input vectoru is not in the zero-gain directions of G 0 . The following two examples show that the conditions given in Theorem 4.1 are not necessary for 60 a specic direction of the disturbance vector. Example 1 (a system with zero gain at a single frequency): Consider the following system (the sampling period is T s = 10 3 sec): G 0 (z) = 1 z 2 0:5z + 0:5 2 6 4 z 2 cos(0:1)z 1 1 z 3 7 5 ; which has a pair of zeros on the unit circle at ! 0 = 0:1 rad/sample (100 rad/sec) and the input direction corresponding to the zero is v 0 = [1;e 0:1j ] T . From Theorem 4.1, complete rejection of periodic disturbances with frequency ! 0 in all directions is impossible for this plant, because rankG 0 (e j! 0 ) = 1 and the sys- tem has zero gain in some direction. However, for some disturbance vectors, say d s (k) = [sin(0:1k + 0:1); sin(0:1k)] T , we can nd a lter of the form (4.6) which completely reject d s , even though the conditions in Theorem 4.1 are not satised. Example 2 (a degenerate system): Consider the following 2-input 2-output de- generate system (the sampling period T s = 10 3 sec): G 0 (z) = 1 z 2 6 4 1 1 1 1 3 7 5 : The normal rank ofG 0 (z) is equal to one, the minimum singular value of the system is zero for all frequencies, and the input direction corresponding to the zero of the system is v 0 = [1;1] T . So, for this system we cannot reject periodic disturbances in all directions. However, for some disturbances, e.g., d s (k) = [sin(0:1k); sin(0:1k)] T , there exists a lter of the form (4.6) for which d s is completely rejected. A disturbance rejecting parameter vector in the ideal case (i.e., when v = 0 and m = 0) leads to a perfect performance and guarantees that the plant output converges to zero exponentially fast. However, in the presence of noise and modeling 61 error, a large gain of G 0 (z)K(z; ) may drastically amplify the noise part of the disturbance; moreover, it may signicantly reduce the stability margin or lead to instability. The situation may get worse if the disturbance has some modesz =e j! i near the transmission zeros of G 0 (z). In such case, G 0 (z) is close to becoming rank decient at the frequencies of the disturbance ! i 's. Indeed, in order to cancel the periodic disturbance terms the controller has to generate periodic terms of the same frequencies which after going through G 0 (z) result in periodic terms identical to that of the disturbance but of opposite sign. Clearly, when the plant has a very low gain at the frequencies of the disturbance, the lter gain must be large enough to make the gain ofG 0 K in the direction of d s close to one. This however may increase kG 0 Kk 1 leading to noise amplication. To design a lter with a satisfactory performance one may use either or both of the following two remedies: 1. Increasing the order of the lter: Increasing the value ofN provides more ex- ibility in selecting the best parameter vector for which the periodic parts of the disturbance are rejected and J =kG 0 (z)K(z; )k 1 is minimized. The minimization of J is equivalent to minimizing the sensitivity function of the output with respect to the random noise in the disturbance. It is clear that once the disturbance characteristics change with time, the minimizer changes accordingly. In Theorem 4.1, if N > 2n f there are innitely many parameter vectors for which complete rejection of the periodic components of the dis- turbance is possible among which we choose the one that minimizesJ, thereby we minimize the eect of noise on the plant output and improve robustness with respect to unmodeled dynamics. It can be easily veried that increas- ing N does not increase min kG 0 (z)K(z;)k, because for N > 2n f , the lter can be written as K N =K N2n f + 1 z N2n f K 2n f , where the subscript of K de- 62 notes the order of the lter. Then by setting parameters of the rst term in the right-hand side (which are due to over-parameterizations) to zero we have kG 0 K N k 1 =k 1 z N2n f G 0 K 2n f k 1 =kG 0 K 2n f k 1 . 2. Pre-ltering modication: If the plant has a small gain at the frequencies of the disturbance and comes close to lose rank at these frequencies, we may have a very poor performance because of possible large value of G 0 K at other frequencies which leads to noise amplication. Moreover, in the presence of high-frequency unmodeled dynamics if the the plant has relatively large gain at high frequencies, with a periodic-disturbance-rejecting controller of the form (4.6), the closed-loop system may have a very small stability margin or may become unstable. Since the plant model G 0 (z) is known, by a proper shaping of the singular value of G 0 (z) both performance and robust stability can be improved. That is, a pre-compensator F 0 (z) is designed such that G 0 0 (z) = G 0 (z)F 0 (z) has a large enough gain (of the order of 1) over the frequency range of the disturbance in all directions, if possible, and have suciently small gain at high frequencies. In the next section, we will discuss this modication and the trade os between performance improvement and robustness with respect to plant modeling uncertainties. The properties of the closed-loop system can be summarized as follows. Assume the conditions given in Theorem 4.1 are satised and let be the desired vector for rejecting the periodic disturbance terms while minimizing the sensitivity function of the output with respect to noise. From the small-gain theorem and input-output stability results [1, 57, 62], ifk m (z)G 0 (z)K(z; )k 1 < 1, then all signals in the 63 closed-loop system are uniformly bounded and lim T!1 sup kT ky(k)k 1 (IG 0 K) (I + m G 0 K) 1 1 v 0 c(1 +kG 0 Kk 1 )v 0 ; where v 0 = sup k jv(k)j and c is a nite positive constant independent of noise level. In addition, in the absence of the noise, the plant output goes to zero exponentially fast with time. The above discussion claries what factors aect the performance of a distur- bance rejecting controller. In the next section by employing a same control structure we propose a robust adaptive control scheme for suppression of unknown periodic disturbances with time-varying characteristics. 4.3 Adaptive Suppression of Unknown Periodic Components In order to counteract the eect of unknown disturbances on the plant output, an adaptive lter is required to adjusts its parameters in the direction of minimizing the norm of the plant output. The structure of the closed-loop system with an adaptive lter is shown in Figure 4.3, where the lter parameter vector (k) is calculated at each sample time. Let (k) be the unknown desired parameter vector which minimizes the output vector norm. If the conditions given in Theorem 4.1 are satised, for any given disturbance vector, there exists an optimum which attenuates the eect ofd ony. Obviously, in case of change in characteristics of the disturbance, varies with time, so the desired parameter vector is time-varying, that is at each sample time k the conditions given in Theorem 4.1 are satised, (k) gives the best performance. By `best performance' we mean complete rejection of periodic terms without amplifying the eect of broadband random noise. 64 Figure 4.3: The structure of the adaptive closed-loop system From (4.6) and Figure 4.3 we have the following relations: u(k) =K(z;)[(k)]; (k) =y(k)G 0 (z)[u(k)]; K(z;) = T ij (z) nuny ; (z) = [z N ;z 1N ;:::;z 1 ] T : (4.7) To generate (the estimate of ), we rst develop a parametric model, then design an appropriate parameter estimator for which the control objective can be achieved. The following lemma presents a parametric model for the closed-loop plant to be used for parameter estimation [1,56]. Lemma 4.1 The closed-loop system (4.1),(4.7) can be parameterized as (k) = T (k) (k) +(k); (4.8) where (k) = [ T 11 (k); T 12 (k);:::; T nuny (k)] T 2 R Nnuny1 is the unknown desired parameter vector, (k) = [ 1 (k);:::; ny (k)] T = y(k)G 0 (z)[u(k)] is measurable, 65 and the regressor matrix is given by T (k) =G 0 (z)[ w T (k)] (4.9) where w(k) = 2 6 6 6 6 4 w(k) . . . w(k) 3 7 7 7 7 5 Nnunynu ; where w(k) = [(z) T [ 1 (k)];:::;(z) T [ ny (k)]] T 2 R Nny1 , and the unknown er- ror term (k) is proportional to the noise level, size of unmodeled dynamics and (k) = (k + 1) (k), and may also depend on exponentially decaying to zero terms. Proof : From (4.7) with = , the ideal control input can be written as u (k) =K(z; (k))[(k)] = (k)(z)[(k)] = (k)w(k) where (k) = 2 6 6 6 6 4 T 11 ::: T 1ny . . . . . . . . . T nu1 ::: T nuny 3 7 7 7 7 5 nuNny (4.10) (z) = 2 6 6 6 6 4 (z) . . . (z) 3 7 7 7 7 5 Nnyny (4.11) 66 and w = (z)[] = [w T 1 ;:::;w T ny ] T and w i =(z)[ i ]. We can write (k) as =G 0 (z)[u ] +G 0 (z)[u ] =G 0 (z)[u ] + ( +G 0 (z)[u ]) = +e ; (4.12) where e = +G 0 (z)[u ], and =G 0 (z)[u ] =G 0 (z)[ w] =G 0 (z) 2 6 6 6 6 4 P ny j=1 T 1j w j . . . P ny j=1 T nuj w j 3 7 7 7 7 5 : Then, the i-th element of is given by i =G 0i1 (z)[ ny X j=1 T 1j w j ]:::G 0inu (z)[ ny X j=1 T nuj w j ]: From the Swapping Lemma, for a transfer function G ij (z) and vectors rs and w t we have G ij (z)[ T rs w t ] = T rs G ij (z)[w t ] +f ijt ( rs ); where f ijt ( rs ) = G c ij (z)[(G b ij (z)[w T t ]) rs ], and rs (k) = rs (k + 1) rs (k). Then, i = ny X j=1 T 1j G 0i1 (z)[w j ]::: ny X j=1 T nuj G 0inu (z)[w j ] ny X j=1 f i1j ( 1j )::: ny X j=1 f inuj ( nuj ): 67 Then, from (4.12) we can write T = T +f T ( ) +e T : (4.13) where = [ T 11 ; T 12 ;:::; T nuny ] T is the unknown parameter vector and = 2 6 6 6 6 4 G 011 (z)[w] ::: G 0ny 1 (z)[w] . . . . . . . . . G 01nu (z)[w] ::: G 0nynu (z)[w] 3 7 7 7 7 5 is the known regressor matrix. The transpose of can be written as T =G 0 (z) 2 6 6 6 6 4 w . . . w 3 7 7 7 7 5 T : The vector f( ) in (4.13) is a function of change in , hence is zero if is constant with time, and the last term in the right hand side of (4.13) is e =G 0 (z)[K(z; )[]]; where is such that at each timeIG 0 (z)K(z; ) has zero gain at the frequencies of the disturbance. Since can be written as = d s +v + m G 0 [u], thenje j is proportional to the noise level, size of unmodeled dynamics and also may depend on exponentially decaying to zero terms. Therefore, the parametric model of the system is given by (k) = T (k) (k) +(k); 68 where the error term (k) =f( ) +e depends on v(k), m (z), and and we can write k(k)k 2 c(v 0 + m + ); (4.14) for some nite constant c > 0 independent of the size of v(k), m (z), and , where m and are positive constants proportional to the size of m (z) and (k), respectively, and v 0 = sup k jv(k)j. It follows from Lemma 4.1 that in the ideal case when there is no noise and modeling error and when the characteristics of the disturbance are xed with time, the error term (k) is zero or goes to zero