University of Southern California Dissertations and Theses

Data-driven H∞ loop-shaping controller design and stability of switched nonlinear feedback systems with average time-variation rate

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Data-driven H∞ loop-shaping controller design and stability of switched nonlinear feedback systems with average time-variation rate

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Content DATA-DRIVENH 1 LOOP-SHAPING CONTROLLER DESIGN AND STABILITY OF SWITCHED NONLINEAR FEEDBACK SYSTEMS WITH A VERAGE TIME-V ARIATION RATE A Dissertation Presented to the Faculty of The Graduate School of UNIVERSITY OF SOUTHERN CALIFORNIA in Candidacy for the Degree of DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) by Yu-chen Sung Dissertation Advisor: Michael G. Safonov May 2019 Copyright c 2019 by Yu-chen Sung All rights reserved. Dedicatedtomyfamily Acknowledgments I would like to express my sincere gratitude to my Ph.D. advisor, Dr. Michael G. Safonov, for his inspiration and guidance. As his Ph. D. student, I learned from him not only his vast knowl- edge in control theory, but also his philosophy of being a researcher. He shows me researches should be motivated by real-world problems that need to be solved. I deeply appreciate his teaching that a researcher must be a contributor with his or her research results that help people. Next, I would like to show my gratitude to my friends and co-workers, Dr. Sagar V . Patil and Dr. Kiran S. Sajjanshetty. I enjoy in working with them on our research projects, and some of works leads to the main results in this thesis. It is satisfactory to solve interesting problems in a good teamwork. I thank my Ph.D. defense committee members, Dr. Ashutosh Nayyar and Dr. Firdaus Udwa- dia for their support. I would also like to thank Dr. Paul Bogdan and Dr. Rahul Jain for giving their professional opinions on my qualifying examination committee. Over the years in my Ph. D. programs, I have been supported by the teaching assistantships offered Dr. Alexander Sawchuk, Dr. Mahdi Soltanolkotabi, Dr.Fariba Ariaei and Dr. Moham- mad Reza Rajati. Serving as a teaching assistant for them is a valuable and educative experience in which I learned how to help students with my humble knowledge. I would like to thank the Ming Hsieh Department of Electrical Engineering, especially Ms. Diane Demetras, Mr. Shane Goodoff, Mr. David Ho and Mr.Tim Boston for assisting me in my Ph. D. program. Finally, I deeply thank my friends and my family in Taiwan for their love that makes the completion of this thesis possible. Yu-chen Sung January 2019 i Abstract DATA-DRIVENH 1 LOOP-SHAPING CONTROLLER DESIGN AND STABILITY OF SWITCHED NONLINEAR FEEDBACK SYSTEMS WITH A VERAGE TIME-V ARIATION RATE Yu-chen Sung 2019 In the first part of the thesis, a data-drivenH 1 loop-shaping controller design approach is inves- tigated in this thesis. The objective is to find controllers which satisfy specifications in terms of sensitivity and complementary sensitivity in the frequency domain by real-time observed plant input-output data. It is found that the proposed data-drivenH 1 loop-shaping controller design is a dual of McFarlane and Glover’s model-based H 1 loop-shaping controller design. Com- pared to McFarlane and Glover’s approach, the proposed approach relaxes the assumption that an upper bound on multiplicative uncertainty of a plant’s Normalized Coprime Factor (NCF) model is known. The proposed data-drivenH 1 loop-shaping controller design returns multiple feasible controller designs (not necessarily active ones) for given sensitivity and complementary sensitivity specifications in contrast to McFarlane and Glover’s modal-basedH 1 loop-shaping controller design that returns one controller design for the specifications, and prunes the con- trollers in real-time if observed plant input-output data prove that those controllers fail given sensitivity and complementary sensitivity specifications, and hence robustness of the adaptive switched feedback system is enhanced. An example is provided to support the improved robust- ness. The proposed data-drivenH 1 loop-shaping controller design implements a cost-function- minimizing scheme with a cost-detectable cost function to adaptively switch the controller in the loop. As a consequence, the adaptive switched system is input-output stable according to the proof derived by Stefanovic, Wang and Safonov [1]. By [1], an adaptive controller eventually stabilizes the adaptive switched systems implemented with cost-function-minimizing controller ii switching algorithm and cost-detectable cost functions given the plant is time-invariant. How- ever, the proof does not assume time-varying plants while plants are inherently subject to slow time variation that might be caused by persistently varying operating environments. Therefore, it is the motivation to find a sufficient condition to preserve the input-output stability given the same adaptive controller that stabilizes the time-invariant frozen-time plant model while consid- ering small and persistent plant perturbations. Zames and Wang [2] have derived a sufficient condition for such an adaptive switched system to be input-output stable, where slow time variation of the plant are expressed in terms of slow time variation of the feedback loop function. Zames and Wang’s sufficient condition gives an upper bound on maximum time-variation rate of the feedback loop function, and hence gives an upper bound on maximum time-variation rate of the slowly time-varying plant. However, Zames and Wang’s sufficient condition does not consider infrequent large perturbations of the feedback loop function, and is unable to conclude input-output stability of the adaptive switched system with infrequent large plant perturbations that might be caused by unexpected component failures. Moreover, Zames and Wang’s sufficient condition considers only the linear feedback loop functions, and hence is only applicable to the linear plants and controllers while real-world plants are inherently nonlinear. Therefore, the second part of the thesis analyzes the robustness against uncertainties in feed- back loop of a general adaptive switched feedback system. A new sufficient condition, expressed in terms of average time-variation rate of the feedback loop function, is derived to preserve the stability of the system without assuming the loop function is stabilizing all the time. The new sufficient condition relaxes three assumptions in Zames and Wang’s sufficient condition on the feedback loop function that (i) the feedback loop function is linear, (ii) the feedback loop func- tion is stabilizing all the time, and (iii) maximum variation rate of the feedback loop function is bounded. It is proved that the new sufficient condition generalizes Zames and Wang’s sufficient condition for a general adaptive switched feedback system to be input-output stable. Moreover, it is found that the new sufficient condition is less conservative than Zames and Wang’s sufficient condition. The new sufficient condition gives a tolerable limit on infrequent time variations or iii slow time-variation rate of a nonlinear MIMO adaptive switched system to preserve its stability. Two examples are provided to support the less conservative sufficient condition. iv Table of Contents List of Figures vii List of Tables ix Chapter 1: Introduction 1 1.1 A Brief Review on Literature on Robust Adaptive Control . . . . . . . . . . . 1 1.2 Unfalsified Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Data-DrivenH 1 Loop-Shaping Controller Design: An UAC System with Empha- sis on Loop-Shape Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Stability of Switched Nonlinear Feedback Systems with Time-Variation . . . . 4 1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Notation and Definitions 8 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 3: Data-drivenH 1 Loop-shaping Controller Design 12 3.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1 Model-basedH 1 Loop-shaping Design Procedure Approach [32, 33] . 15 3.1.2 Issues of Model-basedH 1 Loop-shaping Controller Design Procedure Approach [32, 33] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.3 Data-drivenH 1 Loop-shaping Controller Invalidation . . . . . . . . . 18 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Data-drivenH 1 Loop-shaping Controller Design . . . . . . . . . . . . . . . . 22 3.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6 A Duality between The Proposed Data-DrivenH 1 Loop-Shaping Invalidation Procedure and McFarlane and Glover’s Model-Based H 1 Loop-Shaping Syn- thesis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 4: Stability of Switched Nonlinear Feedback Systems with Average Time- Variation Rate 35 4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 v 4.1.1 Issues of Zames and Wang’s Result [2] . . . . . . . . . . . . . . . . . 36 4.1.2 Fixing the Issues of Zames and Wang’s Result [2] . . . . . . . . . . . . 37 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 Comparison with [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 5: Conclusion 51 Appendix A Lemma A.1 53 Appendix B Lemma A.2 55 Appendix C Proof of Theorem 3.1 56 Appendix D Proof of Lemma 4.1 58 Appendix E Proof of Lemma 4.2 60 Appendix F Proof of Theorem 4.1 61 Appendix G Proof of Lemma 4.4 65 Bibliography 66 vi List of Figures 3.1 Closed-loop systemH = (K;P ). . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Performance and robustness specifications on a loop gainjLj. . . . . . . . . . . 14 3.3 Closed-loop system (K s ;P s ) with shaped plantP s and shaped controllerK s having a transfer functionH s . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Bounds onjSj andjTj in terms of andjL D j =jP s j whenkH s k 1 . . . . . 17 3.5 Bounds onjSj andjTj in terms of andjL D j =jK s j are not guaranteed when kH s k 1 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.6 Fictitious closed-loop system (K s ;P s ) associated with K s = D s 1 N s and P s =W O PW I 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.7 Adaptive switching control system ( ^ K;P ). . . . . . . . . . . . . . . . . . . 23 3.8 Specifications on performance and robustness in term of desired loop shapeL D . 27 3.9 Unfalsified controller set and active controller history. . . . . . . . . . . . . . . 29 3.10 The controllerK 85 satisfies the specification on complementary sensitivity while the controllersK 1 andK 199 violate the specification. . . . . . . . . . . . . . . 30 3.11 The controllers K 1 and K 85 satisfy the specification on sensitivity while the controllerK 199 violates the specification. . . . . . . . . . . . . . . . . . . . . 31 3.12 The reference signal v, control signal u, output y, and tracking error e of the adaptive switching system ( ^ K;P ). . . . . . . . . . . . . . . . . . . . . . . . 31 3.13 Data-driven invalidation design vs. model-based synthesis design. . . . . . . . 34 4.1 A general feedback system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 The feedback system = (IGT ) 1 F . . . . . . . . . . . . . . . . . . . . . 38 4.3 (a)kg t k 1 exceeding d ;1 for allt2 [0; 55]. (b) Ratio of norms of outputx and inputu of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 vii 4.4 (t) for allt2 [0; 55]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5 t i t Q t i j=t+1 (j) for allt2 [t i1 ;t i 1] wheret i1 = 2 andt i = 15. . . . 47 4.6 (a) Average variation rated ;N (G)(t) bounded by d ;N . (b)krg t k 1 exceeding d ;1 (G) att = 3; 99. (c) Ratio of norms of outputx and inputu of the system . 49 A1 The system in terms ofG t andrG t . . . . . . . . . . . . . . . . . . . . . . . 61 viii List of Tables 3.1 Duality and comparison between data-driven and model-based loop-shaping designs. 