UFDC Home  UF Institutional Repository  UF Theses & Dissertations  Internet Archive   Help 
Material Information
Subjects
Notes
Record Information

Table of Contents 
Title Page
Page i Dedication Page ii Acknowledgement Page iii Page iv Table of Contents Page v Page vi Page vii Abstract Page viii Page ix Chapter 1. Introduction Page 1 Page 2 Page 3 Page 4 Page 5 Chapter 2. The Nyquist robust stability margina new metric for the stability of uncertain systems Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Chapter 3. A new perspective on computing robust stability margins for complex parametric uncertainties Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Chapter 4. Variable structure control design for reduced chatter in uncertain state delay systems Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Chapter 5. Sliding mode control for uncertain input delay systems Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Chapter 6. Future work Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Appendix A Page 126 Page 127 Page 128 Page 129 Appendix B Page 130 Page 131 Appendix C Page 132 Appendix D Page 133 Appendix E Page 134 Page 135 Appendix F Page 136 Page 137 Page 138 Appendix G Page 139 Page 140 Bibliography Page 141 Page 142 Page 143 Page 144 Page 145 Page 146 Page 147 Biographical sketch Page 148 Page 149 Page 150 
Full Text 
ROBUSTNESS ANALYSIS OF UNCERTAIN LINEAR SYSTEMS AND ROBUST STABILIZATION OF UNCERTAIN DELAYED SYSTEMS By VARADHARAJAN R. BASKER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 To my parents Smt. Shyamala & Mr. Varadharajan Radhamani ACKNOWLEDGMENTS I express my deep and sincere gratitude to my advisor Professor Oscar Crisalle for his continuing encouragement, guidance, and support throughout the course of my Ph.D. program. Apart from introducing me to the exciting area of control, he enhanced the spectrum of my research activities with a tremendous support system of collaborations with other faculty and talented graduate students in theoretical as well as practical projects, and provided me with outstanding facilities. He has always encouraged me to acquire as diverse an education as possible. It is due to his constant encouragement that I am also acquiring an extra degree from Electrical & Computer Engineering Department. His hardworking nature, quest for perfection, unfailing courtesy, genuine care for the aspirations of others, and outstanding ability to work successfully with people of various temperaments, skills and backgrounds have taught me immensely. I think I could have obtained as good an education in control theory from any good university. But, I sincerely doubt whether I could have secured, as varied an exposure as I received here under his guidance. I wish to thank Professor Haniph Latchman (along with Prof. Crisalle) for teaching me much of the robust control that I know. It was a truly a pleasure to interact with him on the robustness research for uncertain systems. His enthusiasm has been contagious. I cherish the opportunity that I had to work with Professors Fricke (and Crisalle) on the design and construction of the UF pilot plant for the DOE project on evaluation of on line viscometers. Although our applied research work is not part of my dissertation and are published elsewhere, I spent nearly two years of my stay here on the building of the pilot plant and supporting the data acquisition and control operations. Without the practical knowledge that I acquired during the course of this project, my education would have been very much incomplete. His respect for labor, his width and depth of engineering knowledge, and his enthusiasm for work at his age, are truly inspiring. I would like to thank Professor Spyros Svoronos for serving in my supervisory committee. I would also like to thank Professors Richard Dickinson and William Edmonson for serving on the supervisory committee at a short notice. I would also like to thank the support staff in the Chemical Engineering office; particularly Tracy, Janice, Shirley, and Nancy for all the precious help over the years. The financial support from DOE and NSF are gratefully acknowledged. The friendship and support of my colleagues in the lab will always be appreciated. Particularly, I would like to thank Kostas and KeJian for collaborations in Variable Structure control and Mike for endless and fruitful discussions over these years. It was a pleasure to work with Tony Dutka, Tiago, Barbara and Rick Gibbs in the pilotplant. The presence, friendship and the assorted mischiefs of Jon over these years has certainly lightened the stress of working in multiple projects. Many have contributed to my personal growth and enjoyment over this period of my life; it is not possible that I mention everyone by name. However, I would like to specifically mention a few. The company of my best friend, Madhav Durbha, has been invaluable during this period. It has been my pleasure and privilege to have known Professors Ranga and Vasudha Narayanan. The kindness and the compassion that they show to the students and the success and harmony that they have achieved in their lives, has provided me with a model of modern Indian culture that I shall seek to follow in my own life. My brother and sisterinlaw have consistently supported my efforts and they are always in my thoughts. Finally, after all these years of unlimited love and unexpecting affection, thanking my parents would be like thanking myself. Neither of them went to college, but it is due to their numerous sacrifices, small and big, mostly untold, that I have been able to live some of their dreams. Mere words, in any of the languages that I know, are inadequate to express my gratitude. I dedicate this work to them. TABLE OF CONTENTS ACKNOWLEDGMENTS .................................................................................. iii A B ST R A C T ............................................................................................... viii 1. IN TR O D U CTIO N ......................................................................................... 1 2. THE NYQUIST ROBUST STABILITY MARGINA NEW METRIC FOR THE STABILITY OF UNCERTAIN SYSTEMS ........................... 6 2.1 Introduction .......................................................................................... 6 2.2 FrequencyDomain Approach to Robust Stability Analysis ................... 7 2.3 The Critical Direction Method for SISO Systems ................................... 13 2.4 The Critical Direction Method for MIMO Systems ................................. 17 2.5 E xam ples ................................................................................................. 22 2.6 C onclusions .......................................................................................... 28 3. A NEW PERSPECTIVE ON COMPUTING ROBUST STABILITY MARGINS FOR COMPLEX PARAMETRIC UNCERTAINTIES ...... 34 3.1 Introduction ............................................................................................. 34 3.2 Background and Preliminaries ................................................................. 35 3.3 The Critical Direction for a Characteristic Polynomial ........................... 39 3.4 M ain R esults .......................................................................................... 40 3.5 Parametric Robust Stability Margins for Highly Structured U ncertainties ....................................................................................... 43 3.6 Connections with the Classical M A Formalism ................................ 45 3.7 C onclusions .......................................................................................... 47 4. VARIABLE STRUCTURE CONTROL DESIGN FOR REDUCED CHATTER IN UNCERTAIN STATE DELAY SYSTEMS ................... 49 4.1 Introduction .......................................................................................... 49 4.2 Sliding Mode Control Design ................................................................. 51 4.3 Asymptotic Stability with Perturbation Compensation .......................... 58 4 .4 E xam ple ....... ................................................................................ 64 4 .5 C onclusions .......................................................................................... 65 5. SLIDING MODE CONTROL FOR UNCERTAIN INPUT DELAY SY ST E M S ........................................................................................... 69 5.1 Introduction .......................................................................................... 69 5.2 Preliminaries and Problem Formulation ................................................. 71 5.3 The Design of the Sliding Mode Controller ............................................. 72 5.4 Analysis of the Perturbed System ............................................................ 77 5.5 Some Open Issues in Sliding Mode Literature ........................................ 86 5.6 Constraints on the Control Input ............................................................. 99 5.7 Extension to the Case of Multiple Delays ................................................... 101 5.8 Illustrative E xam ple ..................................................................................... 102 5 .8 C onclusion ............................................................................................... 103 6. FU T U R E W O R K ................................................................................................ 106 6.1 Robust Stabilization of Uncertain InputDelay SystemsA Linear Matrix Inequality Approach .................................................................... 106 6.2 Automatic Covariance Resetting for OnLine Recursive Identification ........................................................................................... 119 6.3 Sliding Mode Control of Bilinear Systems .................................................. 124 APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G ...................................... .... ........... oo . .............. ...... 12 6 .......... I .................................... .... ................................. . . 1 3 0 ............................................................................................... 1 3 2 ............... ............. ..................................... .............. ................ 1 3 3 ............................................................................................... 1 3 4 ............................................................................................... 1 3 6 ............................................................................................... 1 3 9 BIBLIO G RA PH Y ............................................................................................... 141 BIO GRAPH ICAL SKETCH .............................................................................. 148 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ROBUSTNESS ANALYSIS OF UNCERTAIN LINEAR SYSTEMS AND ROBUST STABILIZATION OF UNCERTAIN DELAYED SYSTEMS By VARADHARAJAN R. BASKER May 1998 Chairman: Dr. Oscar D. Crisalle Major Department: Chemical Engineering This dissertation focuses on two main aspects. One, developing new tools for the robustness analysis of uncertain linear systems. Two, the synthesis of robust controllers for uncertain delay systems. Traditional methods to analyze the robust stability of linear systems have depended on structured singular value bounds. However, this approach neglects phase dependence of the uncertainty and results in undue conservatism. In this dissertation, the concept of the Nyquist robuststability margin, is introduced for characterizing the closedloop stability of uncertain systems. The approach makes direct use of Nyquist domain arguments and is based on the analysis of the perturbed eigenvalue loci, hence avoiding undue conservatism through the use of singularvalue upper bounds. A key element in the new approach is Critical Direction Theory applied to uncertainties in the Nyquist plane. The critical direction method is based on recognizing that, at any given frequency on the Nyquist plane, there is only one direction of perturbation of relevance to the stability analysis. This allows the characterization of robust stability margins for uncertain systems characterized by irregular perturbation templates, a problem that poses significant challenges to other analysis methods. Examples of practical relevance are given to illustrate the application of the new theory. Using the new approach, the problem of assessing robust stability and computing stability margins for SISO systems with affine complex parametric uncertainties is tackled successfully. Exact analytical results are derived for geometrically simple uncertainty sets such as ellipses and rectangles. In the later part of the dissertation, the synthesis of robust controllers for state delayed and inputdelayed systems is considered. In particular, Sliding Mode Control is chosen as the technique of choice, as it possesses the combination of robustness and performance guarantees that one seeks in a control system. Robust stability to the chosen perturbation characterization is rigorously proven. Practical difficulties in implementation are pointed out and ways to overcome these hurdles are presented. Some open questions in the literature are brought out and theoretical analysis and analytical answers are presented. Finally, ideas and interesting formulations are presented for future work. CHAPTER 1 INTRODUCTION Modeling of any practical system for control design invariably involves uncertainty. Since the exact model of a process may be difficult or impossible to determine, the logical approach is to design a control strategy based on a suitable nominal (i.e., most likely) model. Once a system has been identified and a nominal model and associated uncertainty description is available, there are two main tasks. The first task is the analysis of the uncertain system. The following issues arise naturally: The first, is the question of robustness analysis. What is the measure of the maximum uncertainty that the system can tolerate and still sustain stability and performance? Alternatively, given the size of the uncertainty, one might wish to study the effect of the uncertainty on the stability and performance of the system. A system is said to be robustly stable if it can sustain stability in spite of suffering from uncertainty. The second major issue is the synthesis of robust controller to ensure stability and performance of the physical system. Over the past two decades, a plethora of robust