exponentially fast. In (4.9), w(k) contains the frequency components of the disturbance vector. Since the regressor is obtained by passing w(k) through G 0 (z), then if G 0 (z) has low gains at some frequencies of the disturbance, it may severely attenuate those frequency components of the disturbance in the regressor. This implies that the regressor will carry close to zero information about those frequencies with the con- sequence of making their identication and therefore their rejection dicult if at all possible in some cases; moreover, the corresponding disturbance rejecting controller K(z; ) will have a very high gain at the frequencies of the disturbance which can make the gain of G 0 (z)K(z; ) large at other frequencies. Therefore, we need to shape the frequency response of the plant model by de- signing an LTI lterF 0 (z) = [F 0ij (z)] nunu analytic injzj 1 placed right after the adaptive lter K(z;) as shown in Figure 4.4. The modied closed-loop system is then equivalent to Figure 4.3 whereG 0 (z) is replaced byG 0 0 (z) =G 0 (z)F 0 (z). If par- tial knowledge on the frequency range of the disturbance is available, the lterF 0 (z) must be such that G 0 0 (z) has large enough gains (of the order of 1) in all directions (if possible) over expected disturbance frequency range. This not only increases the 69 partial excitation level of the regressor, but also reduceskG 0 0 (z)K(z; )k 1 , thereby avoid broadband noise amplication. On the other hand, to improve robustness with respect to unmodeled dynamics, theH 1 -norm of m (z)G 0 0 (z) must be as small as possible. Since the size of m (z) over the low frequency range is typically negligible and it may be large at high frequencies, then the maximum singular value of G 0 0 (z) should be small enough at high frequencies. It should be mentioned that the pres- ence of F 0 does not change the previous analyses and results, it just changes the plant model from G 0 (z) to G 0 0 (z). Figure 4.4: The structure of the adaptive controller with a compensator To design lter F 0 (z), the following knowledge may be needed: i) the frequency range where the modeled-part of the plant has high enough accuracy (which is typical at low frequencies); iii) an upper bound for the maximum singular value of the unmodeled dynamics; iii) the frequency range of the dominant part of the disturbance (i.e., the part with all the periodic terms). It should be noted that if the disturbance contains periodic terms with frequencies in the high range where the unmodeled dynamics are dominant their rejection may excite the unmodeled dynamics and adversely aect stability. In practice, however, most high frequency periodic disturbances have a small amplitude and it may be better to ignore them and consider them as part of the noise rather than try to attenuate them. This is one of the trade os of performance versus robustness that is well known in robust control and robust adaptive control [1]. 70 In classical robust control design for MIMO systems, prior to the design of a controller we may need to shape the singular values of the nominal plant to be able to meet the desired specications. The desired shape of the open-loop plant is typically as follows: large enough gain at low frequencies in all directions, low enough roll-o rate at the desired bandwidth (about 20 dB/decade) and higher rate at high frequencies, and very small gain at high frequencies [57, 66, 67]. It is also desired that maximum and minimum gain of the shaped plant to be almost the same, i.e., singular values to be aligned to have a plant with almost the same gain in all directions at each frequency [57, 68]. For an ill-conditioned system (a system with large condition number), however, aligning singular values is not recommended as it may lead to a poor performance and robustness [69]. Several algorithms and procedures have been proposed for singular value shaping which mainly require some trial and error [66,70]. In [71], a more systematic algorithm has been proposed which guarantees that the loop-shape and the singular values and condition number of the shaping weights lie in a pre-specied region. A proper rational transfer function matrixM(z) analytic injzj 1 with no zeros on the unit circle is called inner if it satisesM T (z 1 )M(z) =I (an inner matrix is square or has more rows than columns), and is called outer if it is right invertible with a right inverse proper and analytic injzj 1 (an outer matrix is square or has more columns than rows). Any proper rational transfer function matrix H(z) analytic injzj 1 with no zeros on the unit circle has an inner-outer factorization H(z) = H in (z)H out (z), where H in (z) is inner and H out (z) is outer [72, 73]. A real rational matrixM(z) is called all-pass if it is square and satisesM T (z 1 )M(z) =I. System decomposition techniques such as inner-outer factorization can be also used to design a frequency weighing matrix for well-conditioned square plants. The inner-outer factorization is used in solving problems related to optimal, robust, and 71 H 1 control design and several algorithms have been proposed to calculate an inner- outer factorization of a proper rational matrix [74{76]. Consider a square plant matrix G 0 (z) with stables poles and without any trans- mission zero on the unit circle. By using the algorithm proposed in [77], inner-outer factors of G 0 (z) can be computed. If the plant has some zeros on the unit circle, one can perturb the zeros and move them slightly away from the unit circle and then apply the decomposition procedure. Let G out be the outer factor of G 0 (z) (or perturbed G 0 (z)), then G 0 (z)G 1 out (z) is a proper stable all-pass (or almost all-pass) matrix. Let F 0 (z) = f(z)G 1 out (z), where f(z) = 0 f 0 (z), f 0 (z) is a scalar low-pass lter with dc-gain of one and desired bandwidth and roll-o rate at high frequen- cies, and 0< 0 1 is a design constant. Then G 0 (z)F 0 (z) has gain of 0 over the desired low frequency range and small gain at high frequencies with aligned singular values. After shaping the singular values of the nominal plant, we design a parame- ter estimator for the adaptive lter K(z;(k)). Based on the derived parametric model (4.8), a robust parameter estimator can be developed using the techniques discussed in [1, 56] which guarantees stability and robustness independent of the excitation properties of the regressor (k) in the presence of non-zero error term (k). It should be noted that the number of frequencies of the disturbance n f is unknown and an upper bound for it may be available; moreover, even ifn f is known, we choose N > 2n f to achieve a better performance. This choice of N leads to an over-parametrization and to a regressor (k) with linearly dependent columns which cannot be consistently exciting, no matter what the excitation of the plant is. The lack of persistence of excitation makes the adaptive law susceptible to parameter drift [1] and possible instability. Therefore the adaptive law for estimating must be robust, otherwise, the presence of nonzero error term (k) may cause parameter 72 drift and lead to instability. It is to be noted that for a persistently exciting regressor (with high enough excitation level), even in the presence of a bounded error term the stability is guaranteed and no robustness modication is necessary; then, the parameter error exponentially fast goes to a residual set. In the absence of a persis- tently exciting regressor, several modications have been proposed in the literature to avoid parameter drift in the adaptive law [1]. In this chapter, we used parameter projection to directly restrict the estimate of the unknown parameter vector from drifting to innity. Let (k 1) be the most recent estimate of (k), then the predicted value of (k) is ^ (k) = T (k)(k 1); (4.15) using which a normalized estimation error can be dened as "(k) = (k) ^ (k) m 2 (k) ; (4.16) where m(k) is a normalizing signal to be dened later. We employ a robust pure recursive least-squares algorithm [1,56] to generate the estimate of at time k: P 1 (k) =P 1 (k 1) + (k) T (k) m 2 (k) (k) = proj ((k 1) +P (k)(k)"(k)); (4.17) where P 1 (0) =P T (0)> 0, (0) = 0 , and the normalizing signal is given by m 2 (k) = 1 + 0 trace( T (k)(k)); (4.18) 73 where 0 > 0 is a design constant. To avoid P 1 (k) from growing without bound, covariance resetting modication [1,56] can be used, i.e., set P (k r ) = 0 I, wherek r is the sample time at which min (P ) 1 , and 0 > 1 > 0 are design constants. One may also use the modied least-squares algorithm with forgetting factor [1]. The projection operator proj() in (4.17) guarantees that (k)2 ,8k, where is a known compact set dened as =f2R Nnuny j T 2 max g; where max > 0 is chosen so that the coecient vector of the optimum lter, , belongs to . Since the optimal parameter vector is unknown and time-varying and we do not have a priori knowledge on the its norm, max must be chosen su- ciently large. To avoid parameter drift one may use other modications such as xed -modication that requires no bounds on the set where the unknown parameters belong to [1]. To implement the projection in (4.17),P 1 is decomposed into (P 1 2 ) T (P 1 2 ) at each instantk to guarantee all properties of the corresponding recursive least-squares algorithm without projection: (k) =(k 1) +P (k)(k)"(k) (k) =P 1=2 (k) (k) (k) = 8 > > < > > : (k) if (k)2 ? proj of (k) on if (k)62 (k) =P 1=2 (k)(k); (4.19) where the set is transformed to such that if (k)2 , then (k)2 . Since is chosen to be convex and (k) = P 1=2 (k) (k) is a linear transformation, then 74 is also a convex set [59]. In (4.7), the control signal at time k is produced based on the most recent estimate of (k) at this time, i.e., (k 1), then u(k) =K(z;(k 1))[(k)]: (4.20) The following theorem summarizes the properties of the closed-loop system (4.1), (4.7)-(4.18). Theorem 4.2 Consider the closed-loop system (4.1), (4.7)-(4.18). Then, if the modeling error term is such that c 0 k m (z)G 0 (z)F 0 (z)k 1 < 1 (4.21) where c 0 is a positive constant which depends on max and is independent of the size of unmodeled dynamics, then all signals in the closed-loop system are uniformly bounded. Moreover, the plant output satises lim T!1 sup 1 T k+T1 X i=k ky(i)k 2 2 c ( m + +v 0 ); (4.22) for any k 0, where m and are positive constants proportional to the size of unmodeled dynamics and , respectively, and v 0 = sup k jv(k)j and c > 0 is a nite constant independent of k and the size of modeling error, noise, and . In addition, in the absence of noise, unmodeled dynamics, and variations in the characteristics of the disturbance (i.e., when v 0 = 0, m (z) = 0, and = 0), the adaptive law guarantees the convergence of y(k) to zero. 75 Proof : From (4.6) and (4.20) the control input is given by u(k) =K(z;(k 1))[(k)] =(k)(z)[(k)]; (4.23) where (k) = 2 6 6 6 6 4 T 11 (k 1) ::: T 1ny (k 1) . . . . . . . . . T nu1 (k 1) ::: T nuny (k 1) 3 7 7 7 7 5 nuNny and (z) is dened in (4.11). Then, from (4.23) and Figure 4.3 we have (k) = m (z)G 0 0 (z)[(k)(z)[(k)]] +d(k); then k k k 1 k m (z)G 0 0 (z)k 1 k((z)[]) k k 1 +kd k k 1 c 0 k m (z)G 0 0 (z)k 1 k k k 1 +d 0 ; wherec 0 > 0 is a nite constant whose boundedness is guaranteed due to projection and d 0 is an upper bound forkdk 1 . Then if c 0 k m (z)G 0 0 (z)k 1 < 1, (k) and then all signals in the closed-loop system are uniformly bounded. Since m 2 (k)> 1, then from (4.14) we can write k(k)k 2 m 2 (k) c(v 0 + m + ); (4.24) where c> 0 is a generic symbol used to denote any nite constant. 