33 ix Chapter 1 Introduction 1.1 A Brief Review on Literature on Robust Adaptive Control The first contribution of the thesis is a new data-drivenH 1 loop-shaping controller design that enhances robustness of an adaptive switched feedback system in presence of plant model mis- match, disturbance and noise, and hence is closely related to adaptive control and robust adaptive control. Therefore, a brief review on development in adaptive control and robust adaptive control is provided in the section to highlight the motivation in proposing the new data-drivenH 1 loop- shaping controller design for enhancement of robustness of adaptive control. The cited works in the section are selected from the cited works in the survey paper by Ioannou and Baldi [3] and the survey paper by ˙ Astr¨ om [4]. In adaptive control, controllers’ behavior are changed to conform to varying plant dynamics, disturbance and noise. Research on adaptive control originates in 1950s to solve for the design of autopilots of aircraft that operates in a wide range of altitudes and speeds. The flight dynamics of such aircraft then involves with varying operation conditions, which is impractically well compensated by a constant-gain controller. A more advanced controller design was required to solve the autopilot problem by compensating for variations in flight dynamics. Whitaker et al. [5, 6] proposed Model Reference Adaptive Control (MRAC) to solve the autopilot problem. Kalman [7] also suggested self-tuning regulators (STR) to compensate time-varying plants. In 1960s, ˙ Astr¨ om and Eykhoff [8] made major developments in system identification and in parameter estimation. State space and stability theory based on Lyapunov function approach were introduced into control theory and adaptive control. Park [9] solved instability problems caused by MIT rule in adaptive loops in MRAC by redesigning adaptive loops with Lyapunov 1 function approach. Dynamic programming developed by Bellman [10, 11] and dual control developed by Feldbaum [12, 13, 14, 15] are also important contributions to adaptive control. In 1980, Morse suggested a Lyapunov stability-based approach for adaptively controlling continuous-time SISO plants [16]. Narendra et al. proposed a Lyapunov stability-based approach for adaptively controlling continuous-time and discrete-time plants in MRAC [17]. Goodwin et al. [18] analyzed a general class of discrete-time adaptive control algorithms for MIMO plants, and showed that the algorithms ensure that the system inputs and outputs remain bounded and output tracking error converges to zero. All of the three works considered plants with (i) minimum-phase zeros and (ii) known upper bounds on the relative orders. Also starting from 1979, robustness of adaptive control began to draw more attention of researchers. Rohrs [19] et al. reported a class of adaptive algorithms including MRAC and STR cause instability in the presence of un-modeled dynamics. Egardt [20] showed a case where certain bounded dis- turbances and command signals cause an otherwise globally stable MRAC system to become unstable. Ioannou and Kokotovic [21] reported un-modeled dynamics of plants and bounded disturbances cause instability and suggested a modified adaptive controller with Lyapunov-like function based design to increase robustness. The robustness problems in adaptive control then have been motivating robust adaptive control research, in which signals in adaptive control sys- tems are required to be bounded in the presence of un-modeled dynamics and bounded dis- turbances [20, 21, 22, 23]. Logic based adaptive switching control also known as supervisory adaptive switching control were proposed by Morse [24, 25] to enhance robustness in an adap- tive control framework. In supervisory adaptive switching control, the plant is assumed to be in a plant model set, and the controllers designed for the respective plant models are put in the loop when the plant is close to one of the plant models. Zero tracking error is obtained in the face of constant disturbances, and stability is guaranteed for a class of un-modeled dynamics. However, stability caused by model mismatch were still reported [26] in Morse’s approach. In 1997, Unfalsified Adaptive Control (UAC) was proposed by Safonov et al. [27] as a new class of robust adaptive control which solved plant mismatch and un-modeled plant dynamic issues. A discussion on UAC is given in the next section. 2 1.2 Unfalsified Adaptive Control In Unfalsified Adaptive Control (UAC), a controller is said to be falsified by observed input- output data of the plant if the data imply that the controller fails the given specifications like preservation of closed-loop stability, disturbance rejection and noise response attenuation. UAC prevents falsified controllers from being switched in the loop, and if the current controller in the loop is falsified, a new controller with the minimum cost function value in the candidate con- troller set is switched on [26, 27, 28]. The cost functions implemented in UAC are required to have a property called cost-detectability [1, 27, 29], which means after finite switchings among controllers occur, the cost function value stays bounded if and only if the adaptive controller is stabilizing. It has been proved that with cost-detectable cost functions, UAC preserves closed- loop stability [1, 29]. Moreover, the cost function in UAC is the ratio of norms of filtered or non-filtered input-output data of the plant and fictitious signals [26], where a fictitious signal of a controller is defined as a hypothetical input signal that would have exactly reproduced the observed input-output data of the plant had the controller been in the loop for the entire time period over which the data were collected. As a consequence, the utilization of fictitious signals in the cost functions then enables falsification of controllers whether or not the controllers are in the loop when observed input-output data of the plant are collected. The candidate controller set can be finite or infinite in UAC, and can be designed with various controller synthesis meth- ods based on candidate plant models. When the plant deviates from the plant model much or if the model uncertainties are growing larger than expected uncertainties considered when the controllers are designed, UAC detects these incapable controllers and falsifies them. And hence UAC solves plant mismatch and un-modeled plant dynamic issues. For a UAC system to be stable, the only requirement is that at least one controller in the candidate controller set must be feasible in the sense that it meets given specifications. 3 1.3 Data-Driven H 1 Loop-Shaping Controller Design: An UAC System with Emphasis on Loop-Shape Specifications Cost-detectable cost functions in UAC can be built to accommodate various closed-loop spec- ifications. For example, in the cost function considered in UAC with resetting and bumpless transfer [30], the recent behavior of the plant weights more than the past one does, and UAC with multiple objectives [31] suggests a cost function that considers multiple specifications such as closed-loop stability and tracking errors between the actual output and the reference input. Because loop shapes are closely related to sensitivity and complementary sensitivity of closed- loop systems, while sensitivity and complementary sensitivity are closely related to robustness and performance of systems [32, 33, 34, 35], the first contribution of the thesis considers a loop- shaping specification and thus proposes a cost-detectable cost function to accommodate loop- shape specifications in an UAC framework to enhance its robustness, which leads to an adaptive switched system, data-drivenH 1 loop-shaping controller design [36, 37], that (i) prunes con- trollers that are proved by raw input-output data of the plant to fail to meet given loop-shape specifications, and (ii) puts in the loop an controller whose capability of meeting given loop- shape specifications is consistent with raw input-output data of the plant. The resultant adaptive switched system is found to be a dual to McFarlane and Glover’s model-basedH 1 loop-shaping controller design [33]. 1.4 Stability of Switched Nonlinear Feedback Systems with Time- Variation The proposed new data-drivenH 1 loop-shaping controller design which considers loop-shape specifications in an UAC framework inherently preserves closed-loop stability because any UAC system preserves closed-loop stability if cost-detectable cost functions are implemented [26, 27]. Since the closed-loop stability of the proposed data-drivenH 1 loop-shaping controller design is 4 preserved, the author is motivated to investigate a more general problem that to preserve closed- loop stability of not only UAC systems but also general adaptive switched systems. In a feedback system having an adaptively switching controller and a plant to be compen- sated, switching among controllers essentially leads to switching among loop functions. In other words, this switching feedback system has a time-varying feedback loop function. Hence, the second contribution of this thesis is the discovery of an upper bound on the average variation rate of the loop function in a general adaptive switched system such that the input-output stability of the adaptive switched system is preserved. Desoer [38, 39] has given such upper bounds by con- sidering the system in linear continuous-time and discrete-time state-space forms. Zames and Wang [2] have improved Desoer’s discrete-time result by considering the slightly more general linear case in which the loop function is assumed to be bounded, exponentially stable, and time- varying convolution operator whose all frozen-time linear time-invariant snapshots are bounded and stabilizing. However, for adaptive control applications, it is impractical to assume that either the adaptive loop function is linear or its all frozen-time snapshots are stabilizing. Adaptive controllers are inherently nonlinear. For example, in the Hysteresis Logic switching algorithm [40], the non- linear maximum operator in the switching logic causes the adaptive controller to be nonlinear. Moreover, the switching algorithm in an adaptive switched system may momentarily insert a destabilizing controller in the loop and causes it to be unstable [41]. Also, the results in [2, 38, 39] only consider the worst-case variation rate between two adja- cent frozen-time snapshots of the loop functions, and thus are compatible to slowly time-varying loop functions, but they are not compatible to the loop functions having infrequent and large time-variations that occur with switched adaptive controllers. Therefore the results cannot be used for stability analysis of adaptive switched systems. The author is then interested in expand- ing the results in [2] to the case where the inherently nonlinear adaptive feedback loop func- tion has infrequent and large time-variations. Therefore, it is the motivation for the author to generalize [2] by relaxing the linearity assumption on loop functions and hard constraint on time-variation rate of loop functions. 5 Solo [42] has derived a sufficient condition for the stability of continuous-time linear feed- back system _ x(t) = A(t)x(t) where a time-varying matrix A is allowed to be destabilizing, provided that the time average of absolute largest eigenvalue of A is bounded. Similarly, the second contribution of the thesis aims to derive a sufficient condition to preserve the stability of feedback systems by considering a discrete-time nonlinear case. 1.5 Contributions The contributions of this thesis are presented below. Data-driven H 1 loop-shaping controller design: in continuous time, an adaptive switched system and its switching logic are constructed such that the adaptive switched system (i) prunes and replaces controllers that are proved by raw input-output data of the plant to fail to meet given loop-shaping specifications, and (ii) preserves its closed-loop stability. A sufficient condition in terms of average loop function time-variation rates for general adaptive switched systems to be stable: in discrete time, a general sufficient condition in terms of loop functions variations to preserve stability of a general feedback system is derived by relaxing three assumptions on the feedback loop function that (i) it is linear, (ii) it is stabilizing all the time, and (iii) variation between its adjacent frozen-time snapshots is bounded. 1.6 Organization Notation and definitions are listed in Chapter 2. Chapter 3 presents a continuous-time adaptive switched system with a data-driven H 1 loop-shaping controller, which achieves purely data- driven switchings among candidate controllers to meet given specifications on the sensitivity and the complementary sensitivity of the closed-loop system. A general sufficient condition in terms of loop functions variations to preserve stability of a general discrete-time feedback 6 system is derived in Chapter 4. Chapters 3 and 4 are self-contained and may be read in any order. Conclusions are presented in Chapter 5. 