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Adaptive control: transient response analysis and related problem formulations

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Adaptive control: transient response analysis and related problem formulations

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Content Adaptive Control: Transient Response Analysis and Related Problem Formulations 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 Kiran S. Sajjanshetty Dissertation Advisor: Michael G. Safonov December 2017 Copyright c 2017 by Kiran S. Sajjanshetty All rights reserved. Dedicated to My Mother and My Husband, Prashanth Acknowledgements Firstly, I would like to express my deepest gratitude towards my advisor, Dr. Michael G. Safonov for the motivation, guidance and continued support throughout the course of my studies here at USC. It has truly been a pleasure to work under him. I at- tribute my condence as a researcher to him and I could not have imagined having a better advisor and mentor than him for my Ph.D study. I thank my Ph.D. defense committee members, Dr. Ashutosh Nayyar and Dr. Ketan Savla for their support and cooperation. I would also like to thank Dr. Edmond Jonckheere, Dr. Firdaus Udwadia and Dr. Paul Bogdan for serving on my qualifying examination committee. Over the years I have received a lot of support from the Ming Hsieh Department of Electrical Engineering. I am indeed grateful to Ms. Diane Demetras, Mr. Shane Goodo and Mr. Tim Boston who have been extremely helpful. I am also thankful to all my friends at USC, especially the members of Vidushak who have been a constant source of encouragement and laughter. I thank my colleagues Yu- Chen Sung and Sagar Patil for some of the interesting discussions we have had. I am grateful to my family especially my mother Mrs. Mahadevi Sajjanshetty and my sister Dr. Shalini for their unyielding support and innite love. Lastly, I would like to thank my husband, Prashanth, for his continued and unfailing love, support and understanding makes the completion of this thesis possible. Kiran December 2017 i Abstract Adaptive Control: Transient Response Analysis and Related Problem Formulations Kiran S. Sajjanshetty 2017 A problem that is sometimes associated with adaptive control algorithms is the poor transient response with some plant-controller combinations. The primary contri- bution of this work is the development of a methodology to understand and bound transients in adaptive switching control systems that employ hysteresis type switching algorithms. In particular, two methods to derive bounds on the cost of the adaptive system are proposed. The rst result bounds the transients when the switching al- gorithm starts with an arbitrary controller in the loop. The second result quanties the idea of slow switching or slow adaptation and provides sucient conditions under which transients remain bounded when the initial controller is stabilizing. Plausible solutions to reduce these bounds are provided, which in turn alleviates the problem of poor transient response. The other emphasis of this work is on some of the newer problem formulations in adaptive switching control systems. These problems are addressed within the framework of Unfalsied Adaptive Control. The problem of vector-valued controller cost functions which are solely data-dependent and re ect multiple objectives of a control system is examined. The notions of Pareto optimality of vector valued cost functions and the conditions under which they are cost-detectable is discussed. A sampled data/discrete-time Level-Set controller switching algorithm is investigated which allows for the relaxation of the assumption that the controller cost function be monotonically nondecreasing in time. When an active controller is falsied at the current threshold cost level, the Level-Set switching algorithm replaces it by an eectively unique solution of the weighted Tchebyche method, thus ensuring the selection of an unfalsied Pareto optimal controller. Next, the problem of achieving optimal performance by driving the hysteresis gap to zero is addressed. In this regard, a data-dependent dwell-time switching algorithm is proposed in which the dwell-time is constrained to grow in proportion to the recip- rocal of an adaptively-computed hysteresis variable ^ h(t) that converges to zero. This is in contrast to the Morse-Mayne-Goodwin (MMG) hysteresis switching algorithm in which usually a xed hysteresis constanth> 0 is used. The algorithm is proved to be globally stabilizing assuming feasibility and c -cost-detectability. It has the attractive property that the gap between optimal and achieved unfalsied performance levels are at each time bounded above by the current value of the hysteresis variable ^ h(t), thereby ensuring asymptotically optimal performance. The last part of this work extends an existing result on the multiple lyapunov functions based approach to proving stability of switched systems. A less conservative result is developed which shows that the overall stability of a switched system is preserved with fast yet, small switches and slow longer switches. Theoretical results and simulation examples are provided which validate the eectiveness of the ideas presented in this work. Contents 1 Introduction 1 1.1 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 A Brief Review . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Unfalsied Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Notation and Basic Concepts 8 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 A Switching Adaptive Control System . . . . . . . . . . . . . . . . . 11 3 Transient response Analysis 16 3.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Controller Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 Fictitious reference signal generator . . . . . . . . . . . . . . . 19 3.3 Hysteresis Switching Algorithm . . . . . . . . . . . . . . . . . . . . . 20 3.4 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.6 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 I 3.6.1 Transient Response Bound I . . . . . . . . . . . . . . . . . . . 23 3.6.2 Transient Response Bound II . . . . . . . . . . . . . . . . . . 31 3.6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 35 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Multi-objective Cost-detectable Cost functions 40 4.1 Pareto optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Level-Set algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.1 Cost function example . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Adaptive Dwell-time Switching 57 5.1 Logic Based Switching Algorithm . . . . . . . . . . . . . . . . . . . . 57 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Switching algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.6 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 68 6 Stability of Switched Systems 69 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 II 6.4 Variable Dwell-time Switching . . . . . . . . . . . . . . . . . . . . . . 73 6.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 73 6.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7 Final Conclusions 78 Bibliography 80 III List of Figures 2.1 A general system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 A switching adaptive control system . . . . . . . . . . . . . . . . . . 11 2.3 Operating feedback loop with a switching controller ^ K. . . . . . . . . 12 2.4 Left: Operating closed loop feedback with the switching controller in MFD form. Right: Fictitious closed loop with controller K i in MFD form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Controller realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Switching adaptive control system with MFD realization of controllers 19 3.3 Fictitious reference signal generator of nonactive controller K i . . . . 20 3.4 Bad Transients - Top-left: Controller switching signal, Top-right: Con- trol signal u, Bottom-left: Plant output signal y, Bottom-right Se- quence of ` p norms of data versus reference signal r,8t . . . . . . . 36 3.5 Reducedn s - Top-left: Controller switching signal, Top-right: Control signal u, Bottom-left: Plant output signal y, Bottom-right Sequence of ` p norms of data versus reference signal r,8t . . . . . . . . . . . . 37 3.6 Reduced , N - Top-left: Controller switching signal, Top-right: Control signal u, Bottom-left: Plant output signal y, Bottom-right Se- quence of ` p norms of data versus reference signal r,8t . . . . . . . 38 4.1 Switching adaptive control system tracking a reference model W m . . 50 IV 4.2 Case 1: Controller costs, Threshold cost sequence and the switching signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Case 1: Reference model and Plant output . . . . . . . . . . . . . . . 53 4.4 Case 2: Controller costs, Threshold cost sequence and the switching signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5 Case 2: Reference model and Plant output . . . . . . . . . . . . . . . 55 4.6 Case 2: zoomed out version - Controller costs, Threshold cost sequence and the switching signal . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1 Unfalsied performance cost levels achieved (V ( ^ K;d;t)). . . . . . . . 66 5.2 Controller switching signal (t). . . . . . . . . . . . . . . . . . . . . . 67 V Chapter 1 Introduction 1.1 Adaptive Control An adaptive controller intuitively refers to a controller that can modify its behavior in response to changes in the plant dynamics, disturbance and noise characteristics, especially when these changes cannot be directly measured or predicted apriori. The need for adaptive control was identied in the aerospace community in the 1950s, during the design of autopilot and stability assist systems for high performance air- craft in the presence of highly dynamic operating conditions such as varying wind gusts, sudden maneuver changes etc. Over the years, the eld of adaptive control has evolved, however, much of the traditional techniques have limited applicability, owing partly to the prior assumptions made about the plant model. In the next section, a brief overview of the various techniques in adaptive control literature is provided. 1.1.1 A Brief Review Adaptive control methods are typically classied into continuous adaptive tuning methods and discontinuous/switching control methods. They can further be classied into indirect and direct adaptive methods based on whether the intermediate step of 1 CHAPTER 1. INTRODUCTION 2 plant identication process is included or not. Most of the traditional adaptive tuning methods developed prior to 1980s [1{6] were based on prior plant assumptions like, The plant is minimum phase, The sign of the plant high frequency gain is known, The order of the plant is known, Relative degree of the plant is known, Persistence of excitation condition of inputs. Such assumptions render the system non-robust as shown in [7]. Modications to these methods resulting in robust adaptive tuning techniques were proposed in [8{ 11]. These modications result in stability and performance guarantees in situations when the modeling errors are small. In addition, if unknown parameters enter the process model in complicated ways (not linear), it may be dicult to construct a continuously parameterized family of controllers. To overcome these drawbacks, the idea of adaptive switching control also known as supervisory control was introduced in [12{14]. These methods followed a pre-routed path for switching between controllers without any data based logic, which limit their use in practice. Logic based adaptive switching control algorithms were proposed in [15, 16]. However, these methods are indirect adaptive control methods which involve the intermediate step of plant identication which could sometimes result in model- mismatch instability problems as reported in [17]. A practical solution to the adaptive control problem without making any assumptions beyond feasibility was rst proposed in [18] and is discussed in the next section. CHAPTER 1. INTRODUCTION 3 1.2 Unfalsied Adaptive Control Unfalsied Adaptive Control (UAC) instills a line of research that highlights the im- portance of assumption-free methods based on Popper's unfalsication paradigm in the design of adaptive control algorithms. It paves way for a data-driven control design methodology wherein the controller parameters are directly identied by min- imizing a suitable cost-function expressing the performance goals as a causal function of raw plant input-output data, uninterpreted by plant/noise models or other prior beliefs. A high level controller called the supervisor uses a switching algorithm to insert the cost minimizing controller in feedback with an uncertain plant in response to changes in its dynamics. Data driven implies no plant dependent assumptions are made in the design or evaluation of the adaptive control algorithm itself, which helps prevent model-mismatch problems in which incorrect plant assumptions unintend- edly lead to destabilizing adaptive algorithms [7, 17]. Of course plant assumptions can, and normally do, play an essential role in determining promising candidate con- trollers from which the unfalsied control switching algorithm adaptively selects the active controller. Like most adaptive control methods (continuous adaptive tuning or adaptive switching) [15, 19{21], UAC also employs a cost function and an adap- tive algorithm to order candidate controllers. The literature on UAC [17, 18, 22{25], consists of methods which dier in the way in which each of these components are chosen. UAC puts forth the dogma that an essential property required to prove stability of any adaptive control system, is feasibility. An adaptive control problem is feasible if there exists at least one controller in the candidate controller set that is stabi- lizing. Feasibility is the weakest possible assumption under which it is possible to ensure that an adaptive algorithm is convergent and stabilizing. UAC systems with cost-detectable cost functions have the important property that they always stabilize CHAPTER 1. INTRODUCTION 4 the plant subject only to feasibility, with no other assumptions. However, even with feasibility and a cost-detectable cost function, an adaptive control system could still sometimes have unacceptably large transients as shown by an example in [26]. A care- ful analysis of the transient response of adaptive control algorithms is thus required so as to be cognizant of situations which result in bad transients. This motivates the rst problem that is being addressed in this work. A methodology is developed to understand