76 Since P 1 (0) =P T (0)> 0, from (4.17) we have P 1 (k)P 1 (k 1) = (k) T (k) m 2 (k) 0: (4.25) then P 1 (k) =P T (k)> 0 and with covariance resetting we have P (k) =P T (k)> 0, for anyk. Also,P (k)P (k 1) 0 impliesP (k)P (k 1):::P (0), that isP (k) is bounded and sinceP (k)> 0 andP (k)P (k 1),8k,P (k) converges to a constant matrix as k!1. By pre-multiplying both sides of (4.25) by P (k) and also post-multiplying it by ~ (k 1) =(k 1) (k) we obtain P (k)P 1 (k 1) ~ (k 1) = ~ (k 1)P (k) (k) T (k) m 2 (k) ~ (k 1): (4.26) From (4.8), (4.15), (4.16) we have T (k) ~ (k 1) =m 2 (k)"(k) +(k): (4.27) Then , from (4.26), (4.27) we have P (k)P 1 (k 1) ~ (k 1) = ~ (k 1) +P (k)(k)"(k)P (k) (k)(k) m 2 (k) : (4.28) Consider the following Lyapunov-like function whose boundedness is guaranteed due to projection and the boundedness of P (k): V (k) = ~ T (k)P 1 (k) ~ (k); 77 where ~ (k) = (k) (k + 1). If (k)2 , ~ (k) = ~ (k), and from (4.19) ~ (k) = ~ (k 1) +P (k)(k)"(k) (k) and from (4.28) we have ~ (k) =P (k)P 1 (k 1) ~ (k 1) +P (k) (k)(k) m 2 (k) (k); then V (k),V (k)V (k 1) = ~ T (k)P 1 (k) ~ (k) ~ T (k 1)P 1 (k 1) ~ (k 1) = ~ T (k 1) P 1 (k 1)P (k)P 1 (k 1)P 1 (k 1) ~ (k 1) + 2 ~ T (k 1)P 1 (k 1)P (k) (k)(k) m 2 (k) + T (k) T (k)P (k)(k) m 4 (k) (k) 2 ~ T (k)P 1 (k) (k) T (k)P 1 (k) (k): (4.29) From (4.25) and using the identity (X +YZ) 1 =X 1 X 1 Y (I +ZX 1 Y ) 1 ZX 1 we can write P (k) =P (k 1)P (k 1) (k)Q T (k) m 2 (k) P (k 1) where Q =Q T = (I + T (k)P (k1)(k) m 2 (k) ) 1 > 0. Then P 1 (k 1)P (k)P 1 (k 1)P 1 (k 1) = (k)Q T (k) m 2 (k) : 78 By simplifying (4.29) we obtain V (k) =(m") T Q(m") + (m") T (2Q 2I + 2R)( m ) + ( m ) T ( 2I m 2 Q m 2 2R m 2 +R)( m ) 2 ~ T (k)P 1 (k) (k) T (k)P 1 (k) (k); where R =R T = T (k)P (k)(k) m 2 (k) 0. Then, from the boundedness of (k) m(k) and P (k), it follows that V (k)ck"(k)m(k)k 2 +ck"(k)m(k)k (k) m(k) + c (k) m(k) 2 +c (4.30) where c is a generic symbol for a positive constant and is a positive constant proportional to the size of (k). Using the inequalityax 2 +bx a 2 x 2 + b 2 2a we have V (k)ck"(k)m(k)k 2 +c (k) m(k) 2 +c : (4.31) Therefore, from (4.24), (4.31) V (k)ck"(k)m(k)k 2 +c ; (4.32) where =v 0 + + m . Taking summation on both sides of (4.32) gives 1 T T X k=1 k"(k)m(k)k 2 c + c T (V (0)V (T ))c + c T that is "(k)m(k)2S( ) (i.e., it is -small in mean square sense). Since m 2 (k) 1, 79 we havek"(k)k 2 k"(k)m(k)k 2 , therefore "(k)2S( ). Also, (k)(k 1) =P (k)(k)"(k) =P (k) (k) m(k) "(k)m(k) then k(k)(k 1)k 2 2 max (P (k)) (k) m(k) 2 k"(k)m(k)k 2 ck"(k)m(k)k 2 that is (k)2S( ). By using the procedure in the proof of Lemma 4.1, the plant output can be written as y(k) =G 0 0 (z)[u(k)] +(k) =G 0 0 (z)[K(z;(k 1))[(k)]] +(k) = T (k) ~ (k 1) +(k) +f((k 1)); wheref((k 1)) is a function of (k 1) =(k)(k 1) which is dened in the proof of Lemma 4.1. Then from (4.27), "(k)m(k) = y(k) +f((k 1)) m(k) : Since m(k) is bounded and "(k)m(k); (k)2S( ), then y(k)2S( ) which can be written as (4.22). If (k)62 , then we have V (k) = ( 0 ) T ( 0 ), where 0 (k) is an or- thogonal projection of (k) on . Then, since 2 , ( 0 ) T ( 0 ) ( ) T ( ) = ~ T (k)P 1 (k) ~ (k). This enables us to establish all properties of the adaptive law without projection [59]. It should be noted that the stability condition (4.21) is a sucient condition for BIBO stability of the closed-loop system. Such a norm-bound condition states 80 that the closed-loop system can tolerate nonzero but small-size modeling errors. By proper shaping of the singular values of the plant model (i.e., by an appropriate choice of lter F 0 ), the robustness with respect to modeling uncertainties can be improved and the system may tolerate relatively large size modeling errors at high frequencies. 4.4 Simulation Results This section demonstrates the performance of the adaptive controller applied to a 2-input 2-output plant. We assume that the sampling period is T s = 10 3 sec. Consider the system y = (I + m (z))G 0 (z)[u] +d where G 0 (z) is the known modeled part given by G 0 (z) = 0:01(z 1) R(z) 2 6 4 z 2 z + 0:5 z 2 z + 1 3 7 5 where R(z) = (z 0:75)(z 2 + 1:3z + 0:8). For simulation purposes, the unknown multiplicative uncertainty term m (z) is assumed to be of the form m (z) = 10 5 (z + 0:999) 2 I; which has a negligible size at low frequencies and relatively large size near the Nyquist frequency. The unknown disturbance vector d at each sample timet =kT s 81 is assumed to be d(t) = 2 6 4 d s 1 (t) +v 1 (t) d s 2 (t) +v 2 (t) 3 7 5 where d s 1 (t) = 0:6 sin(30t) + 0:6 sin(95t + 8 ) + 0:6 sin(180t + 6 ); d s 2 (t) = 0:5 sin(20t + 4 ) + 0:5 sin(110t + 2 ) + 0:5 sin(210t + 5 )); and v 1 (t) and v 2 (t) are zero mean Gaussian with standard deviation 0:02 and jv i (t)j 0:1. We assume at t = 25 sec new frequencies are added to the exist- ing disturbance, i.e., d s 1 (t) =d s 1 (t) + 0:9 sin(103t + 7 ) d s 2 (t) =d s 2 (t) + 0:9 sin(128t + 3 ); fort 25 sec. We also assume that at each time the number of distinct frequencies in the disturbance vector is at most 10, then we need to choose N 20. By using the proposed adaptive control scheme we generate a control signal u to counteract the eect of d on y. Figure 4.5 shows the maximum and minimum singular values of the unshaped nominal G 0 (z). From the maximum singular value plot of G 0 (z) at high frequencies (around the Nyquist frequency =T s rad/sec), the closed-loop system seems to be vulnerable to modeling error in the presence of high- frequency unmodeled dynamics. And, from the minimum singular value plot, the plant has very small gains over the low frequency range which may badly aect the performance. In all simulations, we set 0 = 1 and P 1 (0) = 0:01I. The control signal u(k) is 82 Figure 4.5: The singular value plot of the unshaped nominal G 0 (z) applied at t = 10 sec, and at t = 25 sec new frequencies are added abruptly to the disturbance. Figure 4.6 shows the output of the adaptive closed-loop system with N = 20, without pre-ltering modication i.e., F 0 (z) = 1. The poor performance is because of the low gain of the plant over the frequency range of the disturbance. In the absence of modeling uncertainties by scaling up the plant model by using a simple gain, the performance can be improved; however, in practice the presence of inevitable modeling error does not allow us to boost up the plant gain by using a frequency-independent gain, as it may lead to instability of the closed-loop system. To improve the performance as well as robustness with respect to high-frequency unmodeled dynamics, we shape the singular values of G 0 (z) by an appropriate pre- compensation. To design a compensatorF 0 (z) for this plant, we use the inner-outer factorization of plant nominal model. Since the plant has zeros on the unit circle, we apply the algorithm proposed in [77] to the perturbed plant model ~ G 0 (z) obtained by scaling the zeros by factor 0:99 to move them slightly away from the unit circle. Then ~ G 0 (z) = ~ G in (z) ~ G out (z), where ~ G in (z) is a stable proper all-pass lter and ~ G out (z) is a stable proper with stable right inverse. Then, we choose F 0 (z) = 0 f 0 (z) ~ G 1 outer (z), where 0 = 0:1 and f 0 (z) is a scalar low-pass lter with DC gain 83 Figure 4.6: The performance of the adaptive control system with N = 20 without pre-ltering modication. The control is applied at t = 10 sec and at t = 25 sec, new frequencies are added to the existing disturbance. of one designed to compensate the eect of high-frequency modeling uncertainties. We assume f 0 (z) is a third-order Butterworth low-pass lter with cuto frequency of 0:2=T s = 630 rad/sec, f 0 (z) = 0:018099(z + 1) 3 (z 0:5095)(z 2 1:251z + 0:5457) : Then, G 0 0 (z) is a low-pass ler with DC gain of 0 = 0:1 =20 dB and cuto frequency of 630 rad/sec. The selection of the bandwidth off 0 (z) and the gain 0 is based on partial knowledge on the size and the frequency range where the modeling error may be dominant. Large values of these two parameters can adversely aect the stability margin. The singular values of the shaped plant G 0 0 (z) = G 0 (z)F 0 (z) are shown in Figure 4.7. Figure 4.8 shows the performance withN = 20 and ltering modication. After closing the feedback loop att = 10 sec, the controller quickly adjusts its parameters to suppress the output norm by rejecting the sinusoidal part. At t = 25 sec, the controller re-adjusts its parameters to counteract the eect of new periodic terms 84 Figure 4.7: The singular value plot of the shaped plant G 0 0 (z) = G 0 (z)F 0 (z). The singular values are almost aligned. on the plant output. The performance of the proposed scheme with a higher order adaptive lter, N = 60, together with ltering modication is shown in Figure 4.9. Figure 4.8: The performance of the adaptive control system with N = 20 and pre-ltering modication. 85 Figure 4.9: The performance of the adaptive control system with larger adaptive lter order N = 60 together with pre-ltering modication. 86 5 MIMO Continuous-Time Uncertain Systems In this chapter, we design and analyze a robust adaptive control scheme for unknown time-varying periodic disturbance attenuation in uncertain multi-input multi-output (MIMO) continuous-time LTI systems. The trade-o between robust stability and performance improvement and practical design considerations are discussed. It is demonstrated that proper shaping of the open-loop plant singular values as well as over-parameterizing the controller parametric model can signicantly improve performance. Numerical simulations are performed to demonstrate the eectiveness of the proposed scheme. 