controller methods have been investigated and implemented in a wide array of industries. Although, robust analysis and synthesis is an intellectually stimulating and mathematically challenging task, there are some valid criticisms that can be leveled against this approach. Is it a good idea to design a system for worst case uncertainty? If the probability of worst case occurring is very low, why should we sacrifice the performance of the system by designing for the worst case? Should the uncertainties be simply bounded by norms (or) should there be probabilistic ideas inherently associated with the uncertainty descriptions? These are areas of intense interest and importance and the interested reader is referred to the research monograph by Smith and Dahleh [1994] and the references therein. In this dissertation attention will be focused on both the analysis and the synthesis aspects of robust control. The thesis is organized as follows: Chapters 23 address the robustness analysis problem, while Chapters 45 deal with controller design for robustly stabilizing a class of uncertain delay systems. In Chapter 6, future research directions are presented. In Chapter 2, the concept of the Nyquist robuststability margin, for characterizing the closedloop stability of uncertain systems is introduced. The predominant robustness analysis methods typically rely on arguments based on real singular value or magnitude bounds. Such is the case of well known multivariable stability margin km [Safonov, 1982] and the structured singular value ku [Doyle, 1982]. In contrast, the proposed new approach makes direct use of Nyquist arguments and is based on the analysis of the perturbed eigenvalue loci, hence avoiding undue conservatism that may result through the use of singularvalue upper bounds. A key element in the new approach is Critical Direction Theory applied to uncertainties in the Nyquist plane. The critical direction method is based on recognizing that, at any given frequency on the Nyquist plane, there is only one direction of perturbation of relevance to the stability analysis. This critical direction is defined by the oriented line that has its origin at the location of the nominal eigenvalue of the unperturbed system and passes through the critical point l+jO. In fact, from a stability point of view all points on the uncertainty eigentemplate that do not lie on the critical line can be ignored. This allows the characterization of robust stability margins for uncertain systems characterized by irregular perturbation templates, a problem that poses significant challenges to other analysis methods. Other attractive features of the Nyquist robuststability margin theory are that its MIMO (Multiple Input Multiple Output) form is a natural extension of the SISO (Single InputSingle Output) case, and that its versions for continuous and discrete systems are formally identical. A SISO and a MIMO example are given illustrating the concepts. Chapter 3 focuses on the complex parametric uncertainty problem from the perspective of the critical direction paradigm. Considerable attention has been given to the problem of assessing robust stability and computing stability margins for SISO systems with parametric uncertainties. While the general robust stability margin computation problem is known to be NPhard [Braatz et al., 1994], its rankone counterpart results in a convex optimization problem that can be readily computed, and can be often derived as an explicit analytical expression. It is now well known that affine parametric uncertainty problems result in a rankone y formulation. This chapter addresses the case of elementwise complex parametric uncertainties where each individual uncertainty lies in a highly structured domain that is convex along any line from the origin. This is a mild convexity condition, since the actual domain could be highly nonconvex. Exact analytical results are derived for geometrically simple uncertainty sets such as ellipses and rectangles. This approach however, is applicable to a much wider class of systems than the ones considered here. The only information that is needed is the phase dependent magnitude of the uncertainty. The approach presented here is simple, intuitive and yet, mathematically rigorous. For the special case of disk bounded affine uncertainties, the approach recovers, in a transparent manner, relevant results by previous methods. Chapter 4 and Chapter 5 deal with the issue of stabilizing an uncertain timedelay systems (both statedelay and inputdelay are considered). Time delay exists in various branches of engineering. From biological, optical, electric networks, to chemical reaction systems, time delay occurs and affects the stability and performance of a wide variety of systems [See Schell and Ross, 1986, Inamdar et al., 1991]. For example, input delays occur as one of the common sources of time delay, such as in chemical processes, transportation lags and measurement delays, etc. It is well known that the existence of time delay degrades the control performances and makes the closedloop stabilization difficult. Hence, the study of time delay systems has been paid considerable attention over the past years. The description of timedelay systems lead to differentialdifference equations, which require the past values of the system variables. There are no known methods to get necessary and sufficient conditions for the robust stability of such systems. There are many approaches that yield sufficient conditions with varying degrees of sophistication. In Chapter 4, a Variable Structure Control (VSC) design methodology is proposed to robustly stabilize an uncertain statedelay system with nonlinear and possibly mismatched uncertainties, utilizing the concept of perturbation compensation. The study of uncertain statedelay systems has received much attention by researchers in the past decade. One reason for this interest is that time delay is encountered in various engineering systems and can be the cause of instability. Another reason is that practical control systems unavoidably include uncertainty or disturbances due to modeling or measurement errors and other approximations. The sliding mode in VSC possesses well known features that make it very attractive for control systems. These include fast response, insensitivity to parameter variations, and decoupling design procedure, among others. Sufficient robust stability conditions are derived which are independent of the size of the delay; furthermore no matching conditions on the uncertainty are assumed. The main advantages of the proposed approach are that i) a realizable control law is obtained, ii) chattering reduction/elimination is achieved, and iii) guidelines are given for the design of the control matrix involved in the specification of the switching function. Finally the results are illustrated with an openloop unstable example. Chapter 5 proposes an approach for uncertain inputdelayed systems with linear and possibly mismatching uncertainties via Sliding Mode Control. Sufficient robust stability conditions are derived. We consider a transformation to convert the original system into an equivalent system without delay [Artstein, 1982]. This reduction enables the use of known stabilizing methods for linear systems without input delay. The main advantages of the proposed approach are that (i) a realizable control law is obtained, (ii) chattering 5 reduction is achieved since a continuous control law is used, (iii) guidelines are given for the design of the switching function to ensure the asymptotic stability of the normal system in the sliding mode. Certain open issues in the sliding mode literature are brought out and possible solutions are discussed. Finally, future research directions and possible extensions to the presented work are discussed in chapter 6. CHAPTER 2 THE NYQUIST ROBUST STABILITY MARGINA NEW METRIC FOR THE STABILITY OF UNCERTAIN SYSTEMS 2.1 Introduction Many of the now wellknown results on robust stability can trace their origins to the Nyquist stability criterion (SISO case), or the Generalized Nyquist stability criterion (MIMO case) [MacFarlane, 1980). Such is the case of the well known multivariable stability margin k,, [Safonov, 1982] and the structured singular value M [Doyle, 1982]. Although the generalized Nyquist criterion addresses the issue of stability using complex variable (eigenvalue or transfer function) arguments, the predominant robustness analysis methods typically rely on arguments based on real singular value or magnitude bounds. One reason for this choice was clearly the more advantageous conditioning of singular values for numerical calculations. However, another important factor was the historical fact that uncertainty measurements were initially proposed for MIMO systems using singular value bounds, which collapse to magnitude (disk) bounds for SISO systems. In this chapter we introduce the concept of the Nyquist robuststability margin, kN, for characterizing the closedloop stability of uncertain systems. The approach makes direct use of Nyquist arguments and is based on the analysis of the perturbed eigenvalue loci, hence avoiding undue conservatism that may result through the use of singularvalue upper bounds. A key element in the new approach is Critical Direction Theory applied to uncertainties in the Nyquist plane. A basic version of the critical direction concept was first presented in [Guzzella, et al., 1985] as a tool for analyzing the robust stability of polynomials with ellipsoidal uncertainties. In [Latchman and Crisalle, 1995] the critical direction notion is redefined directly on the Nyquist plane to facilitate the derivation of necessary and sufficient robust stability conditions for transfer functions subject to arbitrary perturbations. In this paper we further extend the critical direction theory for the analysis of MIMO systems by formulating the concept in the generalized Nyquist plane, introducing new nomenclature to facilitate the seamless extension from the SISO to the MIMO case, and defining the new concept of the Nyquist robust stability margin. The critical direction method is based on recognizing that, at any given frequency on the Nyquist plane, there is only one direction of perturbation of relevance to the stability analysis. This critical direction is defined by the oriented line that has its origin at the location of the nominal eigenvalue of the unperturbed system and passes through the critical point l+j0. In fact, from a stability point of view all points on the uncertainty eigentemplate that do not lie on the critical line can be ignored. This allows the characterization of robust stability margins for uncertain systems characterized by irregular perturbation templates, a problem that poses significant challenges to other analysis methods. Other attractive features of the Nyquist robuststability margin theory are that its MIMO form is a natural extension of the SISO case, and that its versions for continuous and discrete systems are formally identical. The chapter is organized as follows. Section 2.2 gives a succinct review of robuststability margin results for the SISO and MIMO cases. Section 2.3 formulates the critical direction theory for analyzing the robustness of uncertain SISO systems, and Section 2.4 gives the details of the new critical direction theory for the MIMO case. A SISO and a MIMO example are given in Section 2.5, followed by final remarks and conclusions in Section 2.6. 2.2 FrequencyDomain Approach to Robust Stability Analysis The new Nyquist robust stability margin is an object derived using Nyquist arguments. In order to provide a contextual background for the ensuing discussions, in this section we review relevant SISO and MIMO results on robust stability analysis, placing particular attention on the Nyquist arguments that also lie at the root of the new developments presented here. The transfer function g(s) of a SISO system can be written in terms of an additive perturbation 8(s) about a nominal transfer function go (s) g(s) = go (s) + 8(s), 8(s) r d (2.1) where d represents the set of allowed perturbations. When. all 8(s) E d are considered, then at each frequency the map g(jo) = go(jo) + (jco) defines a region denoted uncertainty template or value set [Barmish, 1993]. It is then possible to visualize the transferfunction uncertainty directly in terms of the classical Nyquist plot as uncertainty templates about the locus of the nominal system go(jCo). The mathematical description for the set of perturbations may be formulated in different spaces. For example, the Laplacefunction domain d is often defined implicitly in terms of a frequencydomain description. Such is the case where the uncertainty templates are specified to be circular at each frequency. This makes the analysis of robust stability particularly straightforward. On the other hand, the templates for a number of useful uncertainty descriptions are not circular but rather irregularly shaped, exhibiting highly directional features. This is the case for parametric uncertainty descriptions of the form 8(s) = g(s,p)g(s,Po), where g(s,po) and Po, respectively, denote the nominal transfer function and the nominal parameter set, and where g(s,p) is characterized by m real, uncertain parameters p de c 91m. In this framework the set d is defined implicitly through the Euclidean domain dp. Typical uncertainty representations for the parameter uncertainty set dp are polytopes and ellipsoids. The special case where the uncertainty templates are ellipses is an interesting example of directional uncertainty templates which also has a tractable mathematical representation. The representation (2.1) is general and adequately encompasses additive perturbations as well as uncertainties that may appear as multiplicative perturbations of the nominal system, as parametric variations on pole/zero locations, and/or variations of the coefficients of the numerator and denominator polynomials in the transfer function. The difference in each case is the shape and orientation of the uncertainty templates associated with 8(s). Of particular interest are three types of SISO uncertainty descriptions. Specifically, we denote the set of uncertainties giving rise to unstructured (circular), elliptical, and structured (arbitrary) templates, respectively, by the notation du, dE, and ds. In the case of uncertain MIMO systems, the matrix transfer function G(s)can be written as an additive perturbation A(s) about a known nominal transfer matrix Go(s) G(s) G, (s)+ A(s), A(s) E D (2.2) where D is the set of allowed perturbations. The uncertainty set D is described in terms of available information about the modeling errors. We use the designation DU to represent Unstructured uncertainties with a single norm bound, DB to denote block diagonal normbounded uncertainties, Dc to represent elementwise circular (disk bounded) uncertainties and DE to represent elementbyelement elliptical uncertainties. Other uncertainty descriptions are defined in terms of errors in the elements of the state space matrices, or errors in the real parameters of the matrix transfer functions. In all these cases the uncertainty description still can be mapped (via eigenvalue inclusion regions) on to frequencyresponse uncertainty templates, in the Nyquist plane which will then form the basis for defining the Nyquist robust stability margin. Classical SISO RobustStability Results The SISO robust stability analysis is concerned with the stability of the system that results when the system g(s) in (2.1) is arranged in a unity negativefeedback configuration. It is normally assumed that (i) the nominal system is stable under unity negativefeedback, and that (ii) the nominal and uncertain system have the same number of openloop unstable poles. These assumptions are adopted throughout this paper. The most studied case is perhaps that where the uncertainty description is unstructured, a(s) e du i.e., when 8(jo) < W(jco), where W(s) is a known function whose frequencyresponse magnitude defines the radius of the circular uncertainty templates. Invoking the Nyquist stability criterion leads to the necessary and sufficient stability requirement that all uncertainty templates exclude the critical point l+jO, i.e., g(jo)= go(jco)+ (jo) i V6(jco) and Vo) (2.3) This condition can be expressed through the inequality [Doyle et al., 1992] W(j(o) + g(Jco) (2.4) or equivalently, lw(i(O) .