7 Chapter 2 Notation and Definitions 2.1 Notation R : Set of real numbers R + : Set of non-negative real numbers Z : Set of integers Z + : Set of non-negative integers jj : Singular value [ ] 0 : Transpose s : Continuous-time differentiator operator 2.2 Definitions Definition 2.1 (signals): A real-valued functionx(t) of timet is said to be a continuous-time (discrete-time) signal mappingt2R (t2Z) tox2R n , wheren2Z + nf0g. Definition 2.2 (discrete-time ` pe norm): Given 1 and p2 [1;1], for any discrete-time signal x and for all t 1 ;t 2 2 Z where t 1 < t 2 , the moving-window fading-memory ` p semi norm [43, 44, 45] is defined as kxk p;[t 1 ;t 2 ] , 8 > > < > > : sup t2[t 1 ;t 2 ] (t 2 t) jx(t)j; ifp =1; P t 2 t=t 1 p(t 2 t) jx(t)j p 1 p ; otherwise. 8 For brevity, the notationkxk p;[t 1 ;t 2 ] is simplified as (i)kxk p;t ift 1 =1 andt 2 =t and (ii) kxk p;[t 1 ;t 2 ] if = 1. The extended space` pe is defined as` pe , n x :kxk p;t <1;8t2Z o Lemma 2.1 For any discrete-time signalx, it holds8p2 [1;1];8 1;82Z; and8t thatkxk p;t t kxk p; . Proof. Refer [30] and [43]. Definition 2.3 (discrete-time systems): Given 1 andp2 [1;1], a discrete-time system or discrete-time operatorH with inputu and outputy is a mappingu2` pe toy2` pe . Definition 2.4 (norms of discrete-time systems): Given 1 and p 2 [1;1], the moving- window fading-memory` pe -semi norm of the discrete-time systemH with inputu is defined as kHk p , sup 2Z kHk p; if the supremum exists, elsekHk p ,1, where kHk p; , sup kuk p; >0 kHuk p; kuk p; is the norm of systemH at time2Z (2R). Definition 2.5 (stability of discrete-time systems): Given 1,p2 [1;1], and an infinite time sequenceft i :t i1 <t i ;i2Zg witht i !1 asi!1, a discrete-time systemH is said to be weakly` pe -stable atft i g if kHk p;t i c; 8i2Z wherec2R + . Ifft i g =Z thenH is said to be` pe -stable which is denoted as` 1e -stability for = 1 andp =1. Definition 2.6 (the backward shift operator and the truncation operator): Given a signalx2 ` pe and n 2 Z, the operatorT is said to be the backward shift operator if for all t 2 Z, 9 (T n x)(t) =x(tn) holds. And the operatorP is said to be the truncation operator if for all 2Z, the following equation holds: (P x) (t) = 8 > > < > > : x(t);8t; 0; otherwise. Definition 2.7 (time-invariant, causal and memory-less discrete-time systems): A systemH is said to be (i) time-invariant (TI) ifHT =TH, (ii) causal ifP t H = P t HP t ,8t2Z, and (iii) memory-less if (P t P t1 )H = (P t P t1 )H (P t P t1 );8t2Z. Definition 2.8 (frozen-time snapshots of discrete-time systems): Given a nonlinear system H with input u2 ` pe and a pair of times t; 2 Z where t , the frozen-time snapshot h t : ` pe 7! R n of H is defined as h t T t u = (Hu) (t), and the unique frozen-time extension H of H at is defined as (H u) (t) , h T t u;8t . The difference between h 1 and h is defined asrh , h 1 h , and the difference between H and H is defined as rH ,HH : Remark 2.1 In Definition 2.8, the frozen-time extensionH is TI according to Definition 2.7. Lemma 2.2 Given a causal nonlinear discrete-time systemH with inputu and a pair of times t;2Z wheret, then8u2` pe , (rH u) (t) = 8 > > < > > : P i=t+1 rh i T t u; t<; 0; t =: Proof. The lemma is a consequence of causality ofH and Definition 2.8. Remark 2.2 For a discrete-time system H, d (H) in [2] is defined as d (H) , sup t2Z;u2`pe kT (Hu) (t) (HTu) (t)k 1 , which is equal to sup 2Z krh k 1 according to our Definition 2.8. 10 In the thesis, a discrete-time systemH is said to be slowly time-varying whenkrh t k 1 is small for allt2Z and it is said to be infrequently varying over some intervalL whenkrh t k 1 has small average for allt2L. The N-width average variation rate of a time-varying nonlinear discrete-time system is defined as follows. Definition 2.9 (N-width average variation rate): Given a causal nonlinear discrete-time system H, theN-width average variation rate ofH is defined as d ;N (H)(t), 1 N t X i=tN+1 krh i k 1 (2.1) and it is said to be bounded by some d ;N (H)2R + ifd ;N (H)(t) d ;N (H);8t2Z. Remark 2.3 A special case of Definition 2.9 havingN = 1 is discussed in [2]. 11 Chapter 3 Data-drivenH 1 Loop-shaping Controller Design 3.1 Introductory Remarks Sensitivity and complementary sensitivity are important for specifying robustness and perfor- mance of closed-loop systems. In a closed-loop system (K;P ) shown in Fig. 3.1, where a SISO plant and a SISO controller are denoted byP andK respectively. The sensitivity is defined asS, (1 +PK) 1 and the complementary sensitivity is defined asT,PK(1 +PK) 1 . The size ofjSj at each frequency! is the factor by which disturbances are attenuated, while 1=jTj is the magnitude of the smallest destabilizing multiplicative plant perturbation. However, due to the identityS +T 1, it is impossible to simultaneously lower bothjSj andjTj arbitrarily. These considerations are the basis for the standard mixed-sensitivity formulation of the robust control problem in terms of inequalities to be satisfied by the magnitude Bode plots ofS andT [46]. - + y d u + + Σ(K,P) Figure 3.1: Closed-loop systemH = (K;P ). 12 Loop shapeL = PK is closely related tojSj andjTj as a consequence of the well-known approximate upper and lower bounds on the loop shape [32, 33, 34, 35]: LS 1 = 1 +PK; ifjLj 1; and LT =PK(1 +PK) 1 ; ifjLj 1: Therefore, at least at frequencies far from the crossover frequency ! c wherejL(j! c )j = 1 , one can specifyjSj andjTj performance objectives for a closed-loop system in terms of loop- shapejLj [47], though of course additional constraints onjTj and/orjSj are needed at and near crossover frequency to ensure adequate stability margins. Specifically, the loop gainjLj is required to be sufficiently large within working bandwidth of a control system to meet perfor- mance specifications such as command tracking. On the other hand, the loop gainjLj is required to be sufficiently small at high frequencies to meet robustness specifications such as attenuation against response due to plant perturbation. The two considerations are illustrated in Fig. 3.2. Several works have been proposed to achieve desired closed-loop objectives, including those expressed in terms ofjSj andjTj. For example, in mixed-sensitivityH 1 control [48, 49, 50], one can find a controller such thatH 1 norm of a matrix of weighted closed-loop functions is smaller than or equal to 1, and then desired closed-loop properties are achieved. TheH 1 loop- shaping design procedure proposed by McFarlane and Glover [32, 33] achieves a specific kind of loop-shaping performance specifications in the frequency domain such that it guarantees bounds onjSj andjTj, and hence can be used to achieve sensitivity and complementary sensitivity shape requirements. These are model-based controller design approaches, in which one synthesizes a controller based on an assumed plant model and plant uncertainties. With unfalsified control theory [1, 26, 27, 51], robustness of a model-based robust control can be enhanced by performing a real-time and data-driven test for whether new data prove that the initial robust design fails to satisfy the loop-shaping objectives for which they are designed, and, 13 10 0 10 1 10 2 10 3 -40 -30 -20 -10 0 10 20 30 40 50 60 Specifications on a loop shape Frequency (rad/s) Singular Values (dB) Figure 3.2: Performance and robustness specifications on a loop gainjLj. if so, immediately switching another as yet unfalsified robust controller into the loop. The thesis examines the implications of unfalsified control theory with respect to McFarlane-Glover loop- shaping performance specifications and develops a necessary theory for finding the candidate controllers achieving desired sensitivity and complementary sensitivity loop-shaping inequalities by pruning the candidate controllers proved to fail desired inequalities. Given a plant model, the McFarlane-GloverH 1 loop-shaping design procedure returns opti- malH 1 controllers having maximal tolerance of unstructured coprime factor uncertainties in the numerator and denominator of a Normalized Coprime Factor (NCF) plant model. However, if the plant model is poor and if the model mismatch variation turns out to be larger than expected, or if an initially small mismatch evolves or drifts or shifts too much with time, then even an optimalH 1 controller may not be capable of robustly maintaining performance. In the thesis, the author is concerned with the problem of deciding at each time which, if any, members of 14 a given set of candidate controllers are provably incapable of meeting McFarlane-Glover type loop-shaping performance specifications using only the past plant input-output data without any assumptions on the plant and thus satisfying the givenjSj andjTj shape requirements. Given a SISO plantP and a SISO controllerK, the closed-loop system (K;P ) with input h v d i 0 and output h y u i 0 as shown in Fig. 3.1 has the following transfer function matrix, H, 2 4 P 1 3 5 (1 +KP ) 1 h K 1 i : The following specifications are considered: S 1 1 maxf1;jL D jg;8!2 [0;1); (3.1) and jTj minf1;jL D jg;8!2 [0;1); (3.2) whereL D is a desired loop shape and > 1 is a given positive number. 3.1.1 Model-basedH 1 Loop-shaping Design Procedure Approach [32, 33] The sensitivity and complementary sensitivity specifications (3:1) and (3:2) are considered by McFarlane and Glover when they solved the following loop-shaping problem in [32, 33]. Problem (Loop-Shaping Synthesis): Given a known plantP , the target > 1, the desired loop shapeL D , and the specifications (3.1) and (3.2), design a controllerK s such that (3.1) and (3.2) hold. McFarlane and Glover have solved the problem with their model-based H 1 loop-shaping controller design. The design procedure consists of two steps. Model-basedH 1 Loop-shaping Controller Design Procedure: 15 P W O v s W I -1 y s d s u s P S =W O PW I -1 W I W O -1 K S =W I KW O -1 u y K - + + + Σ(K S ,P S ) Figure 3.3: Closed-loop system (K s ;P s ) with shaped plantP s and shaped controllerK s having a transfer functionH s . 1) Loop Shaping: Choose stable and minimum-phase weighting functionsW O andW I to con- vert the original closed-loop system (K;P ) in Fig. 3.1 to an equivalent closed-loop system (K s ;P s ) which has a transfer functionH s with input h v s d s i 0 and output h y s u s i 0 as shown in Fig. 3.3, where H s , 2 4 P s I 3 5 (1 +K s P s ) 1 h K s I i ; whereu s , W I u, y s , W O y, d s , W I d, andv s , W O v. The plantP is shaped to be the desired loop shapeL D , e.g.P s =W O PW I 1 =L D ,. 2) Controller Synthesis: compute anH 1 controllerK s such that the following inequality is satisfied [32, 33, 52, 53], kH s k 1 ; (3.3) where > 1 [32, 33] andkk 1 denotes H 1 norm. For a system F with input x, the H 1 norm ofF is defined askFk 1 = sup ! (F (j!)) where is the maximum singular value of F ifF is stable, otherwisekFk 1 =1. In a SISO case,H 1 norm is simplified askFk 1 = sup ! jF (j!)j ifF is stable, otherwisekFk 1 =1. TheL 2e gain ofF is defined askFk = 16 sup kxk 2[0;t] 6=0;t0 kFxk 2[0;t] kxk 2[0;t] withkxk 2[0;t] = q R t 0 x() 0 x()d. If F is linear time-invariant then kFk =kFk 1 (Refer Lemma A.1 in the Appendix). Thus, the equation kH s k 1 = sup " ds vs # 2[0;t] 6=0;t0 2 4 y s u s 3 5 2[0;t] 2 4 d s v s 3 5 2[0;t] provides quantitative interpretations of the performance and robustness of the closed-loop system (K s ;P s ) in the time domain and also interprets the implication of (3:3) on the performance and robustness of the closed-loop system (K s ;P s ). Inequality (3.3) implies mixed-sensitivity loop-shaping bounds [49] as S 1 1 maxf1;jP s jg; 8!2 [0;1); jTj minf1;jP s jg; 8!2 [0;1); and thus the specifications (3.1) and (3.2) are satisfied if the weighting functionsW O andW I are chosen such thatjL D j =jP s j. The facts thatH 1 loop-shaping synthesis procedure achieves (3.1) and (3.2) are illustrated in Fig. 3.4. Singular v alue dB max 1 , | | min , | | | | || Figure 3.4: Bounds onjSj andjTj in terms of andjL D j =jP s j whenkH s k 1 . 17 Remark 3.1 The closed-loop system (K s ;P s ) is internally stable when (3.3) holds [49]. This feature is important as it excludes unstable pole-zero cancellations between the synthesized con- trollerK s and the shaped plantP s . Remark 3.2 TheH 1 loop-shaping synthesis procedure is closely related to the gap metric and the graph metric [35, 52, 54, 55, 56, 57]. 