and bound the transients in switching adaptive control systems while highlighting plausible solutions to reduce these transient bounds. In addition, certain new contributions to the eld of UAC are proposed by introducing cost-detectability of multi-objective controller cost functions, Level-Set algorithm for non-monotone cost functions and Adaptive Dwell-Time controller switching strategy. Lastly, an ex- tension to the result of stability of switched systems given in [27] is provided. The next section provides a brief overview of the contributions of this work. 1.3 Contributions The contributions of this thesis are presented below. A methodology to study the transient response of switching adaptive control systems is developed. This study was motivated in part by the results of [26], which showed that even with feasibility and a cost-detectable cost function, an adaptive control system could sometimes insert a destabilizing controller re- peatedly in the loop before nally stabilizing the system. This in turn produces bad transients. In this work, two bounds on the actual cost of the system i.e., the actual reference to data induced gains are derived. Furthermore, plausible solutions to reduce these bounds are highlighted. Cost functions that appear in the existing methods of UAC focus mainly on CHAPTER 1. INTRODUCTION 5 establishing a stable adaptive control system. These cost functions are cost- detectable, which means that they correctly detect instability of the adaptive system without any plant assumptions. However, there is also a need to in- clude cost functions which re ect performance goals beyond stability, like good tracking of a class of inputs, simultaneous rejection of disturbance having dier- ent characteristics (white noise, bounded energy, persistent) etc. Therefore, a multi-objective formalism of a cost function is proposed wherein, a vector-valued controller cost function is considered. Each element of this vector-valued cost function re ects a particular control goal. The notions of Pareto optimality of vector valued cost functions is discussed. Since stability is the foremost require- ment, the properties that a multiobjective cost function must satisfy in order to be cost-detectable are proposed. A sampled data/discrete-time Level-Set (LS) switching algorithm, which is com- posed of increasing threshold cost levels is proposed. Threshold cost is a con- troller falsication level specied at every instant by the switching logic and is critical in averting false alarms. This algorithm allows for the relaxation of the assumption that the controller cost function be monotonically nondecreasing in time unlike all previous methods in UAC. This opens up the possibility of the use of fading memory cost functions which are nonmonotone. Hysteresis switching [28] and its more recent variant scale-independent hystere- sis switching algorithm [29] is one of the most popular switching algorithms used in adaptive switching control systems. These algorithms preclude the possibil- ity of unbounded chatter by using a xed positive constant h while minimizing controller cost functions. In other words, even if the current controller's cost exceeds the minimum cost value by h, it is still retained in the loop. There- fore, cost-minimization is not exact because of the positive hysteresis constant CHAPTER 1. INTRODUCTION 6 which may result in sub-optimal performance. Chattering instability may occur if the hysteresis constant is set to zero which results in a trade-o. Dwell-time switching algorithm is another popular switching algorithm which shows that if all subsystems in a switched system are exponentially stable with zero as the common equilibrium point, then there exists a scalar d > 0 such that the over- all system remains exponentially stable if the dwell time of every controller is larger than or equal d . The xed dwell-time switching logic was used for linear set-point control of an uncertain process in [15, 16] where it was assumed that d is prespecied. Fixed dwell-time is not suitable for adaptive switching con- trol of nonlinear systems since it may lead to nite escape of the closed loop in some cases. To overcome the disadvantages of these algorithms, a self-adjusting, data-dependent controller switching strategy called Adaptive Dwell-time (ADT) switching algorithm is proposed which constrains the dwell-time of a controller to grow in proportion to the reciprocal of an adaptively-computed hysteresis variable ^ h(t) that decreases asymptotically to zero. It is proved to be globally stabilizing assuming only feasibility and cost-detectability. It has the attractive property that the gap between optimal and achieved unfalsied performance levels are at each time bounded above by the current value of the hysteresis variable ^ h(t), thereby ensuring asymptotically optimal performance. Multiple Lyapunov functions based approach to proving stability of switched systems was proposed in [27]. It showed that a switched system is stable if all the individual subsystems are stable and the switching is suciently slow. In this work, an extension to the result of [27] is provided that is much less conservative. In addition, it is shown that the stability of switched systems is ensured with faster switching between systems that are closer and slow switching between systems that are farther. CHAPTER 1. INTRODUCTION 7 1.4 Organization This thesis is organized as follows: Chapter 2 provides an introduction to UAC theory, preliminary concepts and notation. A method to analyze transient response of switching adaptive systems along with solutions to alleviate poor transient response is derived in Chapter 3. The idea of multi-objective cost-detectability is presented in Chapter 4. This chapter also discusses the Level-Set algorithm along with theoretical results and examples. The ADT switching algorithm is discussed in detail in Chapter 5. Theoretical results along with simulation examples are provided in this chapter. The result on the stability of switched systems is provided in Chapter 6. Each of the chapters 3, 4, 5 and 6 is self-contained and may be read in any order. Conclusions are presented in Chapter 7. Chapter 2 Notation and Basic Concepts General notation and concepts that have been used throughout this work are intro- duced in this chapter. 2.1 Notation R + : Set of non-negative real numbers Z + : Set of non-negative integers x : Real valued vector sequence with x(t)2R n for all t2Z + j:j : Absolute value d:e : Ceiling function [: ] T : Transpose z : Discrete time forward shift operator s : Continuous time dierentiator operator 8 CHAPTER 2. NOTATION AND BASIC CONCEPTS 9 2.2 Denitions Denition 2.2.1. (` p -norm): The ` p -norm of x, truncated at time t is dened as kxk p;[0;t] = 8 > > < > > : p p t =0 n i=1 jx i ()j p ; if p2 [1;1) max 2[0;t] max i2[1;n] jx i ()j; if p =1 (2.2.1) where x i denotes the i-th component of x. Ifkxk p;[0;t] <1 for t2 [0;1), then x2` n pe and if lim t!1 kxk p;[0;t] <1, then x2` n p . Denition 2.2.2. (` p -norm): The ` p -norm of x, is dened as kxk p;[0;t] = 8 > > < > > : p p t =0 p(t) f n i=1 jx i ()j p g; if p2 [1;1) max 2[0;t] (t) fmax i2[1;n] jx i ()jg; if p =1 (2.2.2) where is called the fading memory parameter and 2 (0; 1]. The ` p -norm satises the following property, kxk p;[0;t] t+1 kxk p;[0;1] +kxk p;[;t] ; 8t (2.2.3) It can be veried that (2.2.3) is an immediate consequence of (2.2.2) and triangle inequality property of norms. The denition of-stability, -unfalsication, ` p gain of a system and a coercive function are given below. Denition 2.2.3. (-stability): The system : ` n pe ! ` m pe , as shown in Figure (2.1) with input r of size n and output of size m, is -stable if for every input r2 ` n pe , there exist constants 0, 0 such that for all t 0, kk p;[0;t] krk p;[0;t] +: (2.2.4) CHAPTER 2. NOTATION AND BASIC CONCEPTS 10 Γ Γ1 Figure 2.1: A general system . When = 1, we simply say is stable. Denition 2.2.4. (-unfalsication): Given a particular input r and output , the stability of system is said to be -unfalsied by the data pair (r; ) if there exist constants 0, 0 such that, for all t 0, we have kk p;[0;t] krk p;[0;t] + (2.2.5) where r2` n pe . Otherwise, the stability of is -falsied by (r; ). When = 1, we simply say that stability is unfalsied if (2.2.5) holds, or falsied if (2.2.5) does not hold. Remark 2.2.1. The notion of -stability is equivalent to exponential stability of sys- tems. Suppose as shown in Figure (2.1) is a nite dimensional Linear Time Invariant (LTI) system with transfer function (z), then is-stable i every pole of (z) has magnitude strictly less than . Also, -stability of (z) is equivalent to stability of (z). Denition 2.2.5. (` p gain): The` p gain of the system as shown in Figure (2.1) is dened as kk p = sup krk p;[0;t] 6=0;t0 kk p;[0;t] krk p;[0;t] (2.2.6) where = r, p2 [1;1] and 2 (0; 1]. When = 1, its simply the ` p gain of . It follows from the above denition that the ` p gain of a system , satises the CHAPTER 2. NOTATION AND BASIC CONCEPTS 11 following property, krk p;[0;t] kk p krk p;[0;t] (2.2.7) for all t 0. Denition 2.2.6. (Coercive function): A functionf :R n !R is said to be coercive [30] if lim k~ xkp!1 f(~ x) = +1: (2.2.8) Such functions are also said to be radially unbounded 2.3 A Switching Adaptive Control System ! ! 1 ! 1 ! 1 ! 1 1 y 1 ℎ ℎ Γ ! ! ! ! Figure 2.2: A switching adaptive control system . A typical switching adaptive control system, : ` n pe ! ` m pe , comprising of a high level controller called the supervisor and a nite candidate controller set K = CHAPTER 2. NOTATION AND BASIC CONCEPTS 12 fK 1 ;K 2 ;::::;K M g in feedback with an uncertain plant P looks as shown in Figure (2.2). The input to the system is r2 ` n pe which denotes the reference signal and the output is = [u y] T 2 ` m pe , where u and y denote the control signal and plant output signal respectively. The set D = GraphfPg :=f = (u;y)jy = Pug. The supervisor along with the candidate controller set forms an adaptive switching con- troller. The supervisor orchestrates the switching of controllers and orders them using a cost function denoted asV (K;;t), which is dened as a causal-in-time map- ping V : KDZ + ! R + [1. Given a specic , we may for brevity denote V i =V (K i ;;t). The active controller at timet is denoted as ^ K(t) and the closed loop switched system is denoted as ( ^ K;P ). The controller switching signal is denoted as , therefore 2f1; 2;:::Mg. The controller switching instants are denoted as ft k g k2Z + and thek-th switching interval is denoted asT k =ft k ;t k + 1;:::;t k+1 1g, over which remains constant. The number of controller switches till the current time t is denoted as n t and the total number of controller switches is denoted as n s . Denition 2.3.1. (Fictitious/virtual Reference Signal (FRS)): Given the plant input- output data = [u; y] T from an experiment and a nonactive controller K i , the c- titious reference signal ~ r i corresponding to K i is dened as a signal that would have reproduced exactly the measured data, had the controllerK i been in the loop during the entire time the data was collected. $ − + Figure 2.3: Operating feedback loop with a switching controller ^ K. If the data, = [u; y] T was generated as shown in the above Figure (2.4), then CHAPTER 2. NOTATION AND BASIC CONCEPTS 13 the ctitious reference signal ~ r i corresponding to K i is given by, ~ r i =K 1 i u +y (2.3.9) Fictitious reference signals enable the evaluation of performance of non-active con- trollers simultaneously, even before any of these controllers is ever inserted in the feedback loop. It is therefore required that the map 7! ~ r exists and is causal and incrementally stable. A controller for which such a map exists is called Stably Causally Left Invertible (SCLI). A necessary condition for the controller to be SCLI is to be minimum phase. However, in case of non minimum phase controllers (non SCLI controllers), the map 7! ~ r is unstable and would involve numerical problems. In order to include even the non SCLI controllers, the use of a more general ctitious reference signal ~ v i in place of ~ r i was proposed in [31] by using the Matrix Fraction Description (MFD) realization of controllers. Let the ordered pair (N i ;D i ) denote the left Matrix Fraction Description (MFD) of a controller K i 2K. Therefore, the following holds, K i =D 1 i N i (2.3.10) where factors N i and D i are possibly nonlinear, stable, causal and D i has a causal inverse. Then the new ctitious reference signal ~ v i corresponding to controller K i as shown in Figure (2.4) is given by, ~ v i =D i u +N i y: (2.3.11) Denition 2.3.2. (-cost-detectability): Consider the switching adaptive control sys- CHAPTER 2. NOTATION AND BASIC CONCEPTS 14 # $% − + # # $% − + # # - Figure 2.4: Left: Operating closed loop feedback with the switching controller in MFD form. Right: Fictitious closed loop with controller K i in MFD form tem as shown in Figure (2.2). A cost function and controller pair (V i ;K i ) is said to be cost- detectable if the following two statements are equivalent: (1) V i 2` 1 . (2) Stability of (K i ;P) is -unfalsied by the input-output data (r; ). When = 1, we simply say that the cost-controller set pair (V i ;K i ) is cost- detectable if the above statements are equivalent. Denition 2.3.3. (Robust cost): Given a particular plant P and controller K i , the robust cost of controller K i is denoted as V rsp (K i ) and is dened as V rsp (K i ) = sup 2D;t2Z + V i . A robust optimal controller is the one that minimizes the robust cost over all controllers in the candidate controller set. Let rsp = min K i 2K V rsp (K i ) denote the robust cost of the system. Denition 2.3.4. (-feasibility): The adaptive stabilization problem is considered to be-feasible if there exists at least one controller in the candidate controller setK such that the closed loop system is-stable. In other words, the problem is-feasible if rsp is nite when the controller cost function, V i re ects -stability. When = 1, we simply say that the adaptive stabilization problem is feasible if there exists at least one controller inK such that the closed system is stable. Denition 2.3.5. (Falsication at a level): Given a cost-controller set pair (V i ;K i ), with K i 2K and a scalar 0, we say that a controller K i 2K is falsied at time t at cost level by past measurement data if V (K i ;;t)> . Otherwise it is said CHAPTER 2. NOTATION AND BASIC CONCEPTS 15 to be unfalsied at level . In order to interpret the ideas presented in the rest of the chapters, a basic un- derstanding of concepts introduced in this chapter is