< I +Wg(jo) A+go (jo) <1 (2.5) The robustness analysis for uncertainty descriptions with non circular templates is more challenging. For this reason it is quite common for elliptical or arbitrarily shaped uncertainty templates, arising for example from parametric uncertainties, to be circumscribed by an appropriate circle [Bhattachrya et al., 1993]. Although this approach yields sufficient conditions for robust stability, it is nevertheless inherently conservative. Classical MIMO RobustStability Results As in the SISO case, the robustness analysis for MIMO systems makes use of the generalized Nyquist stability criterion [MacFarlane, 1980], where avoidance of the critical point l+jO is at the center of interest from the point of view of absolute stability assessment. This has been the major focus of much research interest in the development of robust multivariable stability margins. Consider the uncertain transfer matrix G(s) in (2.2) with nominal model Go(s) affected by an uncertainty A(s). Again it is assumed that (i) the nominal system is stable under unity negativefeedback, and that (ii) the nominal and uncertain system have the same number of openloop unstable poles. The conditions under which no eigenvalue of the uncertain system Go(jco)+ A(jo) is equal to 1+jO, i.e., A(Go(jco)+A(jao)) I VA(jco) and Vco (2.6) is readily shown to lead to the determinantal condition det(I + M(jco)A(jco)) 0 VA(jco) and Vco (2.7) where M(jo) := (I + Go(jc))1. Condition (2.7) is of course equivalent to the eigenvalue condition (M(jco)lA(jo))#1 VA(jo) and Vo (2.8) The multivariable stability margin km and the structured singular value Y. were independently defined by Safonov [1982] and Doyle [1982], respectively, in terms of the determinantal stability condition (2.7), rather than the eigenvalue conditions (2.6) or (2.8). For an uncertainty of a given description D, the multivariable stability margin km is defined as the matrix 2norm of the smallest destabilizing uncertainty in the given class, namely: k..(o)) = min{5(A(jo)): det(I + M(jo)A(jco)) =O} AeD (2.9) This definition corresponds to the reciprocal of the robustness measure Y. defined in [Doyle, 1982], which is defined as the inverse of the norm of the smallest destabilizing uncertainty in the class, namely: u(co) = "minjU(A(j0)): det(I + M(jo))A(jo))=O}1 AED (2.10) Note that if no uncertainty in the allowable class destabilizes the system (i.e.,, makes an eigenvalue of Go(jo)+A(jco) equal to 1 at any frequency) then km = and U =0. These margins provide a measure of tolerable uncertainty size. A number of powerful robuststability results can be obtained in terms of the robust stability margins (2.9)and (2.10). Consider, for example, the following case of interest. An uncertainty description D is said to be closed under contraction and rotation if for any A(s) E D, then yejoA(s) E Dfor all 0 < y < 1 and for all 0 < 0 < 27.. Starting from the eigenvalue stability condition (2.8), we can state the following lemma. Lemma 2.1 Consider an uncertainty description D that is closed under contraction and rotation. Then the uncertain system G(s) = Go(s)+ A(s) is stable for allA(s) ED if and only if sup p(M(jo)A(joO)) < 1 VO AED (2.11) Proof: Since the eigenvalue condition(2.8) is necessary and sufficient for robust stability, it is clear that (2.11) immediately constitutes a sufficient stability condition. That (2.11) is also necessary is established by contraposition as follows: Consider an uncertainty A,(s) c D such that p(MAo) = []' > 1, then due to the closure under contraction we can always find an uncertainty A1 c Dwith A = A4,; 0 < y < 1, such that p(MA1 ) = 1. Since the phase of the uncertainty class is arbitrary, due to the closure under rotation we can assign a scalar phase multiplier to 4 to get A2 such that A(MA2)=1, which of course implies instability. Thus, it follows that for the description D specified in the Lemma, condition (2.11) is both necessary and sufficient for robust stability. V For the classes of uncertainties covered by Lemma 2.1, the spectral radius condition (2.11) is in fact equivalent (after appropriate normalizations) to the definitions of y and k.. and thus it naturally follows that u is equal to the left hand side of (2.11). Hence the conclusion of Lemma 2.1 can be equivalently formulated as km > 1 Vo (2.12) or 'U The k.. and a stability margins serve to provide valuable characterization of destabilizing uncertainties. Specific procedures and results have been developed for computing these stability margins (or at least good upper and lower bounds) for various useful classes of uncertainties. Additionally, the u problem formulation has produced an extensive set of results associated with the class of block diagonal bounded uncertainties, and has thus attained wide acceptance for robustness analysis, and more recently, also for controller synthesis using the usynthesis method. In all these applications the computation of the stability margin is effected by calculating singularvalue upper bounds which give sufficient, and in some cases necessary and sufficient, stability conditions. Unfortunately, however, the use of normbounds, especially singular value bounds, in the characterization of destabilizing uncertainties often causes the rich structural properties, including phase and directionality to be ignored. On the other hand, as we show below significant advantages can be gained by studying the robustness problem directly from an eigenvalue point of view, especially since necessary and sufficient stability conditions correspond precisely to an eigenvalue condition via the generalized Nyquist criterion. 2.3 The Critical Direction Method for SISO Systems In this section we present an exact robust stability result for the case of SISO systems using the critical direction theory. We adopt a nomenclature that permits the extension of the criticaldirection concepts to the MIMO case. We also introduce the definition of the Nyquist robust stability margin for SISO systems in a form which readily extends to the MIMO case. Figure 2.1 shows a typical Nyquist diagram for a SISO system illustrating the nominal frequency response go(jco) and an irregularly shaped uncertainty template. We define the critical line at a given frequency o as the directed line which originates at the nominal point go(jo) and passes through the critical point 1+jO. The figure is also useful for identifying the entities defined below: 1. The crtitical direction d(jo)= 1 + go (jo) l + g(jo)) (2.14) which may be interpreted as the unit vector that defines the direction of the critical line. 2. The uncertainty template T(co) := { g(jo) I g(jo) = go(jo) +6(jo), 8(s) e d} (2.15) 3. The critical template T'c(cO):={zcT(o) z=go(jo) +cd(jco) for some aE9E+ } (2.16) namely, the set of perturbed frequencyresponses lying along the critical direction d(jco). 4. The critical perturbation radius pc(o) := max {a z = go (jco) +Mat(jo)E Tc)} (2.1 ac:91+ (2.17) As an illustration of the previous definitions note that the critical line at frequency co = o, is readily identified in Figure 2.1 as the directed line with origin at go (Jo, ) and passing through the point l+jO. The critical direction d(jcol) is simply the unitlength vector that characterizes the direction of the critical line. In addition, the critical radius Pc(col) is shown as the distance between the nominal Nyquist point go(Jwl) and the point where the critical line intersects with the boundary of the template. Finally, the critical template T(co) is also readily characterized in Figure 2.1 as the subset of the uncertainty template T(co) that intersects with the critical line. By definition Tc (co) is a subset of the critical line, thus it follows that T (Co) is either a single straightline segment, or the union of such segments. The critical template may also contain isolated points should the boundary of the template T(w) be tangent to the critical direction. Figure 2.1 shows the case where the critical template is a continuous segment, and hence Tc (a)) is a convex set even though the entire template is highly nonconvex. The critical template is discontinuous when it is made up of the union of distinct segments. In this case T(o)) is not convex, but each of its member segments is a convex subset. With these definitions in hand we can proceed to state the following robust stability theorem. For ease of exposition we assume that the critical template T(co) is a convex set for all frequencies. This restriction can be relaxed through obvious modifications to account for each of the convex segments of T (o). Theorem 2.1 Let the nominal SISO system go(s)be subject to an uncertainty 8(s) : d. Then the uncertain closedloop system remains stable under unity feedback if and only if P ( ) < 1 1 + /c (g(Juo))) (2.18) The proof of the theorem is omitted since all details are given in [Latchman and Crisalle, 1995]. V Motivated by (2.18) we now propose the following definition of the Nyquist RobustStability Margin for SISO systems: kNC) =1+ A'C(goJc) (2.19) From Theorem 2.1 it follows that kN(CO) < 1 V0) (2.20) is a necessary and sufficient condition for robust closedloop stability. Furthermore, the quantity y = [kN(CO)]1 specifies the amount by which the uncertainty template should be increased or decreased to attain the limiting case of stability. The SISO critical direction result enables an exact assessment of stability for systems with directional uncertainty templates, without having to circumscribe the templates with larger circular uncertainties. The calculation of the critical perturbation radius p,(o) involves determining the intersection of a straight line (the critical direction) with a curve (the boundary of the template T(o)). Section 2.5 shows an example where the critical radius can be calculated analytically. It is also worth noting that the critical perturbation radius also gives a systematic methodology for determining the uncertainty weights which can be used as one of the inputs for various robust synthesis methods. Of particular interest in this regard are cases where there exists a Critical Weighting Function, We(s), such that its magnitude satisfies the interpolation condition 1Wc(s) = pc(0) Vo) (2.21) When (2.2 1) can be exactly satisfied, Theorem 2.1 leads to the familiar H_ condition WC(s) < 1+ go (s) (2.22) which is of the form (2.4). If (2.22) cannot be satisfied exactly but can be approximated using standard frequencydomain regression methods, then the resulting approximate weight can still be used as the basis for practical robustsynthesis design for systems with highly structured templates. It should be noted that Theorem 2.1 recovers as a special case the situation where the uncertainties are circular or diskbounded. The interested reader is referred to [Latchman and Crisalle, 1995] for further details on this point. In the following section we show how the critical direction approach and the Nyquist robuststability margin may be generalized to the case of uncertain MIMO systems. 