3.1.2 Issues of Model-based H 1 Loop-shaping Controller Design Procedure Approach [32, 33] McFarlane and Glover’s model-basedH 1 loop-shaping controller design relies on the following two assumptions: The plant’s NCF model is known. TheH 1 norm of multiplicative uncertainties of the denominator and the numerator of the plant’s NCF model are bounded by a real number with respect to time. However, if the plant model mismatch occurs or the model uncertainties are larger than expected, then the optimalH 1 controllerK s returned by McFarlane and Glover’s design procedure may not be capable of robustly satisfying the specifications (3.1 ) and (3.2) as the above two assump- tions are invalid. 3.1.3 Data-drivenH 1 Loop-shaping Controller Invalidation Considering the issues of McFarlane and Glover’s model-based H 1 loop-shaping controller design, the first contribution of the thesis aims to solve the following problem in order to find the controllers which satisfy (3.1) and (3.2) without the model ofP and assumption on uncertainties onP but merely with raw plant input-output data (u;y): Problem (Loop-Shaping Invalidation): Consider the closed-loop system (K;P ) in Fig. 3.1. Given a controllerK, an unknown plant P , observed data (u;y), > 1, the desired loop shapeL D and the specifications (3.1) and (3.2), 18 determine at each time t if available past data implies that the controller K fails to meet the specifications (3.1) and (3.2). If inequality (3.3) does not hold, this implies that (3.1) and (3.2) are not guaranteed if jK s j =jL D j as shown in Fig. 3.5. Therefore, one should design weighting functionsW I and W O such thatK s =W I KW O 1 andjK s j =jL D j. Then one should detect if (3.3) is violated so that (3.1) and (3.2) are not guaranteed withjL D j =jK s j. Thus, given a controllerK, the thesis is interested in using real-time data (u;y) to determine whether K can be proved to violate (3.3) irrespective of whetherK is in the loop or not. One solution to the proposed loop-shaping invalidation problem is the following procedure. Singular value dB max 1 , | | min , | | | | || || Figure 3.5: Bounds onjSj andjTj in terms of andjL D j =jK s j are not guaranteed when kH s k 1 > . H 1 Loop-Shaping Invalidation Procedure: 1) Loop Shaping: Choose stable and minimum-phase weighting functions W O and W I such thatK s =W I KW O 1 havingjK s j =jL D j. 2) Shaped Controller Invalidation: Use data (u;y) only to determine ifK s violates (3.3). If so, then (3.1) and (3.2) are not guaranteed. The controllerK is said to be invalidated by the data 19 (u;y). Therefore, the loop-shaping invalidation problem is simplified as the shaped controller inval- idation problem, which is formulated in the next section. The remainder of the chapter is organized as follows. The formulation of the shaped con- troller invalidation problem is given in section 3.2, followed by the main result in section 3.3. An application of the main result, the data-drivenH 1 loop-shaping controller design, is proposed in section 3.4. A simulation example is presented in section 3.5. A duality between the McFarlane- Glover model based loop-shaping design procedure and the proposed data driven loop-shaping design is discussed in section 3.6, and the chapter concludes with the concluding remarks in section 3.7. 3.2 Problem Formulation This chapter of the thesis considers continuous time signals and systems. The extendedL 2 space is denoted byL 2e =fxjkxk 2[0;t] <1; 8t 0g. A closed-loop system in Fig. 3.3 is considered, where an unknown plantP :L 2e !L 2e and a linear time-invariant controller K are given. The specifications on the sensitivityjSj, (1 +KP ) 1 and complementary sensitivityjTj, P (1 +KP ) 1 K are given as in (3.1) and (3.2) respectively, where > 1 and L D are also given. Let the reference input signal, plant input signal and the plant output signal are denoted byv;u andy respectively. The measured data from the plantP are denoted by = [yu] 0 . For the controllerK,W O 1 andW I are the invertible stable minimum-phase pre and post weighting functions respectively. The weighting functionsW O andW I are designed based on the desired loop shapeL D such that the shaped controllerK s =W I KW O 1 satisfiesjK s j =jL D j. This chapter of the thesis considers a special structure for every shaped controller. The pair (N s ;D s ) is a normalized left-coprime Matrix Fraction Description (MFD) of the linear shaped controllerK s such thatK s =D s 1 N s whereN s andD s are stable andD s is invertible [29]. 20 Cost function V is a data-driven cost function, which is a non-negative real-valued function, for the linear time-invariant shaped controllerK s =W I KW O 1 , it is given by V (K;W;;t) := kWk 2[0;t] +k~ vk 2[0;t] ; (3.4) where> 0,W := 2 4 W O 0 0 W I 3 5 is a stable minimum-phase invertible weighting function, and ~ v := [N s D s ]W. Remark 3.3 The signal ~ v is called the fictitious input signal of the shaped controllerK s as in Fig. 3.6. It is a hypothetical input signal that would have exactly reproduced the measured data had the controlleru s =D s 1 (~ vN s y s ) been in the loop as shown in Fig. 3.6 for the entire time period over which the data were collected [27]. u y = − 1 + - − 1 − 1 ṽ Figure 3.6: Fictitious closed-loop system (K s ;P s ) associated withK s = D s 1 N s andP s = W O PW I 1 . Remark 3.4 The small constant is included in V (K;W;;t) to account for non-zero plant initial states, e.g.6= [0 0] 0 for zero input ~ v = 0. Now the following formal problem formulation is stated: Problem 3.1 (Shaped Controller Invalidation Problem): Given a candidate controllerK with a weighting functionW , a target cost , a desired loop shapeL D and measured data over the interval [0;t], determine if the past data available at timet prove thatK fails to satisfyH 1 inequality (3.3) at timet. 21 3.3 Main Result This thesis uses the following assumptions to establish the main result about the proposed inval- idation of a shaped controller using the time domain cost functionV . Assumption A1: ControllerK is linear time-invariant. Assumption A2: Weighting functionsW O andW I are invertible, stable, and minimum-phase. Theorem 3.1 (Data-driven Shaped Controller Invalidation): Consider the measured plant data = [yu] 0 over the interval [0;t]. Consider a given controllerK with A1 and the corresponding given weighting functionW with A2. Then for the given target , past data available at timet prove thatK violates (3.3) if max t V (K;W;;)> : (3.5) Proof: Refer Appendix C Remark 3.5 According to [27], in unfalsified control, hypotheses are falsified or invalidated if collected data prove that they are not true. Theorem 1 is a special case of unfalsified control, where plant input-output data are used to invalidate the hypothesis that (K;P ) satisfies the specification (3.3). The controllerK is invalidated by the data if inequality (3.5) holds at time t 0, because by Theorem 1, inequality (3.5) implies the violation of inequality (3.3). 3.4 Data-drivenH 1 Loop-shaping Controller Design In this section, the main result in the previous section, data-driven shaped controller invalida- tion procedure, is applied to adaptive control for robustness enhancement. A new data-driven controller design, data-drivenH 1 loop-shaping controller design, is then proposed. Consider the adaptive switching system ( ^ K;P ) in Fig. 3.7, which consists of an unknown plantP :L 2e !L 2e , and a setK =fK 1 ;K 2 ;::::::;K N g ofN linear time-invariant candidate controllers. The active controller at timet is denote by ^ K(t)2K, wheret 0. The measured 22 data from the plant P are denoted by = [y u] 0 . The reference signal is denoted by v. The specifications (3.1) and (3.2) are given and the desired loop shapeL D are given, where > 1. For a controllerK i 2K, supposeW Oi 1 andW Ii are the invertible stable minimum-phase pre and post weighting functions respectively. The weights W Oi and W Ii are designed based on the desired loop shape L D such that the shaped controller K si = W Ii K i W Oi 1 satisfies jK si j =jL D j;8i2f1; 2;:::;Ng. The pair (N si ;D si );i2f1; 2;:::;Ng is a normalized left-coprime MFD of linearK si such thatK si =D si 1 N si whereN si andD si are stable andD si is invertible. v u y + - 1 cost function computation and switching algorithm switches ෡ ( ) P Σ ( ෡ ,P ) Figure 3.7: Adaptive switching control system ( ^ K;P ). Let V be a data-driven cost function which is a non-negative real valued function. For a linear time-invariant shaped controllerK si , it is given by V (K i ;W i ;;t) := kW i k 2[0;t] +k~ v i k 2[0;t] ; 23 where> 0,W i := 2 4 W Oi 0 0 W Ii 3 5 and ~ v i := h N si D si i W i . Define H si , 2 4 P si 1 3 5 (1 +K si P si ) 1 h K si 1 i ;8i2f1; 2;:::;Ng; whereP si ,W oi PW Ii 1 . LetK unf (t) be the set ofK i 2K whose ability to satisfy specification (3.3) has been unfal- sified by the data at timet 0. In other words, ifK i = 2K unf (t), thenkH si k 1 > has been proved by the past plant data at timet. With V (K i ;W i ;;t) > 0 as the criterion for controller falsification, one can use the standard unfalsified control algorithm [27] to switch a new unfalsified controller into the feed- back loop whenever data collected up to the current time t prove the current controller ^ K(t) meets the invalidation criterion (3.5). For any given 2R, the algorithm is as follows. Algorithm 1 (Unfalsified Control): Step 1: Lett = 0. Choose > 0 and t > 0. Let ^ K(t) = K 1 , whereK 1 2K. Initialize the unfalsified controller setK unf (0) =K. Step 2: For eachK i 2K unf (t), computeV (K i ;W i ;;t) and ifV (K i ;W i ;;t)> , removeK i fromK unf (t). Step 3: If ^ K(t) = 2 K unf (t), ^ K(t + t) K j , whereK j 2 K unf (t), otherwise ^ K(t + t) ^ K(t). Step 4:t t + t. Step 5: Go to Step 2. By [29], the cost functionV is cost-detectable becauseV is of MFD form. It means when ^ K(t) is computed using the unfalsified control algorithm, the resultant switched control system ( ^ K(t);P ) has following two properties: There are at most (N 1) switching times amongK in ( ^ K;P ). 24 For all t2 R + and8r2L 2e , there exist c2 R + and 2 R + such thatkk 2[0;t] ckvk 2[0;t] + holds. In other words, ( ^ K(t);P ) is stable after the final switch,8r2L 2e . Furthermore by [29], the system is closed-loop stable provided the following feasibility assump- tion holds: Assumption A3 (Feasibility): There exists at least oneK i 2K satisfyingkH si k 1 . In Step 3 of Algorithm 1, when the active controller ^ K(t) is invalidated by the data at time t, the replacing and as yet unfalsified controllerK j 2K unf (t) may be selected by the following criterion: Criterion C1 (Selection by minimum cost function): At a switching timet, when ^ K(t) is inval- idated by, ^ K(t) is replaced with the controllerK j = arg min K2K unf (t) V (K;W;;t). In other words, the controller that has the minimum cost function value is selected. Because with respect to the data, the controllerK j is most likely to meet the specification (3.3) and thus the specifi- cations (3.1) and (3.2) for having the smallest lower bound ofkH s k 1 , which isV (K j ;W j ;;t). However, because Algorithm 1 guarantees an active controller is switched off and prevented from being active again if the controller’s ability to meet the specification (3.3) is invalidated by the data regardless of how a replacing controller is selected, the criterion C1 is not mandatory. Remark 3.6 It is possible a controller that does not meet the specifications (3.1) and (3.2) is switched on by Algorithm 1. However, the active incapable controller is switched off and pre- vented from being active by Algorithm 1 immediately once the controller meets the invalidation criterion (3.5) and is invalidated by the collected data. The simulation example in Section 3.5 illustrates an initially active incapable controller is replaced by Algorithm 1 after short dwell time. Remark 3.7V (K i ;W i ;;t) enables tests of the hypothesis that (K i ;P ) guarantees specifica- tions (3.1) and (3.2) for allK i 2K, because for eachK i 2K, is collected andV (K i ;W i ;;t) is computed in real time while eitherK i is active or other controllers are active, and by Theorem 3.1, ifV (K i ;W i ;;t)> , then the hypothesis that (K i ;P ) satisfies the specification (3.3) and thus guarantees the specifications (3.1) and (3.2) is invalidated. Robustness of ( ^ K;P ) is then 25 improved by Algorithm 1, because the active controllers and the candidate controllers proved by the data to fail to guarantee the performance specifications (3.1) and (3.2) are removed from the set of unfalsified controllerK unf and are prevented from being active respectively. 