required. Chapter 3 Transient response Analysis 3.1 Introductory Remarks Adaptive control algorithms, whether continuous adaptive tuning or adaptive switch- ing [15, 19{21] are inherently associated with a cost function that orders candidate controllers and an adaptive algorithm that selects controllers which reduce the cost. Unfalsied Adaptive Control (UAC) [17, 22{25, 32{34] is an important real-time, data driven approach to adaptive switching control that ts the aforementioned framework. Data driven implies no plant dependent assumptions like `minimum phase property of plant' or `tunability' are made in the design or evaluation of the adaptive control algorithm itself, which helps prevent model-mismatch problems that can sometimes cause the adaptive algorithms to converge to destabilizing controllers [7, 17]. Of course plant assumptions can, and normally do, play an essential role in determining promising candidate controllers from which the unfalsied control switching algorithm adaptively selects the active controller. An essential property required to prove stability of an adaptive control system, is feasibility. An adaptive control problem is feasible if there exists at least one controller in the candidate controller set that is stabilizing. Feasibility is the weakest 16 CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 17 possible assumption under which it is possible to ensure that an adaptive algorithm is convergent and stabilizing. UAC systems with cost-detectable cost functions have the important property that they always stabilize the plant subject only to feasibility, with no other assumptions. However, even with feasibility and a cost-detectable cost function, an adaptive control system could still sometimes have unacceptably large transients as shown by an example in [26]. In this chapter, a theory to analyze the transient response of the class of adaptive control systems that employ hysteresis type switching algorithms [28, 35] and ` 2e gain type cost detectable cost functions is developed. Upper bounds on the actual cost function of the adaptive system are derived followed by a bound on the transient response of the plant input-output signals with respect to the ref- erence input signal. Furthermore, plausible solutions to reduce these upper bounds are discussed, thus alleviating the problem of poor transient response. Simulation examples validating the proposed ideas are also provided. Some of the techniques used to develop the theory in this paper have been inspired from [25, 36], wherein a dierent but, theoretically closely related problem of adaptive control resetting was analyzed. This chapter begins with a discussion of the controller realization and the multi- controller structure in the switching adaptive control system. A brief overview of the hysteresis switching algorithm is then provided. This is followed by a detailed analysis of the transient response of the system and transient response bounds followed by simulation results. CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 18 3.2 Controller Realization As given in Denition 2.3.1, let the ordered pair (N i ;D i ) denote the left Matrix Fraction Description (MFD) of a controller K i 2K. Therefore, the following holds, K i =D 1 i N i (3.2.1) where N i = [N r i N y i ]; factors N i and D i are possibly nonlinear, stable, causal and D i has a causal inverse [31]. The realization of a controller given by Equation (3.2.1) is shown in Figure (3.1), $ $ $ ' 1 $ ( 1 + + − + $ $ ̃$ + Figure 3.1: Controller realization Therefore the output of controller K i is given by, u i =v i (D i 1)uN y i y (3.2.2) With the above controller realization in place, the structure of the switching adap- tive system is as shown in Figure (3.2), CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 19 # $ 1 % $ 1 & $ 1 1 Γ 1 ℎ ℎ # # ? 1 % % ? 1 & & ? 1 1 # % & A B # B % B & B A + + − + + + Figure 3.2: Switching adaptive control system with MFD realization of controllers Therefore, the control signal u is given by, u =v (D 1)uN y y (3.2.3) 3.2.1 Fictitious reference signal generator The ctitious reference signal given in Equation (2.3.11) is slightly modied in accor- dance with the controller realization of Equation (3.2.1) where N i = [N r i N y i ]. The ctitious reference signal ~ v i is now given by, ~ v i =D i u +N y i y (3.2.4) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 20 Let, ;i = [D i N y i ] (3.2.5) Therefore, ~ v i = ;i (3.2.6) where = [u y] T . The ctitious reference signal generator corresponding to con- troller K i is shown in Figure (3.3). # # % 1 + + Σ *,# ̃ # Figure 3.3: Fictitious reference signal generator of nonactive controller K i Remark 3.2.1. When N r i = N y i = I and K i = D 1 i , then ~ v i equals the conven- tional ctitious reference signal ~ r i in [18]. The use of the more general ~ v i from [31] in place of the ~ r i of [18] allows for the use of non-minimum phase controllers and the non-uniqueness of the stable matrix fraction description (N i ;D i ) provides addi- tional exibility that can be used to incorporate frequency-dependent weights in cost functions as well. 3.3 Hysteresis Switching Algorithm The supervisor in the switching adaptive control system orchestrates the switching of controllers using a switching logic/algorithm. It is a hybrid dynamical system whose input is the controller cost function V and output is the switching signal . The CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 21 algorithm which the supervisor uses to generate is called the switching algorithm. Hysteresis switching algorithm [28, 35, 37] is an example of a switching algorithm and is one of most popular switching algorithms used in adaptive control literature. To specify the hysteresis switching algorithm, it is rst necessary to pick a positive number h> 0, which is called a hysteresis constant. The steps of the algorithm are provided below, Hysteresis Algorithm A1 1. Initialize: Let t = 0; choose h > 0: Let ^ K(t) = K 0 ; K 0 2 K be the rst controller in the loop. 2. t =t + 1, If V ( ^ K(t 1);;t) min K i 2K V (K i ;;t) +h, then ^ K(t) = arg min K2K V (K;;t) Else ^ K(t) = ^ K(t 1) 3. Go to Step 2. Hysteresis switching algorithm requires the controller cost functionV i to be mono- tonically increasing in time. The total number of controller switches n s for this algo- rithm is bounded above by, n s l M rsp h m (3.3.7) where rsp is nite if the adaptive control problem is-feasible (from Denition 2.3.4). Therefore, the bound in Equation (3.3.7) is nite if the adaptive control problem is -feasible. CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 22 3.4 Cost function The following controller cost function is used for analysis in this chapter, V i (t) = max 2[0;t] kk p;[0;] +k~ v i k p;[0;] (3.4.8) where 0. It can be veried that this cost function is monotonically increasing in time and is also -cost detectable. 3.5 Problem Description When-feasibility holds, hysteresis switching algorithm along with a-cost-detectable cost function ensures convergence and -stability of the switching adaptive control system. However, in some cases, a destabilizing controller may be inserted repeat- edly in the loop before nally stabilizing the system. This results in excessively large transient control and plant output signals. In order to understand and alleviate this poor transient response problem, there is a need to develop a theoretical framework for transient response analysis of the switching adaptive control system. In the next section, such a theoretical framework is developed. Furthermore, theoretical bounds on the actual cost of the system i.e., the actual reference to data induced gains such as those given by Equations (3.5.9, 3.5.10) have been derived. These bounds given an insight into the parameters that can enable the designer to control the transient response of a switching adaptive control system. V (t) = kk p;[0;t] +kv k p;[0;t] (3.5.9) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 23 W (t) = kk p;[0;t] + max 2[0;t] kv k p;[0;] (3.5.10) 3.6 Main Results In this section, two transient response bounds have been derived. The rst result bounds the transients when the switching algorithm starts with an arbitrary controller in the loop. The second result provides sucient conditions under which transients remain bounded when the initial controller in the loop is stabilizing. 3.6.1 Transient Response Bound I This section begins with three lemmas which are based on the work in [25, 36], wherein several very similar results were derived in conjunction with the dierent, but theoretically closely related problem of adaptive control resetting. Lemma 3.6.1. Consider the controller realization as shown in Figure (3.2). Then, ~ v (t) (t) =v (t) (t); 8t (3.6.11) Proof. The following holds from the switching adaptive controller realization of Figure (3.2), u(t) =v (t) (t) (D (t) 1)u(t)N y (t) y(t); 8t ,u(t) =D (t) 1 (v (t) (t)N y (t) y(t)); 8t (3.6.12) From Equation (3.2.4) and (3.2.6), ~ v (t) (t) = ;(t) (t) =D (t) u(t) +N y (t) y(t); 8t (3.6.13) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 24 Substituting for u from (3.6.12), the following holds, ~ v (t) (t) =v (t) (t); 8t (3.6.14) Lemma 3.6.2. The following holds for t2T k , k~ v (t k ) k p;[0;t] tt k +1 k ;(t k ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t] (3.6.15) Proof. Since remains constant over each switching intervalT k =ft k ;t k +1;:::;t k+1 1g, the following holds, (t) =(t k ); 8t2T k (3.6.16) Combining (3.2.6), (3.6.11) from Lemma 3.6.1 and (3.6.16), the following is true, ~ v (t k ) (t) = 8 > < > : ;(t k ) (t) for t<t k v (t k ) (t) for t2T k (3.6.17) From (2.2.3), the following holds for t2T k , k~ v (t k ) k p;[0;t] tt k +1 k~ v (t k ) k p;[0;t k 1] +k~ v (t k ) k p;[t k ;t] (3.6.18) Substituting (3.6.17) in the above equation, the following is true for t2T k , k~ v (t k ) k p;[0;t] tt k +1 k ;(t k ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t] (3.6.19) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 25 Let, k := min K i 2K V i (t k+1 1) +h (3.6.20) where V i is given by (3.4.8). Lemma 3.6.3. Consider the cost function given by (3.4.8). If hysteresis switching algorithm is used, then the following holds for t2T k , kk p;[0;t] tt k +1 k k ;(t k ) k p kk p;[0;t k 1] + k kv (t k ) k p;[t k ;t] + : (3.6.21) Proof. Using the switching condition of hysteresis switching algorithm for t2T k , V (t k ) (t) k ; 8t2T k , max 2[0;t] kk p;[0;] +k~ v (t k ) k p;[0;] k ; 8t2T k , kk p;[0;t] +k~ v (t k ) k p;[0;t] k ; 8t2T k ,kk p;[0;t] k k~ v (t k ) k p;[0;t] + ; 8t2T k (3.6.22) Using Lemma 3.6.2 in the above equation, the following holds for8t2T k , kk p;[0;t] tt k +1 k k ;(t k ) k p;[0;t k 1] + k kv (t k ) k p;[t k ;t] + (3.6.23) The following holds from (2.2.7), k ;(t k ) k p;[0;t k 1] k ;(t k ) k p kk p;[0;t k 1] (3.6.24) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 26 Therefore, (3.6.23) can be written as, kk p;[0;t] tt k +1 k k ;(t k ) k p kk p;[0;t k 1] + k kv (t k ) k p;[t k ;t] + ; 8t2T k (3.6.25) Rewriting the result of Lemma 3.6.3 for t2T k1 =ft k1 ;t k1 + 1;:::;t k 1g, kk p;[0;t] tt k1 +1 k1 k ;(t k1 ) k p kk p;[0;t k1 1] + k1 kv (t k1 ) k p;[t k1 ;t] + ; 8t2T k1 (3.6.26) Since t k 12T k1 , Equation (3.6.26) can be written as follows, kk p;[0;t k 1] t k t k1 k1 k ;(t k1 ) k p kk p;[0;t k1 1] + k1 kv (t k1 ) k p;[t k1 ;t k 1] + ; 8t2T k1 (3.6.27) Let, x k =kk p;[0;t k 1] (3.6.28) A k1 = t k t k1 k1 k ;(t k1 ) k p (3.6.29) B k1 = k1 (3.6.30) u k1 =kv (t k1 ) k p;[t k1 ;t k 1] + (3.6.31) Therefore, (3.6.27) can be written as follows: x k A k1 x k1 +B k1 u k1 (3.6.32) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 27 Let, (k;l) = 8 > < > : A k1 A k2 :::A l if k>l 0 1 if k =l (3.6.33) Then, (3.6.32) can be written as follows, x k (k; 0)x 0 + k1 X i=0 (k;i + 1)B i u i (3.6.34) The rst main result of this chapter which calculates the transient response bound for the cost in Equation (3.5.9) is given below. Theorem 3.6.4. (Main Result): Consider the switching adaptive control system shown in Fig.(3.2) with hysteresis switching algorithm and the -cost-detectable cost function given by (3.4.8). Let, = rsp +h (3.6.35) = max i k ;i k p (3.6.36) = (3.6.37) where rsp is the robust cost as given by Denition 2.3.3. The following holds for all t2Z + , kk p;[0;t] +kv k p;[0;t] 8 > < > : h 1 n t 1 i if 6= 1 n t if = 1 (3.6.38) where n t is the number of controller switches till time t. Proof. Writing (3.6.27) in the form of (3.6.34), kk p;[0;t k 1] k2 X i=0 k1 Y j=i+1 t j+1 t j j k ;(t j ) k p i kv k p;[t i ;t i+1 1] + + k1 kv k p;[t k1 ;t k 1] + (3.6.39) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 28 The following is obtained after substituting Equation (3.6.39) in the result of Lemma 3.6.3, kk p;[0;t] tt k +1 k k ;(t k ) k p h k2 X i=0 k1 Y j=i+1 t j+1 t j j k ;(t j ) k p i kv k p;[t i ;t i+1 1] + + k1 kv k p;[t k1 ;t k 1] + i + k kv k p;[t k ;t] + ; 8t2T k (3.6.40) The above equation can further be written as, kk p;[0;t] k1 X i=0 k Y j=i+1 j k ;(t j ) k p i tt i+1 +1 kv k p;[t i ;t i+1 1] + + k kv k p[t k ;t] + ; 8t2T k (3.6.41) It can easily be shown that the following holds for all t2T k and i2 [0;k 1], kv k p;[t i ;t i+1 1] t i+1 t1 kv k p;[0;t] (3.6.42) kv k p;[t k ;t] kv k p;[0;t] (3.6.43) Therefore, (3.6.41) can be written as follows, kk p;[0;t] k1 X i=0 k Y j=i+1 j k ;(t j ) k p i kv k p;[0;t] + + k kv k p;[0;t] + (3.6.44) where t2T k . CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 29 Given (3.6.36), (3.6.37) and that j for all j, the following holds, kk p;[0;t] kv k p;[0;t] + + kv k p;[0;t] + k1 X i=0 k Y j=i+1 = kv k p;[0;t] + + kv k p;[0;t] + k1 X i=0 k Y j=i+1 8t2T k = kv k p;[0;t] + k1 X i=0 i (3.6.45) The following is true about the summation term in the above equation, k1 X i=0 i = 8 > < > : 1 k 1 if 6= 1 k if = 1 (3.6.46) Therefore, Equation (3.6.45) can be written as follows, kk p;[0;t] +kv k p;[0;t] 8 > < > : h 1 k 1 i if 6= 1 (k) if = 1 (3.6.47) wherek represents the number of switches until timet2T k . Therefore for anyt2Z + , the following holds true, kk p;[0;t] +kv k p;[0;t] 8 > < > : h 1 n t 1 i if 6= 1 n t if = 1 (3.6.48) Corollary 3.6.5. Let, N = max i kN r i k p (3.6.49) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 30 then the following holds true for all t2Z + , kk p;[0;t] 8 > < > : h 1 n t 1 i +N krk p;[0;t] if 6= 1 n t +N krk p;[0;t] if = 1 (3.6.50) Proof. From Theorem 3.6.4, the following is true, kk p;[0;t] +kv k p;[0;t] 8 > < > : h 1 n t 1 i if 6= 1 n t if = 1 (3.6.51) for all t2Z + . However from Figure (3.2), v =N r r (3.6.52) ,kv k p;[0;t] kN r k p krk p;[0;t] ; From Equation (2.2.7) (3.6.53) From Equation (3.6.49), the following holds, kv k p;[0;t] N krk p;[0;t] (3.6.54) Therefore, kk p;[0;t] 8 > < > : h 1 n t 1 i +N krk p;[0;t] if 6= 1 n t +N krk p;[0;t] if = 1 (3.6.55) The terms h 1 n t 1 i N and N n t in the above equation provide a bound on the transient response of plant input output data with respect to the input reference signal r. Of the parameters that appear in these bounds, in , n t and N can be changed by the designer. In the section on simulation results, the implications of CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 31 changing each of these designer controlled parameters on the transient response of the system are discussed. 