2.4 The Critical Direction Method for MIMO Systems For MIMO systems we focus attention on the effect of uncertainties on the eigenvalue loci along the critical direction. This treatment has several desirable properties. First of all we derive necessary and sufficient robust stability conditions based upon eigenvalue relationships obtained directly from eigenvalue uncertainty templates. Thus, in principle this method yields a stability margin measure even in cases where singular value conditions fail to give necessary and sufficient stability conditions, provided that a method can be found to define tight inclusion regions for the uncertain eigenvalues. Furthermore by considering only the subset of the eigenvalue inclusion region that lies along the critical direction, we significantly reduce the computation involved in using the boundary of the eigentemplates (See Econtours [Daniel and Kouvaritakis, 1985] ) for stability assessment. Consider the generalized Nyquist plots in Figure 2.2 which (for illustrative purposes only) show two irregularly shaped eigenvalue inclusion regions (templates) about two of the n eigenvalues Xi(Go(jco)), i =1, 2,...,n. In analogy with the SISO case, for each of the n eigenloci we define at each frequency an associated critical line, defined as the directed line originating at the location of nominal eigenvalues 2Ai(Go(jao)) and passing through the critical point l+jO. Also in analogy with the SISO case discussed in Section 2.3 we define the following entities: 1. The critical directions di (jo) 1 + X i(Go (jco)) (.3 di(jc) := _1 l + Xi(Go(jco)) (2.23) which may be interpreted as the unit vectors which define the direction of each critical line. 2. The uncertainty eigentemplates Ti (o) :={2L(G(jco)) Xi (G(jo)) = 2Li(Go(jco) + A(jo)), A(s) e D} (2.24) 3. The critical eigentemplates T'fi(o) := {z E T(co) I z = /i(G,(jco)) +xdi(jo) for some ax e 9I 1(2.25) 4. The critical perturbation radii Pci(co):= max {ai z = Ai(Go (jo)) + cidi (jo) (= T (O)} (2.26) The entities defined above have identical interpretations to their SISO counterparts defined in Section 2.2. Hence, their geometrical interpretation can be obtained directly from the SISO Nyquist diagram given in Figure 2.1, provided that the nominal frequencyresponse plot go(jo) is substituted by an eigenvalue plot Go(jco). Obviously, the tight Econtour templates for unstructured and structured uncertainty descriptions utilized in [Daniel and Kouvaritakis, 1985; Kouvaritakis and Latchman, 1985] are respectively equivalent to the uncertainty templates TI(co). Using these definitions we can now state the following robust stability theorem. As in the SISO case, for simplicity of exposition we assume that the critical uncertainty eigentemplates Tj (o) are convex sets at all frequencies. Theorem 2.2 Let the nominal MIMO system G(s) be subject to an uncertainty A(s) e D. Then the uncertain closedloop system remains stable under unity feedback if and only if max 1+ = PC (G )) < 1 (2.27) i=1,2,...nit (Go(jO)) 1 + 2y(G0(jcv))< where pc () and Ac (.) are respectively used to denote the critical perturbation radius and the eigenvalue associated with the eigenloci resulting from the maximization over all i = 1,2,...n in (2.2 7). Proof. Assuming nominal closed loop stability and that the nominal and perturbed open loop systems have the same number of open loop unstable poles, the Generalized Nyquist stability criterion guarantees that the uncertain closed loop system is stable if and only if I(Go(jwo)+A(jwo)) # 1 Vi and Vco (2.28) Using the defining equation for the eigenvalues of G, (jco) + A(jco), namely det(Go (jco) z + A(jo))= 0 (2.28) at each frequency co we can parametrize the uncertain eigenvalues by z = Xi (Go(jco)) + piejoi (2.29) where Oi = Oi(co) varies in the range 0 < &i < 2r, and for each value of 0i the scalar pi = pi (co) varies in the range 0 pi < i where k i(Go (jco)) + jpiej i is an eigenvalue corresponding to the boundary of the eigentemplate 'i(co). Then condition (2.28) can be written as l+2Li(Go(jco))+ pieJ' # 0 Vi (2.30) Since the term 1 + Xi (Go(jco)) is fixed, the only possibility for violating the stability condition (2.30) is for piejei to be oriented along the critical direction, namely pieJoi = ai(co)di(jo) (2.31) where 0 < ao(o) < Ac. Thus, for robust stability we have the necessary and sufficient stability condition l+_ i(Go(jo))+oai(o)di(jo).# 0 Vi and Vo (2.32) Invoking definition (2.23) it follows that a sufficient stability condition is given by a<(co) 1l+ i(Go(jo)) <1, O ci(co) Furthermore, since along the critical direction a1 (o) < p_ (co) stability is ensured if Pci() 1 + 2j(Go(jo)) (2.33b) The proof is completed by taking the maximum over all i and noting that, because of the convexity assumption on the critical templates 'T (co) conditions (2.33a)(2.33b) are also necessary for stability. V In complete parallel with the SISO case kN (0))= Pei 0)) N( 1) :1+ (Go (jo))j (2.34) defines the MIMO Nyquist RobustStability Margin, and the scalar 7 = [kN(o)i]' again specifies the amount by which the uncertainty eigentemplate should be increased or decreased to attain the limiting case of stability. Consequently, from Theorem 2.2 it follows that a necessary and sufficient condition for robust stability is given by the Nyquistderived constraint kN(w)) < 1 V co (2.35) The Nyquist robust stability condition (2.35) provides an exact answer to the robust stability margin problem, much in the same vein as the margins k (o) and i(co). A most pleasing feature of the new stability measure is that the SISO version (2.20) and the MIMO version (2.35) of the Nyquist robust stability margin are formally identical. Much more importantly, the kN(o) formalism adheres to the classical generalization of SISO system properties to the MIMO case via the vehicle of the eigenvalues of the frequency response matrix. Relationship between km (o),(O)), and kN(a)) In light of Theorem 2.2 and the remarks above, it is clear that there are strong equivalences between the Nyquist stability margin kN(wO) and the margins k,, (co)) and j,(co). To see this recall that the definition (2.9) for kin(co) is equivalent to finding at each frequency the smallest destabilizing A such that 1+ ,i(M(jco)A(jco)) # 0 Vi (2.36) which is precisely the condition exploited in Theorem 2.2 to get the necessary and sufficient stability condition l + ,i(Go(jo)))+Oi(o))di(jc))# 0 Vi and Vo (2.37) For uncertainties which satisfy the condition of Lemma 2.1, from (2.12)(2.13) it follows that kN(CO) < 1 <*/ (CO) < 1 #* ki,(o) > 1 * sup p(MA) < 1 AED (2.38) In fact, even for uncertainties for which Lemma 2.1 does not hold, it is still true that for all A(s) E D l+A (M(jco)A(jo)) 0# kN(cO) The practical utility of Theorems 2.1 and 2.2 requires the computation of the critical perturbation radius pc(co). Several methods are presently being developed to exploit the critical direction theory to compute P (o) and hence kN (co) for the case of affine and multiaffine parametric uncertainties in SSO and MIMO systems. The results thus far are very encouraging, and have shown significant computational and algorithmic simplifications due to the focus on the critical direction. For the case of structured and unstructured MIMO uncertainties, tight eigenvalue inclusion regions may be obtained using the singularvalue based Econtours method [Daniel and Kouvaritakis, 1993; Kouvaritakis and Latchman, 1985], with similarity or nonsimilarity scaling deployed to remove or reduce conservatism. 2.5 Examples SISO Example This section presents an example of a SISO system with an uncertainty description motivated from statistical parameterestimation techniques. The uncertainty considered is an ellipsoidal parameterspace model which, in addition to its intrinsic merits, facilitates the direct analysis in the Nyquist plane permitting an explicit characterization of the Nyquist robuststability margin kN(W0),thus illustrating the application of the critical direction theory proposed in this paper. Furthermore, a discretetime representation of the model is deliberately chosen to emphasize the fact that the critical direction method and its attendant Nyquist robuststability margin concept are applicable to both the continuous and discrete domains. Let us consider a general SISO system model given by the discretetime transfer function q k H(z;p) = Yhkz k 1 (2.40) with q parameters defined by h = [hj,h2,.. .hq ]T. The nominal system, denoted H,(z), is obtained when h assumes the nominal values ho = h,h. h so that the real system is modeled as H(z) = Ho (z) + 6H(z). Let h in 9jq be such that h = ho + 5h. Then we define the uncertain ellipsoidal parametric uncertainty description M = h ho E Dh (2.41) Di, M E 9q s.t 6h TQh(%; Q1h =QhT >} (2.42) We argue that, in addition to the remarkable mathematical tractability which we shall show later, the ellipsoidal parametric description of uncertainties is quite natural in many applications and offers, in contrast to hyperrectangular descriptions, the further advantage of allowing the dependence among various system parameters to be taken into explicit account. Ellipsoidal models often arise quite naturally, as for example, whenever linear regression or leastsquares analysis is used in model estimation [Guzzella, et al., 1991; Kosut et al., 1992]. Clearly the nominal parameter vector ho defines the nominal transfer function Ho(z) = YXhzk. Consider now the frequency response H(e') k = I q kh o 0 . o T b l n i g t t e whereH(z) = Xhkz contains parameters ho= ,..ha1 belonging toth kI parameter ellipsoid. Under these conditions the following lemma shows that the parameter space ellipsoid maps precisely to ellipses at all frequencies except for co = 0 and co = ir. Let Re (z) and Im (z) represent respectively the real and imaginary components of a complex number z. Lemma 2.2 The parameter space ellipsoid defined by (2.42) maps to the elliptical uncertainty template T(co) = {z(co)= x1 (co) )= 0,1 xl(cO) xl0(o)< VR(O) QhVR(O)),CJ =0 c)= /}(2.43) where X(co)=[X(co) x2(co)]TE 2 Xo(Co):=[xO(o) X0(co)]T E {2 X((O:=[X(CO X2 IJ such that FRe He X(CO)= Im H(eJw)] = V(co)h Re H(eJC) V(o)h Im Ho(eiW) =) Q0o :=V(o)Qh1V(o)T e 12x2 F coso cos2co ... cosqo] [ sinco sin2o ... sinqco_ At co = 0 and co = 7r, matrix V(co) is singular and the uncertainty template is entirely real and given by T(o) {Z(CO) = X1 ()) 1 X2 (CO) = 0, 1 1(CO) 1(w) 1vR(O)QAvR(CO),CO= 0 0) } where VR(CO) := [cos CO cos 20 ... cos qo Proof. The proof makes use of a result given in [Guzzella, et al., 1985]. Details are omitted. V Our intention is to characterize the Nyquist robuststability margin for the unity feedback system comprised of the nominal discretetime system and its associated parametric uncertainty ellipsoid (2.42). Note that the robust stability of the closedloop system cannot be analyzed using the 12 results in [Tsypkin and Polyak, 1991] because their method is applicable only to continuoustime systems and to ellipsoidal uncertainty descriptions where the matrix Qh is diagonal (i.e., the principal directions are aligned with the coordinate axes). In fact, none of the conventional methods for analyzing robust stability of SISO systems appears capable of treating in a systematic fashion the case of ellipsoidal uncertainties Dh considered in this example. Theorem 2.3 Under the assumptions of nominal closedloop stability and that the nominal and perturbed systems share the same number of openloop unstable poles, the unity negative feedback system with openloop transfer function (2.40) and uncertain parameters (2.42) is stable for all h e Dh if and only if kN(CO) < 1 V 09 (2.44) where kN(CO)1 for co E (0,zr) N) T(c)Qd (o) (2.45) kN ()) dT for o = (0,r) (2.46) and d (CO) := ]1 V(co)h o 2.7 L0J (2.47) Proof. Inequality (2.44) is a direct result of applying the necessary and sufficient condition (2.18) of Theorem 2.1 in the Nyquist robustness margin form (2.20). It then suffices to prove (2.45) and (2.46). First consider the case where o E (0,7r). Using twodimensional vector analysis and the definitions given in Section 3, it is clear that the components of d,.(o) given by (2.47) are, respectively, the unnormalized real and imaginary parts of the critical direction d(jco) as defined in (4.1). Now from Lemma 2, at each co the frequencyresponse H(ejc) is located inside an elliptical template with center at the point Ho(eJo). Clearly, the elliptical template is a convex set because it is continuous along any ray. A fundamental result from twodimensional coordinate geometry gives the length of the line joining the center of an ellipse to the point of intersection along the vector dc (co) as d (co)d (co) PC (O) TCO)Q0 ld (w)(2.48) Recognizing that l + Ho (eJO)j = dT (co)dc(o) (2.49) it readily follows that kN(co)= =~ w 1 + Ho(ej)) (2.50) is of the form (2.45). Analogously, the proof for the singular cases  0, 7r is derived noting that from Lemma 2.2 it follows that at these frequencies pC(co) = 1vR(cO)T QhvR(co) from which (2.46) is readily established. In summary, for the case of ellipsoidal parametric uncertainties, the critical direction theory permits the exact characterization of the Nyquist robuststability margin, and the derivation of the exact necessary and sufficient condition (2.44) for robust stability. V MIMO Example Consider the MIMO nominal transfer function __30 3 1 (s + 1)(s + 2)(s + 3) (s +4) Gos)=_05_ 10 (s+5) (s+l) j and the associated elementbyelement elliptical uncertainty description. DE = {A(s) s.t. Aik(jo)) e Eik(jwo) VO)}\ (2.51) where the boundary dEik (JO) of each elliptical domain Eik (j) is given by the map dEik(jwo) = 2AikeJOi cosOik + j 2BikeJtiksineik, i,k=1,2 (2.52) and where the coefficients Aik and Bik are the elements of the frequencydependent matrices A(o) and B(co), and the majoraxis orientation parameters qik are the elements of the matrix 0()). The objective is to illustrate the calculation of the Nyquist robust stability margin kN (co) at the frequency 01 = 1.21, where A(col) = L0.1264 0.0359] = 0.0246 0.02780(0)1) = 2.1799 4.57591 1 0.0680 0.3185] L0.0537 0.2707 L1.4906 3.8865 Note that formulations based on the multivariable stability margin km(co) or the structured singular value (co) cannot currently offer an obvious approach for calculating the stability margins for the uncertainty description DE. At each frequency of interest we propose the following calculation procedure. First, the nominal eigenvalues 2 i(G,(j0))) and the critical directions di(O),i= 1,2 are computed. Second, for each eigenvalue, z satisfying zi = Xi + pidi is found. In the above expression, D is a diagonal similarity scaling matrix, P is the diagonal matrix where each diagonal element is the radius of the circle circumscribing the elliptical uncertainties. E1 and E2 are derived from the diagonalization of the uncertainty [Kouvaritakis and Latchman, 1985] and i represents the eigenvalues of Go. This yields the upper bound for Xi (G, + A) in the critical direction. Then an optimization is carried out over all Oik minimizing the function 2di  11 + +i(Go A) (2.53) Equation (2.53) gives the eigenvalue of (Go + A) in the critical direction, which is closest to the upper bound z. The basic principle is similar to that described in [Kouvaritakis et al., 1991]. At the selected frequency o), = 1.21, the nominal eigenvalues for the system considered in this example are Al(Go(jcol))=0.2005jl.2313, and 112(Go(jwO1))= 1.9332j2.4525 and have the associated critical directions d1(jco1)=0.5446+j0.8387 and d2(jC0)=0.7672+j0.6414. Then using the structured Econtour method for the fixed critical directions d1 (j1) and d2 (Jc01) yields the critical radii p,1 (o1) =0.3140 and P'2 (co) = 0.6445. Finally, from (2.27) and (2.34) it is found that k{E(c =max{ Poic(o) Pc2(0)1) max{0.2139,0.1686} (2.54) Hence, the Nyquist robust stability margin is kNE (ot1) < 1, and it is concluded that the system satisfies the necessary and sufficient condition for robust stability at this particular frequency. Figure 2.3 shows a Nyquist diagram with the eigenplots for the nominal system considered in this example. The eigentemplates T1(0)1) and T2(c)1) are not shown in the figure because for the purpose of determining the Nyquist robuststability margin it is not necessary to calculate the entire templates. However, for reference the figure shows the Optimal Dscaling bounds obtained using the scalings given in [Kouvaritakis and Latchman, 1985]. The actual eigentemplates are bounded by these Optimal Dscaling contours. The points denoted zl(Jwl) and z2(J)l) are on the boundary of the critical templates T,1 (o1) and T,2 (co1), respectively, and were obtained using the Econtour method. Note that for this example the Optimal Dscaling bounds provide a good estimate for the boundary point z, (j)i), but yield a poor estimate for z2 (J0)y). Note also that the critical eigentemplates T,, (o)1) and T,, (o),) can be readily identified from the figure as the straightline segments joining each nominal eigenvalue ,i(G, (j0)1)) with its corresponding boundary point zi (J01). 