5.1 Problem Statement Consider an uncertain multi-input multi-output LTI plant with an additive output disturbance vector: y(t) =G(s)[u(t)] +d(t) = (I + m (s))G 0 (s)[u(t)] +d(t); (5.1) where y(t)2R ny is the measurable output, u(t)2R nu is the control input, d(t)2 R ny is an unknown and unmeasurable bounded disturbance. The n y n u transfer 87 function matrix of the plant isG(s) = (I + m (s))G 0 (s), whereG 0 (s) is the known modeled part of the plant which is stable and possibly non-minimum phase, and m (s) is a multiplicative output uncertainty term which is unknown but is such that m (s)G 0 (s) is proper and stable and is small in some norm sense. In many applications unwanted vibrations are often modeled as the sum of some sinusoidal terms with unknown frequencies, magnitude and phase, corrupted by broadband noise as presented by the following equation for each output channel: d k (t) =d s k (t) +v k (t) = n f k (t) X i=1 a ik (t) sin(! ik (t)t +' ik (t)) +v k (t); (5.2) for k = 1;:::;n y , where d s k (t) is the dominant part of the disturbance for channel k with n f k (t) distinct frequencies at time t, v k (t) is a zero-mean bounded random noise whose amplitude is much smaller than that of d s k (t), and the parameters a ik , ! ik , and ' ik are all unknown and may vary with time. The number of distinct frequencies in the disturbance vector is also unknown and time-varying. We write the disturbance vector as d(t) =d s (t) +v(t) and assume that an upper bound n max for the maximum number of distinct frequencies of the periodic components of the disturbance vector and an upper bound ! max for its largest frequency are known. These bounds are based on a priori information and can be conservative. The control objective of rejecting the periodic disturbance components involves the design of the control signal vector u to suppress the eect of d on y as much as possible. In this chapter, we use the structure shown in Figure 5.1 for the controller where the stable LTI lter F (s) and the adaptive lter K(s;t) are to be designed such that all signals in the closed-loop system are uniformly bounded and the norm of the output signal y is minimized. We assume theij-th element of then u n y transfer function matrixK is of the 88 Figure 5.1: The structure of the controller and the closed-loop system form K ij (s;t) = N1 X k=0 ij k (t) Nk (s +) Nk ; (5.3) where ij = [ ij 0 ; ij 1 :::; ij N1 ] T 2R N is the parameter vector of the ij-th element ofK, and the scalar> 0 and integerN > 0 are design parameters. The controller parameter vector (t) = [ T 11 (t); T 12 (t);:::; T nuny (t)] T 2 R Nnuny is to be adjusted at each time t by an adaptive law to achieve the controller objective. The following lemma gives a necessary and sucient condition under which the periodic components of the disturbance vector can be rejected. Lemma 5.1 Let! 1 ;:::;! n f be the distinct frequencies of the disturbance vectord s . There exists a control input vector u which completely rejects periodic disturbances in any direction if and only if rank(G(j! i )) =n y n u , for i = 1;:::;n f . Proof : From the input-output relationship of the system, it is obvious that the periodic disturbances are rejected if and only if G(s)[u(t)] =d s (t), that is there exists a transfer function matrix H(s) such that u(t) =H(s)[d s (t)], where at the frequencies of the disturbance H(j! i ) =G + (j! i ), where G + (j! i ) denotes the right inverse ofG(s) at frequency! i . From linear algebra, such a transfer function matrix 89 exists if and only if G(s) is full-row rank at the frequencies of the disturbance, or equivalently rank(G(j! i )) =n y n u , fori = 1;:::;n f . This completes the proof. It should be mentioned that the condition given in Lemma 5.1 is necessary and sucient for rejection of periodic disturbances in `any' direction independent of the phase and magnitude of the disturbance vector components. Obviously, for a specic disturbance vector the condition is not necessary. With the controller structure shown in Figure 5.1, we have y(t) = I G 0 (s)K(s) [ d(t)]; (5.4) where G 0 (s) =G 0 (s)F (s) and d(t) = I + m (s) G 0 (s)K(s) 1 [d(t)]. From Lemma 5.1, if G 0 (s) is full-row rank at the frequencies of the disturbance, there exists ann u n y transfer function matrixK such that at the frequencies of the disturbance,K(j!) is equal to the right inverse of G 0 (j!) denoted by G + 0 (j!). Letn f be the total number of distinct frequencies in d(t), then with the lterK of the form (5.3) and by writing the equation G + 0 (j! i ) = K(j! i ), i = 1;:::;n f , as a system of linear equations, it follows that if N 2n f , the equation is always solvable, hence there exists a con- troller parameter vector for which the transfer function from d toy has zero gain in any direction at the frequencies of the disturbance. Then, from the small-gain the- orem [57, 62], if the unmodeled dynamics is such thatk m (s) G 0 (s)K(s;)k 1 < 1, then it is guaranteed that d and therefore all signals in the closed-loop system are uniformly bounded and that the eect of the periodic components of d(t) on y is completely rejected. It is to be noted that with N > 2n f , there exists an innite number of parameter vectors for which the gain of the transfer function from d toy at the the frequencies of the disturbance is zero. We are interested in the parameter vector that minimizes the norm of G 0 K, improves the stability margin, and avoids 90 the amplication of the noise part of the disturbance. In other words, overparame- terization provides the exibility to select the controller parameter vector which in addition to rejecting the periodic disturbance terms, it does not amplify the out- put noise and improves robust stability. We will discuss later that a proper design of the stable LTI lter F (s) can provide a good compromise between performance improvement and robust stability. From the above discussions, with the structure of the controller shown in Fig- ure 5.1 and the lterK of the form (5.3) withN 2n f , if the condition in Lemma 5.1 is satised the control objective can be achieved. In the next section, we propose a robust adaptive control scheme for suppression of unknown periodic disturbances with time-varying characteristics. 5.2 Robust Adaptive Rejection of Periodic Disturbances We consider the control structure of Figure 5.1 with lter (5.3): u(t) =F (s)[K(s;(t))[z(t)]]; z(t) =y(t)G 0 (s)[u(t)]; K(s;(t)) = [ T ij (t)(s)] nuny ; (s) = [ N (s +) N ;:::; s + ] T ; (5.5) where the controller parameters are calculated at each time t to minimize the norm of the plant output y(t). The existence of a desired parameter vector has been discussed in the previ- ous section in the case of known frequencies. In the case of unknown frequencies, an adaptive law has to be designed to estimate online. The desired parameter vector for which the best performance is achieved depends on the unknown char- 91 acteristics of the disturbances which may vary with time. This implies that is a function of time. We assume the time derivative of is small in the mean square sense, which implies the time variations of the unknown desired closed-loop system are small in the average sense. To estimate (t) online, we need to express it in the form of a parametric model [1, 64]. The ideal control input vector for which the best performance is achieved is given by u (t) =F (s)K(s; )[z(t)]; (5.6) then the signal vector z(t) dened in (5.5) can be written as z(t) =G 0 (s)[u (t)] + (z(t) +G 0 (s)[u (t)]) =w(t) + 0 (t); (5.7) where w(t) =G 0 (s)[u (t)]; 0 (t) = (IG 0 (s)F (s)K(s; )) [z(t)],T (s; )[z(t)]; z(t) =y(t)G 0 (s)[u(t)] =d(t) + m (s)G 0 (s)[u(t)]: The ideal parameter vector is such that the transfer function matrix T (s; ) is pointwise stable, slow-varying in the mean square sense, and has zero gain at the frequencies of the disturbance. Also, m (s)G 0 (s) is assumed to be stable and proper with boundedH 1 -norm. We will show that if m (s)G 0 (s) satises a norm-bound condition, 0 (t) is a bounded signal whose size is proportional to the noise level and the size of unmodeled dynamics term. It also depends on exponential decaying to zero terms due to nonzero initial condition of the plant which are assumed to be 92 zero or negligible at the time instant we close the loop. The signal w(t) in (5.7) can be written as w(t) =G 0 (s)[u (t)] =G 0 (s)F (s)K(s; )[z(t)] =G 0 (s)F (s)[ (t)q(t)]; (5.8) where (t)2R nuNny dened in (5.9) contains the unknown vectors ij (t), q(t) = Q(s)[z(t)] = [q T 1 (t);:::;q T ny (t)] T 2 R Nny1 is a known signal vector available for measurement,q i (t) = (s)[z i (t)]2R N , (s) is dened in (5.5), andQ(s)2R Nnyny is a known transfer function matrix given by Q(s) = 2 6 6 6 6 4 (s) . . . (s) 3 7 7 7 7 5 ; = 2 6 6 6 6 4 T 11 ::: T 1ny . . . . . . . . . T nu1 ::: T nuny 3 7 7 7 7 5 : (5.9) Let g ij (s) be the ij-th element of G 0 (s) = G 0 (s)F (s); then the i-th element of w(t)2R ny1 is given by w i = g i1 (s)[ ny X j=1 T 1j q j ] + g i2 (s)[ ny X j=1 T 2j q j ] +::: + g inu (s)[ ny X j=1 T nuj q j ]; i = 1;:::;n y : (5.10) By applying the Swapping Lemma to each term in the right hand side of (5.10), we have w i = ny X j=1 T 1j g i1 (s)[q j ] + ny X j=1 T 2j g i2 (s)[q j ] +::: + ny X j=1 T nuj g inu (s)[q j ] + 1i ( _ ); (5.11) 93 where 1i ( _ ) = ny X j=1 h g i1 (q j ; _ 1j ) + ny X j=1 h g i2 (q j ; _ 2j ) +::: + ny X j=1 h g inu (q j ; _ nuj ): It should be noted that for a constant (i.e., when disturbance characteristics do not change with time), 1i is zero for any i. Then from (5.7) and (5.11) we can write z i =w i + 0i = ny X j=1 T 1j g i1 (s)[q j ] + ny X j=1 T 2j g i2 (s)[q j ] +::: + ny X j=1 T nuj g inu (s)[q j ] + 1i ( _ ) + 0i : (5.12) Equation (5.12) can be expressed in a matrix form as z(t) = T (t) (t) +(t); (5.13) where = [ T 11 ; T 12 ;:::; T 1ny ; T 21 ; T 22 ;:::; T 2ny ;:::; T nu1 ; T nu2 ;:::; T nuny ] T 2R Nnuny1 is the unknown parameter vector, (t)2R Nnunyny is the known regressor matrix given by (t) = (G 0 (s)F (s)[ q(t)]) T ; (5.14) where q(t) = 2 6 6 6 6 4 q T (t) . . . q T (t) 3 7 7 7 7 5 2R nuNnuny ; q(t) =Q(s)[z(t)]2R Nny1 ; 94 = 1 ( _ ) + 0 = [ 11 ( _ ) + 01 ; 12 ( _ ) + 02 ;:::; 1ny ( _ ) + 0ny ] T 2R ny is treated as an error term whose size is proportional to the noise level, size of unmodeled dynamics, and _ (t). The purpose of introducing the stable LTI lter F (s) is to shape the singular values of the plant model G 0 (s). When partial knowledge about the frequency range of the disturbances is available, the lter F (s) should be chosen such that G 0 (s)F (s) has large enough gains in all directions (if possible) over the expected disturbance frequency range, and very small gains at high frequencies where the unmodeled dynamics may be dominant. Large enough gains of G 0 (s)F (s) at the frequencies of the disturbance increases the excitation level of the regressor at those frequencies and can signicantly improve the performance; moreover, very