3.6.2 Transient Response Bound II This section begins with two lemmas and the analysis here takes into consideration the distance between subsequent controllers in the switching sequence. Interesting conclusions can be derived with this bound, as discussed towards the end of this section. Lemma 3.6.6. The following holds for t2T k , k~ v (t k ) k p;[0;t] tt k +1 k ;(t k ) ;(t k1 ) +~ v (t k1 ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t] ; 8t2T k (3.6.56) Proof. Combining (3.2.6), (3.6.11) from Lemma 3.6.1 and (3.6.16), the following is true, ~ v (t k ) (t) = 8 > < > : ;(t k ) (t) for t<t k v (t k ) (t) for t2T k (3.6.57) The above equation can be rewritten as follows, ~ v (t k ) (t) = 8 > < > : ;(t k ) (t) ;(t k1 ) (t) + ~ v (t k1 ) (t) for t<t k v (t k ) (t) for t2T k (3.6.58) From (2.2.3), the following holds for t2T k , k~ v (t k ) k p;[0;t] tt k +1 k~ v (t k ) k p;[0;t k 1] +k~ v (t k ) k p;[t k ;t] (3.6.59) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 32 After substituting (3.6.58) in the above equation, the following is obtained fort2T k , k~ v (t k ) k p;[0;t] tt k +1 k ;(t k ) ;(t k1 ) +~ v (t k1 ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t] ; 8t2T k (3.6.60) Lemma 3.6.7. Consider the cost function given by (3.4.8). If hysteresis switching algorithm is used and, t k+1 t k = d k (3.6.61) k ;(t k ) ;(t k1 ) k p;[0;t k 1] M k kk p;[0;t k 1] (3.6.62) then the following holds, +k~ v (t k ) k p;[0;t k+1 1] ( d k M k k1 + d k )( +k~ v (t k1 ) k p;[0;t k 1] ) + ( +kv (t k ) k p;[t k ;t k+1 1] ) (3.6.63) Proof. The result of Lemma 3.6.6 can be written as, k~ v (t k ) k p;[0;t] tt k +1 k ;(t k ) ;(t k1 ) k p;[0;t k 1] + tt k +1 k~ v (t k1 ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t] ; 8t2T k (3.6.64) Given Equations (3.6.61), (3.6.62), the above equation can be further written as follows for t =t k+1 1, k~ v (t k ) k p;[0;t k+1 1] d k M k kk p;[0;t k 1] + d k k~ v (t k1 ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t k+1 1] (3.6.65) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 33 Using the hysteresis switching condition at t k 1 in the above equation, k~ v (t k ) k p;[0;t k+1 1] d k M k k1 ( +k~ v (t k1 ) k p;[0;t k 1] ) + d k k~ v (t k1 ) k p;[0;t k 1] +kv (t k ) k p;[t k ;t k+1 1] (3.6.66) Therefore, +k~ v (t k ) k p;[0;t k+1 1] ( d k M k k1 + d k )( +k~ v (t k1 ) k p;[0;t k 1] ) + ( +kv (t k ) k p;[t k ;t k+1 1] ) (3.6.67) Theorem 3.6.8. (Main Result) Let, k = d k (M k k1 + 1) (3.6.68) and = rsp +h. If hysteresis algorithm is used and = max k k < 1, then, kk p;[0;t] + max 2[0;t] kv k p;[0;] 1 1 (3.6.69) for all t2Z + . Proof. Iterating on the result of Lemma 3.6.7, the following is obtained, +k~ v (t k ) k p;[0;t k+1 1] k Y j=0 ( d j M j j1 + d j ) + k1 X i=0 k Y j=i+1 ( d j M j j1 + d j )( +kv k p;[t i ;t i+1 1] ) + ( +kv k p;[t k ;t k+1 1] ) (3.6.70) CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 34 Given Equation (3.6.68) and that = max j j , the following holds, +k~ v (t k ) k p;[0;t k+1 1] k Y j=0 + k1 X i=0 k Y j=i+1 ( +kv k p;[t i ;t i+1 1] ) + ( +kv k p;[t k ;t k+1 1] ) (3.6.71) The above equation can further be written as follows, +k~ v (t k ) k p;[0;t k+1 1] ( + max i2[0;k] kv k p;[0;t i+1 1] ) k+1 X i=0 i (3.6.72) The following is true from the hysteresis switching condition, kk p;[0;t k+1 1] ( +k~ v (t k ) k p;[0;t k+1 1] ) (3.6.73) On substituting Equation (3.6.72) in the above equation, kk p;[0;t k+1 1] ( + max i2[0;k] kv k p;[0;t i+1 1] ) k+1 X i=0 i (3.6.74) If < 1, then, kk p;[0;t] + max 2[0;t] kv k p;[0;] 1 1 (3.6.75) for all t2Z + . When< 1, the cost of Equation (3.6.69) is bounded above by 1 1 . The term is given by, = max j j = max j d j (M j j1 + 1): (3.6.76) In the above equation,f( d 0 ;M 0 ); ( d 1 ;M 1 ); ( d 2 ;M 2 );:::; ( d k ;M k );:::g denotes a se- quence of pairs of a controller dwell-time and a measure of its distance to next con- troller in the sequence. Each of these pairs satises the condition on given by the CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 35 above equation. Therefore, < 1, d j log 1 M j +1 log (3.6.77) To keep < 1, it is required that either M j be small or d j be large. In other words, < 1, implies slow adaptation or slow switching. If the initial controller is stabilizing, then slow switching (< 1) is a sucient condition for the actual cost of the system to be bounded above by 1 1 . 3.6.3 Simulation Results 3.6.3.0.1 Bad Transients Suppose the unknown plant shown in Fig. 3.2 has a transfer function, P (z) = (e 0:03 1)z ze 0:03 , which is a discretized version of the plant used in [26], with a sampling interval of 0:03 seconds and the controller set isK =fK 1 = 2;K 2 = 2g, whereN r 1 =N y 1 =2,N r 2 =N y 2 = 2 andD 1 =D 2 = 1. If the reference signal is chosen to be r(t) = 1;8t, the hysteresis algorithm A1 is used with h = 0:01 and the cost function (3.4.8) is used with = 0:99, p = 2 and = 1, the simulation results show that n t = 9. It can be veried 1 that N = 2, = 2, rsp 1:81, therefore, = rsp +h 1:82 and 3:6. Therefore, the coecient ofkrk p;[0;t] in the bound given by (3.6.50) is , h 1 nt 1 i N 1:4 10 5 : The simulation results are shown in Fig. (3.4) which indicate large transients, max t ju(t)j 175, max t jy(t)j 95 and max t kk p;[0;t] 800. 1 The ` p gain of an LTI system , i.e.,k(z)k p is equivalent to the ` p gain of (z). When p = 2, the ` 2 gain and H 1 gain are equal as shown in [38, 39]. Therefore, the MATLAB command norm can be used to calculate the ` 2 gain ask(z)k 2 = norm((z), Inf). CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 36 0 200 400 600 800 1000 0 1 2 3 time (1 unit = 0.03 seconds) Controller switching signal σ(t) 0 200 400 600 800 1000 −200 −100 0 100 200 time (1 unit = 0.03 seconds) Control signal u u(t) 0 200 400 600 800 1000 −20 0 20 40 60 80 100 time (1 unit = 0.03 seconds) Plant output y y(t) 0 200 400 600 800 1000 0 200 400 600 800 time (1 unit = 0.03 seconds) ∥r∥ λp,[0,t] ∥ζ∥ λp,[0,t] Figure 3.4: Bad Transients - Top-left: Controller switching signal, Top-right: Control signal u, Bottom-left: Plant output signal y, Bottom-right Sequence of ` p norms of data versus reference signal r,8t 3.6.3.0.2 Reducingn s Based on Theorem 1, one strategy to reduce the transient response is to modify the hysteresis constanth, so as to reduce the upper bound on the number of controller switchesn s given by (3.3.7). The bound onn s can be drastically reduced by increasing the hysteresis constant h. If h = rsp , then, for the example given in the previous subsection, it is true that, n s l M rsp h m = l 2 rsp rsp m = 2 If all other parameters are the same as in the previous subsection, then = rsp +h 3:62 and = 7:24. Therefore, the coecient ofkrk p;[0;t] in the bound given by (3.6.50) is , h 1 ns 1 i N 59:4 The simulation results are shown in Fig. (3.5) and it can be seen that the transients CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 37 0 200 400 600 800 1000 0 1 2 3 time (1 unit = 0.03 seconds) Controller switching signal σ(t) 0 200 400 600 800 1000 −10 −5 0 5 10 time (1 unit = 0.03 seconds) Control signal u u(t) 0 200 400 600 800 1000 −3 −2 −1 0 1 2 3 time (1 unit = 0.03 seconds) Plant output y y(t) 0 200 400 600 800 1000 0 5 10 15 20 25 30 35 time (1 unit = 0.03 seconds) ∥r∥ λp,[0,t ] ∥ζ∥ λp,[0,t ] Figure 3.5: Reduced n s - Top-left: Controller switching signal, Top-right: Control signal u, Bottom-left: Plant output signal y, Bottom-right Sequence of ` p norms of data versus reference signal r,8t have reduced signicantly. 3.6.3.0.3 Reducing , N The bound in (3.6.50) can be reduced further by choosing dierent co-prime realizationsN i ;D i for the candidate controllersK i , which in turn reduces the transients. If each of the controllers inK of (3.6.3.0.1) is realized using the following co-prime factor descriptions, N r 1 (z) =N y 1 (z) = 2(0:33z 0:26) z 0:86 N r 2 (z) =N y 2 (z) = 2(0:33z 0:26) z 0:86 D 1 (z) =D 2 (z) = 0:33z 0:26 z 0:86 then, = 1:04,N = 1:02, rsp 3:5, therefore = rsp +h 5:31 and = 5:52. The coecient ofkrk p;[0;t] in (3.6.50) is, CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 38 0 200 400 600 800 1000 0 1 2 3 time (1 unit = 0.03 seconds) Controller switching signal σ(t) 0 200 400 600 800 1000 −3 −2 −1 0 1 2 3 time (1 unit = 0.03 seconds) Control signal u u(t) 0 200 400 600 800 1000 −1 0 1 2 3 time (1 unit = 0.03 seconds) Plant output y y(t) 0 200 400 600 800 1000 0 5 10 15 time (1 unit = 0.03 seconds) ∥r∥ λp,[0,t ] ∥ζ∥ λp,[0,t ] Figure 3.6: Reduced , N - Top-left: Controller switching signal, Top-right: Con- trol signal u, Bottom-left: Plant output signal y, Bottom-right Sequence of` p norms of data versus reference signal r,8t h 1 ns 1 i N 36: The simulation results with the reduced andN are shown in Fig.(3.6) and it can be seen that the transients have reduced further when compared to that in (3.6.3.0.2). 3.7 Concluding Remarks A theoretical framework for the transient response analysis of the class of adaptive control systems with hysteresis switching algorithm and` 2e gain type cost detectable cost functions was proposed. Two upper bounds on the transient response of the actual cost of the adaptive system were obtained. The rst bound is useful when initial controller in the loop is arbitrary while the second bound provides a much tighter bound, albeit, with an initial stabilizing controller. Parameters that can CHAPTER 3. TRANSIENT RESPONSE ANALYSIS 39 lower these bounds were highlighted. Simulation results were provided that indicated improvement in the transient response with the proposed ideas. Chapter 4 Multi-objective Cost-detectable Cost functions Cost functions that appear in the existing methods of UAC focus mainly on estab- lishing a stable adaptive control system. These cost functions are cost-detectable, which means that they correctly detect instability of the adaptive system without any plant assumptions. However, there is also a need to include cost functions which re ect performance goals beyond stability, like good tracking of a class of inputs, simultaneous rejection of disturbance having dierent characteristics (white noise, bounded energy, persistent) etc. Therefore, in this chapter, a multi-objective formal- ism of a cost function is proposed wherein, a vector-valued controller cost function is considered. Each element of this vector-valued cost function re ects a particular control goal. Since stability is the foremost requirement, the primary objective of this chapter is to establish the properties that a multi-objective cost function must satisfy in order to be cost-detectable. Multi-objective control problems have been studied extensively in the past using plant models (e.g., [40] [41] [42] [43]). Nonetheless, it is discussed here in a completely data-driven framework. This chapter also examines the Level-Set (LS) switching algorithm, which is com- 40 CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS41 posed of increasing threshold cost levels. Threshold cost is a controller falsication level specied at every instant by the switching logic and is critical in averting false alarms. When an active controller is falsied at the current threshold cost level, it is switched out of the loop and replaced by a Pareto optimal controller by solving the multi-objective cost minimization problem constraining to the unfalsied set of controllers. In this work, the multi-objective cost minimization problem is formu- lated as a weighted Tchebyche problem (or min-max formulation) [44] [45] [46] and an eectively unique controller is selected, which guarantees that it is Pareto opti- mal. Like Hysteresis Algorithm (HA) [28], the LS algorithm is assured of nite time convergence, irrespective of any nite dwell time constraints. However, unlike the HA algorithm, the LS algorithm opens up the possibility for using nonmonotone cost functions, where the controller costs need not be monotonically increasing in time to guarantee convergence. LS is inspired by the switching logic in [43]. Motivated from Model Reference Adaptive Control (MRAC), a brief overview of evaluating tracking performance of a controller using ctitious outputs from reference models is presented [47]. This is particularly important because the performance of all the controllers can be assessed without actually inserting them in the feedback loop. A simulation example is provided which demonstrates the use of this performance measure as one of the elements in the vector-valued controller cost function. It is shown that the switching algorithm converges to an optimal controller if the adaptive control problem is feasible, which means that the plant is stabilizable and at least one controller exists in the candidate controller set that achieves the specied control goal for the plant. The notion of feasibility renders the adaptive control problem as a standard constrained optimization problem. A similar feasibility notion can be found in [12]. The next section provides an overview on the preliminaries of a multi-objective CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS42 minimization problem. 