2.6 Conclusions In this paper we have proposed the Nyquist Stability Margin, kN (o) as a new metric for robustness analysis of SISO and MIMO systems. The definition of new stability margin is based on the Critical Direction Theory which provides a single framework for robustness analysis for SISO and MIMO systems. The analysis methodology makes direct use of the generalized Nyquist diagram, and in contrast to the prevalent approaches which emphasize singularvalue perturbations, it focuses attention on eigenvalue perturbations. The main advantage of the criticaldirection theory is that it provides necessary and sufficient conditions for robust stability in the presence of highly structured uncertainties with phase and directionality constraints. Other approaches to these problems either do not have the inherent capability to deal with these structural details, or the directionality and phase constraints are deliberately ignored, giving rise in either case to sufficientonly conditions such as those associated with singularvalue theory. On the other hand the new method explicitly exploits the detailed directionality and phase constraints of the uncertainties as these are manifested in the frequency domain uncertainty templates. Thus, the new method is applicable to a number of uncertainty descriptions for which other methods fail, such as the case of elementbyelement ellipsoidal uncertainties in the transferfunction matrix, and other uncertainty descriptions with highly directional frequencydomain templates. The new criticaldirection technique opens up new avenues for robustness analysis and could lead to novel approaches for robust control synthesis. There is significant promise for fruitful new results in this area where the computational efforts are concentrated on a single and welldefined frequencydependent directed line. Nomenclature for Chapter 2 s Laplace variable dc(jo) direction in the complex plane g0(jo) Nominal SISO transfer function g(jo) Uncertain SISO transfer function Go(jo) Nominal MIMO transfer function G(jco) Uncertain MIMO transfer function kN(o)) Nyquist robust stability margin km (co) Multivariable robust stability margin M(jco) Augmented system matrix W (s) SISO weight Greek Letters w) Frequency Aji(G(jo)) Tth eigen value A(s) Uncertainty description PC (co) Critical perturbation radius Y(wo) Structured singular value dF(.) Maximum singular value p(.) Spectral radius T(co) Eigen template T, (co) Critical eigen template 31 critical line Imz g(jo)) ____' Re g(jco) go,(J),)+pc(t,)d(J ) (d Figure 2. 1. An irregularly shaped uncertainty template at frequency 0), (shaded area) and its critical perturbation radius p,(w1). The critical template 'i.(o)) (solid line) is the subset of template points lying on the line segment with end points g0 (w1) and g0 (co) + p, (o1 )d(jcol). d(Jc1) Im (Go(jo)) /LI (G, (jco)) Figure 2.2. Nyquist plot of the eigenvalues of a 2x2 MIMO system showing two irregularly shaped eigen templates Tl (wo) and T2(coj) (shaded areas) at frequency CO = col. 3o ...... 4 5 . . . 2 1 0 1 2 3 4 5 Re 1I(G(ja)) Figure 2.3. Nyquist plot of the MIMO Example. The contours represent the optimal Dscaling bounds for the eigentemplates T1 (col) and T2 (o1) at the frequency o) = 1.21 The points zi(o1) and z2(co1) lie on the intersection of the boundary of their respective eigentemplates with the critical direction CHAPTER 3 A NEW PERSPECTIVE ON COMPUTING ROBUST STABILITY MARGINS FOR COMPLEX PARAMETRIC UNCERTAINTIES 3.1 Introduction Considerable attention has been given to the problem of assessing robust stability and computing stability margins for SISO systems with parametric uncertainties. It is now well known that affine parametric uncertainty problems result in a rankone 9 formulation. While the general robust stability margin computation problem is known to be NPhard [Braatz et al., 1994], its rankone counterpart results in a convex optimization problem that can be readily computed, and can be often derived as an explicit analytical expression. This aspect of the problem is amply documented in the literature [Qiu and Davison, 1989][Hinrichsen and Pritchard, 1992] [Chen et al., 1994a][Chen et al., 1994b]. Of particular interest are references [Chen et al., 1994b] and [Young, 1994], which establish connections between frequencybased stability conditions and polynomial conditions in the spirit of Kharitonov's theorem [Kharitonov, 1979]. This chapter focuses on the complex parametric uncertainty problem from the perspective of the recently proposed critical direction paradigm [Latchman and Crisalle, 1995] [Latchman et al., 1997]. The approach followed in critical direction theory consists of first mapping the parameter space uncertainties into the Nyquist plane in the form of value sets or uncertainty templates, and then invoking specific directionality and phase properties to infer stability. The method is based on recognizing that at any given frequency on the Nyquist plane, the only uncertainties of relevance to the stability analysis are those which lie on a well defined critical direction. In fact, from a stability perspective all points on the uncertainty template that do not lie in the critical direction can be ignored. This approach makes it possible to solve problems with highly directional uncertainties, such as SISO elliptical uncertainties, for which, at this time, approaches based on the more traditional structured singular value methods are not obvious. This chapter addresses the case of elementwise complex parametric uncertainties where each individual uncertainty lies in a highly structured domain that is convex along any line from the origin. This is a mild convexity condition, since the actual domain could be highly nonconvex. Exact analytical results are derived for geometrically simple uncertainty sets such as ellipses and rectangles. For the special case of diskbounded affine uncertainties, the approach recovers in a transparent manner relevant results by previous methods [Hinrichsen and Pritchard, 19921][Chen et al., 1995b] . Section 3.2 of the chapter provides background information and mathematical preliminaries, presents the uncertainty descriptions considered, and gives a brief review of the main elements of the critical direction theory. The main results are presented in Section 3.3, and connections with the classical approach based on MD structures is explored in Section 3.4. Concluding comments are given in Section 3.5. 3.2 Background and Preliminaries Consider a SISO system with an openloop transfer function g(jco,q) = n(jco, q) (3.1) d(jco, q) where n(jco,q) and d(jco,q) are complex polynomials that depend on an uncertainty vector q = [q(jw),q2(00) j E Q c (3.2a) whose components qk (jco) belong to complex domains Qk, i.e., qk(jwo) G Qk c C, (3.2b) and hence the uncertainty domain is the artesian product space Q = Q1 x Q2x... x Qm" Figure 3.1 illustrates a case of a radially convex domain Qk. Note that the entire domain is nonconvex. The uncertainty class considered in this s are described by elementwise uncertainty sets Qk, k = 1, 2, ..., m, that are closed under contraction (i.e., are radially convex), and are unconstrained in phase. Letting 9Qk denote the boundary of domain Qk, then the boundary elements dqk(jo) E dQk are of the form dqk= 7k(CO, Ok)ejk, k = 1, 2 ..., m (3.2c) where 0 < 0k < 2r is an unconstrained phase angle, and Yk (CO Ok) = qk(jo)) 0 is the frequency and phasedependent magnitude of the boundary element. Im qk Qk r(t) y Qk Re q A, Figure 3.1 Radially convex Figure 3.2 Unity negative uncertainty domain Qk for feedback configuration. complex component qk The elements of the uncertainty vector q are assumed to appear in an affine fashion in (3.1), i.e., the uncertain model is of the form m n(jc, O) + nk (jco)qk (jo) g(jco,q)= k=1 m d(jo, O) + dk (jCo)qk (jCO) k=1 (3.3) Under the unity negativefeedback arrangement shown in Figure 3.2, the characteristic polynomial for system (3.3) is given by the polynomial sum p(jo, q) = n(jco,q) + d(jw, q) (3.4) It follows that the robust stability of the closed loop with respect to the uncertainty set Q can be evaluated by studying the uncertain polynomial (3.4). It is convenient to decompose (3.4) into its nominal and uncertain parts p(jco,q)= po(jco)+S(jco,q), where po (jo) = n(jco, 0) + d(jco, 0) (3.5a) and m 3(jco,q):= Ypk(jo))qk(jCo) (3.5b) k=1 where pk(jo)) := nk(Co) + dk(jco) = pk(jco) ejePk (jC) (3.6) The variable 0Pk (jco) is simply the frequencydependent phase of polynomial Pk (ico). Hence, p(jco,q) = po(jco) + Y pk(jO)qk(jco) (3.7) k=1 Definition. The Parametric Robust Stability Margin is the entity defined by the minimization expression a (co)= min 1OEp(jco,xq) for some q E Q} a + (3.8) The parametric robust stability margin is a nonnegative real scalar that can be interpreted as the minimal magnification a*(co)> 1 or contraction a*(co) < 1 of the uncertainty set Q that brings the closedloop system to the edge of stability. Geometrically, the parametric robust stability margin represents the minimum tolerable blowup factor. Note that the parametric robust stability margin is defined for each frequency. Specific Parameter Uncertainty Descriptions Considered Three types of uncertainty descriptions of the form (3.2c) are considered, namely the (i) circular, (ii) elliptical, and (iii) rectangular elementwise complex domains illustrated in Figure 3.3. (a) hn qk (b) Im1qk (c) Imqk )k Bk Ak Ak Ak \ ~kBk Reqk > Reqk Re qk Figure 3.3 Three types of elementwise complex parametric uncertainty regions with unconstrained phase: (a) circular, (b) elliptical, and (c) rectangular domains. Circular Uncertainties As depicted in Figure 3.3a, this is the familiar case of circular (diskbounded) uncertainties of radius Ak (c) where Yk(coOk)>Ak(co) Vok (3.9) Elliptical Uncertainties Let Ak (co) > 0, Bk (co) > 0, and Ok (co) respectively represent the frequency dependent semimajor axis, semiminor axis, and the orientation with respect to the real axis of the an ellipse centered at the origin of the complex plane as shown in Figure 3.3b. Then the boundary of each elliptical domain is given by the map dqk (cO) = Ak (co)ejik (0))cos( Ok Ok (co)) + jBk (co)eJ 'k (C0)sin( Ok Ok (cv)) from which it follows that Yk(CO, Ok) =[Ak(Co)2COS2(Ok k(o)) + Bk(cO)2sin2(Ok k(c)))]2 (3.10) Obviously, the circular uncertainty case (3.9) is recovered by (3.10) after setting Ak(co) = Bk(co). Rectangular Uncertainties Let Ak((o) > 0 and Bk(o) > 0 respectively represent the frequencydependent half width and the halfheight of a rectangle centered at the origin of the complex plane as shown in Figure 3c. Consider the orientationphase definition k (co) := arctan Bk (co) (3. 1 a) Ak(O) and the associated phase sets 0ak :=[0, Ok) U [k' )r + k) U"[21r Ok,27r] (3.11b) '9k:= [kIrO)U7r+ O,2rk)(3.1 Ic) The shaded and the plain areas in the Figure 3.3c corresponds to Obk and Oak, respectively. Then an analytical expression for the magnitude of any point on the boundary of the rectangular regions can be compactly written in the form [ Ak (w0) for Ok E Oak "k(oek) = COS(0) (3.12) 7k(Colt~k)Bk (() for Ok E= 'bk sin(Ok) 3.3. The Critical Direction for a Characteristic Polynomial The critical direction theory proposed for rational systems in [Latchman et al., 1997] can be readily modified for the case of polynomials through the adoption of the definitions given below. (i) The Critical Direction d(jo)(:= jO) e_ jec(o)) (3.13) P (J) _= is the unit vector which defines the direction from the nominal point Po(jw) towards the origin. The critical direction is uniquely identified by the critical phase angle e, (co). (ii) The Uncertainty Template (or value set) T(o):= {p(jco,q) e C Ip(jco,q) =po(jco) +8(jco,q), q EQ } (3.14) (iii) The Critical Template Tc (o) := { p(jo,q) T(co) I p(jco,q) = po(jo) + r d(jco), for some r e 91+}(3.15) (iv) The Critical Perturbation Radius p,(co):= max {r Z = po(jco) +rd(jo) T (co)} (3.16) (v) The Nyquist Robust Stability Margin kN(0o) := PC ()) (3.17) Note that the Nyquist robust stability margin kN(o) defined in (3.17) is a metric that characterizes the distance on the Nyquist plane to the point of instability (i.e., the point l+jO). In a complementary fashion, the parametric robust stability margin a*(Co) defined in (3.8) characterizes the "distance" on the parameter space Q to the limit of stability. 