small gains of G 0 (s)F (s) at high frequency range compensate the eect of possible high- frequency unmodeled dynamics. To design F (s), one may use the pre-compensator design techniques proposed in [66] or system decomposition approaches such as inner-outer factorization [77]. The estimated model of (5.13) is obtained by replacing (t) by its estimate(t) at time t and ignoring the unknown error term (t) as ^ z(t) = T (t)(t) (5.15) where ^ z(t) is the predicted value ofz(t) based on(t) at timet. In order to measure the dierence between(t) and (t) at each time, we dene the estimation error as "(t) = z(t) ^ z(t) m 2 s (t) ; m 2 s (t) = 1 + 0 trace( T (t)(t)); (5.16) where m 2 s (t) is a normalizing signal and 0 > 0 is a design constant. It is obvious that in the presence of overparameterization and modeling uncertainties, a robust 95 adaptive law is required to generate the parameter estimate(t), because withN > 2 n max , the regressor cannot be persistently exciting and traditional adaptive laws designed for plants with no uncertainties may lead to parameter estimates that drift to innity as shown with simple examples in the adaptive control literature such as in [1]. Following the design procedure of [1,56], we consider the robust least squares algorithm to generate the estimates of (t): _ P (t) =P (t) (t) T (t) m 2 s (t) P (t) _ (t) = proj (P (t)(t)"(t)); (5.17) where(0) = 0 andP (0) = 0 I > 0 is a design constant matrix. Some modications should be made in (5.17) to avoid covariance wind-up, i.e., to prevent P (t) from becoming close to singularity. One may use the covariance resetting modication [1] to keep the minimum eigenvalue of the covariance matrix greater than a pre-specied small positive constant at each time, i.e., set P (t + r ) = 0 I, where t r is the time at which min (P ) 1 , where 0 > 1 > 0 are some design constants and min denotes the minimum eigenvalue. It should be noted that if the value 1 is not small, frequent resetting may adversely aect the performance and if it is too small, the rate of adaptation in some direction may become very small which weakens the ability of the adaptive scheme to track time-variations of the disturbance characteristics. One may also use the least squares with forgetting factor [1] to avoid covariance wind-up. The projection operator proj() in (5.17) is used to ensure that (t) belongs to a known compact set dened as S =fjg(), T 2 max 0g, where max > 0 is chosen such that (t)2 S,8t. Since is unknown and no a priori knowledge about the size of its norm is available, max must be chosen conservatively large to ensure that 2S. Some alternatives to projection and other robust modications can be found in [1]; for example, one may use the xed -modication that requires 96 no bounds on the set where the unknown parameters belong to. The projection operator may be implemented as _ (t) = 8 > > > > > > < > > > > > > : P (t)(t)"(t) if 2S 0 or if 2S and (P ") T rg 0; P (t)(t)"(t)P (t) rgrg T rg T P (t)rg P (t)(t)"(t) otherwise (5.18) whereS 0 =fjg() = T 2 max < 0g andS =fjg() = T 2 max = 0g denote the interior and the boundary of S. Theorem 5.1 Consider the closed-loop system (5.1), (5.5) with adaptive law (5.17) with N > 2 n max . If the plant modeling error is such that M k m (s)G 0 (s)F (s)k 1 < 1 (5.19) where M is a positive constant proportional to max 2S kk 1 and is independent of the size of unmodeled dynamics, then all signals in the closed-loop system are uniformly bounded. Moreover, the plant output signal vector y(t) satises lim T!1 sup 1 T Z t+T t ky()k 2 2 dc ( m + _ +v 0 ); (5.20) for any t 0, where m and _ are positive constants proportional to the size of unmodeled dynamics and _ , respectively, and v 0 = sup t jv(t)j and and c > 0 is a nite constant independent of t and the size of modeling error, noise level, and _ . In addition, in the absence of noise, unmodeled dynamics, and variations in the characteristics of the disturbance (i.e., when v 0 = 0, m (s) = 0, and _ = 0), the adaptive law guarantees the convergence of y(t) to zero. 97 Proof : The control input vector can be written as u(t) =F (s)K(s;(t))[z(t)] =F (s)(t)Q(s)[z(t)]; where the transfer function matrix Q(s) is dened in (5.9) and matrix (t) is the estimate of in (5.9) at time t. Then we write the signal vector z(t) as z(t) =y(t)G 0 (s)[u(t)] = m (s)G 0 (s)[u(t)] +d(t) = m (s)G 0 (s)F (s)(t)Q(s)[z(t)] +d(t): Letd 0 = sup t kd(t)k 1 , then using the properties of theL 1 -normk() t k 1 [1] we have kz t k 1 k m (s)G 0 (s)F (s)k 1 k(Q(s)[z]) t k 1 +d 0 k m (s)G 0 (s)F (s)k 1 M kz t k 1 +d 0 ; where M is a positive constant proportional to max 2S kk 1 . Then if M k m (s)G 0 (s)F (s)k 1 < 1; z(t) and therefore all signals in the closed-loop are uniformly bounded. From The- orems 4.3.4 and 4.3.5 in [1], the robust adaptive law (5.17) guarantees that: "(t), "(t)m s (t), (t), and _ (t) are bounded and "(t);"(t)m s (t); _ (t)2S( 2 ), where is an upper bound forj(t)=m s (t)j which is proportional to the size of modeling error, noise level, and _ . It is to be noted that if the condition given in Lemma 5.1 is satised and N > 2 n max , the existence of a parameter vector for which the peri- odic terms of the disturbance are completely rejected is guaranteed. Following the 98 procedure (5.8)-(5.13), we have G 0 (s)[u(t)] = T (t)(t) + 1 ( _ (t)); then from (5.13), y(t) =G 0 (s)[u(t)] +z(t) = T (t)(t) 1 ( _ (t)) + T (t) (t) +(t) = T (t) ~ (t) +(t) 1 ( _ (t)): (5.21) Then from (5.16) and (5.21), we have "(t)m s (t) = z(t) ^ z(t) m s (t) = T (t) ~ (t) +(t) m s (t) = y(t) + 1 ( _ (t)) m s (t) : (5.22) Sincem s (t) is bounded and"(t)m s (t); _ (t)2S( 2 ), theny(t)2S( 2 ) which implies (5.20). This completes the proof. The proposed control scheme can provide a satisfactory performance and stabil- ity margin if the design parameters are chosen properly. In any practical control design, dierent types of uncertainties and modelling errors demand some sort of trade-o between robust stability and performance, and to achieved a compromise solution some partial knowledge of the uncertainties such as bounds on the size of error terms may be needed. Here we summarize the design parameters which contribute to the performance and robustness with respect to unmodeled dynamics. From (5.14), (5.16), and (5.17), the lterF (s),,N, 0 , and the initial value of the covariance matrix P are the design parameters aecting the excitation level of the regressor and the rate of adaptation. And from (5.19), the margin of stability de- pends upon the selection of the lterF (s) and the value ofN and max . As explained 99 earlier, we choose the design parameters based on available a priori knowledge of the maximum number of distinct frequencies of the disturbance, frequency range of the disturbance, and the frequency range over which the unmodeled dynamics may be dominant. 5.3 Numerical Simulations We demonstrate the eectiveness of the proposed scheme using a 2 2 plant model of the form (5.1), where the modeled part of the plant is given by G 0 (s) = 0:02 s 2 + 2s + 5 2 6 4 s 3 2(s + 4) 2(s 1) 3s 3 7 5 : We assume the unknown unmodeled dynamics term is given as m (s) =0:001sI 2 , where I 2 is the 2 2 identity matrix. The magnitude of m (s) is very small at low frequencies, but it increases with frequency and becomes large at high frequency. Figure 5.2(a) shows the singular value plot of the plant model G 0 (s) (the solid curves) and the singular value plot of m (s)G 0 (s) is shown in Figure 5.2(b) (the solid curves). We assume n max = 8 and ! max = 500 rad/sec. Since the gain of the plant model G 0 (s) is small over the expected frequency range of disturbance (i.e., ! < ! max ), we need to design a pre-compensator F (s) such that G 0 (s)F (s) has large enough gain for ! < ! max and very low gain at high frequencies to compensate for the eect of high-frequency modeling errors. For this example, we use the inner-outer factorization [77] of the plant model and write G 0 (s) =G in (s)G out (s), where G in (s) is a stable proper all-pass lter (inner factor) and G out (s) is a stable proper with 100 Figure 5.2: The singular value plots of (a) G 0 (s) and the shaped plant G 0 (s)F (s), and (b) m (s)G 0 (s) and m (s)G 0 (s)F (s). The singular values of G 0 (s)F (s) and m (s)G 0 (s)F (s) are aligned. stable right inverse (outer factor). Then we choose F (s) =k 0 2 (s +) 2 G 1 out (s) where k 0 = 0:05 and = 500 are design constants which specify the maximum gain and the bandwidth of the shaped plant G 0 (s)F (s), respectively. The dashed curve in Figure 5.2(a) shows the singular value plot of the shaped plant G 0 (s)F (s), where the singular values are aligned. The singular value plot of m (s)G 0 (s)F (s) is shown in Figure 5.2(b) (the dashed curve). By reducing the design parameters k 0 and , theH 1 -norm of m (s)G 0 (s)F (s) decreases and the stability margin is 101 improved; however, it lowers the regressor level and degrades the performance. The other design parameters are P (0) = 1000I 4N , (0) = 0, 0 = 1, and = 500. The unknown disturbances are assumed to be d 1 (t) = sin(! 1 t + 3 ) + 0:8 sin(! 2 t + 8 ) + 0:9 sin(! 3 t + 6 ) +v 1 (t); d 2 (t) = sin(! 4 t + 4 ) + 0:9 sin(! 5 t + 2 ) + 0:9 sin(! 6 t + 5 ) +v 2 (t) (5.23) where ! 1 = 80, ! 2 = 153, ! 3 = 275, ! 4 = 60, ! 5 = 180, and ! 6 = 310 rad/sec, and v 1 ;v 2 are zero-mean Gaussian noise with standard deviation 0:02. Figure 5.3 shows the plant output vectory(t) = [y 1 (t);y 2 (t)] T forN = 2 n max = 16 andN = 40, where the control input u(t) is applied at time t = 10 sec. Increasing the value of N improves the performance. Now assume from t = 20 to t = 30 sec, the frequency ! 1 varies linearly with time from 80 to 100 rad/sec, and ! 4 varies linearly from 60 to 80 rad/sec. Also, at t = 30 sec, ! 3 abruptly changes from 275 rad/sec to 350 rad/sec and ! 6 abruptly changes from 310 rad/sec to 200 rad/sec. Figure 5.4 shows the output signals y(t) = [y 1 (t);y 2 (t)] T and the estimate of the controller parameters (t) for N = 40. When the disturbance parameters change, the controller readjusts its parameters to counteract the eect of disturbances on the plant outputs. 102 Figure 5.3: The performance of the proposed scheme with (a) N = 16, (b) N = 40. The control input is switched on at t = 10 sec. 103 Figure 5.4: (a) The plant output signals for N = 40. The control input is switched on at t = 10 sec. From t = 20 to t = 30 sec, ! 1 varies linearly with respect to time from 70 to 100 rad/sec and ! 4 changes from 60 to 80 rad/sec and at t = 30 sec, ! 3 abruptly changes from 275 rad/sec to 350 rad/sec and ! 6 changes from 310 rad/sec to 200 rad/sec. (b) The estimate of the controller parameters (t). 