4.1 Pareto optimality Consider the following multi-objective cost minimization problem, min K2K V 1 K ;V 2 K ;:::;V n K (4.1.1) where V j :R n !R + ;j = 1;:::;n represent n objective functions, K represents the decision variable andK represents the set of all feasible decisions. Denition 4.1.1. (Superior solution)[45]: A feasible solution K 1 2K is said to be superior to another feasible solution K 2 2K if, V i K 1 V i K 2 for all i and V j K 1 <V j K 2 for some j: (4.1.2) Denition 4.1.2. (Pareto optimality) [45]: A feasible solution K 2 K is Pareto optimal (or ecient/non-inferior) if there is no feasible solution superior to K . In other words, a Pareto optimal solution is a feasible decision that cannot be improved upon in one objective without being degraded in another objective. There are usually a lot (innite number) of Pareto optimal solutions which form the Pareto optimal set. Denition 4.1.3. (Weakly Pareto optimal) [44]: A feasible solution K 2 K is weakly Pareto optimal if there does not exist another feasible solution K2 K such that V j K <V j K for all j = 1;:::;n. CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS43 A Pareto optimal solution is weakly Pareto optimal but the converse is not true. One way to solve the multi-objective problem of Equation (4.1.1) is to examine the following weighted metric formulation [44], min K2K X j (w j jV j K z j j p ) 1 p (4.1.3) where z j denotes the goal or the desired value for objective V j K , the weights w j > 0 for all j and p is a positive integer. Whenz j min K V j K for allj, the solution to (4.1.3) is Pareto optimal for 1p< 1 [44]. The results dier for the casep =1 which is termed as weighted Tchebyche problem. Since V j K is non-negative and z j min K V j K for all j, the absolute value sign is redundant. WLOG, we can consider z j = 0 for all j and write the weighted Tchebyche problem as follows, min K2K max j w j V j K (4.1.4) If the Tchebyche problem (4.1.4) has a unique solution, it is Pareto optimal [44]. If the solutions are non-unique, then there is atleast one Pareto optimal solution and the rest of them are weakly Pareto optimal. Additionally, every Pareto optimal solution can be generated by changing the weights w j . Denition 4.1.4. (Eective Uniqueness) [45] The solution of problem (4.1.4) is said to be eectively unique if no other solution of the same problem is superior. Having dened eective uniqueness, Theorem 1 in [45] provides a sucient condi- tion for Pareto optimality of solutions to the problem (4.1.4). The theorem essentially says that, if K is an eectively unique solution of problem (4.1.4), then it is Pareto optimal. CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS44 Uniqueness in objective or the solution space implies eective uniqueness, but not conversely. If there are multiple solutions to (4.1.4) each with a dierent objective value, it is required to discard the ones which have solutions superior to them, thus retaining only eectively unique solutions which guarantees their Pareto optimality. 4.2 Problem Description In this section, the problem addressed in this chapter is formally stated. The super- visor of Figure (2.2) evaluates the cost of each of the controllers in real time using a plant data-driven multi-objective cost function, V K (), which is a causal mapping of signals 2 ` m pe to signals V i 2 ` n 1e . When specic is given, the cost function corresponding to controller K i 2 K is denoted V i and its unfalsied cost at time t is denoted as V i (t)2R n + . The j-th element of the vector V i is denoted as V j i . The elements of the vector V i may be chosen to re ect dierent and possibly competing control goals, (e.g., disturbance attenuation, stability, noise tolerance, control signal energy, etc.) Problem (P2): Design a data driven switching algorithm with a vector valued multi-objective cost function and a threshold cost sequence such that the switching adaptive control system is stable and satises the chosen performance criteria. 4.3 Level-Set algorithm The LS algorithm consists of increasing threshold cost levels, denoted by . The algorithm starts with an arbitrary controller in the feedback loop. At discrete time instants, the supervisor tests if the currently active controller ^ K(t) is falsied at the current threshold cost level. When the active controller is falsied at the current threshold cost level , it is immediately replaced by another as yet unfalsied Pareto CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS45 optimal controller from the set K, by solving the multi-objective, vector-valued cost minimization problem. The multi-objective cost minimization problem is formulated as a weighted Tchebyche problem (4.1.4). We select unity weights w j = 1 for all j and select only eectively unique solutions to the weighted Tchebyche problem (4.1.4). The falsied active controller is replaced by another as yet unfalsied Pareto optimal controller so long as the set K is not empty; otherwise, the threshold cost level is increased and the setK is reinitialized with all the controllers. The notation K = Kn ^ K(t) means the \exclusion of the controller ^ K(t) from the set K". Since at any given time, the set K contains only those controllers which are unfalsied at the current threshold cost level, we call the algorithm the Level-Set algorithm. A pseudo-code of the Level-Set algorithm is given below. Level Set Algorithm A2 1. Initialization: Set 0 > 0, = 0 , h> 0, t = 0, ^ K(t) =K 1 ,K =K 1 ;K 2 ;::::K N . 2. If max j V j ^ K(t) (t)> then for i = 1 :M If max j V j i (t)> then K =KnK i ; endif ; end ; CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS46 If K =, then K =fK 1 ;K 2 ;:::K M g ; = maxf +h; min K i 2K max j V j i (t)g ; endif ; ^ K(t + 1) = arg min K i 2K max j V j i (t) ; else ^ K(t + 1) = ^ K(t); endif 3. t =t + 1, go to Step 2. 4.4 Main Results The results of stability, convergence and niteness of controller switchings are pre- sented in this section. Lemma 4.4.1. Consider the feedback adaptive control system in Figure 2.2, together with the LS switching algorithm. If the problem P2 is feasible, then the number of controller switches is uniformly bounded above by some nite M2N. Proof. WLOG, assume thatK 1 is a feasible controller with the robust costV rsp (K 1 ). From A2, it is clear that once a controller is falsied at a given threshold cost level, at most M controller switches can occur before the set of unfalsied controllers at that cost level becomes empty, at which time the cost level is incremented by at least h. Hence, the number of controller switches at a given cost level is at most M. The CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS47 number of threshold cost levels is at most, Vrsp(K 1 ) 0 h + 1. Thus the overall number of controller switches is bounded above by, M =M V rsp (K 1 ) 0 h + 1 (4.4.5) Remark 4.4.1. It is important to note that there is no assumption made about the monotonicity property of the cost functions in V K . Each of the cost functions inV K need not be monotonic as specied in some of the switching algorithms to guarantee convergence. This opens up avenues for the usage of fading memory cost functions which are nonmonotonic. Hysteresis switching algorithm [28] is a special case of Algorithm 2 with monotone scalar cost functions. Remark 4.4.2. The total number of controller switches using LS algorithm is nite as proved in Lemma 4.4.1. This result is very similar to the one proved in [17] using hysteresis algorithm, however LS is more general in the sense thatK always contains only those controllers which are unfalsied at the current threshold cost level. Theorem 4.4.2. Suppose the adaptive control problem with the candidate controller setK is feasible, the pair (V i ;K i ) is cost-detectable for all K i 2K. Then with the LS algorithm, the switching will stop and the system (P; ^ K(t)) is stable. Proof. WLOG, assume thatK 1 is a feasible controller with the robust costV rsp (K 1 ). From Lemma 4.4.1, it is known that the total number of controller switches is nite. Let K f denote the nal controller. Since K 1 is feasible, its robust cost is nite and the following holds, max j V j K f <V rsp (K 1 )<1 (4.4.6) CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS48 From the above equation, it is clear thatV K f is uniformly bounded. Since (V i ;K i ) is cost-detectable for all K i 2K, Equation (4.4.6) implies that stability of the adaptive control system is unfalsied by the input-output data (r;). Since this holds true for all possible inputs (as feasibility and LS algorithm always guarantee nite switchings and convergence), the adaptive system is stable. In Theorem 4.4.2, the pair (V i ;K i ) is assumed to be cost-detectable for allK i 2K. In the following theorem, the conditions under which this is possible is proved. Theorem 4.4.3. Suppose for each K i 2K, the multi-objective cost function V K i () is a stable causal map of into V i 2 ` n 1 . If (V j i ;K i ) is cost-detectable for some j, then (V i ;K i ) is also cost-detectable. Proof. In order to prove the cost-detectability property of (V i ;K i ), it suces to show that the two conditions given in Denition 2.3.2 are equivalent when = 1. ((1) =) (2)) : Since V i 2 ` n 1 , it holds for all j = 1;:::;n that V j i 2 ` 1 . Further, since (V j i ;K i ) is cost-detectable for some j, the condition V j i 2 ` 1 implies that the stability of the adaptive system is unfalsied by the input-output data. ((2) =) (1)) Since stability of the system is unfalsied by the input-output data, is stable. Considering V K i () is a stable causal map of into V i 2` n 1 , stable implies that V i 2` n 1 . The set of conditions under which a general functional f(:) is cost-detectable is discussed in the following theorem. Theorem 4.4.4. Consider a general function f : R n ! R and the pair (f(V i );K i ) for each K i 2K. Suppose (i) for all t2Z + ;2D, the function f(:) is continuous and coercive on V i , and (ii) for each K i 2K, the multi-objective cost function V K i () is a stable causal map CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS49 of into V i 2` n 1 . If (V j i ;K i ) is cost-detectable for some j, then (f(V i );K i ) is also cost-detectable. Proof. In order to prove the cost-detectability property of (f(V i );K i ), it suces to show that the two conditions given in Denition 2.3.2 are equivalent when = 1. ((1) =) (2)) : Since f(V i )2` 1 and since f(:) is a coercive function of V i , it holds for allj = 1;:::;n thatV j i 2` 1 . Further, since (V j i ;K i ) is cost-detectable for somej, the condition V j i 2` 1 implies that the stability of the adaptive system is unfalsied by the input-output data. ((2) =) (1)) If stability of the system is unfalsied by the input-output data, con- dition (ii) implies that V i 2 ` n 1 . Further, since f(:) is continuous, this implies f(V i )2` 1 . 4.4.1 Cost function example Consider a switching adaptive control system as shown in Figure 4.1, where the closed loop system ( ^ K;P) tracks the reference model W m . The output of the reference model is denoted as y m . Suppose ~ y mi is ctitious output obtained by driving the reference model with the ctitious input signal ~ v i , corresponding to a non-active controllerK i . The error signal ~ e mi = ~ y mi y represents the ctitious tracking error had the controller K i been in the loop when the plant data was collected. Consequently, the following cost function serves as an example of a multi-objective cost function re ecting stability and the reference model tracking error as a performance measure. V i (t) = kk 1 p;[0;t] +k~ v i k 1 p;[0;t] ;k~ e mi k 2 p;[0;t] T (4.4.7) CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS50 $ ' 1 ( 1 ) 1 * 1 1 y 1 ℎ ℎ Γ ' ) ( ? $ 1 Figure 4.1: Switching adaptive control system tracking a reference model W m where is a small positive constant. It was shown through simulation examples in [17] that the cost V 2 i can lead to mismatch instability when used with the hysteresis switching algorithm, suggesting that it might not be cost-detectable. However, V 1 i is cost-detectable [17] and the mappingsV K i () of intoV i are stable and causal. Since all the conditions of Theorem 2 are satised, the pair (V i ;K i ) is cost detectable for all K i 2K. 4.5 Simulation Examples The eectiveness of the ideas presented in this chapter are demonstrated using a very simple illustrative example in this section. Suppose the unknown plant shown in Fig- ure 2.2 has a transfer functionP (z) = 0:01 z1:01 , with a sampling interval of 0:01 seconds, CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS51 the candidate controller set isK =f2;1:4; 1:5; 3; 4; 5g and the reference model is W m (z) = 0:03 z0:98 . There is perfect matching of the closed loop system (P;K 4 = 3) with the reference modelW m . Suppose the plant input signal is corrupted by a distur- bance input modeled as uniformly distributed on [0:5; 0:5]. Additionally assume that there is sensor noise which is also modeled as uniformly distributed on [0:2; 0:2]. Two cases are presented, the rst with the scalar valued cost functionV 1 i (t) = kk 1 p;[0;t] +k~ v i k 1 p;[0;t] which just re ects stability and the second with multi-objective cost function V i (t) given by Equation (4.4.7), which re ects both stability and performance. The pa- rameters used for simulation are 1 = 1, 2 = 0:99, p = 2, 0 = 2;h = 1; = 1, the initial controller placed in the loop is K 1 =2. The reference input is a sinusoidal signal of frequency 0:4 radians/sec which is also sampled with a sampling interval of 0:01 seconds. 4.5.1 Case 1 In this case, theK i -th controller cost function isV 1 i , which just re ects stability. The controller costs, threshold cost sequence t and the controller switching signal are shown in Figure 4.1. The controller switching signal which is generated using the LS algorithm for Case 1 shows that the nal controller is K 3 = 1:5, which is stabilizing but is not optimal. The reference model output and the plant output are also shown along with the reference model tracking error y m y in Figure 4.2 . 4.5.2 Case 2 The result for Case 2 with vector-valued multi-objective cost function as given by Equation (4.4.7) is shown in Figures 4.4-4.5. For every controller in the chosen can- didate set K, the maximum of the entries of its vector valued cost along with the threshold cost sequence t is shown in the rst part of Figure 4.4. The second part of CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS52 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 time (1 unit = 0.01 seconds) Cost functions Controller costs γ K 1 K 2 K 3 K 4 K 5 K 6 0 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 time (1 unit = 0.01 seconds) Switching signal Controller switching signal Figure 4.2: Case 1: Controller costs, Threshold cost sequence and the switching signal Figure 4.4 shows the controller switching signal generated using the LS algorithm. It converges to K 4 = 3 and results in perfect matching of the closed loop system with the reference model. The tracking error, as seen in Figure 4.5 is not zero because of the presence of disturbance and noise signals. However, the amplitude of the tracking error is much smaller than that of Case 1. As a result, the use of the multi-objective cost function is justied. A zoomed out version of Figure 4.4 is shown in Figure 4.6. This gure is shown just to indicate that the controller costs corresponding to stabilizing controllers are bounded, thereby, conrming that the chosen vector valued cost function is cost-detectable. Comparing Case 1 with Case 2, we see that the addition of the performance termk~ e mi k 2 p;[0;t] to the cost vector in Case 2 dramatically improved tracking error CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS53 0 100 200 300 400 500 600 700 800 900 1000 −4 −2 0 2 4 time (1 unit = 0.01 seconds) Amplitude Reference and Plant model output y−plant output ym − Reference Model output 0 100 200 300 400 500 600 700 800 900 1000 −3 −2 −1 0 1 2 3 time (1 unit = 0.01 seconds) Amplitude Tracking error Figure 4.3: Case 1: Reference model and Plant output performance. 