3.4. Main Results This section develops a technique for deriving analytical expressions for the parametric robust stability margin. The approach makes use of the criticaldirection theory elements presented in Section 3.2. The developments are specialized here for the case of uncertain polynomial systems. Lemma 3.1. Consider the uncertain characteristic polynomial (3.4), and assume that nominal stability is attained under unity feedback, and that the critical templates T(wO) are convex at all frequencies. Then the uncertain system remains stable for all parameter uncertainties q ( Q if and only if kN(t))< 1 VO) (3.18) Proof. The proof follows from an application of the zero exclusion principle and is analogous to the development given in [Latchman et al., 1997] for transfer functions, and details are omitted for brevity. V Note that Lemma 3.1 is quite general and holds for multiaffine, and complex and/or real uncertainties. The major issue is the calculation of the critical radius p,(o)) needed for the characterization of kN(Co) in (3.17); however, as shown below in Theorem 3.1, the computation of the critical radius for complex affine parametric uncertainties is particularly straightforward. Theorem 3.1. Consider the uncertain system (3.3) with the uncertainty description (3.2a)(3.2c), and its corresponding closedloop uncertain characteristic polynomial (3.7). Then the critical perturbation radius is given by the expression Pc())= I Pk(Jfl)) Yk(C,OC(0) 0,k (c)) (3.19) k=1 Proof. The points p(jo),q) on the value set T(co) that belong to the critical template T, (co) must satisfy the condition p(jco,q) = po(jco) + 8(jco,q) = po(jco) + rd(jco) (3.20) or i YPk (jo)) qk (o)) = reJec) (3.21) k=1 Due to the radial convexity of the uncertainty set, it suffices to consider only boundary uncertainties dqk(CO); then, utilizing (3.2c) the preceding equality can be rewritten in the form Pk (ico)) Yk (c, Ok)eJ(Ok +OPkOc(cv)) e (3.22) k=1 From the definition (3.16) it follows that p,(o) is equal to the maximal value of r in (3.22) over all possible uncertainty phases 0 < Ok < 21r, i.e., p~cO)= 0 max k(jCO) )k(c ,Ok) eJ(Ok +01k e(co)) (3.23) Clearly, the maximum is obtained for Ok = (c) Op (no), leading to the expression (3.19). V Theorem 3.2. Consider the uncertain system (3.3) with the uncertainty description (3.2a)(3.2c), and the corresponding critical perturbation radius pc(CO). Then the parametric robust stability margin is given by a *(C) Po (Jo) (3.24) Proof. From equation (3.4) p(jco,q) = po(jo))+ IXPk(jco) qk(jco), q eQ (3.25) k=1 and m p(jc9,aq) = po(jco)+ t XPk(jcO) qk(jco), q e Q (3.26) k=1 Note that the map (3.25) yields the uncertainty template T(co) defined in (3.14). As for the question of stability, note that only those uncertainties that give rise to points p(jco,q) that are aligned with the critical direction are relevant. Then, using the definition (3.13) for the critical direction, all the points of relevance for the stability analysis must be of the form p(jco,q) = p,(jo) + rd(jco) (3.27) Equating (3.25) and (3.27) yields m mYPk(jCo) qk(jo) = rd(jco) (3.28) k=1 Note that if qk E Q is such that p(jco,q) is aligned with the critical direction, then for any scalar a > 0, (3.26) implies that p(jo,aq) also is aligned with the critical direction. Hence, making use again of the definition of the critical direction given in (3.13), and from (3.25) and (3.27) it is concluded that p(jco, a q) = p,(jco) + a r d(jco) = d(jo)(arp,(jo) ) (3.29) from which if clearly follows that p(jo,aq)=O a Po (jo) (3.30) r Since by definition the maximum possible value of r is p,(co) ,it follows that a*(co) is obtained when r = p,(co) and hence a* (co) = po(jco)l/p,(co). V Combining Theorem 3.1 and Theorem 3.2, it follows that the parametric robust stability margin (3.8) can be written in the form )= p (JCO)l (3.31) X Pk(JCO) Yk(0), O,(CO)OPk (co)) k=1 Hence, (3.31) gives an analytical expression for the parametric robust stability margin for all complex elementwise parametric uncertainties that are closed under contraction of the form (3.2a)(3.2c) for systems with the affine structure (3.3). 3.5. Parametric Robust Stability Margins for Highly Structured Uncertainties The connection between the Nyquist robust stability margin kN(aO) and the parametric robust stability margin a* (wo) is the reciprocal relationship =(co) = 1 (3.32) kN (CO) which follows from Theorem 3.2 and the definition (3.17). Invoking Lemma 3.1, it then concluded that the closedloop system is robustly stable if and only if a**(co) < 1, VcO. This result holds for all elementwise complex parametric uncertainty descriptions that are closed under contraction. This section focuses on obtaining analytical expressions for parametric robustness margin for the three specific types of uncertainty descriptions depicted in Figure 3.3 Theorem 3.3 Consider the uncertain system (3.3) with the characteristic polynomial (3.7), and the circular, elliptical, and rectangular complex parametric uncertainty descriptions whose elementwise magnitudes are respectively given by (3.9), (3.10) and (3.12). The robust stability margins for each of these descriptions are given by the analytical expressions: (3. i) For the case the circular uncertainty description a (0)) = m P() (3.33a) I Ak(wO) pk(jo)l k=1 (ii) For the case the elliptical uncertainty description (CO) = [po (jco) I[Ak ())2Cos 2 (OC(0)_pk (O) Ok((O))+ Bk(W)2sin2(Oc(O))Ok (cO) O k())) 12 Pk(jo)[ k=1 (3.33b) (iii) For the case the rectangular uncertainty description a* ( ko) = [(J) ) iOe ) (3.33c) 'n Ak (0)) IN (J C0) Bk (0)) PkJA 0 CO] k=1 jcos(Oc(CO)OPk (CO) ) o +sin(O (CO)) ON (CO) ) where (ak = 1 if Ok G (ak, otherwise 8ak = 0, and where 3bk = 1 if Ok E (lbk, otherwise abk = 0. Proof. Equations (3.28b)(3.28a) simply follow by substituting the expressions for the phasedependent uncertainty magnitudes (3.9), (3.10), and (3.12) into the analytical expression (3.27) for the parametric robust stability margin. V The generalized structured singular value approach [Chen et al., 1995b] has previously been used to derive the result (3.33a) for the case of circular uncertainties. In principle, the general formalism presented in [Chen et al., 1995b] can also be used to derive the results (3.33b) and (3.33c); however, the critical direction theory extracts exact analytical results through very simple and straightforward geometric arguments. 3.6 Connections with the Classical MA Formalism In this section expression (3.31) for the parametric robust stability margin is rederived utilizing the framework of the classical M A formalism where robust stability is characterized by a determinantal constraint. The approach follows wellestablished developments, except that we simplify the derivations by invoking a criticaldirection concept, namely, that a specific orientation in the Nyquist plane is the only direction of relevance for stability analysis Lemma 3.2. The unity negativefeedback configuration of the SISO uncertain system (3.3) with affine uncertainty elements can be transformed into an equivalent MIMO M A structure via a linear fractional transformation [Zhou et al., 1996] where the uncertainty matrix A(s) is diagonal and the structural matrix M = (Mik) is of rankone and has rational elements given by Mik(s) Pk(S) = 1, 2, ..., m; k = 1, 2, ..., m. po(S)' Proof. The proof is an exercise in blockdiagram algebra. Details are given in the Appendix. V For the MIMO M A structure in question it is typical to define the parametric robust stability margin in terms of a determinantal condition, as follows a *(co) min {a det(I + aM(jco)A(jco))=O } (3.34) a 9 + which is formally analogous to the definition of the parametric robust stability margin for a SISO system given in (3.8). In fact, the theorem below shows that the definitions (3.8) and (3.34) lead to identical expressions for the parametric robust stability margin for the case where the uncertainties appears affinely in the model. Theorem 4. Consider the system (3.3) with its associated affine uncertainty description (3.2) and suppose that the nominal system is stable under unity negative feedback, and that the nominal and the uncertain systems have the same number of open loop unstable poles. Then the parametric robust stability margin at each frequency is given by = P (Jco) (3.35) 1 Yk(O C~O) p,(0)) 7r) jk0)l kI Proof. From Lemma 3.1, the uncertain system with unity negative feedback can be transformed into an equivalent MIMO M A structure. Using standard arguments, the robust stability condition for the transformed MIMO system is given by the determinantal condition det(I M(j o)A(j0o))) # 0 Vco (3.36) Introducing the parametric robust stability margin a c 9 + interpreted as a blowup factor for the uncertainties, the determinantal condition is rewritten as det(I aM(jo)A(jco)) 0. Since from Lemma 3.1, matrix M is of rank one, it is straightforward to establish the inequality det(I aM(jco)A(jcO)) = 1 + I Pk(j(0) qk (jw)] (3.37) Since the aim is to find the minimum scalar a ( 3Z+ that makes (3.37) identically zero, equating (3.37) to zero and solving for a yields a MPk(JO) k(co,Ok)e k+Ok) (3.38) k=1 Clearly, the minimum real nonnegative value of (3.37) will be realized when all terms in the summation in the denominator (3.37) are aligned with respect to each other and with respect to the numerator, i.e., using the notation po (jo) = p, (jco) ejepI and recalling the fact that the uncertainty phase is 0k unrestricted, the minimum is attained when Ok = Opo(o))Op, (co) +7r, k = 1, 2,...,m (3.39) From the definition (3.13) it follows thatOp (co) = 0,(co) r, hence the minimizer is realized at Ok = 09 ((o) ON (Co), k = 1, 2,.., m (3.40) Substituting (3.40) into (3.38) gives the minimal value a = a*(co) which is of the form (3.30). V 3.7 Conclusions In this chapter we demonstrated the significance and ease of application of the critical direction theory to the complex affine uncertainty problem. The development is simple and intuitive and for the special cases of interest recovers the exact results by the previous approaches. The case of multiaffine, complex and/real uncertainties may also be handled using the critical direction arguments. However, for all these cases the computation of the critical radius remains the major theoretical challenge; fortunately, for a number of uncertainty descriptions, such as the ones considered in this chapter it is possible to obtain analytical expressions for the critical radius. Nomenclature for Chapter 3 S Laplace variable q Uncertainty vector n(jo,q) Uncertain numerator polynomial d(jo),q) Uncertain denominator polynomial p(jco,q) Uncertain characteristic polynomial d, (jo) Critical direction in the complex plane kN (Co) Nyquist robust stability margin k, (co) Multivariable robust stability margin Greek Letters a (o) Blow up factor 0) Frequency P, (a)) Critical perturbation radius Y(co) Structured singular value dF(.) Maximum singular value p(.) Spectral radius T(o) Eigen template T, (co) Critical eigen template CHAPTER 4 VARIABLE STRUCTURE CONTROL DESIGN FOR REDUCED CHATTER IN UNCERTAIN STATE DELAY SYSTEMS 4. 1 Introduction The study of uncertain statedelay and inputdelay systems has received much attention by researchers in the past decade [cf. Zaveri and Jamshidi, 1987 and Hale and Lunel, 1993 for background and extensive references]. One reason for this interest is that time delay is encountered in various engineering systems and can be the cause of instability. Another reason is that practical control systems unavoidably include uncertainty or disturbances due to modeling or measurement errors and other approximations. Several authors have investigated the problem of stabilization of uncertain statedelay systems. The main strategies for stabilization of delay systems include designing state or output feedback controllers (using pole assignment, Lyapunov or LQ theory), and variable structure controllers (VSC). Cheres et al., [1989] construct a minmax controller from the knowledge of the upper bound on the delay. Shen et al., [1991] propose a memoryless linear state feedback based on Riccati equation approach [also see Wang and Lin, 1988]. Phoojaruenchanachai and Furuta [1992] also construct a memoryless state feedback, but consider a larger class of systems. Trinh and Aldeen, [1994] consider interconnected systems and propose memoryless state feedback controllers to stabilize the uncertain state delay systems. Niculescu et al., [1996] design memoryless state feedback controllers to stabilize an uncertain state delay system with constrained input via ricatti approach. Recently, sliding mode control has been employed as a tool for stabilization of uncertain statedelay systems. The sliding mode control possesses wellknown features that make it very attractive for control systems. These include fast response, insensitivity to parameter variations, and decoupling design procedure, among others. Recently Shyu and Yan [1993] and Oucheriah [1995] have used VSC to guarantee the stability of uncertain delay systems, deriving sufficient conditions that depend on the size of the delay. However, the ShyuYan method suffers from the disadvantage that the control synthesis procedure involves unknown matrices; hence the control law can not be implemented. Oucheriah proposes a dynamic switchingsurface control scheme based on pole assignment. Both of the above approaches involve strong discontinuous control across the switching surface to overcome the effects of the uncertainty, and hence suffer from severe chattering. Chattering is undesirable because it might excite the unmodeled highfrequency components of the system, and it may lead to premature wear or tear of the actuators. The recent technique of perturbation compensation has been shown to provide effective chatter suppression for delayfree systems [Kim, 1992; Chan, 1996]. This method exploits the specific structure of VSC to estimate and compensate for the effect of perturbation, and thus results in reduction or removal of chattering. This chapter proposes a VSC design methodology to robustly stabilize an uncertain statedelay system with nonlinear and possibly mismatched uncertainties, utilizing the concept of perturbation compensation. Sufficient robust stability conditions are derived which are independent of the size of the delay; furthermore no matching conditions on the uncertainty are assumed. The main advantages of the proposed approach are that (i) a realizable control law is obtained which does not need the bounds on the system uncertainty, (ii) chattering reduction/elimination is achieved, (iii) delay independent sufficient robust stability conditions which can accommodate uncertainty in the delay are derived, and (iv) guidelines are given for the design of the control matrix involved in the specification of the switching function. Finally the results are illustrated with an openloop unstable example. 