3.5 Simulation Example This MATLAB example demonstrates how the proposed data-driven H 1 loop-shaping con- troller design prunes and replaces controllers proved by data not to guarantee the given specifica- tions on sensitivity and complementary sensitivity in the form of (3.1) and (3.2) when Algorithm 1 is implemented. Consider the following the unknown plantP [58] in this simulation example, P = s + 1 s(s 3) : The input and the output of P are denote by u and y, respectively. A family of 980 PID controllers, K = fKjK = c p + c i 1 s + c d s 0:005s+1 g, are given, where c p 2 C p , f:1;:2;:4;:6;:8; 1; 2; 4; 6; 8; 10; 20; 40; 60g, c i 2 C i ,f:1;:2;:5; 1; 2; 5; 10; 20; 50; 100g, and c d 2C d ,f:1;:4;:7; 1; 1:4; 1:7; 2g. The adaptive switching system is as shown in Fig. 3.7 with N = 980. For the adaptive switching system, the following specifications on the loop shapeL of the closed-loop system are given and shown in Fig. 3.8: (Performance specification 1) For adequate bandwidth of the closed-loop system, the crossover frequency! c must be at least 30 rad/sec. (Performance specification 2) To maintain good command tracking of the closed-loop system, the magnitude of the loop shape must be bigger than 70 dB at the low frequency ! = 0:01 rad/sec and must have a roll-off slope not less than20 dB/dec at low frequen- cies! 10 rad/sec. 26 10 -2 10 -1 10 0 10 1 10 2 10 3 -20 0 20 40 60 80 100 Specifications on the loop shape L Frequency (rad/s) Singular Values (dB) 10 110 30 Figure 3.8: Specifications on performance and robustness in term of desired loop shapeL D . (Performance specification 3) To maintain good command tracking of the closed-loop system, the magnitude of the loop shape must be bigger than 10 dB at low frequencies ! 10 rad/sec. (Robustness specification 1) To maintain good plant perturbation rejection, the loop gain must be smaller than 0 dB and has a roll-off slope20 dB/dec at high frequencies! 110 rad/sec. Based on the above four loop-shape requirements and the specifications (3.1) and (3.2), the target loop shapeL D is computed: L D = 60 s 27 and = 1:667: Theorem 3.1 is used to eliminate and replace the candidate controllers proved by past data to fail to guarantee the specifications (3.1) and (3.2). To this end, each controller is shaped such that the shaped controllerK s ,W I KW O 1 hasjK s j =jL D j;8K2K by choosing the weighting function W = 2 4 W O 0 0 W I 3 5 with W O = 1 andjW I j = L D K 1 , and next the set of the candidate controllers that meet the invalidation criterion (3.5) is eliminated at each timet 0 with the data = h y u i 0 collected from the closed-loop system in Fig. 3.7. The experiment time is fromt = 0 tot = 30 seconds. Let the initially active controller ^ K(0) be the controller K 980 that has the parameters c p = 0:1, c i = 0:1 and c d = 0:1; note K 980 is a destabilizing controller. Let the reference signalv = sin 2:5t;8t2 [0; 30]. Let the constant = 0:01 in the cost functionsV (K;W;;t);8K2K. Algorithm 1 is implemented in the adaptive switching system with the controller selection criterion C1. Upon each switch, the state of the new controller is re- initialized to maintain continuity and smoothness of the control signalu by using the bumpless transfer method [30, 59]. Fig. 3.9 shows that the initially active and destabilizing controller K 980 is switched off, and the controller K 1 with the parameters (c p ;c i ;c d ) = (60; 100; 2) is switched on at the first switching time ST 1 = 0:09 seconds. At the second switching time ST 2 = 15:58 sec- onds, the active controller K 1 is switched off, and the controller K 85 with the parameters (c p ;c i ;c d ) = (60; 100; 0:1) is switched on and remained active for the rest of the experiment. It is concluded that the initially active controllerK 980 is detected as a destabilizing controller by the data at the switching timeST 1 = 0:09 seconds, and the controllerK 1 ’s ability to meet the McFarlane-Glover loop-shaping specification (3.3) is invalidated while being active by the data at the switching time ST 2 = 15:58 seconds. Thus it is concluded by Theorem 1 that the controllerK 1 violates McFarlane-Glover loop-shaping specification (3.3), and thus does not 28 0 5 10 15 20 25 30 1 100 200 300 400 500 600 700 800 900 980 Time (sec) Controller index Unfalsified controller set and active controller history Unfalsified controller indices Active controller index Figure 3.9: Unfalsified controller set and active controller history. guarantee the specifications (3.1) and (3.2). Indeed, Fig. 3.10 shows the complementary sensi- tivity specification (3.2) is violated by the controllerK 1 , whereT 1 , K 1 P (1 +K 1 P ) 1 . On the other hand, the final controllerK 85 (which is active after timeST 2 = 15:58 sec) has been unfalsified by the data with respect to McFarlane-Glover loop-shaping specification (3.3), and thus has been unfalsified by the data with respect to the specifications (3.1) and (3.2) for the entire experiment timet2 [0; 30] seconds. Indeed, controllerK 85 satisfies the two specifications (3.1) and (3.2) as shown in Fig. 3.10 and Fig. 3.11. The evolving set of the unfalsified controllers are shown in Fig. 3.9 as time increases from 0 to 30. Each of the 980 horizontal line segments shows the length of the time before being invalidated by the data for each of the 980 candidate controllers. As shown in Fig. 3.9, 973 out of 980 controllers are invalidated by the data without being active during the experiment. It is concluded that these controllers satisfy (3.5) and are thus invalidated by the data even before ever becoming active in the feedback loop. By Theorem 3.1, it is concluded that these controllers 29 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 −60 −40 −20 0 20 40 60 80 100 Complementary sensitivity of Σ(K i ,P),∀ i ∈ {1,85,199} and complementary sensitivity specification Frequency (rad/s) Singular Values (dB) T 85 T 1 T 199 min{γ L D ,γ} Figure 3.10: The controllerK 85 satisfies the specification on complementary sensitivity while the controllersK 1 andK 199 violate the specification. violate the McFarlane-Glover loop-shaping specification (3.3). For example, the controllerK 199 with the parameters (c p ;c i ;c d ) = (20; 20; 2) violates the specifications (3.1) and (3.2) as shown in Fig. 3.11 and Fig. 3.10, whereS 199 , (1 +K 199 P ) 1 andT 199 ,K 199 P (1 +K 199 P ) 1 . The history of the reference v, the control signal u, the output y and the tracking error e, vy of the adaptive switching system ( ^ K;P ) are shown in Fig. 3.12, where the upper plot shows the overall history of the signals of the system, and the lower plot focuses on the transient response of the system near the first switching time ST 1, and indicates the initially active and destabilizing controllerK 980 is replaced shortly after the rapidly increasing control signalu is detected. To summarize, the experiment results illustrate that in this example, the proposed data-driven H 1 loop-shaping controller design was able to eliminate the initial destabilizing controller, then quickly converge with negligible transients to a stabilizing controller meeting all performance specifications. Interestingly, with the data acquired with just two switches and three of active controllers, the algorithm was also able to eliminate several hundred of controllers that would have failed to meet the specifications (3.1) and (3.2) had they been used. 30 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 −60 −40 −20 0 20 40 60 80 100 Sensitivity of Σ(K i ,P),∀ i ∈ {1,85,199} and sensitivity specification Frequency (rad/s) Singular Values (dB) S 85 −1 S 1 −1 S 199 −1 max{L D /γ,1/γ} Figure 3.11: The controllersK 1 andK 85 satisfy the specification on sensitivity while the con- trollerK 199 violates the specification. 0 5 10 15 20 25 30 −4 −2 0 2 4 Time (sec) (a) time response (0 ≤ t ≤ 30). Reference signal v, control signal u, output y, and tracking error e v u y e 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −4 −2 0 2 4 Time (sec) (b) transient response (0 ≤ t ≤ 5). v u y e Figure 3.12: The reference signal v, control signal u, output y, and tracking error e of the adaptive switching system ( ^ K;P ). 31 3.6 A Duality between The Proposed Data-Driven H 1 Loop- Shaping Invalidation Procedure and McFarlane and Glover’s Model-BasedH 1 Loop-Shaping Synthesis Procedure A duality between our data-drivenH 1 loop-shaping invalidation procedure and the McFarlane and Glover’s model-basedH 1 loop-shaping synthesis procedure is discussed here. The goal of both the approaches is to find controllers which achieve the specifications (3.1) and (3.2). In McFarlane and Glover’s model-based approach, weighting functions W O and W I are chosen such that shaped plant P s = W O PW I 1 has the desired loop shape, e.g. jP s j =jL D j, then controller K s is synthesized such that (3.3) holds. On the other hand, in the proposed data- driven invalidation approach where P is unknown, the weighting functions W O and W I are instead chosen such that shaped controller K S = W I KW 1 O has the desired loop shape, e.g. jK s j =jL D j. Next, the real-time data is used to test whether (3.3) holds for the entire time period during which were collected. If (3:3) does not hold, then the given specifications (3.1) and (3.2) are not guaranteed. Therefore, by the invalidation process one can find the feasible candidate controller satisfying the specifications (3.1) and (3.2) by eliminating provably inca- pable controllers based on the past data. Consequently, the shaped controller in the proposed data-drivenH 1 loop-shaping invalidation procedure plays a role that is dual to the role played by the shaped plant in McFarlane and Glover’s model-basedH 1 loop-shaping controller syn- thesis procedure. It is also worth mentioning that there is a duality among weighting functions in both the approaches, since the pre weighting functionW O 1 in the invalidation procedure is dual to the pre weighting functionW I 1 in the synthesis procedure, and the post weighting func- tionW I in the invalidation procedure is dual to the post weighting functionW O in the synthesis procedure. Table. 3.1 summarizes dualities between the two approaches and Fig. 3.13 illustrates the design procedures of the both approaches. 32 Table 3.1: Duality and comparison between data-driven and model-based loop-shaping designs. Approach McFarlane and Glover’s Sung, Patil and Safonov’s Input P ,jL D j and ,jL D j and Category of method Model-based synthesis Data-driven invalidation Loop shaping jL D j = W O PW I 1 jL D j = W I KW O 1 Output The synthesizedK The synthesized ^ K(t) 3.7 Concluding Remarks A data-driven loop-shaping design for controllers to meet McFarlane-Glover loop-shaping spec- ifications has been developed. The proposed design enables pruning and replacing candidate controllers when new data prove that they no longer meet specifications. Because switching only occurs when new data prove that the controller in the loop fails to meet performance speci- fications, the proposed design enhances robustness. An interesting discovery is that the shaped controller K s in our data-driven loop-shaping design plays a role that is dual to the role played by the shaped plant P s in McFarlane and Glover’s model-based loop-shaping design. 33 Given: the specification (3.1) and (3.2), the desired loop shape and γ > 0. Given: an unknown plant P with a controller set = . Given: an plant model and an u pper bo u nd of plant N CF m od el u ncertainty . Shape controller such that the shaped controller = with weighting function . Shape such that the shaped plant = . Measure plant input-output data ζ = ′ and co mpute ( , , ζ , ). is synthesized. ( 3 . 3 ) is satisfied → 3 . 1 and 3 . 2 are sati sf ied . If , , ζ , > γ f o r some ≥ 0 → ( 3 . 3 ) is violated. → 3 . 1 and 3 . 2 are not guar anteed . ℎ is invalidated, and thus is invalidated. Synthesize an optimal ∞ co nt roller . If the currently active controller ෠ ( ) has been invalidated, switch to another controller that is not yet invalidated. ( Mo d el − b as ed a pp r o ach ) ( D at a − d r iven a pp r o ach ) Figure 3.13: Data-driven invalidation design vs. model-based synthesis design. 