104 6 Plants with Unknown Parameters In this chapter, the objective is to design an output-feedback adaptive controller by employing robust adaptive pole placement technique to reject the eect of unknown additive output disturbances in linear systems with unknown parameters. The cal- culation of the controller parameters involves solving a system of algebraic equations which may become singular or close to singularity during adaptation. To avoid an ill-conditioned system, it is assumed that the plant model is minimum-phase with known relative degree, but unknown parameters, then an all-pole approximate model of the plant is used in control design. 6.1 Problem Statement In many industrial applications, undesired narrow-band disturbances, dominated by a sum of a nite number of sinusoidal terms, adversely aect regulation/tracking performance. The disturbance suppression problem is dened as follows: the output of an open-loop stable system is corrupted by an unknown/time-varying additive disturbance. The plant model is uncertain and may occasionally vary with time. A control input signal must be applied to the plant to counteract the eect of the 105 disturbance on the plant output. We consider plants of the form y(k) =G(z)[u(k)] +d(k) =G 0 (z)(1 + m (z))[u(k)] +d(k); (6.1) where y(k) is the observed output, u(k) is the control input, d(k) is an unknown (directly non-measurable) bounded disturbance at time instant t = kT s sec. The discrete-time plant transfer functionG(z) is stable strictly proper and its dominant modeled part is of the form G 0 (z) = Z 0 (z) R 0 (z) = k 0 (z m +b m1 z m1 +::: +b 0 ) z n +a n1 z n1 +::: +a 0 ; (6.2) where parametersk 0 6= 0,b i , anda i are unknown,n = deg(R 0 ) andm = deg(Z 0 ) are known, andmn1. The term m (z) in (6.1) represents an unknown unstructured multiplicative uncertainty; a wide class of modeling errors can be expressed in the form of (6.1) [1]. In this chapter, we assume all zeros of the modeled part of the plant, G 0 (z), are injzj < 1. We will discuss later the generalization of the control scheme for possibly non-minimum phase systems with unknown relative degree. The term d(k) in (6.1) is bounded narrow-band disturbance. In many applica- tions, the dominant part of narrow-band vibrational disturbances can be modeled as a sum of a nite number of sinusoidal terms d(k) = n f (k) X i=1 A i (k) sin(! i (k)T s k +' i (k)) +(k) =d s (k) +(k); (6.3) where the real scalarsA i , ! i , ' i , and the number of distinct frequencies n f are 106 all unknown and may very with time, and the term is a bounded broad-band noise disturbance. The sum of sinusoidal terms (i.e., the dominant part of d(k)) is denoted byd s (k). We assume that upper bounds for the highest frequency ofd s and for the maximum number of distinct frequencies at each time are available. Any a priori knowledge on the expected frequency range and frequency components of the disturbance can be used for the selection of some design parameters to achieve a desirable performance. The control objective is to produce a control input signal u(k) for which the eect of d(k) on the plant output y(k) is minimized. 6.2 Controller Structure In this section, we propose a control structure for suppression of unknown distur- bances for unknown plants. First, we consider the non-adaptive case and assume that the plant model and disturbance characteristics are known and xed with time. We study conditions under which the control objective is achievable as well as the limitations imposed by the control structure. Then, based on the certainty equivalence principle, an adaptive algorithm is proposed to handle the unknown plant/disturbance case, i.e., the unknown parameters are replaced with their esti- mates. 6.2.1 Non-adaptive Scenario: Known-parameter case We dened the generating polynomial of the disturbance to be an N th -order monic polynomial Q d (z) =z N +q N1 z N1 +::: +q 1 z +q 0 (6.4) 107 where N 2n f , such that the magnitude of signal d (k) dened as d (k), Q d (z) z N [d(k)] = (1 +q N1 z 1 +::: +q 1 z 1N +q 0 z N )[d(k)] (6.5) at steady state is minimized. For complete rejection of sinusoidal terms in (6.3), Q d (z) must have zero gain at ! i 's, i.e., it must have Q n f i=1 (z 2 2 cos(! i T s )z + 1) as a factor. In the absence of noise (i.e., when (k) = 0), d (k) is an exponentially decaying to zero term, but in the presence of noise, its amplitude at steady state is of the order of the noise level. We over-parameterize Q d (z), i.e., choose N > 2n f , to have some free parameters for making the maximum gain of Q d (z)=z N as close as possible to 1 in order to avoid broad-band noise amplication [55]. We use the pole-placement technique [1, 56] to generate the control signal u(k) as follows: u(k) = P (z) L(z)Q d (z) [y(k)] (6.6) where the polynomialsP (z) andL(z) are solution to the following polynomial equa- tion L(z)R 0 (z)Q d (z) +P (z)Z 0 (z) =A (z); (6.7) whereR 0 (z),Z 0 (z), andQ d (z) are polynomials of degreen,mn1, andN 2n f , respectively (as dened in (6.2), (6.4)), P (z) = p N+n1 z N+n1 +p N+n2 z N+n2 + ::: +p 1 z +p 0 is of degree N +n 1 and L(z) =z n1 +l n2 z n2 +::: +l 1 z +l 0 is a monic polynomial of degree n 1. The roots of polynomial A (z) are the desired closed-loop system poles; for simplicity we choose A (z) =z 2n+N1 . 108 From (6.1) and (6.6), the closed-loop system can be expressed as y(k) = 1 1 + 0 (z) L(z)R 0 (z)Q d (z) z 2n+N1 [d(k)] = 1 1 + 0 (z) L(z)R 0 (z) z 2n1 [ d (k)] ; where 0 (z) = z (2n+N1) P (z)R 0 (z)G 0 (z) m (z) is due to the plant modeling un- certainty, and signal d (k) is dened in (6.5). Using the small-gain theorem [1,57], ifk 0 (z)k 1 < 1, the closed-loop system is stable and all signals are bounded. Remark 6.1 If a priori knowledge about the frequency range where the high fre- quency unmodeled dynamics may be dominant is available, by introducing a stable input lterF 0 (z) in (6.1), the eect of modeling uncertainties at high frequencies can be compensated and the stability margin can be improved. In this case, the new plant model is given byG 0 0 (z) =G 0 (z)F 0 (z), whereF 0 (z) has low gain at high frequencies. The polynomial equation (6.7) however may be unsolvable or ill-conditioned which may cause the failure of the approach. To ensure that this equation is well- conditioned, some assumptions aboutR 0 ,Z 0 , andQ d must be made. It is well known [78] that for given polynomials A;B;C, the polynomial equation AX +BY = C is solvable if and only if the greatest common divisor of A and B divides C. In practice, however, to avoid numerical problems in solving the equation we need no common divisor of A and B to be close to violate the solvability condition. Then, for equation (6.7) to be solvable, we make the following assumption. Note that the desired closed-loop poles are chosen to be all at z = 0, i.e., A (z) =z 2n+N1 . Assumption 6.1 The polynomials R 0 (z)Q d (z) and Z 0 (z) are strongly coprime, or if their greatest common divisor is of the form z M , for some integer M, the polyno- mials z M R 0 (z)Q d (z) and z M Z 0 (z) are strongly coprime. 109 The assumption that R 0 (z) and Z 0 (z) are strongly coprime is realistic in many applications; also, that Z 0 (z) must not have any root on the unit circle at the frequencies of the disturbance! i 's (i.e., atz i =e j! i Ts which are the roots ofQ d (z) for complete rejection of the sinusoidal terms of the disturbance) is a necessary condition to achieve the control objective independent of the control law [55], and is not a restrictive assumption. In order to avoid noise amplication, Q d (z) is chosen to be of high degree [55] (even if the number of frequenciesn f is known). Therefore the main concern is the possible closeness of the roots ofQ d (z) andZ 0 (z), especially around the unit circle. In this chapter, we show that this problem can be resolved for minimum-phase plants as explained below. It is well known that stables zeros can be approximated by multiple poles [79]. Since 1az 1 = 1= P i a i z i , ifjaj < 1, a minimum-phase zero-pole model can be approximated by an all-pole model, i.e., a model which contains only trivial zeros (at z = 0) and nontrivial poles by truncating the innite series to a nite one. So, if all zeros of the plant model G 0 (z) are inside the unit circle, we can write G 0 (z) = Z 0 (z) R 0 (z) = k 0 (z m +b m1 z m1 +::: +b 0 ) z n +a n1 z n1 +::: +a 0 k 0 z n(nm) z n +c n1 z n1 +::: +c 0 , Z 0 (z) R 0 (z) = G 0 (z); (6.8) where n > n is the order of the estimated all-pole model. By choosing a large enough n, the accuracy of the estimate can be improved; particularly if G 0 (z) has zeros close to the unit circle, a large enough n may be needed to have a suciently high modeling accuracy. Several algorithms have been proposed for the calculation of parameters of the all-pole model [79], e.g., the autocorrelation method which minimizes the total squared estimation error and determines the coecients c i 's 110 for a given lter order n. The autocorrelation method approximates any stable minimum-phase model with a stable all-pole model. In our problem, the error between G 0 (z) and G 0 (z) is considered as part of the modeling error which has a negligible size for a large enough n. The use of the all-pole approximate model for minimum-phase systems has been also studied in [80] to resolve the zero-pole cancellation problem in adaptive systems. The following example claries how the conversion of nontrivial stable zeros to multiple poles can eliminate the concern that the plant may have zeros close to the roots of Q d (z). Example 1 : AssumeT s = 10 3 sec,Q d (z) = (z 2 2 cos(0:1)z + 1)(z + 0:67) (the rst term has zero gain at! = 100 rad/sec, and the second term is for pushing down the peak magnitude of Q d (z)=z 3 ), and consider a plant with Z 0 (z) = 0:01(z + 0:67) and R 0 (z) = (z + 0:9)(z 0:4)(z 0:8). Obviously, equation (6.7) for this system is not solvable for an arbitrary A (z) including our preferred choice A (z) = z 8 , unless we restrictA (z) to have (z + 0:67) as a factor. If we approximate the stable minimum-phase part of a plant by an all-pole model, we will get rid of any nontrivial zero inside the unit circle. One can use Matlab `lpc' function (linear prediction lter coecients) which employs the autocorrelation method to nd the coecients of an all-pole approximate model. By applying this function to the impulse response of G 0 (z) = Z 0 (z)=R 0 (z) we nd its best all-pole approximate of order n = 4 in the least squares sense. The approximate model is given by G 0 (z) = Z 0 (z) R 0 (z) = 0:01(z + 0:67) (z + 0:9)(z 0:4)(z 0:8) 0:01z 2 (z 2 1:816z + 0:831)(z 2 0:8353z + 0:4022) = Z 0 (z) R 0 (z) = G 0 (z): Figure 6.1 shows the frequency response ofG 0 (z) and G 0 (z). By increasing the lter order n, the size of the modeling error decreases. With the approximate model, the 111 equation L(z) R 0 (z)Q d (z) +P (z) Z 0 (z) =z 10 is solvable. Figure 6.1: The bode plots of the plant modelG 0 (z) and its forth-order approximate G 0 (z) (Example 1). The approximate model G 0 (z) does not have any (nontrivial) zero. The approximation can be arbitrarily improved by increasing the lter order n. If the plant model has a known xed zero on the unit circle at z = 1 (which acts as a dierentiator), one can approximate the plant by an all-pole model over medium and high frequencies by introducing an input lter with a stable pole very close toz = 1. For example, letG 0 (z) =k 0 (z 1)=R 0 (z), then by introducing input lter W (z) =k 1 =(z), where 0 < 1, the compensated plant model is G 0 =G 0 (z)W (z) = k 1 k 0 z (z + (1))R 0 (z) (1 + m (z)); where m (z) =(1)=(z(z)) represents the approximation error. By choosing close to 1, the size of the modeling error is small at medium and high frequency range, but at low frequencies is quite large. In the next subsection, we consider the adaptive scenario and study the chal- lenges in calculation of the estimate of the unknown parameters of the plant and 112 disturbance, where both the plant parameters and parameters of the internal model of the disturbance are unknown. 