4.2 Sliding Mode Control Design Consider an uncertain statedelay system subject to uncertainty and/or external perturbations governed by the equations i(t) = Ax(t) + Adx(t h) + Bu(t) + f(x(t), t) + fd (x(t h), t) x(O)=qO) 0e[h, 0] (4.1) where x(t) r R' is the state vector which is assumed measurable and has the initial state x(o) = xo, u(t) e Rm is the input vector; A e RlX and B E Rnxm are constant matrices; O(t) is a continuous vectorvalued initial function; and h > 0 is the time delay. The vector functions f(x(t),t)( R' and fd(x(th),t)c R' represent nonlinear perturbations that depend on the current state x(t) and on the delayed state x(th) of the system, respectively. It is assumed that the modeling uncertainties satisfy the bounds f(x(t),t) < k Ix(t)ll (4.2) lfd (x(th),t)JJ k llx(th)ll where k and kd are known positive real constants, and 11 II is any vector norm or corresponding induced matrix norm. Notice that no matching assumptions are used. It should be noted that these bounds are required not for controller design, but for robust stability analysis. Two definitions are useful for the ensuing developments. The matrix measure [Vidyasagar, 1993] is defined as the function #:C"' > R" /(A) = lim III + eAll 1 E)O+ E The matrix measure is also known as the logarithmic norm. For an excellent discussion on logarithmic norms, see [Strom, 1975]. Also, a class of matrix pairs (A, B) is introduced for the stabilization problem, defined through the set membership U := {(A,B) 3 F E Rnxm: u(A+ BF) < 0} (4.3) where U.I clearly represents the class of systems for which there exists a statefeedback law u(t) = Fx(t) such that !u(A + BF) < 0. The objective is to design a control law that assures reachability of the sliding mode within a finite time and asymptotic stability of the system in the sliding mode, even in the presence of uncertainties and/or external disturbances. The design involves two major phases: The first phase is the selection of a switching surface manifold that has desirable dynamics, and the second phase is the determination of an appropriate control law that guarantees the existence and reachability of the sliding mode and the effective rejection of disturbances. Switching Function and Control Design This section discusses the selection of an appropriate switching function. In the sliding mode the system must not only be stable but also be robust to any kind of bounded perturbations as specified in (4.3). The procedure is as follows. First, select matrices F and Fd such that matrix A = A BF is a stable matrix, and that matrix Ad = Ad BFd satisfies AdJ < AdJ At this point we anticipate that the role of Fd is to reduce the effect of delay, since the stability conditions derived later in this chapter involve AdJ Using the matrices described above the system equation (4.1) can be rewritten in the form i(t)=( + BF )x(t)+ (Ad + BFd )x(th)+ Bu(t)+ f(x(t),t)+ fd(x(th),t)(4.4) An integral switching function [Fernandez and Hedrik, 1987] is adopted in the modified form proposed by Shyu and Yan [1993] namely S(t) = Cx(t) f{C~ix(z)+CAdx(zh)}d'r 0 (4.5) where matrix C c R" is a control design matrix that is chosen such that the product CB is nonsingular. The reason for choosing such a switching function instead of a traditional combination of the states is that, this choice results in a desirable switching dynamics. This will be clear after the design of the controller. The goal is to find a VSC controller that drives the system to the switching surface and ensures a sliding mode condition. The controller structure adopted is of the form U(t) = Ueq(t) + Ur(t)+ up(t) which contains three terms. The first term in turn is of the form Ueq(t) = (Fx(t) + Fdx(th)) (4.6) and thus represents the modelbased equivalent control for the nominal system in the absence of perturbations, obtained by setting i(t) = 0 [DeCarlo et al., 1988]. The second term Ur(t) = (CB)"1 (Ks(t) +d sgn[s(t)]) (4.7) is the reaching control law where the symmetric positivedefinite matrix K and the scalar d > 0 are design parameters. The final term is intended to cancel the effect of perturbations on the dynamics of the switching function [Chan, 1996] and is given by up(t) = (CB)Tp(t) (4.8) with Tp(t) =(t) CB (Ur(t) +up(t T)) (4.9) where Tp(t) is designed to reject the effects of the perturbation signal. In order to avoid an algebraic loop in calculation, a deliberately delayed value up(t T) is used. The value of the computational time delay T is chosen to be small. As long as the variations in the system dynamics are slower than the computational delay, this will be good approximation. However, it should be admitted that regardless of the value of T, up (t) up (t T). However, as long as the estimation error is less than the actual perturbation, we only gain by using perturbation compensation. Also a finite difference scheme is used to estimate i(t). This could also be a source of errors if the measurements are noisy. The ideal VSC approach assumes that it is possible to switch infinitely fast across the switching surface. The phenomenon of nonideal but fast switching is labeled as chattering [DeCarlo et al., 1988]. Interestingly, the name comes from noise generated by the relays. Chattering is undesirable for two reasons: (i) It might excite the unmodeled high frequency plant dynamics which could lead to unforeseen dynamics. Note that in certain applications it is not of concern as the frequency of switching is far above the structural frequencies of the mechanical system [DeCarlo et al., 1988]. However, in most applications this will be a major concern, (ii) It might lead to the premature wear and tear of the actuators. It should be noted here that it is the discontinuous control, which in turn leads to chattering, enables one to guarantee the robustness features of sliding mode control. In a sense, we gain robustness but pay for it with chattering. To alleviate chattering, many approaches have been proposed in the literature. Here, we give a brief review of the past efforts. Although, they have not been used for sliding mode control of statedelayed systems, they are nevertheless useful as a concept. One of them is the "boundary layer approach" [see Slottine and Sastry, 1983] which uses the continuous control law and time varying sliding surfaces. Although this succeeds in eliminating chattering, it is no longer possible to assure the asymptotic stability of the system. Rather, one has to be content with uniform ultimate boundedness of the trajectories within a neighborhood of the origin depending on the boundary layer. Ambrosino et al., [1984] cascaded a lowpass filter after the VSC to attenuate the high frequency switching component. In this case chattering is alleviated at the expense of robustness. Espana et al., [1984] propose a zone in the vicinity of the switching surface in which the feedback gain is adjusted in order to avoid chattering. Morgan and Ozguner [1985] propose to decrease the control gain in order to minimize chattering, but the disadvantage is the increased time that it takes to achieve sliding mode. Chang et al., [1990] proposed an adaptive alleviation algorithm which requires recursive prediction at each step. Their procedure involves computing the derivative of the states, but they assume that they are either available or can be computed without errors. Elmali and Olgac [1992] do an online estimation of the effect of the perturbations based on the measurements of the state dynamics. They assume that the error in estimation is bounded within a scalar multiple of the estimation. Also see an experimental verification of their approach in Elmali and Olgac [ 1996]. Chan [ 1996] proposed an perturbation compensation approach utilizing the specific structure of the variable structure control. The approach seeks to estimate the effect of perturbations on the switching dynamics with the assumption that the relative error in the estimation of the switching dynamics is bounded and known. One of the major advantages of the perturbation compensation schemes is that one need not have a priori bounds on the uncertainties for controller design. Note that with the application of the control law (4.6) (4.9) and the choice of the switching function (4.5), the derivative function (t) is given by (t) = CB (Ur (t) + up (t)) + C(f(x(t), t) + fd (x(t h), t)) i(t)= (Ks(t) + d sgn[s(t)])+[C(f(x(t),t)+ fd(x(th),t)) Tp] It is clear that i(t) is driven by the estimation error, and the purpose of control is to counter the effect of perturbations on the switching surface. The usual approach is to overpower the uncertainty by discontinuous control. For example, in this case the value of d should be chosen such that d > Cl(k Jlx(t)l + kdllx(t h)11). In fact, a choice of this type is made by [Oucheriah, 1995]. For proving the reaching condition, this is indeed a clever selection of parameters. However, there are some significant problems with this choice. Firstly, we are assuming it is always the worst case that is affecting the system. Secondly, there is no method to estimate even conservatively, the norm of the states when the system oscillates near the switching surface. Hence, while we are guaranteed to suffer from chattering, we have no available worst case upper bound. In contrast, the effect of up (t) is to ease the burden on the discontinuous control by canceling the effect of perturbations on the switching dynamics. The discontinuous control is now utilized, to combat the residual error in estimation rather than the whole perturbation. Irrespective of the actual value of the perturbation, as long as the estimation scheme works well, this scheme will outperform the conventional VSC. One point needs to be clarified here. Estimating i(t) is in fact, equivalent to saying that we are estimating the perturbation. Our contention here is that i(t) can be estimated from the measurable states of the system via (4.5). It goes without saying that we are introducing a new parameter here, namely the bound on the estimation error. Apart from the measurement error in the states, there is also the danger of noise. Since s(t) depends on x(t), it is assumed that proper filtering strategies will be employed in practice. In the worst case, one has to cut off the perturbation compensation and has to live with the chattering. By worst case, we mean that the computed perturbation signal is greater than the maximum norm of the allowed perturbation, i.e., jT(t) < C(f(x(t),t)+ fd(x(th),t)) < Cl(klx(t)l+kdlx(th)l) However, this decision can only be made depending on the application at hand. For theoretical development, in order to prove the reaching condition within a finite time, we need to assume certain error bounds on the derivative estimation. We assume that the relative error in the estimation of i(t) is bounded, ~(t) est (t)< ~<3 Sest(t) with the positive scalar bound 8 being known. An estimate of 3 can be obtained via simulation studies with appropriately assumed perturbations. Then with the choice d(t) > &est(t) + e, e > 0, and K the reaching condition can be satisfied and in addition, the transient behavior can be shaped. Theorem 4.1. The uncertain delay system (4.1) with modeluncertainty and bounded measurement uncertainty given by (4.2) and (4.10) achieves the desired sliding mode within a finite time if the control law u(t) = Ueq(t) + Ur(t)+ Up(t) is applied, where the individual components are given by (4.6)(4.9). Furthermore, the sliding mode is attained within t = tr, where tr 2 min() In + srmin (K) ( (O)s(O) and the upper bound on the norm of the switching surface is given by Is min (K)] A min (K) Proof: Choose a positive definite function V(t) = 1 s(t) Ts(t). Then 2 differentiating V(t) along the trajectories of the system and using (4.5)(4. 10) yields: V(t) =s(t)Ti(t) =s(t)T [Ks(t) ds(t) + (c(f(x(t),t)+fd(x(th),t)) Tp(t))](4.11) In the rest of the proof the time argument is dropped for convenience of notation. Noting that Ai = C(f(x(t), t) + fd (x(t h), t)) TP (t) and using the bound on the estimation error (4.10) the equation (4.11) can be used to derive a bound on V(t) as follows: V(t) = sT Ks dsT sign(s) + sT Ai We note that since s(t) is a vector, s Tsign(s) = Isl > Is12. Hence, V(t) Ks dlsl +lslJlAJ Since d = est(t)l + E, and AJ 5[iest (t)l V(t) < (Amin(K))[s12 E[SI V(t) ( min(K))V eV The above is a scalar differential equation and the solution is given by WV(t) I (0) +Vkmin(+)] 8 Amin (K) Amin (K) 1E )] Ain(K)t 2e S (t)l+_ s(0) min (K) e 2 Amin (K) Note that first term is always positive and decreasing, and the second term is a positive constant and hence, their difference will attain zero within a finite time. It is straightforward to compute the upper bound on the reaching time. A detailed proof is given in Appendix F. Thus the system is guaranteed to reach the sliding mode within a finite time since the reaching condition s(t)T(t) < 0 for s(t) # 0 is satisfied. V Remarks: 1. The rate at which the switching surface is reached is determined by the controller parameters K and d. Therefore by adjusting these parameters the designer may select an appropriate reachingtransient behavior. 2. Chattering in the input variable is unavoidable for any positive value of d, because ur(t) + _d when the system oscillates around s = 0. To reduce chattering, the parameter d must be chosen as small as possible. In some cases .(t) is measured directly, therefore d can be set equal to zero without any loss. In other cases Sst (t) can be compared with simulated i(t) (for a given set of perturbations) to aid in choosing a suitable the parameter d. Alternatively, one can choose to set d = 0 and specify a large K to overpower the estimation errors if any are present. However in the last case the penalty to be paid is high initial controller gains. 4.3 Asymptotic Stability with Perturbation Compensation Once in the sliding mode the system equations reduce to i(t) = Ax(t) + ,dx(t h) + f(x(t),t) + fd(x(t h),t) B(CB)1Tp(t) (4.12) which is obtained from (4.4) after substituting (4.6)(4.8). The intent now is to derive conditions that establish the asymptotic stability of system (4.12). First, we note that if perturbation compensation is not employed, then (4.12) reduces to .i(t)= Ax(t) +Adx(t h)+f(x(t),t) +fd(x(t h),t) (4.13) With perfect perturbation compensation, the system equations become i(t) = Ax(t)+Adx(t h) + y(f(x(t),t)+ fd(x(t h),t)) (4.14) where y = [I B(CB)'c] There are basically two approaches in the literature for deriving conditions for the asymptotic stability of these systems. One is to follow Lyapunovtype of arguments [Hale and Lunel, 1993], and the other is to use normbased inequalities [Mori et al., 198 1]. Both approaches result in sufficientonly stability conditions. Lemma 4.1. The delay system (4.13) with nonlinear uncertainties (4.2) is asymptotically stable if the following condition is satisfied y(A)+ AIjd +k+kd Proof. See Appendix B. V Theorem 4.2 The delay system (4.14) with nonlinear uncertainties (4.2) is asymptotically stable if the following condition is satisfied (A7) + d I +klll+kdjl j< 0 (4.16) Proof. The proof is identical to the proof of the previous lemma. V Remarks 1. It is possible that different choices of a given norm and the corresponding matrix measure give different conclusions about stability. This is consistent with the fact that the conditions derived are only sufficient. 2. In order to improve the sharpness of the stability condition, a similarity transformation z(t) = Tlx(t) can be introduced so that the system equation (4.14) becomes i(t) = T1ATz(t) + T1AdTz(t h) + y (f(x(t), t) + fd (x(t h),t)) (4.17) where T(I B(CB)'c). In this case the stability condition reads ,(TAT)+ 1 TAdT +kd lTl y1 +kjjTj j; < 0 (4.18) The usual approach is to find a transformation matrix T to diagonalize TXAT. 4. Under perfect estimation and the assumption of matched uncertainties [Shyu and Yan, 1993] the perturbation error is (f+fd)+BuP = 0. The stability condition then reduces to y(T1A T) + rTdT < 0. Of course, there is a penalty for imperfect estimation and this is reflected in the reduction of the size of the perturbations the system can withstand and still remain stable. 