34 Chapter 4 Stability of Switched Nonlinear Feedback Systems with Average Time-Variation Rate 4.1 Introductory Remarks Zames and Wang [2] derived a sufficient condition for a time-varying linear MIMO feedback system to be stable provided the feedback loop has sufficiently small time-variation. To preserve the stability, the condition gives a tolerable variation rate of a time-varying linear MIMO adaptive switched system with respect to the worst time-variation in the feedback loop. The basic objective of various adaptive switching control systems mentioned below is to switch suitable controllers in the loop so that the stability is preserved, while simultaneously meeting additional performance specifications. A switched system with Hysteresis Logic (HL) switches controllers based on their real-time data-driven performance evaluations [40], while a model-based switched system with HL takes into account plant uncertainties or noise [60, 61, 62]. A HL based switched system with additional reset feature safely discards past evaluated controller performances [43]. A switched system [30] generalizes [43] by considering nonlinear plants and controllers and adding bumpless switching feature. A switched system [37] considers real-time and data-driven controller performance based on loop-shape specifications. Generally speaking, in an adaptive system having an adaptively switching controller and plant to be compensated, switching among controllers essentially leads to switching among loop functions. In other words, this switching feedback system has time-varying feedback loop. 35 Therefore, the thesis considers a general discrete-time MIMO feedback system in Fig. 4.1, whereG is a time-varying nonlinear loop function,F is a time-varying nonlinear system, andT is a backward shift operator. The chapter of the thesis considers a problem that to find an upper + + Figure 4.1: A general feedback system. bound on variation rate of G such that the input-output stability [63] of the system is pre- served. Desoer [38, 39] has given such upper bound considering the system in linear state-space forms _ x =G(t)x andx(i + 1) =G i x i , which are equivalent to (i)G being a memoryless and time-varying matrix and (ii) F being the identity matrix in continuous time and discrete time respectively. Zames and Wang [2] have improved Desoer’s result by considering the slightly more general linear case in which both G and F are assumed to be bounded, exponentially stable, and time-varying convolution operators whose all frozen-time linear time-invariant snap- shots are bounded and stabilizing. Their work [2] has obtained a sufficient condition, expressed in terms of the worst-case time-variation rate between two adjacent frozen-time snapshots of the loop functionG, for the system to be closed-loop stable all the time. 4.1.1 Issues of Zames and Wang’s Result [2] However, Zames and Wang’s result cannot be used for analysis of closed-loop stability of general adaptive control systems due to the limitations caused by the following assumptions in [2]: Adaptive loop functions are linear: in [2], it is considered the loop functionG is linear. However, adaptive controllers are inherently nonlinear in many adaptive control systems due to the switching logics and algorithms used. For example, in the HL switching algo- rithm [40], the nonlinear maximum operator that appears in the switching algorithm causes the adaptive controller to be nonlinear. 36 Adaptive loop functions are stabilizing: in [2], it is considered that the frozen-time model of loop functionG is stabilizing for all time. However in many cases, switching algorithms may momentarily insert a destabilizing controller in the loop causing the loop functions to be unstable as shown in [41] Only slowly time-varying adaptive loop functions are considered: the sufficient condition derived in [2] is expressed in terms of the worst-case time-variation rate between two adjacent frozen-time snapshots of the loop functionG. This sufficient condition does not apply to cases where the loop function G has large and infrequent time variations, which are caused by switching between controllers in general adaptive control systems. 4.1.2 Fixing the Issues of Zames and Wang’s Result [2] Hence, the results in [2] are compatible to slowly time-varying loop functionG, but they are not compatible toG having infrequent and large time-variations that occur with switched adaptive controllers. Therefore the results cannot be used for stability analysis of adaptive switched sys- tems. The chapter of the thesis expands results in [2] to the case where the inherently nonlinear adaptive feedback loopG has infrequent and large time-variations whereG not required to be stabilizing for all time. Solo [42] has derived a sufficient condition for the stability of continuous-time linear feed- back system _ x(t) = A(t)x(t) where a time-varying matrix A is allowed to be destabilizing, provided that the time average of absolute largest eigenvalue of A is bounded. Similarly, the chapter of the thesis aims to derive a sufficient condition to preserve the stability of feedback system by considering a discrete-time nonlinear case. The main result of the chapter of the thesis is the derivation of the generalized sufficient condition in terms of tolerable time variation of nonlinear feedback loop functionG to preserve stability of the system without assuming the loop function is stabilizing all the time. The chapter also derives another sufficient condition by considering tolerable average time-variation rate ofG as a special case of the main result. 37 4.2 Problem Formulation The general feedback system in Fig. 4.1 can be described as = (IGT ) 1 F as shown in Fig. 4.2. The main problem is formulated as follows. − − 1 Figure 4.2: The feedback system = (IGT ) 1 F . Problem 4.1 Consider 1. Consider the nonlinear feedback system in Fig.2 where G : ` 1e 7! ` 1e and F : ` 1e 7! ` 1e havingkFk 1 < 1. Given time sequence ft i :t i1 t i ;i2Zg, find a sufficient condition such that, for alli2Z, inequalitykk 1;t i c holds wherec2R + . Remark 4.1 Problem 4.1 considers a general case where the system is weakly` 1e -stable at given time sequenceft i g. Ifft i g =Z, then the system is` 1e -stable by Definition 2.5. 4.3 Main Results Lemma 4.1 Consider; 0 2R + where 1 < 0 . Consider two causal nonlinear systems H andK. Then8t2Z the following inequality holds kh t rK t k 1;t kh t k 0 1;t c ; 0 (K;t); (4.1) where c ; 0 (K;t), sup i1 2 4 0 i t X q=ti+1 krk q k 1 3 5 : (4.2) Proof: Refer Appendix D. 38 Corollary 4.1 Consider Lemma 4.1. Then kh t rK t k 1 kh t k 0 1 c ; 0 (K;t): Proof :The corollary is a consequence of Definition 2.4 and Lemma 4.1. The following lemma is derived for systems with boundedN-width average variation rate defined in Definition 2.9. Lemma 4.2 Consider; 0 2R + where 1 < 0 . Consider a causal nonlinear systemsK having d ;N (K)2R + for someN2Z + . Then for allt2Z the following inequality holds c ; 0 (K;t)c ;N (K) where c ;N (K), e ln 0 1 0 N1 d ;N (K): (4.3) Proof: Refer Appendix E. The following corollary is derived immediately by Lemma 4.1 and Lemma 4.2. Corollary 4.2 Consider; 0 2R + where 1< 0 . Consider two causal nonlinear systems H andK having d ;N (K)2R + for someN2Z + . Then8t2Z the following inequality holds kh t rK t k 1;t kh t k 0 1;t c ;N (K) and kh t rK t k 1 kh t k 0 1 c ;N (K) Proof :The corollary is a consequence of Lemma 4.1, Lemma 4.2 and Definition 2.4. 39 Remark 4.2 In [2], it is derived thatkh t rK t k 1 kh t k 0 1 1 e ln 0 1 d (K). By Remark 2.2 and Definition 2.8,d (K) = sup 2Z krk k 1 . Therefore Corollary 4.2 general- izes the result in [2] by consideringN 1. A solution for Problem 4.1 is proposed as follows. Theorem 4.1 Consider; 0 2 R + where 1 < 0 , 2 R + , andft i :t i1 t i ;i2Zg. Consider the nonlinear feedback system in Fig. 4.2 whereF :` 1e 7!` 1e havingkFk 1 < 1 andG : ` 1e 7! ` 1e . Leth t andl t be the TI frozen-time snapshots of (IG t T ) 1 and (IG t T ) 1 G t T respectively. If t i t t i Y j=t+1 (j); 8t2 [t i1 ;t i 1];8i2Z; (4.4) then for alli2Z the following inequality holds kxk 1;t i t i t i1 kxk 1;t i1 +kuk 1;t i ; (4.5) where (t), max n kl t k 0 1;t c ; 0 (G;t); 1 o : (4.6) andc ; 0 (G;t) is defined in (4:2). Furthermore, if2 1 ; 1 , then for alli2Z inequality kk 1;t i c holds where c, t1 1 ; (4.7) t, sup i2Z (t i t i1 ); (4.8) , tkFk 1 sup i2Z max t2[t i1 +1;t i ] kh t k 1;t t i t : (4.9) Proof: Refer Appendix F. 40 Remark 4.3 To hold condition (4:4), it is not necessary that (IG t T ) 1 G t T to be stable for all time. The frozen-time snapshotl t can be unstable at some times other thanft i g as long as (4:4) holds, if both Q t i t kl t k 0 1;t and Q t i t c ; 0 (G;t) are small enough for allt2 [t i1 + 1;t i ] and for alli2Z. The following corollary is immediately derived by Theorem 4.1 and Lemma 4.2 given the N-width average variation rate ofG is bounded. Corollary 4.3 Consider; 0 2 R + where 1 < 0 ,2 R + , andft i :t i1 t i ;i2Zg. Consider the nonlinear feedback system in Fig. 4.2 whereF :` 1e 7!` 1e havingkFk 1 < 1 and G : ` 1e 7! ` 1e with d ;N (G)2 R + for some N 2 Z + . Let h t and l t be the TI frozen-time snapshots of (IG t T ) 1 and (IG t T ) 1 G t T respectively. If t i t t i Y j=t+1 N (j); 8t2 [t i1 ;t i 1];8i2Z; (4.10) then for alli2Z inequality (4:5) holds, namely kxk 1;t i t i t i1 kxk 1;t i1 +kuk 1;t i ; where N (t), max n kl t k 0 1;t c ;N (G); 1 o ; (4.11) and c ;N (G) is defined in (4:3). Furthermore, if 2 1 ; 1 , then for all i2 Z inequality kk 1;t i c holds, wherec is defined in (4:7). Proof: By Lemma 4.2, inequalityc ; 0 (G;t) c ;N (G) holds for allt2Z. By (4:6) and (4:11), (4:4) holds if (4:10) holds sincec ; 0 (G;t) c ;N (G) for allt2Z. By Theorem 4.1, if (4:4) holds then (4:5) holds. Therefore if (4:10) holds then (4:5) holds. Furthermore, by Theorem 4.1, if (4:4) holds and2 1 ; 1 holds thenkk 1;t i c for alli2Z. Therefore 41 if (4:10) holds and2 1 ; 1 holds thenkk 1;t i c for alli2Z. Hence, the corollary is proved. Like Zames and Wang’s sufficient condition [2, inequality (2:22)] for system to be` 1e - stable, Theorem 4.1 considers the frozen-time snapshotsl t , but it does not consider assumptions thatF andG are linear,G is stabilizing, andft i g =Z. Therefore Theorem 4.1 is a generalization of [2, inequality (2:22)]. The following Lemma 4.3 is derived to give a sufficient condition for the system to be ` 1e -stable for all time. Lemma 4.3 Consider; 0 2R + where 1< 0 and2 ( 1 ; 1). Consider the nonlinear feedback system in Fig. 4.2 whereF : ` 1e 7! ` 1e havingkFk 1 <1 andG : ` 1e 7! ` 1e . Let h t and l t be the TI frozen-time snapshots of (IG t T ) 1 and (IG t T ) 1 G t T respectively such that sup t2Z kh t k 1 <1 and sup t2Z kl t k 0 1 <1. If for allt2Z, c ; 0 (G;t) kl t k 0 1;t ; (4.12) then for allt2Z inequalitykk 1;t c holds, where c, kFk 1 sup t2Z kh t k 1 1 : (4.13) Proof: Note that if (4:12) holds, then (4:4) holds too for the special caset i t i1 = 1 by (4:2) and (4:6). By Theorem 4.1,kk 1;t c;8t2 Z. By (4:7), (4:8), (4:9), (4:13) and by noting t = 1, = tkFk 1 sup t2Z kh t k 1 , equationc = c holds. Thereforekk 1;t c;8t2Z. The following corollary is immediately derived by Lemma 4.3 given theN-width average variation rate ofG is bounded. Corollary 4.4 Consider ; 0 2 R + where 1 < 0 and 2 ( 1 ; 1). Consider the nonlinear feedback system in Fig. 4.2 where F : ` 1e 7! ` 1e havingkFk 1 <1 and G : ` 1e 7! ` 1e with d ;N (G)2R + for someN2Z + . Leth t andl t be the TI frozen-time 42 snapshots of (IG t T ) 1 and (IG t T ) 1 G t T respectively such that sup t2Z kh t k 1 <1 and sup t2Z kl t k 0 1 <1. If for allt2Z, d ;N (G) d ;N ; (4.14) then for allt2Z inequalitykk 1;t c holds, where c is defined in (4:13) and d ;N , 0 1N e ln 0 sup t2Z kl t k 0 1 1 : (4.15) Proof: Given (4:14) holds, then d ;N (G) d ;N ,c ;N (G) sup t2Z kl t k 0 1 1 (By (4:3) and (4:15)) ,c ;N (G) kl t k 0 1;t 1 ;8t2Z )c ; 0 (G;t) kl t k 0 1;t 1 ;8t2Z (by Lemma 4.2) and (4:12) holds. By Lemma 4.3, inequalitykk 1;t c holds. Hence, the corollary is proved. Remark 4.4 For the system to be ` 1e -stable, Zames and Wang’s sufficient condition [2, inequality (2:22)] is krg t k 1 d ;1 ;8t2Z (4.16) where d ;1 , e ln 0 sup t2Z kl t k 0 1 1 (4.17) 43 with < 1. In [2], Zames and Wang