4.6 Summary The idea of a data-driven Pareto optimal multi-objective unfalsied control design was examined and the conditions under which it correctly detects instability of the adap- tive switching system were derived. A discrete-time/sampled-data switching algo- rithm called the Level-Set (LS) algorithm was introduced. Unlike standard hysteresis switching algorithm, this algorithm does not require monotonicity or any restriction on the cost functions to guarantee convergence; the existence of a feasible controller is the only requirement for convergence. The LS algorithm works by computing a Pareto optimal controller from the unfalsied level set and inserts it in the loop to CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS54 0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 time (1 unit = 0.01 seconds) Cost functions Controller costs γ K 1 K 2 K 3 K 4 K 5 K 6 0 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 time (1 unit = 0.01 seconds) Switching signal Controller switching signal Figure 4.4: Case 2: Controller costs, Threshold cost sequence and the switching signal replace currently active controller when it is falsied at the current threshold cost level. Simulation results demonstrate the advantage of using a multi-objective cost detectable cost function over the use of cost functions which focus solely on stability. CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS55 0 100 200 300 400 500 600 700 800 900 1000 −4 −2 0 2 4 time (1 unit = 0.01 seconds) Amplitude Reference and Plant model output y−plant output ym − Reference Model output 0 100 200 300 400 500 600 700 800 900 1000 −3 −2 −1 0 1 2 3 time (1 unit = 0.01 seconds) Amplitude Tracking error Figure 4.5: Case 2: Reference model and Plant output CHAPTER 4. MULTI-OBJECTIVE COST-DETECTABLE COST FUNCTIONS56 0 500 1000 1500 2000 2500 3000 0 20 40 60 80 100 time (1 unit = 0.01 seconds) Cost functions Controller costs γ K 1 K 2 K 3 K 4 K 5 K 6 0 500 1000 1500 2000 2500 3000 1 2 3 4 5 6 time (1 unit = 0.01 seconds) Switching signal Controller switching signal Figure 4.6: Case 2: zoomed out version - Controller costs, Threshold cost sequence and the switching signal Chapter 5 Adaptive Dwell-time Switching 5.1 Logic Based Switching Algorithm Dwell-time switching and hysteresis switching are two of the most popular switching logics introduced in the adaptive switching control literature. Morse-Mayne-Goodwin (MMG) proposed the hysteresis switching algorithm [28] with a xed positive hystere- sis constant h which precludes the possibility of unbounded chatter while switching. Subject to certain `tunability' assumptions on the plant, MMG proved stability and convergence to a controller with nite cost after at most nitely many switches pro- vided that such a controller exists in the candidate controller set. Later, Wang et. al. [22] improved this slightly by identifying a non-convex class of cost-detectable cost functions for which the MMG tunability assumptions can be relaxed. Nonethe- less, for hysteresis type switching logic, the cost-minimization is not exact because of the positive hysteresis constant. Chattering instability may occur if the hysteresis constant is set to zero which results in a trade-o. The concept of dwell-time switching was proposed in [48] and shows that if all subsystems in a switched system are exponentially stable with zero as the common equilibrium point, then there exists a scalar d > 0 such that the overall system 57 CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 58 remains exponentially stable if the dwell time of every controller is larger than or equal d . The xed dwell-time switching logic was used for linear set-point control of an uncertain process in [15, 16] where it was assumed that d is prespecied. Fixed dwell-time is not suitable for adaptive switching control of nonlinear systems since it may lead to nite escape of the closed loop in some cases. It was shown in [49] that it does not matter if one occasionally has a dwell time smaller than the critical number d as long as the average dwell time is not less than this critical number. Hysteresis switching [28, 35] and its more recent scale-independent hysteresis switching logic [29] produce switching signals with an average dwell-time equal to d . However, certain plant dependent assumptions are made and the hysteresis constant is chosen accordingly in order to guarantee that the average dwell-time is equal to d . Moving in a dierent direction, Alhajri [50] proved that for strictly convex cost functions the convergence and stability properties of the MMG hysteresis algorithm are preserved in the limiting case where the hysteresis constant h is set to zero, but that in this limiting case the number of switches may no longer be nite and dwell- times may approach zero and switching becomes continuous. In this chapter, a self adjusting dwell-time switching algorithm is proposed which is a hybrid of hysteresis and dwell-time switching logics in which the dwell-time is constrained to grow in proportion to the reciprocal of an adaptively-computed hys- teresis variable ^ h(t) that asymptotically converges to zero. The algorithm is proposed under the framework of UAC [17, 18, 22{25, 31, 51] wherein c -cost-detectable cost functions are used. For a switched adaptive system, c -cost-detectability means that the controller cost functions correctly detect instability of the adaptive system with- out any plant assumptions as long as the dwell-time of the controllers is greater than or equal to c . The proposed switching logic has some interesting and attractive prop- erties. First, it results in asymptotically optimal performance since ^ h(t) approaches CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 59 zero, which in turns results in exact controller-cost minimization. The dwell-time approaches innity as the hysteresis variable ^ h(t) approaches zero which prevents chattering. Second, the dwell-time is not xed and is completely data-dependent. The dwell-times are chosen to ensure that the critical value of c is reached in nite time. Third, the controller cost functions can be non-convex in contrast to the ones in [50] providing greater exibility. In this chapter, controller cost functions which are monotonically non-decreasing in time are used. Feasibility, c -cost-detectability and asymptotic convergence of ^ h(t) to zero results in a stable optimal adaptive system. 5.2 Preliminaries Denition 5.2.1. (Minimum Cost): Given a cost function V (K;;t) and a plant input-output data , the minimum cost at time t denoted by V m (t) is dened as V m (t) = min K i 2K V (K i ;;t). A hysteresis variable at time t dened as h(K i ;;t) = V (K i ;;t)V m (t) corre- sponding to controller K i is introduced. For brevity, the hysteresis variable corre- sponding to the active controller for a specic is denoted as ^ h(t). Denition 5.2.2. ( c -cost-detectability): Given the cost candidate controller set pair (V;K), we say c -cost-detectability holds if there exists a scalar c > 0 such that if the dwell-time of an active controller ^ K(t) is greater than or equal to c , the following two statements are equivalent: (1) V ( ^ K(t);;t)2` 1 (2) is stable Denition 5.2.3. (Admissible dwell-time) A sequence of dwell-times d k wherek2N is said to be admissible if the following properties hold, 1. d k 0 for all k2N. CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 60 2. d k !1 as k!1. 5.3 Switching algorithm Following are some of the plant-independent assumptions made, Assumptions 1. The adaptive control problem is feasible. 2. c -cost-detectability holds. 3. Monotone cost function:8K i 2K,V (K i ;;t)V (K i ;;),8t> and2D. The problem addressed in this chapter is formally stated as follows, Probem (P3): Given that the above stated assumptions hold, design a stabilizing switching algorithm along with a suitable cost function such that the system shown in Figure (2.2) with an input r2` m pe converges to an optimal unfalsied controller. The proposed adaptive dwell-time switching algorithm is described as follows, Given a non-negative, monotonically non-decreasing, radially unbounded function f : R! R, the k-th active controller is constrained to stay in the loop as long as its dwell time d k is smaller than f( 1 ^ h(t) ) where ^ h(t) = V ( ^ K(t 1);;t)V m (t) and t2 [t k ;t k +1:::;t k+1 1]. When d k grows larger thanf( 1 ^ h(t) ), the active controller is discarded and replaced by a cost-minimizing controller. Consequently, the dwell-time of an active controller is completely data-dependent since it depends only on ^ h(t), which measures the oset of the active controller's cost value from the minimum cost value at time t. Since f is monotonically non-decreasing, a small ^ h(t) results in a large dwell-time and vice-versa. Given that feasibility holds, it can be shown that ^ h(t) converges asymptotically to zero while the switching algorithm simultaneously drives the dwell-time to innity since f is radially unbounded, which in turn avoids chattering instability. The switching algorithm is presented below, CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 61 Algorithm A3 1. Initialization: Set t = 0, k = 0, t k = 0, ^ K(t) = K 1 , K =fK 1 ;K 2 ;::::K M g. Choose a non-negative, monotonically non-decreasing, radially unbounded f : R!R. 2. t = t + 1. Collect data r, u, y. Calculate V (K i ;;t) and h(K i ;;t) for all K i 2K 3. If tt k f( 1 ^ h(t) ) then d k =tt k ^ K(t) =argmin K i 2K V (K i ;;t) k =k + 1 t k =t else ^ K(t) = ^ K(t 1) endif 4. Go to Step 2. 5.4 Main Results Theoretical results depicting the asymptotic convergence to an unfalsied optimal controller are presented in this section. Assuming feasibility, we show that the hys- CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 62 teresis variable ^ h(t) corresponding to active controller approaches to zero asymptot- ically. Next, it is shown that when f is radially unbounded and non-negative, the admissibility of the dwell-time sequence can be established. Finally proofs of stability and convergence to unfalsied optimal controller are given. Lemma 5.4.1. The sequence ^ h(t) is such that, 1. ^ h(t) h(t), where h(t) monotonically decreases to zero. 2. lim t k !1 ^ h(t k )! 0, whereft k g k2N denote the controller switching instants. Proof. Since Assumption 1 holds, V m (t) rsp ; 8t 0 Since the controller costs are monotonically increasing in time (Assumption 3 ),V m (t) is also monotonically increasing in time. Therefore, the sequenceV m (t) converges [52] and the following holds, lim t!1 V m (t) =V f rsp (5.4.1) Since K is nite and the controller costs are monotonically increasing in time, for t suciently large, say for all tt > 0, we have V ( ^ K(t 1);;t)V f . Thus, ^ h(t) =V ( ^ K(t 1);;t)V m (t) V f V m (t) = h(t) Since V m (t)V f and Equation (5.4.1) holds, it can be concluded that h(t) tends to zero monotonically. Therefore, ^ h(t) h(t) (5.4.2) CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 63 where h(t) monotonically decreases to zero. Therefore, lim t!1 ^ h(t)! 0 (from Equa- tion (5.4.2)). Since every subsequence of a convergent sequence is convergent [52], we have, lim t k !1 ^ h(t k )! 0: (5.4.3) Using the above lemma, it is shown that the sequence of controller dwell-times d k generated by the switching algorithm A3 is admissible if the function f is non- negative and radially unbounded. From the switching algorithm, it is evident that the dwell-time of k-th active controller is given by d k =tt k f( 1 ^ h(t k ) ). Lemma 5.4.2. Consider a general function f : R! R. If f is non-negative and radially unbounded, then d k is an admissible dwell-time sequence. Proof. 1. The term, 1 ^ h(t k ) is always non-negative since ^ h(t k ) =V ( ^ K(t k 1);;t k ) V m (t k ) and V ( ^ K(t k 1);;t k ) V m (t k ). If f is a non-negative function, then f( 1 ^ h(t k ) ) 0. Hence, d k =tt k f( 1 ^ h(t k ) ) 0 2. From Lemma 5.4.1 (2), we know that, lim t k !1 ^ h(t k )! 0) 1 ^ h(t k ) !1. Since f is radially unbounded, we can conclude that f( 1 ^ h(t k ) )!1. Hence, d k !1. Hence the sequence d k is an admissible dwell-time sequence. Lemma 5.4.3. Consider a general functionf :R!R. Iff is non-negative, radially unbounded and monotone non-decreasing, then the dwell-time sequence d k approaches c in nite time. Proof. From Lemma 5.4.1(1), we have, ^ h(t) h(t); CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 64 where h(t) monotonically decreases to zero. Therefore, 1 ^ h(t) 1 h(t) ) 1 ^ h(t k ) 1 h(t k ) ; 8t k where 1 h(t k ) !1 monotonically. Sincef is monotonically non-decreasing and radially unbounded, f( 1 h(t k ) )!1 monotonically. Therefore, there exists a nite time T > 0 such that for all t k T , f( 1 h(t k ) ) c . Since, 1 ^ h(t k ) 1 h(t k ) ; 8t k and f is monotone non-decreasing, we have f( 1 ^ h(t k ) )f( 1 h(t k ) ) c for all t k T . Therefore, d k f( 1 ^ h(t k ) ) c ) d k c ; 8t k T: The main result of this chapter is stated below. Theorem 5.4.4. (Main result): The adaptive dwell-time switching algorithm A3 results in a stable optimal switched adaptive system. Proof. The dwell-time approaches c in nite time T > 0 (from Lemma 5.4.3) and from Assumption 3, c -cost-detectability holds. Since K is nite and the controller cost functions are monotonically increasing in time, we have after suciently long time t > 0, V ( ^ K(t 1);;t) lim t!1 V m (t) =V f CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 65 which implies, V ( ^ K;;t)2L 1 (5.4.4) Therefore, for all t maxft ;Tg, since c -cost-detectability holds, Equation (5.4.4) implies is stable. This in turn implies that the adaptive switched system is sta- ble. Since lim t!1 ^ h(t)! 