6.2.2 Adaptive Scenario: Unknown-parameter case To deal with the problem of unknown disturbance attenuation for unknown plants, we employ the same control structure discussed in the previous subsection and use the certainty equivalence approach [1] to generate the control signal. Figure 6.2 shows the architecture of the closed-loop system. The adaptive controller has three main components: (i) an estimator for the generating polynomial of the disturbance Q d (z), (ii) an estimator for the parameters of the modeled part of the plant G 0 (z), and (iii) the control law which uses the estimate of Q d (z) and G 0 (z) at each time t =kT s and produces the control signal u(k). The saturation block is used to limit the control signal u in order not to violate control actuator constraints. Figure 6.2: The structure of the adaptive closed-loop system 113 6.2.2.1 Estimation of Q d (z) From (6.1) and (6.2) we have y(k) = Z 0 (z) R 0 (z) [u(k)] +G 0 (z) m (z)[u(k)] +d(k): (6.9) By operating on both sides of (6.9) by Q d (z)=z N and R 0 (z)=z n , we can write Q d (z) z N R 0 (z) z n [d(k)] = Q d (z) z N R 0 (z) z n [y(k)] Z 0 (z) z n [u(k)] Q d (z) z N R 0 (z) z n G 0 (z) m (z)[u(k)]: (6.10) It is to be noted that due to the boundedness of the coecients of R 0 and Z 0 , the signalz n R 0 (z)[d(k)] is a linear combination ofd(k);:::;d(kn);z n R 0 (z)[y(k)] is a linear combination ofy(k);:::;y(kn); andz n Z 0 (z)[u(k)] is a linear combination of u(kn +m);:::;u(kn). Now, we dene monic polynomials Q y (z) and Q u (z), each of degree K, such that the amplitude of signals y (k) =z K Q y (z)[y(k)] and u (k) =z K Q u (z)[u(k)] are minimized. Then, we let Q d (z) =Q u (z)Q y (z) to be the disturbance generating polynomial of degree N = 2K. Such a polynomial guarantees that the left hand side of (6.10) is minimized. Let Q y (z) =z K + K1 z K1 +::: + 0 and Q u (z) =z K + K1 z K1 +::: + 0 , then we can write the following parametric models for estimation of the parameters of Q y and Q u : y(k) = y T y (k) + y (k) u(k) = u T u (k) + u (k); (6.11) where y = [ K1 ;:::; 0 ] T , u = [ K1 ;:::; 0 ] T , y (k) = [y(k 1);:::;y(k 114 K)], and u (k) = [u(k 1);:::;u(kK)]. Any robust adaptive law [1,56] can be used for the estimation of y and u . Let ^ Q y (z;k) and ^ Q u (z;k) be estimates of Q y and Q u at time t =kT s , respectively. Then, the estimate of Q d at time t =kT s is given by ^ Q d (z;k) = ^ Q y (z;k) ^ Q u (z;k). Based on the parametric models (6.11), we use pure least-squares algorithm [56] to estimate y and u as follows: P y (k) =P y (k 1) P y (k 1) y (k) T y (k)P y (k 1) m 2 y (k) + T y (k)P y (k 1) y (k) " y (k) = y(k) T y (k 1) y (k) m 2 y (k) y (k) = proj( y (k 1) +P y (k)" y (k) y (k)); (6.12) and P u (k) =P u (k 1) P u (k 1) u (k) T u (k)P u (k 1) m 2 u (k) + T u (k)P u (k 1) u (k) " u (k) = u(k) T u (k 1) u (k) m 2 u (k) u (k) = proj( u (k 1) +P u (k)" u (k) u (k)); (6.13) where P y (0) = P T y (0) > 0, P u (0) = P T u (0) > 0 are design parameters, m 2 y (k) = 1 + T y (k) y (k), m 2 u (k) = 1 + T u (k) u (k) are normalizing signals, and projection operator proj() [1,56] is used to guarantee the boundedness of y (k) and u (k),8k. It is to be mentioned that a high rate of adaptation during transient period may cause an undesirable peak in the estimation error which may yield a large control signal and destroy the stability properties of the closed-loop system. The use of projection can prevent such a destabilizing phenomenon. Lemma 6.1 The adaptive laws (6.12) and (6.13) guarantee that " u (k), " y (k) , " u (k)m u (k), " y (k)m y (k), u (k), y (k) are bounded, and " u (k), " u (k)m u (k),j u (k) 115 u (k 1)j2S( 2 u =m 2 u ), and " y (k), " y (k)m y (k),j y (k) y (k 1)j2S( 2 y =m 2 y ). Proof : See [56], Chapter 4. From Lemma 6.1, the coecients of ^ Q u (z;k) and ^ Q y (z;k) are bounded and their rate of variation is small in the mean; also, we have " u (k)m u (k) = ^ Q u (z;k)[u(kK)] m u (k) 2S( 2 u =m 2 u ) " y (k)m y (k) = ^ Q y (z;k)[y(kK)] m y (k) 2S( 2 y =m 2 y ): 6.2.2.2 Estimation of G 0 (z) Let ^ Q d (z;k) = ^ Q u (z;k) ^ Q y (z;k) be the estimate of the generating polynomial of the disturbance at time t =kT s . We can estimate the plant parameters as follows. Operating on both sides of (6.9) by ^ Q d (z;k)=z N and R 0 (z)=z n we obtain R 0 (z) z n ^ Q d (z;k)[y(kN)] = Z 0 (z) z n ^ Q d (z;k)[u(kN)] + R 0 (z) z n ^ Q d (z;k)G 0 (z) m (z)[u(kN)] + R 0 (z) z n ^ Q d (z;k)[d(kN)]: (6.14) We then write R 0 (z) z n [ y(k)] = Z 0 (z) z n [ u(k)] + G (k) + d (k); (6.15) where y(k) =z N ^ Q d (z;k)[y(k)], u(k) =z N ^ Q d (z;k)[u(k)], and G (k) = R 0 (z) z n ^ Q d (z;k)G 0 (z) m (z)[u(kN)] d (k) = R 0 (z) z n ^ Q d (z;k)[d(kN)]: (6.16) 116 The error term G depends on the plant unmodeled dynamics m (z), and d ; ^ q depends on the properties of ^ Q d (z;k). In the ideal case (i.e., when m = 0, = 0, and ^ Q d =Q d ), these error terms are zero at steady state. Then, a parametric model for the plant is given by p (k) = p T p (k) + G (k) + d (k); (6.17) where p (k) = [ y(k1);:::; y(kn); u(kn+m);:::; u(kn)] T is the regressor, p (k) = y(k), and p = [ T R 0 ; T Z 0 ] T = [a n1 ;:::;a 1 ;a 0 ;k 0 ;k 0 b m1 ;:::;k 0 b 0 ] T is the unknown parameter vector of the plant. Based on the parametric model (6.17), any robust estimator [1, 56] can be employed to estimate the plant parameters online. The stability and robustness of such estimators have been established in the presence of error terms G and d independent of the excitation properties of the regressor [1,56]. 6.2.2.3 Control law To produce the control signal, we use the control law (6.6) and replace the unknown parameters by their online estimates. We can then implement the adaptive control law as u(k) = z N+n1 ^ Q d (z;k) ^ L(z;k) z N+n1 [u(k)] ^ P (z;k) z N+n1 [y(k)]; (6.18) whereN = deg( ^ Q d ),n = deg( ^ R 0 ), and ^ L, ^ P are solutions to the following polynomial equation ^ L(z;k) ^ R 0 (z;k) ^ Q d (z;k) + ^ P (z;k) ^ Z 0 (z;k) =z 2n+N1 ; (6.19) 117 where deg( ^ L) =n1, deg( ^ P ) =N +n1, and polynomials ^ L, ^ R 0 , ^ Q d are monic. It should be noted that ^ Q d ^ L is a monic polynomial of degreeN+n1 and the rst term in the right-hand side of (6.18) is a linear combination ofu(k1);:::;u(kNn+1). As we discussed earlier for the non-adaptive case, equation (6.7) is well-conditioned if Assumption 1 is satised. Clearly, in the adaptive case where the coecients of ^ R 0 ^ Q d and ^ Z 0 are time-varying, it is highly possible that during adaptation equation (6.19) becomes ill-conditioned and the adaptive controller fails. Similar to the non- adaptive case, for minimum-phase systems this issue can be resolved by replacing the plant model with its all-pole approximate (6.8). It should be mentioned that the following assumptions about the plant model G 0 (z) are made to prevent a possible ill-conditioned polynomial equation. Assumption 6.2 All zeros of G 0 (z) are always injzj< 1. The relative degree of G 0 (z) (i.e. nm) is known and xed. The sign and a lower bound on the magnitude of the leading coecient ofZ 0 (z) (i.e., k 0 ) is known. Under the above assumptions, the polynomial equation (6.19) with the approximate all-pole model (6.8) is given by ^ L(z;k) ^ R 0 (z;k) ^ Q d (z;k) + ^ P (z;k) ^ k 0 (k)z n(nm) =z 2 n+N1 ; (6.20) where deg( ^ R 0 ) = n, deg( ^ L) = n 1, deg( ^ P ) = N + n 1. Since k 0 6= 0, if we keep ^ k 0 (k) bounded away from zero, equation (6.20) is always solvable even if ^ R 0 (z;k) ^ Q d (z;k) has some roots at z = 0. By using parameter projection and based on some a priori knowledge about the gain k 0 (sign and a lower bound on its magnitude) we can ensure that its estimate stays bounded away from zero [1,56]. 118 With the approximate all-pole model (6.8) we have y(k) = Z 0 (z) R 0 (z) [u(k)] + ( a (z) +G 0 (z) m (z))[u(k)] +d(k); (6.21) where a (z) =G 0 (z) G 0 (z) is a stable error term due to the all-pole approximation. Then, similar to (6.14)-(6.17), the plant can be parameterized as p (k) = p T p (k) + p (k); (6.22) where p (k) = [ y(k); y(k 1);:::; y(k n)] T is the regressor, p (k) = u(kn +m), and p = [1=k 0 ;c n1 =k 0 ;:::;c 0 =k 0 ] T is the unknown parameter vector of the plant, and p (k) = G (k) + d (k), where G (k) = ^ Q d (z;k) z N R 0 (z) z n ( a (z) +G 0 (z) m (z))[u(k)] d (k) = ^ Q d (z;k) z N R 0 (z) z n [d(k)]: (6.23) Based on the parametric model (6.22), the estimate of the plant parameters can be calculated as P p (k) =P p (k 1) P p (k 1) p (k) T p (k)P p (k 1) m 2 p (k) + T p (k)P p (k 1) p (k) " p (k) = p (k) T p (k 1) p (k) m 2 p (k) p (k) = proj( p (k 1) +P p (k)" p (k) p (k)); (6.24) wherem 2 p (k) = 1 + T p (k) p (k) is the normalizing signal and p (k) is the estimate of p at timet =kT s . The projection operator in (6.24) in addition to guaranteeing the boundedness of p (k) ensures thatj ^ k 0 (k)j is bounded away from zero. The following lemma summarizes the properties of the adaptive law (6.24). 