considered the sensitivity function (IGT ) 1 and proved (IGT ) 1 1 sup t2Z khtk 1 1 if (4:16) holds. On the other hand, sys- tem = (IGT ) 1 F is considered in the thesis, and it is proved in Lemma 4.3 that (IGT ) 1 F 1 c = kFk1 sup t2Z khtk 1 1 if the sufficient condition (4:12) holds. In the following lemma, the relation between our sufficient condition (4:4) and and Zames and Wang’s [2] sufficient condition (4:16) is discussed. Lemma 4.4 Consider; 0 2R + where 1< 0 and2 ( 1 ; 1). Consider the nonlinear feedback system in Fig. 4.2 whereF : ` 1e 7! ` 1e havingkFk 1 <1 andG : ` 1e 7! ` 1e . Let h t and l t be the TI frozen-time snapshots of (IG t T ) 1 and (IG t T ) 1 G t T respectively. Then the sufficient condition (4:4) holds whenever Zames and Wang’s sufficient condition (4:16) holds, and there exist cases where Zames and Wang’s sufficient condition (4:16) does not hold when the sufficient condition (4:4) holds. Proof: Refer Appendix G. Remark 4.5 By Lemma 4.4, the system is ` 1e stable when G varies with periodic large- variation such that for all q 2 Z,krg t k 1 2 d ;1 ;N d ;N i ;8t = qN, andkrg t k 1 = 0;8t6=qN. 4.4 Comparison with [2] The reference [2] proved that kHrKk () 1 e ln 0 1 kHk ( 0 ) d (K); (4.18) where both H and K are causal and linear with boundedkHk 0 1 andkKk 1 . By [2], kHk ( 0 ) = sup 2Z kh k 0 1 . By Remark 2.2, d (K) = d ;N (K) when N = 1. By [2] and Definition 2.8,kHrKk () = sup t2Z kh t rK t k 1;t . Therefore (4:18) is a special case of 44 our Lemma 4.1 when the worst-case variation rate ofK is bounded, andH andK are causal, sta- ble, and linear. Therefore, Lemma 4.1 generalizes (4:18) by considering nonlinear and unstable H andK and by relaxing the assumption that the worst-case variation rate ofK is bounded. Theorem 4.1 generalizes Zames and Wang’s sufficient condition (4:16) by relaxing assump- tions thatF andG are linear and the worst-case variation rate ofG is bounded. Theorem 4.1 allows unstable (IGT ) 1 GT such that its the frozen-time snapshot is not necessarily` 0 1e - stable for all time. Furthermore, Theorem 4.1 considers weakly` 1e -stability of the system at given time sequenceft i g, which generalizes Zames and Wang’s sufficient condition (4:16) where ` 1e stability of the system is considered for all time. By Lemma 4.4, Zames and Wang’s sufficient condition (4:16) is a special case of Theorem 4.1. 4.5 Simulation The following MATLAB simulation Examples 1 and 2 are demonstrated to support the Theorem 4.1 and Lemma 4.4 respectively. Example 1: Consider the system in Fig. 4.1. LetF be an identity matrix. Let the loop functionG be equal to a memory-less time-varying nonlinearH t where with input [v 1 v 2 ] 0 and output [w 1 w 2 ] 0 is a dead-zone operator such that for alli2f1; 2g, w i = 8 > > > > > > < > > > > > > : v i 0:5; ifv i 0:5; 0; ifv i 2 (0:5; 0:5); v i + 0:5; ifv i 0:5: andH t is a time-varying 2 2 real-matrix such that (i) ift2f0; 3g,j max (H t )j> 1; otherwise, j max (H t )j< 1 and (ii)H i 6=H j ;8i;j2Z + andi6=j. The above system in MATLAB is simulated with zero initial conditions and u = exp t 20 cos t 2 h 1 1 i 0 fort2 [0; 55]. The constants are considered: = 1:4, 0 = 1:54, and = 0:9361. The simulation results are shown in Fig. 4.3. 45 Fig. 4.3 (a) shows abrupt variations inG att = 1,t = 3, andt = 4. c ; 0 (G)(t) and (t) are computed for allt2 [0; 55] by (4:2) and by (4:6) respectively. Fig. 4.4 shows (t) for all t2 [0; 55]. Therefore, by the sufficient condition (4:4) of Theorem 4.1, the weak` 1e -stability 0 5 10 15 20 25 30 35 40 45 50 55 0 0.2 0.4 0.6 0.8 1 1.2 (a) 0 5 10 15 20 25 30 35 40 45 50 55 time (s) 0 0.2 0.4 0.6 0.8 1 1.2 (b) Figure 4.3: (a)kg t k 1 exceeding d ;1 for allt2 [0; 55]. (b) Ratio of norms of outputx and inputu of the system . of the system is preserved for all timet2 [0; 2][ [15; 55], since (i)kxk 1;0 = 0 because of the zero initial conditions, (ii) (4:4) holds for allt i 2 [1; 2][ [16; 55] andt i1 =t i 1, and (iii) (4:4) holds whent i = 15 andt i1 = 2, namely (t i1 ;t i ) = (2; 15), t i t Q t i j=t+1 (j) holds for allt2 [t i1 ;t i 1] as shown in Fig. 4.5. Next, by (4:7),c = 2702 is computed. Therefore, by Theorem 4.1,kk 1;t c and kxk1;t kuk1;t c for allt2 [0; 2][ [15; 55] as shown in Fig. 4.3 (b). Therefore, by Theorem 4.1,kk 1;t c and kxk1;t kuk1;t c for allt2 [0; 2][ [15; 55] as shown in Fig. 4.3 (b). On the other hand, sincej max (H t )j > 1 att = 0 andt = 3, the frozen-time snapshotl t of (IG t T ) 1 G t T is unstable, and sup t2Z kl t k 0 1 =1 according to Definition 2.4. Zames and Wang’s sufficient condition (4:16) for the system to be` 1e -stable iskg t k 1 d ;1 for 46 0 5 10 15 20 25 30 35 40 45 50 55 time (s) 0 5 10 15 20 25 Figure 4.4: (t) for allt2 [0; 55]. 2 4 6 8 10 12 14 time (s) 0 0.2 0.4 0.6 0.8 1 Figure 4.5: t i t Q t i j=t+1 (j) for allt2 [t i1 ;t i 1] wheret i1 = 2 andt i = 15. allt2Z. By (4:17), d ;1 = 0 is computed. But,kg t k 1 > 0 for allt2 [0; 55] as shown in Fig. 4.3 (a). Therefore, Zames and Wang’s sufficient condition (4:16) does not hold for allt2 [0; 55] as shown in Fig. 4.3 (a) and so it does not conclude that the system is` 1e -stable. This proves for this example that our sufficient condition (4:4) is less conservative than Zames and Wang’s sufficient condition (4:16); ie, (4:4) holds while Zames and Wang’s sufficient condition (4:16) does not hold. 47 Example 2: Consider the system in Fig. 4.1. LetF be an identity matrix. Let the loop functionG be equal to a time-varying linearG t whereG t is a time-varying 22 real-matrix such that (i)j max (G t )j < 0:3 for allt2Z + and (ii)G i 6= G j ;8i;j2Z + andi6= j. The frozen- time snapshotl t of (IG t T ) 1 G t T satisfies sup t2Z + kl t k 0 1;t = 1:3613 with the designed G t . The above system in MATLAB is simulated with zero initial conditions and u = [cos(t=2) cos(t=2)] 0 for t2 [0; 160]. The constants are considered: = 1:05, 0 = 1:155, = 0:98, andN = 10. The simulation results are shown in Fig. 4.6. By Definition 2.9, d ;N (G)(t) is bounded by d ;N (G) = 0:0787 for all t 2 [0; 160] as shown in Fig. 4.6 (a). By (4:15) and (4:17), d ;N = 0:0791 and d ;1 = 0:1903 are computed respectively. By (4:13), c = 103:33 is computed. Therefore, by Lemma 4.3 and Corollary 4.4, the` 1e -stability of the simulated system is preserved,kk 1;t c and kxk1;t kuk1;t c for all t2 [0; 160] since d ;N (G) d ;N as shown in Fig. 4.6 (a). Indeed, it is true that kxk1;t kuk1;t c for allt2 [0; 160] as shown in Fig. 4.6 (c). On the other hand, Zames and Wang’s sufficient condition (4:16) does not hold since krg t k 1 > d ;1 att = 3 andt = 99 as shown in Fig. 4.6 (b). Therefore, Zames and Wang’s sufficient condition (4:16) does not conclude that the ` 1e -stability of the simulated system is preserved. This proves for this example that our sufficient condition (4:4) is less conservative than Zames and Wang’s sufficient condition (4:16); ie, (4:4) holds while Zames and Wang’s sufficient condition (4:16) does not hold. 48 0 20 40 60 80 100 120 140 160 0 0.2 0.4 0.6 (a) 0 20 40 60 80 100 120 140 160 0 0.2 0.4 0.6 (b) 0 20 40 60 80 100 120 140 160 time (s) 0 1 2 (c) Figure 4.6: (a) Average variation rate d ;N (G)(t) bounded by d ;N . (b)krg t k 1 exceeding d ;1 (G) att = 3; 99. (c) Ratio of norms of outputx and inputu of the system . 49 4.6 Concluding Remarks In this chapter of the thesis, the input-output stability of a general time-varying MIMO nonlin- ear feedback system has been investigated by generalizing the results in [2]. A general sufficient condition to preserve stability of the feedback system has been derived by relaxing three assump- tions [2] on the feedback loop function that (i) it is linear, (ii) its frozen-time snapshot is stabiliz- ing all the time, and (iii) variation between its adjacent frozen-time snapshots is bounded. The sufficient condition gives a tolerable limit on past time-variation of a MIMO nonlinear adap- tive switched system to preserve its ` 1e -stability. Compared to the sufficient condition given in [2], our general sufficient condition is less conservative than the sufficient condition given in [2]; ie, whenever the sufficient condition given in [2] holds, our general sufficient condition holds, and the sufficient condition given in [2] does not hold when the adaptive switched system has infrequent large time variations while our general sufficient condition holds. Because our general sufficient condition is less conservative and has relaxed the three assumptions that are considered in the sufficient condition given in [2], our general sufficient condition is more prac- tical than the sufficient condition given in [2] to conclude stability of adaptive switched systems that are inherently nonlinear and subject to infrequent large variation that might be caused by unexpected component failures. As a special case of our general sufficient condition, another sufficient condition for the time-varying nonlinear feedback system to be stable has also been derived. This condition is that the loop function of the system should have a bounded average time-variation rate over every interval of fixed length. It has been proved that the sufficient condition given in [2] is a special case of the proposed sufficient condition when the length is equal to 1. 50 Chapter 5 Conclusion The thesis focuses on stability and robustness of adaptive control systems by solving two prob- lems. In the first contribution of the thesis, a purely data-driven controller design has been derived to accommodate loop-shape specifications in adaptive control systems given the plants are subject to uncertainties and model mismatch. By preventing controllers with previously fal- sified feasibility of meeting given loop-shape specifications from being in the loop, the loop shapes of adaptive control systems eventually meet given specifications in consistency with real- time observed` 2 norm of loop functions of the systems. Since specifications on the robustness of a closed-loop system can be represented in terms of sensitivity and complementary sensitivity of the system, which, in turn, can be represented in terms of loop-shape specifications, meet- ing loop-shape specifications implies meeting specifications on sensitivity and complementary sensitivity, and hence meeting specifications on the robustness of the system. Therefore, the pro- posed data-driven controller design enhances the robustness of adaptive control systems when the plants are subject to uncertainties and model mismatch. Interestingly, the proposed data- driven controller design has been found to be a dual of McFarlane and Glover’s model-based H 1 loop-shaping controller design procedure. In the proposed data-driven H 1 loop-shaping controller design, controllers are shaped to be the target loop shape according to given specifica- tions expressed in terms of sensitivity and complementary sensitivity, while the plant is shaped to be the target loop shape in McFarlane and Glover’s model-basedH 1 loop-shaping controller design procedure. Therefore, the shaped controller in the proposed data-drivenH 1 loop-shaping controller design plays a role that is dual to the role played by the shaped plant in McFarlane and Glover’s model-basedH 1 loop-shaping controller design procedure. The second contribution of the thesis investigated input-output stability of adaptive control systems with adaptively switching controllers and time-varying plants where controllers and 51 plants are MIMO nonlinear. By generalizing adaptive control systems with a general feedback system framework along with nonlinear time-varying loop functions, a general sufficient condi- tion for adaptive control systems to be input-output stable has been derived in terms of average time-variation rates of loop functions. The derived sufficient condition has been proved to gen- eralize Zames and Wang’s sufficient condition by extending to cases where loop functions (i) are destabilizing, (ii) are nonlinear, and (iii) have infrequent large time variations. It has been proved that our general sufficient condition is less conservative than the sufficient condition given in [2]; ie, whenever the sufficient condition given in [2] holds, our general sufficient condition holds, and our general sufficient condition holds for adaptive switched systems having infrequent large time variations while the sufficient condition given in [2] does not hold. Therefore, our sufficient condition is more practical than the sufficient condition given in [2] to analyze stability of real- world adaptive control systems with nonlinearity and infrequent large time variations that might be caused by unexpected component failures. 