0, the switching algorithm ensures convergence to a cost- minimizing controller. Therefore, the switched adaptive system is optimal with re- spect to the cost function used. 5.5 Simulation Example The eectiveness of the ideas presented in this chapter are demonstrated by comparing the results of the proposed switching algorithm with that of hysteresis switching algorithm. Suppose the unknown plant shown in Figure (2.2) has a transfer function, P (z) = 0:01 z 1:01 with a sampling interval of 0:01 seconds and the candidate controller set is given by, K =f2; 0:5; 2; 2:3; 2:5g: It is easy to verify that K 1 =2 and K 2 = 0:5 are destabilizing controllers and the rest of them are stabilizing. The reference signal is a unit step input and the controller cost function used is given by, V (K i ;;t) = max t0 k~ e i k p;[0;t] +kuk p;[0;t] +k~ v i k p;[0;t] (5.5.5) CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 66 where ~ e i = ~ v i y, = 1, = 1 and p = 2. The optimal controller corresponding to the chosen cost function is K 5 = 2:5. For the proposed algorithm, the following function is used for the switching condition, f( 1 ^ h(t) ) = 1 ^ h(t) k (5.5.6) where k denotes the number of controller switches till the current time t. Since the controller switches k increase with time, it can be veried that f in Equation (5.5.6) satises all the properties specied in Lemma 5.4.3. For the hysteresis algorithm, a positive hysteresis valueh = 0:65 is used. The costs of the active controller generated using hysteresis algorithm and the proposed adaptive dwell-time switching algorithm are shown in Figure (5.1) and the controller switching signals for the two algorithms are shown in Figure (5.2). Figure 5.1: Unfalsied performance cost levels achieved (V ( ^ K;d;t)). The minimum costV m (t) converges to a nal valueV f = 2:3219 which is indicated CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 67 Figure 5.2: Controller switching signal (t). in Figure (5.1). It is evident from Figure (5.1) that V ( ^ K;;t) using the proposed algorithm converges exactly to V f (i.e. ^ h(t)! 0). This indicates convergence to an optimal controller, which corresponds to ^ K(t) =K 5 = 2:5 as shown by the switching signal in Figure (5.2). However, using the hysteresis algorithm (h = 0:65), V ( ^ K;;t) converges to a value of 2:9448. This implies convergence to a sub-optimal controller K 3 = 2 as shown in Figure (5.2). In hysteresis algorithm, an active controller is not switched out of the loop as long as the following condition is satised, V ( ^ K(t);d;t)V m (t) +h V f +h For the given problemV f +h = 2:3219 + 0:65 = 2:9719 and the nal cost of controller K 3 is 2:9448 which is less than 2:9719. Hence, once K 3 is selected by the hysteresis CHAPTER 5. ADAPTIVE DWELL-TIME SWITCHING 68 switching algorithm, it continues to stay in the loop forever even though the nal cost ofK 5 is lesser than the nal cost ofK 3 . However, the proposed algorithm converges to the optimal controllerK 5 . Thus the proposed adaptive dwell-time switching algorithm results in an asymptotically optimal performance. 5.6 Discussion and Conclusion A data-dependent dwell-time switching algorithm which is a hybrid of the standard hysteresis and dwell-time switching algorithms has been proposed in this chapter. The dwell-time of the currently active controller is chosen as a function of the reciprocal of its hysteresis variable ^ h(t) which asymptotically converges to zero. Chattering is avoided by constraining the dwell-times to approach innity as ^ h(t) approaches zero. The advantage of the proposed approach over the traditional xed hysteresis algorithm is that asymptotically optimal performance is achieved. The performance improvement of the proposed algorithm has been demonstrated using a simulation example. Chapter 6 Stability of Switched Systems 6.1 Introduction Consider a linear plant in feedback with a switching linear controller such that the closed loop system comprising the plant in feedback with every controller is stable. Overall, this system is analogous to a switching system wherein all the individual subsystems are stable. However, the overall switched system may be unstable if the switching between subsystems is unconstrained. Therefore, in this chapter, the ques- tion of how often is it safe for a switch to occur such that the overall switched system remains stable is addressed. It should be noted that this problem is not considered from an adaptive control point of view where the plant knowledge is limited. One of the popular methods to address this issue is the called the dwell-time switching technique which was proposed in [48] and shows that if all subsystems in a switched system are exponentially stable with zero as the common equilibrium point, then there exists a scalar dm > 0 such that the overall system remains exponentially stable if every subsystem remains unswitched for a time larger than or equal dm . The stability of the switched system using dwell-time switching logic was proved using multiple lyapunov functions in [27]. 69 CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 70 In this chapter, a variable dwell-time switching logic method is proposed which ensures faster switching between individual subsystems than the dwell-time switching logic. For a switched system comprising of linear and stable subsystems, it is shown that variable dwell-time switching logic ensures smaller dwell-times for subsystems which are closer and larger dwell-times for subsystems which are farther. Positive denite symmetric matrices P i are rst calculated by solving the Lyapunov equation corresponding to each individual subsystem whose system matrix A i is assumed to be known. The variable dwell-time switching logic then determines the dwell-times in real time using a reasonable measure of distance between subsystem pairs. The multiple lyapunov functions approach as in [27] is used to show that the proposed switching logic ensures stability of the overall switched system. The advantages of the proposed method are twofold. First, since the dwell-times are based on the distances between subsystems, the results obtained are much less conservative than the logic shown in [27]. Second, the variable dwell-time strategy ensures zero dwell-time when two systems share a common lyapunov function as opposed to a non-zero dwell-time value as shown in [27]. This chapter is organized as follows. The theory of Lyapunov stability is discussed in Section 2. Notation, preliminaries and background are provided in Section 3. The main result is provided in Section 4. Section 5 summarizes the results of this chapter. 6.2 Lyapunov Stability Lyapunov stability theorems are discussed in this section. These theorems are used to prove stability of switched systems with the proposed switching logic. Consider an autonomous system [53], _ x =f(x) (6.2.1) CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 71 wheref :D!R n is a locally lipschitz map from a domainDR n intoR n . The following theorem makes conclusions about the stability of (6.2.1). Theorem 6.2.1. [53] Let x = 0 be an equilibrium point for (6.2.1). Let V :D!R be a continuously dierentiable function on a neighbourhood D of x = 0, such that, V (0) = 0 and V (x)> 0 in D 0 _ V (x) 0 in D (6.2.2) Then, x = 0 is stable. Moreover, if _ V (x)< 0 in D 0 (6.2.3) then x = 0 is asymptotically stable. A continuously dierentiable function V (x) satisfying the above conditions is called a Lyapunov function. The following theorem simplies the above theorem to a linear system, _ x =f(x) = Ax, Theorem 6.2.2. [53] A matrix A is a stability matrix, that is Re( i ) < 0 for all eigenvalues of A, if and only if for any given positive denite symmetric matrix Q, there exists a positive denite symmetric matrix P that satises the Lyapunov equa- tion, A T P +PA =Q (6.2.4) Moreover, if A is a stability matrix, then P is the unique solution of (6.2.4). CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 72 6.3 Preliminaries Consider a nite set of linear stable systems as follows, _ x(t) =A i x(t) (6.3.5) where i2 P =f1; 2; 3;:::;Mg. The system matrix A i 2 R nn is hurwitz for all i. A switched system generated by the above set of linear systems can be described as follows: _ x =A (t) x (6.3.6) where : [0;1)! P is called the switching signal. Let t 0 ;t 1 ;:::t k ;::: denote the switching times and let d k denote the dwell-time of the subsystem that switches on at switching time t k . Since A i is hurwitz for all i, for every symmetrix positive dente Q, there exists symmetric positive denite matrices P i , such that the following lyapunov function holds for all i (from Theorem 6.2.2), A T i P i +P i A i =Q (6.3.7) Let V i (x(t)) = x(t) T P i x(t) denote the Lyapunov function corresponding to system with system matrix A i . Since the individual subsystems are linear and stable, each of the Lyapunov functions, V i (x(t)) has an exponential rate of decay. Let i denote the least upper bound on the rate of exponential decay of V i . We dene a new term, `Lyapunov gain' for a pair of Lyapunov functions as follows, Denition 6.3.1. Lyapunov Gain: The Lyapunov gain corresponding to the pair CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 73 (V i ;V j ) is dened as a positive constant ij such that, V i (x) ij V j (x) 8x2R n (6.3.8) It should be noted that, ij 6= ji . The matrix pencil corresponding to two matricesP i andP j is denoted as (P i ;P j ). The maximum eigenvalue of the pencil (P i ;P j ) is denoted as max (P i ;P j ) and the minimum value is denoted as min (P i ;P j ). 6.4 Variable Dwell-time Switching 6.4.1 Problem Statement : Given a reasonable measure of distance between the subsystems of a switched system, derive a switching logic such that the dwell time d i is small for subsystems that are closer and is large for subsystems that are farther while ensuring stability of the switched system. 6.4.2 Results Before the variable dwell-time switching logic is introduced in this section, few im- portant lemmas are discussed as follows: Lemma 6.4.1. The supremum of rate of exponential decay of V i is given by i = min (Q;P i ) Proof. Lyapunov function corresponding to subsystem i is given by, V i (x) =x T P i x CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 74 Using (6.3.7), the derivative of V i is given by, _ V i =x T Qx (6.4.9) Since i denotes an upper on the exponential rate of decay of V i , we have, _ V i i V i (6.4.10) , _ V i =x T Qx i x T P i x , i x T P i xx T Qx 0 ,x T (P i 1 i Q)x 0 , (P i 1 i Q) 0 , 1 i max (Q 1 P i ) , 1 i max (P i ;Q) (6.4.11) Let 1 i = max (P i ;Q), therefore i = min (Q;P i ). As a consequence of Lemma 6.4.1, V i (t)e min (Q;P i )t V i (0) (6.4.12) Lemma 6.4.2. The Lyapunov gain of the pairV i andV j is given by ij = max (P i ;P j ). Proof. From Denition 6.3.1, we have, V i (x) ij V j (x) 8x2R n ,V i (x) ij V j (x) CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 75 ,x T P i x ij x T P j x 0 ,x T (P i ij P j )x 0 ,P i ij P j 0 , ij max (P 1 j P i ) , ij max (P i ;P j ) (6.4.13) Hence, ij = max (P i ;P j ). Consider two arbitrary switching times t k and t k+1 . Let (t) =i for t2 [t k ;t k+1 ) and (t) =j for t2 [t k+1 ;t k+2 ). The following theorem calculates the dwell-time se- quence d k that ensures that the peaks in the Lyapunov function at switching instants t k and t k+1 are decreasing. Theorem 6.4.3. (Main Result) Let (t) = i for t 2 [t k ;t k+1 ) and (t) = j for t2 [t k+1 ;t k+2 ). If V j (x(t k+1 ))V i (x(t k )) (6.4.14) then, d k ln( max (P j ;P i ) min (Q;P i ) (6.4.15) Proof. Given that(t k ) =i and(t k+1 ) =j, the following is true from Lemma 6.4.1, V i (x(t k+1 ))e min (Q;P i ) d k V i (x(t k )) (6.4.16) Using Lemma 6.4.2, V j (x(t k+1 )) max (P j ;P i )V i (x(t k+1 )) (6.4.17) CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 76 From (6.4.16), the following is true, V j (x(t k+1 )) max (P j ;P i )e min (Q;P i ) d k V i (x(t k )) (6.4.18) If V j (x(t k+1 ))V i (x(t k )), then using (6.4.18), it is sucient if, max (P j ;P i )e min (Q;P i ) d k V i (x(t k ))V i (x(t k )) , max (P j ;P i )e min (Q;P i ) d k 1 0 , d k ln( max (P j ;P i ) min (Q;P i ) (6.4.19) Remark 6.4.1. If each d k is chosen such that the result of Theorem 6.2.1 holds, then the peaks in the Lyapunov function at switching instants are continuously decreasing. This ensures stability of the overall switched system. 6.5 Discussion The advantages of the current result with respect to the one in [27] are as follows, If P i = P j , then d k = 0. This is contrast to the result of [27] where d k is non-zero for this case. The case where the subsystems share a common Lyapunov function is taken into consideration in this result. In other words, d k = 0 when subsystems share a common Lyapunov function which is not the case in [27] The dwell-time of the current controller depends only on the controller it is going to switch to, as opposed to the entire switching sequence in [27]. CHAPTER 6. STABILITY OF SWITCHED SYSTEMS 77 It can be seen that d k is small forP i P j and is large whenP j >P i , indicating fast switching for closer systems and slow switching for farther systems. Chapter 7 Final Conclusions In this work, a theoretical framework to understand the transient response for the class of switching adaptive control systems with hysteresis type switching algorithms and` 2e gain type cost detectable cost functions was presented. In particular, bounds on the transient response of the actual cost of the system i.e. the actual reference to data induced gains were obtained. The parameters that reduce these bounds were identied which in turn help alleviate the problem of poor transient response seen in some cases. Simulation results were provided which validated the advantages of the proposed framework. This work also proposed some new contributions to the eld of UAC. First, the idea of a data-driven Pareto optimal multi-objective unfalsied control design was examined and the conditions under which it correctly detects instability of the adap- tive switching system were derived. A discrete-time/sampled-data switching algo- rithm called the Level-Set (LS) algorithm was introduced. Unlike standard hysteresis switching algorithm, this algorithm does not require monotonicity or any restriction on the cost functions to guarantee convergence; the existence of a feasible controller is the only requirement for convergence. When an active controller is falsied at the current threshold cost level, the LS algorithm replaces it by a Pareto optimal con- 78 CHAPTER 7. FINAL CONCLUSIONS 79 troller from the unfalsied set by solving the weighted Tchebyche formulation of the multi-objective cost minimization problem. Simulation results demonstrate the advantage of using a multi-objective cost detectable cost function over the use of cost functions which focus solely on stability. Following this, a data-dependent dwell-time switching algorithm which is a hybrid of the standard hysteresis and dwell-time switching algorithms was proposed. 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Abstract (if available)