119 Lemma 6.2 The adaptive law (6.24) guarantees that " p (k), " p (k)m p (k), p (k) are bounded, and " p (k), " p (k)m p (k),j p (k) p (k 1)j2S( 2 p =m 2 p ). Proof : See [56], Chapter 4. It should be mentioned that in the parameterization (6.17), the regressor p (k) carries information for adaptation only during the transient period. At steady state, the regressor signal is mainly pure noise with almost no information about the plant input-output relation. Therefore, we may need to combine the adaptive law (6.24) with a small-size dead zone to prevent undesirable bursting phenomena [1,56]. The theorem below presents the properties of the closed-loop system. Theorem 6.1 Consider the closed-loop system (6.1), (6.12), (6.13), (6.24), (6.18) and assume the conditions given in Assumption 6.2 are satised. Then, there exists a constant > 0 such that ifkG 0 (z) m (z) + a (z)k < , then all signals in the closed-loop system are bounded and the plant output y(k) converges to the residual set R = ( y(k) lim T!1 sup 0<T T 1 T T1 X k=0 jy(k)jc( G + ) ) ; (6.25) where c > 0 is a generic constant, and G , are positive constants proportional to the size of plant modeling error and the noise level, respectively. Furthermore, in the absence of modeling error and noise the plant output converges to zero. Proof : The proof is similar to that of Theorem 7.4.1 in [56] and is omitted. Theorem 6.1 implies that if the modeling error satises a norm-bound condition, the output signal is of the order of the modeling error and the noise level in the mean-square sense. 120 Remark 6.2 Since input disturbances can be viewed as part of the output distur- bance (i.e., y =G(z)[u +d i ] +d =G(z)[u] +d 0 , where d 0 =G(z)[d i ] +d), for better plant parameters identication one may apply some sinusoidal terms to the plant input as additive input disturbances. The adaptive controller will reject the eect of the manually added disturbances as well as the unknown output disturbances on the plant output; however we need to use a higher order controller which increases the computational cost. 6.3 Numerical Simulation Consider the following third-order plant model G 0 (z) = Z 0 (z) R 0 (z) = k 0 (z 2 +b 1 z +b 0 ) z 3 +a 2 z 2 +a 1 z +a 0 = 0:5(z 0:6) (z + 0:7)(z 2 0:7z + 0:3) ; (6.26) where the parameters ofG 0 (z) are unknown. We assume that the following informa- tion about the plant is given: the plant model is minimum-phase of order three and relative degree two, and 0:0001<k 0 < 100. In control design instead of the above model we use its all-pole approximation of order n = 6, that is for plant parameter estimation the structure of the plant is assumed to be G 0 (z) = k 0 z 4 z 6 +c 5 z 5 +c 4 z 4 +c 3 z 3 +c 2 z 2 +c 1 z +c 0 (6.27) The sampling period is T s = 10 3 sec, and the unknown disturbance signal that we like to reject is d(t) = 0:3 sin(50t) + 0:4 sin(200t) 0:6 sin(500t) +(t) (6.28) 121 where is a zero-mean Gaussian noise with standard deviation 0:02 (j(t)j 0:1). A control command limiter is used to keepju(k)j 100 and prevent a very large control signal to be generated during the transient period. The other design parameters are P u (0) = P y (0) = 10I, P p (0) = 10I, u (0) = y (0) = 0, p (0) = [1; 0;:::; 0] T , and K = 30. The following plots show the performance of the closed-loop system, where the control input is applied at t = 10 sec. In Figure 6.3, at t = 20 sec new sinusoidal terms 0:7 sin(100t) + 0:8 sin(300t) are abruptly added the existing disturbance (6.28). The controller at this time re- adjusts its parameters to counteract the eect of new disturbance terms on the plant output. In Figure 6.4, at t = 20 sec the unknown plant gain k 0 abruptly changes from 0:5 to 0:05. This sudden change in the characteristic of the plant dynamics leads to signicant changes in the estimate of the unknown paraments which generate large control signal during the transient. The use of control command limiter keeps the control signal within the pre-specied limits. 122 Figure 6.3: The performance of the adaptive control system. The control is applied at t = 10 sec and at t = 20 sec a new disturbance term 0:7 sin(100t) + 0:8 sin(300t) is abruptly added to the existing disturbance. Figure 6.4: The performance of the adaptive control system. The control is applied at t = 10 sec and at t = 20 sec the unknown plant gain k 0 abruptly changes from 0:5 to 0:05. 123 7 Conclusion and Directions for Future Work The problem of attenuating unknown narrow-band disturbances in the presence of broad-band random noise and plant unmodeled dynamics for SISO and MIMO plants has been examined in both continuous and discrete-time formulations. We showed that by using proper plant pre-ltering, an over-parametrization the con- troller parameters, and a robust adaptive law for parameter estimation we can achieve the following: Guarantee stability provided the unmodeled dynamics are small in the low frequency range, Attenuation of the periodic components of the disturbance despite the presence of noise, unmodeled dynamics and time varying frequencies of the periodic disturbance terms Improve performance as well as stability margin especially in cases where the zeros of the plant are close to the zeros of the internal model of some of the disturbance terms. For plants with unknown parameters, a robust adaptive control algorithm based on the internal-model principal and the robust adaptive pole-placement has been 124 proposed for discrete-time minimum-phase systems which guarantees global stability and convergence. 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Abstract (if available)

Abstract Attenuation of unwanted sound and vibrations is a key enabling technology in a vast array of aerospace and industrial control applications. In many defense, industrial, and medical applications, unwanted sound and vibrations which are primarily caused by rotating or reciprocating components are typically dominated by a number of harmonics. Effective suppression of these types of disturbances is essential to achieve accurate control. Examples include optical/laser jitter suppression, vibration reduction in helicopters, precise pointing/tracking of spacecraft, acoustic noise reduction in powertrain and aerial vehicles. In these and other applications, disturbance attenuation requires adaptation both in the estimation and rejection of the jitter source as well as the determination of the system itself. Vibrational disturbances often have multiple narrow-band components combined with broadband random noise which can be modeled as a sum of noise and some sinusoidal terms with unknown and possibly time-varying frequencies, phases, and amplitudes. Moreover, there always exists discrepancy between the actual dynamical system and its identified model. Therefore, a controller designed for attenuation of disturbances must be able to handle types of uncertainties. The proposed research focuses on output-feedback adaptive control of uncertain systems. The main objective is to design and analyze a robust adaptive control scheme which ensures globally asymptotic suppression of unknown unmeasurable disturbances with time-varying characteristics for some classes of uncertain plants. In almost all past work in this area, the problem of robustness with respect to plant unmodeled dynamics and bounded noise disturbances has not been adequately addressed and it is not clear that how the performance can be improved. In fact, most adaptive laws proposed in the literature for the problem of disturbance rejection were shown not to be robust back in the early 80's. These issues indicate the importance of establishing the robustness of these schemes with respect to inevitable modeling errors. ❧ In some applications, the plant model is almost linear and time-invariant (LTI) and can be identified off-line with relatively high accuracy over some frequency range. In the first step, we consider known single-input single-output (SISO) LTI plants in the presence of unknown unmodeled dynamics. We examine robust stability and performance of a robust adaptive control scheme and discuss the trade-off between robust stability and performance improvement and practical design considerations. We show that over-parameterization of the internal model of the disturbances provides structural flexibility to reduce the sensitivity of the plant output with respect to noise and provides further improvements. We demonstrate that pre-filtering of the plant input together with over-parameterization significantly improves the performance especially when the zeros of the internal model of the disturbance are close to those of the plant. We generalize the idea to multi-input multi-output (MIMO) LTI plants for both discrete-time and continuous-time systems and show that how the proposed control scheme designed for SISO plants can be modified to be applicable to MIMO plants with significant cross-couplings. ❧ In some other applications, the plant model may have large parametric uncertainties and/or the parameters may vary with time. For instance, change in flight condition in a flight vehicles, partial failure in some components of the system, or change in environment may lead to significant change in the plant model. Therefore, the controller must be able to adjust its parameters to counteract these variations and at the same time try to minimize the effect of vibrational disturbances. In such cases, both disturbance and plant model are unknown, ensuring the stability and performance of such a nonlinear and time-varying closed-loop system is very complicated as there is coupling and a nonlinear relation between the unknown parameters of the internal model of the disturbance and the unknown parameters of the plant. This makes separate estimations of the plant model and the internal model of the disturbance problematic. In spite of efforts done on this problem, the case of unknown plant model and unknown disturbance remains an open problem as no solution with guaranteed global stability has been yet proposed for practical implementation. We propose a practical solution to the problem of attenuation of unknown narrow-band disturbances acting on unknown linear systems in the presence of unstructured unmodeled dynamics.

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Creator Jafari, Saeid (author)

Core Title High-accuracy adaptive vibrational control for uncertain systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 03/03/2016

Defense Date 08/18/2015

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

Tag disturbance suppression,modeling uncertainties,OAI-PMH Harvest,robust adaptive control,unknown narrow-band disturbances

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ioannou, Petros (committee chair), Bogdan, Paul (committee member), Savla, Ketan (committee member)

Creator Email saeid.jafari.1984@gmail.com,sjafari@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-175384

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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