52 Appendix A Lemma A.1 Lemma A.1 Consider a continuous time, stable, causal and linear time-invariant systemG with the transfer functionG(s), inputx2L 2e and outputy with zero initial conditions. Then kGk 1 =kGk = sup x6=0;t0 kyk 2[0;t] kxk 2[0;t] : Proof. Letx t () denotes the truncation ofx at timet 0, such that x t () = 8 > > < > > : x();82 [0;t]; x t () = 0; otherwise. 53 Denotey t the response ofG tox t with zero initial conditions. By the causality ofG,y t () = y(); 82 [0;t]: Let the Laplace transforms ofx,x t ,y andy t be ^ x(s), ^ x t (s), ^ y(s) and ^ y t (s) respectively. Then kyk 2 2[0;t] = Z t 0 jy()j 2 d = Z t 0 jy t ()j 2 d (By causality ofG) Z 1 0 jy t ()j 2 d = 1 2 Z 1 1 j^ y t (j!)j 2 d! (By Parseval’s Theorem) = 1 2 Z 1 1 jG(j!)^ x t (j!)j 2 d! 1 2 Z 1 1 kGk 2 1 j^ x t (j!)j 2 d! = 1 2 kGk 2 1 Z 1 1 j^ x t (j!)j 2 d! =kGk 2 1 Z 1 0 jx t ()j 2 d (By Parseval’s Theorem) =kGk 2 1 Z t 0 jx t ()j 2 d =kGk 2 1 Z t 0 jx()j 2 d =kGk 2 1 kxk 2 2[0;t] : Therefore, sup x6=0;t0 kyk 2[0;t] kxk 2[0;t] kGk 1 is well-known and was proved by Sandberg [45]. To prove the equality, let ! = arg max !2[0;1) (G(j!)), and letx(t) = Re(ve j !t ), where ( ) is the maximum singular value and the constant vectorv is the associated right singular vector of G(j !). Then as t!1, y(t)! Re(G(j !)ve j !t ), and hence lim t!1 kyk 2[0;t] kxk 2[0;t] = kGk 1 as stated in [65, 66]. If arg sup !0 (G(j!)) = 1, let x(t) = lim !!1 Re(ve j!t ) where the constant vector v is the associated right singular vector of lim !!1 G(j!), then as t!1,y(t)! lim !!1 Re(G(j!)ve j!t ), and hence lim t!1 kyk 2[0;t] kxk 2[0;t] =kGk 1 . 54 Appendix B Lemma A.2 Lemma A.2 Let (N;D) be a normalized left coprime MFD ofK s such thatK s =D 1 N. Then 2 4 P s 1 3 5 (1 +K s P s ) 1 D 1 1 = 2 4 P s 1 3 5 (1 +K s P s ) 1 h K s 1 i 1 : Proof. Denote [] as complex conjugate. A transfer function matrix G(j!) is called coin- ner if GG = I;8! 2 [0;1): Since (N;D) is a normalized left coprime MFD of K s , h N D ih N D i =I, and thus h N D i is coinner. Consequently, 2 4 P s 1 3 5 (1 +K s P s ) 1 D 1 1 = 2 4 P s 1 3 5 (1 +K s P s ) 1 D 1 h N D i 1 = 2 4 P s 1 3 5 (1 +K s P s ) 1 h K s 1 i 1 ; since theH 1 norm is invariant under right multiplication by a coinner function [35]. 55 Appendix C Proof of Theorem 3.1 Let (N s ;D s ) be a normalized left-coprime MFD ofK s =W I KW O 1 . As in Remark 3.3, let ~ v be the fictitious signal ofK s for the given measured data over [0;t]. Then from Fig. 3.6, the following equations hold: y =W O 1 P s D s 1 (1 +N s P s D s 1 ) 1 ~ v =W O 1 P s (1 +D s 1 N s P s ) 1 D s 1 ~ v W O y =P s (1 +K s P s ) 1 D s 1 ~ v; (1) and u =W I 1 D s 1 (1 +N s P s D s 1 ) 1 ~ v =W I 1 (1 +D s 1 N s P s ) 1 D s 1 ~ v W I u = (1 +K s P s ) 1 D s 1 ~ v: (2) 56 Let time2 [0;t]. Therefore using (1) and (2) in the cost functionV (3.4), it is true that V (K;W;;) = kWk 2[0;] +k~ vk 2[0;] sup k~ vk 2[0;] 6=0;2[0;t] kWk 2[0;] +k~ vk 2[0;] sup k~ vk 2[0;] 6=0;0 2 4 W O y W I u 3 5 2[0;] +k~ vk 2[0;] sup k~ vk 2[0;] 6=0;0 2 4 W O y W I u 3 5 2[0;] k~ vk 2[0;] (*> 0) = 2 4 P s 1 3 5 (1 +K s P s ) 1 D s 1 1 (By Lemma A.1 in Appendix A, (1), (2)) = 2 4 P s 1 3 5 (1 +K s P s ) 1 h K s 1 i 1 (By Lemma A.2 in Appendix B) =kH s k 1 (3) Therefore from (3) it is true that kH s k 1 =) V (K;W;;) ;82 [0;t]: Hence the claim follows by logical equivalence, e.g. max t V (K;W;;)> =) kH s k 1 > : 57 Appendix D Proof of Lemma 4.1 Consider the systemh t rK t with inputu2 ` 1e and outputy(t)2 R n , then by Lemma 2.2, 8t2Z it is true that y(t) =h t rK t u t =h t 2 6 6 6 6 6 6 6 4 . . . P t i=t1 rk i T 2 u rk t Tu 0 3 7 7 7 7 7 7 7 5 : 58 Next, jy(t)jkh t k 0 1;t 2 6 6 6 6 6 6 6 4 . . . P t i=t1 rk i T 2 u rk t Tu 0 3 7 7 7 7 7 7 7 5 0 1;t =kh t k 0 1;t 2 6 6 6 6 6 6 6 4 . . . 2 0 P t i=t1 rk i T 2 u 1 0 rk t Tu 0 3 7 7 7 7 7 7 7 5 1 kh t k 0 1;t 2 6 6 6 6 6 6 6 4 . . . 2 0 2 P t i=t1 krk i k 1 0 krk t k 1 0 3 7 7 7 7 7 7 7 5 1 kuk 1;t (4) kh t k 0 1;t sup i1 2 4 0 i t X q=ti+1 krk q k 1 3 5 kuk 1;t (5) where (4) is by Lemma 2.1, and (5) is by the definition of` 1 -norm. Hence, the claim is proved by (5) and Definition 2.4. 59 Appendix E Proof of Lemma 4.2 Consideri2Z + nf0g. Since8t2Z,9j2Z + nf0g such thatti+12 [tjN +1;t(j1)N] andi2 [(j 1)N + 1;jN]. Thus by Definition 2.9, 0 i t X q=ti+1 krk q k 1 0 (j1)N+1 j X l=1 t(l1)N X q=tlN+1 krk q k 1 0 1N d ;N (K) 0 jN jN 0 1N d ;N (K) e ln 0 1 | {z } c ;N (K) ; (6) where (6) is due to sup x0 xy x (e ln(y)) 1 ;8y > 1 [2]. Hence, the claim is proved by (6) and (4:2). 60 Appendix F Proof of Theorem 4.1 Lett2 [t i1 + 1;t i ] for somei2Z. By Definition 2.8,GT =G t T +rG t T , and so the system can be depicted as in Fig. A1. + + + + ∇ Figure A1: The system in terms ofG t andrG t . Let I be an identity operator such that Ix = x. Then according to Fig. A1, x = Fu + G t Tx +rG t Tx 1 and thus x = (IG t T ) 1 Fu + (IG t T ) 1 rG t Tx = (IG t T ) 1 Fu + h I + (IG t T ) 1 G t T i rG t Tx = (IG t T ) 1 Fu +IrG t Tx + h (IG t T ) 1 G t T i rG t Tx: (7) 1 According to Definitions 2.1 and 2.3 F , Gt andrGt are not necessarily real matrices, and hence is not necessarily a system in the state-space representation. In a special case whereF , Gt andrGt are memory-less systems and thus can be represented as real matrices, is a system expressed in the state-space representation. 61 Letk t be the frozen-time snapshot ofIrG t , thenk t Tx = 0 by the fact thatI is memory-less and by Definition 2.8. Next, sinceh t andl t are the frozen-time snapshots of (IG t T ) 1 and (IG t T ) 1 G t T respectively, then by (7), x(t) =h t Fu +k t Tx +l t rG t Tx =h t Fu +l t rG t Tx; jx(t)jkh t k 1;t kFk 1;t kuk 1;t +kl t rG t k 1;t kTxk 1;t kh t k 1;t kFk 1 kuk 1;t +kl t rG t k 1;t kxk 1;t1 kh t k 1;t kFk 1 kuk 1;t +kl t k 0 1;t c ; 0 (G;t)kxk 1;t1 ; (8) where (8) is by Lemma 4:1. Next, the property of ` 1 -semi norm kxk 1;t max n 1 kxk 1;t1 ;jx(t)j o along with (8) are used to get kxk 1;t maxf 1 kxk 1;t1 ;kh t k 1;t kFk 1 kuk 1;t +kl t k 0 1;t c ; 0 (G;t)kxk 1;t1 g kh t k 1;t kFk 1 kuk 1;t + (t)kxk 1;t1 ; (9) 62 where (t) is defined in (4:6), and (9) is by (4:6). Next, by applying the Gronwall-Bellman Lemma [64] on (9), it is true that kxk 1;t i 0 @ t i Y t=t i1 +1 (t) 1 A kxk 1;t i1 +kh t i k 1;t i kFk 1 kuk 1;t i + t i 1 X t=t i1 +1 2 4 t i Y j=t+1 (j)kh t k 1;t kFk 1 kuk 1;t 3 5 t i t i1 kxk 1;t i1 +kh t i k 1;t i kFk 1 kuk 1;t i + t i 1 X t=t i1 +1 h t i t kh t k 1;t kFk 1 kuk 1;t i i (10) t i t i1 kxk 1;t i1 +kh t i k 1;t i kFk 1 kuk 1;t i + t i 1 X t=t i1 +1 max t2[t i1 +1;t i ] t i t kh t k 1;t kFk 1 kuk 1;t i t i t i1 kxk 1;t i1 + (t i t i1 ) max t2[t i1 +1;t i ] t i t kh t k 1;t kFk 1 kuk 1;t i t i t i1 kxk 1;t i1 +kuk 1;t i ; (11) where (10) is by (4:4) and bykuk 1;t kuk 1;t i 8t t i , and (11) is by (4:8) and (4:9). By considering2 1 ; 1 and applying the Gronwall-Bellman Lemma [64] on (11),8a<ji it is true that kxk 1;t j t j ta kxk 1;ta + j X k=a+1 t j t k kuk 1;t k ; t j ta kxk 1;ta + 1 kuk 1;t k ; (12) t j ta kxk 1;ta + 1 kuk 1;t i ; (13) 63 where (12) is by t j t k and (13) is bykuk 1;t k kuk 1;t i respectively. Next, by choosing a such thatkxk 1;ta = 0, the following inequality holds: kxk 1;t j 1 kuk 1;t i : (14) Therefore for allt2 [t j1 + 1;t j ], (t j t) jx(t)j 1 kuk 1;t i ; (15) ) jx(t)j (t j t) 1 kuk 1;t i ; ) kxk 1;[t j1 +1;t j ] t1 1 kuk 1;t i ; (16) ) kxk 1;[t j1 +1;t j ] ckuk 1;t i ; (17) ) kxk 1;t i ckuk 1;t i ; (18) where (15) is by (14) and Definition 2.2, (16) is by (4:8) and Definition 2.2, (17) is by (4:7), and (18) is bykxk 1;t i sup ji kxk 1;[t j1 +1;t j ] respectively. The theorem follows by noting x = u. 64 Appendix G Proof of Lemma 4.4 Since the sufficient condition (4:14) is a special case of the sufficient condition (4:4) with sup t2Z kh t k 1 <1, sup t2Z kl t k 0 1 <1,ft i g =Z, and theN-width average variation rate ofG is bounded, it is true that (4:14)) (4:4). Therefore, to prove that (4:4) holds whenever (4:16) holds and there exist cases where (4:4) holds while (4:16) does not hold, it suffices to prove (i) (4:16)) (4:14), and (ii) (4:14); (4:16). (i) Let N = 1, and thus d ;N = d ;1 by (4:15) and (4:17). Choose d ;N (G) = sup t2Z d ;N (G)(t), and thus (4:16) , (4:14) by d ;N = d ;1 and Definitions 2.8 and 2.9. Therefore it is true that (4:16)) (4:14). (ii) Consider a case whereN > 1,N 0 1N > 1, and a time2Z such that krg k 1 2 d ;1 ;N d ;N i and t X i=tN+1 krg i k 1 N d ;N ;8t2Z: Since N 0 1N > 1, it is true that N d ;N > d ;1 and d ;1 ;N d ;N i is not empty. Next, bykrg k 1 > d ;1 (4:16) does not hold. On the other hand, by choosing d ;N (G) = sup t2Z d ;N (G)(t) and by P t i=tN+1 krg i k 1 N d ;N , inequality (4:14) holds. Then it is true that (4:14); (4:16). By (i), (ii), and (4:14)) (4:4), it is true that (4:16)) (4:4) and (4:4); (4:16), and thus the lemma is proved. 65 Bibliography [1] M. Stefanovic, R. Wang, and M. Safonov, “Stability and convergence in adaptive systems,” in Proc. 2004 IEEE American Control Conference (ACC’04), Boston, MA, pp. 1923-1928, June 2004. [2] G. Zames and L. 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Abstract (if available)

Abstract In the first part of the thesis, a data-driven H∞-shaping controller design approach is investigated to find controllers which satisfy specifications in terms of sensitivity and complementary sensitivity in the frequency domain by real-time observed plant input-output data. The proposed data-driven approach is found to be a dual of McFarlane and Glover’s model-based H∞ loop-shaping controller design while it relaxes McFarlane and Glover’s assumption that an upper bound on multiplicative uncertainty of a plant’s Normalized Coprime Factor (NCF) model is known. By pruning the controllers in real-time if observed plant input-output data prove that those controllers fail given sensitivity and complementary sensitivity specifications, the proposed approach returns multiple feasible controller designs for given specifications. In contrast, McFarlane and Glover’s approach only returns one controller design for the specifications. Because the proposed approach finds feasible controllers only by raw real-time observed plant input-output data instead of relying on assumptions on plant multiplicative uncertainty, the robustness of the adaptive switched feedback system against uncertainties is enhanced. In the second part of the thesis, a new sufficient condition is derived for a general MIMO switched nonlinear feedback system that has uncertainties in feedback loop to be input-output stable. The sufficient condition, expressed in terms of average time-variation rate of the feedback loop function, is derived to preserve the stability of the system. The new sufficient condition relaxes three assumptions in Zames and Wang’s sufficient condition for the general adaptive switched feedback system to be input-output stable that (i) the feedback loop function is linear, (ii) the feedback loop function is stabilizing all the time, and (iii) maximum variation rate of the feedback loop function is bounded. It is proved that the new sufficient condition generalizes Zames and Wang’s sufficient condition. Moreover, it is found that the new sufficient condition is less conservative than Zames and Wang’s sufficient condition.

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Creator Sung, Yu-chen (author)

Core Title Data-driven H∞ loop-shaping controller design and stability of switched nonlinear feedback systems with average time-variation rate

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 03/13/2019

Defense Date 03/07/2019

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

Tag control theory,data-driven,feedback control,H∞ control,H-infinity control,input-output stability,loop shaping,MIMO,nonlinear system,OAI-PMH Harvest,time-varying systems

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Language English

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Advisor Safonov, Michael George (committee chair), Nayyar, Ashutosh (committee member), Udwadia, Firdaus (committee member)

Creator Email ycsong216@gmail.com,yuchens@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-130999

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