Abstract A problem that is sometimes associated with adaptive control algorithms is the poor transient response with some plant-controller combinations. The primary contribution of this work is the development of a methodology to understand and bound transients in adaptive switching control systems that employ hysteresis type switching algorithms. In particular, two methods to derive bounds on the cost of the adaptive system are proposed. The first result bounds the transients when the switching algorithm starts with an arbitrary controller in the loop. The second result quantifies the idea of slow switching or slow adaptation and provides sufficient conditions under which transients remain bounded when the initial controller is stabilizing. Plausible solutions to reduce these bounds are provided, which in turn alleviates the problem of poor transient response. ❧ The other emphasis of this work is on some of the newer problem formulations in adaptive switching control systems. These problems are addressed within the framework of Unfalsified Adaptive Control. The problem of vector-valued controller cost functions which are solely data-dependent and reflect multiple objectives of a control system is examined. The notions of Pareto optimality of vector valued cost functions and the conditions under which they are cost-detectable is discussed. A sampled data/discrete-time Level-Set controller switching algorithm is investigated which allows for the relaxation of the assumption that the controller cost function be monotonically nondecreasing in time. When an active controller is falsified at the current threshold cost level, the Level-Set switching algorithm replaces it by an effectively unique solution of the weighted Tchebycheff method, thus ensuring the selection of an unfalsified Pareto optimal controller. ❧ Next, the problem of achieving optimal performance by driving the hysteresis gap to zero is addressed. In this regard, a data-dependent dwell-time switching algorithm is proposed in which the dwell-time is constrained to grow in proportion to the reciprocal of an adaptively-computed hysteresis variable ^h(t) that converges to zero. This is in contrast to the Morse-Mayne-Goodwin (MMG) hysteresis switching algorithm in which usually a fixed hysteresis constant h > 0 is used. The algorithm is proved to be globally stabilizing assuming feasibility and $\tau_c$-cost-detectability. It has the attractive property that the gap between optimal and achieved unfalsified performance levels are at each time bounded above by the current value of the hysteresis variable ^h(t), thereby ensuring asymptotically optimal performance. ❧ The last part of this work extends an existing result on the multiple lyapunov functions based approach to proving stability of switched systems. A less conservative result is developed which shows that the overall stability of a switched system is preserved with fast yet, small switches and slow longer switches. Theoretical results and simulation examples are provided which validate the effectiveness of the ideas presented in this work.

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Creator Sajjanshetty, Kiran Somashekar (author)

Core Title Adaptive control: transient response analysis and related problem formulations

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/19/2017

Defense Date 05/18/2017

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

Tag adaptive control,data driven control,OAI-PMH Harvest,robust control,transient bounds,transient response

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Safonov, Michael G. (committee chair), Nayyar, Ashutosh (committee member), Savla, Ketan (committee member)

Creator Email kiransajjanshetty@gmail.com,sajjansh@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-446332

Unique identifier UC11265448

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Legacy Identifier etd-Sajjanshet-5844.pdf

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Document Type Dissertation

Rights Sajjanshetty, Kiran Somashekar

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

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