5. The conditions (4.15) and (4.16) are delay independent. This could be advantageous in cases where delaydependent stability conditions cannot assure stability. The example in the chapter illustrates this fact. However, since all conditions reported to date in the literature (including this one) are sufficient only, no general claims can be made. It is also possible to derive delay dependent conditions for the problem considered in the chapter. Optimal Choice of C The stability condition (4.18) and the condition derived in Shyu and Yan [1993] involves the matrix y = T1 (IB(CB)1C). Then the matrix C is chosen to minimize T1(I B(CB)'C) F" There appears to be no documented systematic method for optimizing the left hand side of (4.18) (or similar stability conditions for other approaches) over all allowable matrices T and C. To get the tightest possible stability condition for the result in Shyu and Yan [1993] the matrix C must be chosen to minimize the norm of y. We propose a new systematic approach to design matrix C. The following definition and inequalities concerning Frobenius norm are useful. Let A E= 9t'xn, and let II and ar represent the Frobenius norm and 'i'th singular value, respectively. Then [Golub and VanLoan, 1989] The following relationships are useful IIA112 F = U2 + +. Co2; p =minfm,n} I + }(4.19) Ilal12 [[AIIF  Vn1II12 Lemma 4.2. Let N E 91 nPm and M E 9'. Then the best approximate solution to the problem min 11 M NX F X F mxn is Xopt = (NTN)I NTM. Proof: See Barnett [1971]. The definition of Xopt implies that for all X, X E 91'mxn either 11 M NX IIF > M NXopt F or M NX IF = M NXopt F and ]X, o It is instructive to note that (NTN) IN is the MoorePenrose generalized inverse of matrix N. Theorem 4.3. The best approximate choice of matrix C for the following problem min TI(I B(CB)1C) F C E 911LX ) is Copt =a(BTTTT1), where a isascalar. Proof. Define the auxiliary matrix X = (CB)'C. The minimization problem reduces to min TI(I BX) X C gr7tmxn F Substituting M = T1 and N = T1B in Lemma 4.2 the best approximate solution is found to be Xopt= (BTTTT1B)lBTTTT1 =(CCptB/1 Copt It is obvious that Copt = BTTTT1. Note that multiplying by a scalar a does not change the value of the functional being minimized, hence Copt = a(BTTTT 1) V Remarks: 1. Note that Theorem 4.3 gives the optimal solution in the sense of the Frobenius norm. Using (4.19), it is easy to see that for the case where B E 9'x<, the choice Cpt = a(BTTTT1) is optimal in the sense of the 2norm as well. 2. The Matrix C enters in the last three terms of the stability condition (4.18). An alternative method for finding the optimal matrix C is to carry out a numerical optimization. However, noting that the contribution from the last term will be small for a well performing estimation strategy, the last term can be ignored for analysis purposes. Then, from triangular inequality it is easy to verify that the result of Theorem 4.3 provides a sub optimal upper bound in the sense of the Frobenius norm. Implementation Issues 1. The estimation error depends on the approximation scheme used. Although there is no a priori information on the estimation error, different estimation strategies can be compared by assuming appropriate perturbations and tracking the estimation error. 2. Also, note that the main challenge in the proposed approach is that of approximating i(t) by Sest(t). In general, a finite difference estimate is used. However, this may cause a problem near the switching surface where some chattering combined with a small step length may contribute to large estimation errors. Also, there is the unavoidable nuisance of noise, and the danger of differentiating it. Hence it is necessary to employ filtering. Although the switching dynamics with perturbation compensation is complex, one can still perform a low pass filtering on s(t) for the purpose of estimating Sest(t) by finite difference [Elmali and Olgac, 1996]. Note that the actual value of s(t) can be used elsewhere in the controller. Hence, this does not destroy the guarantee of asymptotic stability. The cutoff frequency for the filter can be chosen based on the maximum frequency component that we wish to process in the state dynamics. For example, if temperature in a reactor is the state variable that is used in the construction of the switching function, a cutoff frequency of 100 Hz is acceptable, whereas if pressure in a flowing line is the variable then one needs to go for higher frequencies. With the advances in digital signal processing these decisions can be made scientifically depending on the application at hand. 4.4 Example Consider the uncertain timedelay system studied by Shyu and Yan [1993] 1.5] x(t)+ Lo 1 x(th)+ L u + f(x(t),t) + fd(x(th),t), t>O [=0.3 20 011] The delay is h = 2, and the unknown nonlinear perturbations are of the form f(x(t),t) = 0.2x(t)sin(x(t)) and fd(x(th),t) = O.lx(th)sin(x(th)) The design objective is to asymptotically stabilize the uncertain system. The initial condition is specified as x(t) = [1,3]T h < t < 0. Note that for the delay considered, the sufficient stability condition of Shyu and Yan [1993] is violated. Also note that the homogeneous part of the system is unstable because A has a positive eigenvalue A = 1.1432. Choosing the same control matrices investigated in Shyu and Yan (1993), namely F =[0.3 0.15] Fd = [0 0.1] leads to = Li 2.15' Ad = 0.] The particular choice F adopted makes A diagonal (and hence T = I), and the choice of F,1 ensures Ad < Ad1. As per Theorem 4.3, Cpt = a (BTT2). In this design we choose K = 2, and d = 0. Then choosing a = 0.1 leads to C = [1 0.1]. If a positive value for d is chosen, it will result in some chattering. From (4.16) we can find the maximum tolerable estimation error. For the 1norm and infinity norm we find that asymptotic stability is assured if e < 1.4445, while for the 2norm asymptotic stability is assured if e 1.6085. The delay introduced in the computation of up(t) is 0.01 seconds. The simulation results are presented in Figures 4.1 4.4. Figure 4.1 shows the evolution of state norm as a function of time, reflecting the asymptotic stability of the system. The switching function and its evolution towards the sliding mode is shown in Figure 4.2. The sliding mode is reached rather quickly, and most importantly, a chattering free behavior is observed. Figure 4.3 shows that the input variable is also smooth and free of chattering. The jump in the input around t = 2 reflects the fact that control variable is responding to the delayed states coming into effect. Figure 4.4 shows the evolution of the estimation error as a function of time, which also reaches the value of zero asymptotically. From Figure 4.4, it is obvious that the estimation error in the sliding mode is much less than 1.6085, thus implying robust stability via equation (4.16). In this example, there is no chattering, and the system is asymptotically stable. 4.5 Conclusions This chapter considered the robust stabilization of uncertain timedelay systems with mismatched nonlinear uncertainties. A practically implementable control design methodology is proposed and an illustrative design example is given. The present method can significantly reduce or completely eliminate chattering as compared to previous approaches. 3.5 3 2.5 S2 1.5 0.5 0 0 1 2 3 t (see) 4 5 67 Figure 4.1. State norm Vs Time 4 3.5 3 2.5 , 2 1.5 1 0.5 0 0 1 2 3 4 5 6 t (sec) Figure 4.2. Switching function Vs Time 0 1 2 3 4 5 6 7 t (sec) Figure 4.4. Compensation error Vs Time 0.2 0 0.2 0.6 0.8 0 1 2 3 4 5 6 7 t (see) Figure 4.3. Control input Vs Time I, Nomenclature for Chapter 4 A State matrix Ad State delay matrix B Control matrix C Switching function matrix d Magnitude of discontinuous control f State dependent nonlinear perturbation fd Delayed state dependent nonlinear perturbation F State feedback matrix Fd Delayed state feedback matrix h delay K Positive definite matrix k Bound on state dependent uncertainty kd Bound on state delay dependent uncertainty s(t) Switching function T Transformation matrix Tp Estimated perturbation signal u(t) Control input x(t) State z(t) Integral equation with deviating argument Greek Letters (5 Bound on the derivative estimation error e Derivative estimation error in the sliding mode ig Bound on the derivative estimation error in the sliding mode a Scalar multiplier 0 time index 68 0(0) Initial condition y(A) Matrix Measure of A CHAPTER 5 SLIDING MODE CONTROL FOR UNCERTAIN INPUT DELAY SYSTEMS 5. 1 Introduction Time delay exists in various branches of engineering. From biological, optical, electric networks, to chemical reaction systems, time delay occurs and affects the stability and performance of a wide variety of systems [See Schell and Ross, 1986, Inamdar et al., 1991]. Input delays occur as one of the common sources of time delay, such as in chemical processes, transportation lags and measurement delays, etc. It is well known that the existence of time delay degrades the control performances and makes the closedloop stabilization difficult. Hence, the study of time delay systems has been paid considerable attention over the past years. The description of timedelay systems lead to differential difference equations, which require the past values of the system variables. There are no known methods to get necessary and sufficient conditions for the robust stability of such systems. There are many approaches that yield sufficient conditions with varying degrees of sophistication. The case of perfectly known (i.e., no uncertainty) timedelay system is well studied. It is known that a stabilizable timedelay system [see Kamen et al., 1985, Logemann, 1986] can always be stabilized by a finite dimensional controller. The study of state uncertain delay systems has been paid much attention. However, with regard to the control of uncertain input delay systems, only a few studies have been reported in the literature. Based on Riccatiequation approach, a robust controller is derived in Kojima and Ishjiama [1995]. Yan et al., [1997] propose a functional observer and state feedback mechanism using the factorization approach. For a LQG/LTR method see Lee et al., [1988]. Wu and Chou [1996] propose a control algorithm from inputoutput feedback linearization by means of a parametrized coordinate transformation. Recently, an LMI based approach was given in Niculescu et al., [1997] for delaydependent closed loop stability of inputdelay systems. However, the authors allowed uncertainty only in the input delay. One of the common methods used in chemical engineering applications, for processes with timedelay, is to use the idea of the Smithpredictor scheme to cancel the effect of time delay [Stephanopoulos, 1984]. However, if the model is imperfect (uncertain) then Smith predictor is known to give poor performance. Some approaches have been developed for timedelay compensation based on prediction strategies [See Henson and Seborg, 1994]. In this chapter we propose an approach based on the Sliding Mode Control [Hung et al., 1993; Utkin, 1977, Decarlo et al., 1988]. The sliding mode control approach possesses many advantages, e.g., fast response, good transient performances, robustness to the variation in plant parameters and external disturbances. The authors in [Shyu and Yan, 1993; Oucheriah, 1995; Luo and Sen, 1993, Basker et al., 1997] study the sliding mode control for uncertain systems with state delay. Also see Luo et al., [1997] for a sliding mode approach for uncertain time delay systems with internal and external point delays via various types of feedback. This chapter proposes a SMC controller to robustly stabilize an uncertain inputdelay system with linear and possibly mismatching uncertainties. Sufficient robust stability conditions are derived. No matching conditions on uncertainties are assumed. The rest of the chapter is organized as follows: In Section 5.2, a transformation is adapted to convert the original system with bounded input delay into one without any delay. The stabilization of the modified system implies the stability of the original system as long as the control law is bounded [see Kwon and Kim, 1980; Artstein, 1982]. Section 5.3 gives the guideline to design the sliding mode controller which consists of the switching function and the control law. In Section 5.4, the analysis of the perturbed system is carried out, and the reaching condition and the asymptotic stability of the sliding mode are proven. In Section 5.5 some open issues in the sliding mode literature are raised and answers are provided. Section 5.6 considers constraints on the controller input. Section 5.7 gives an illustrative example with an open loop unstable plant. Conclusions and future work are discussed in Section 5.8. 5.2 Preliminaries and Problem Formulation Consider the uncertain input delay system i(t) = Ax(t) + Bu(t) + Bdu(t h) + f(x,t) (5.1) where x(t) E R is the state, u(.) : Rm is the control, A, B and Bd are constant matrices with appropriate dimensions, h is a known constant time delay. The total plant uncertainty is bounded by If (x, t)l !lx (q1 (5.2) where 1.1 denotes a vectornorm and Il1 denotes the associated induced matrixnorm. Without loss of generality, the vector 2norm 11.112 is used. The positive scalar boundfi is assumed to be known. In rest of the development, without loss of generality, we assume that m < n. Note that we do not assume any matching condition on the uncertainty. The method described here can be trivially extended to the case of multiple known delays. Assumption 1: The pair [A, B] is controllable and all the states are measurable. Transformation into a delay free system We consider a transformation to convert the original system (5.2) into an equivalent system without delay. This reduction enables the use of known stabilizing methods for linear systems without input delay. See Artstein [1982], for extensive theoretical discussions on the validity and the applications of this type of reduction and also see Fiagbedzi and Pearson [1986], for further generalizations. Define the transformation Z(t) = x(t) + h e A(tr)Bdu(T)dt (5.3) Assumption 2. The solution Z(t) is not affected by a change in u(t) on a set of zero Lebesgue measure [Artstein, 1982]. For an arbitrary bounded control u(t), the system (5.2) is transformed to (see Appendix D for details) i(t) = Az(t) + Bu(t) + f(x, t) (5.4) Where B = B + eAhBd. It is assumed that B has full rank m. Then, via PBH rank test [Kailath, 1980] it is trivial to show that the system (5.4) is also controllable. Note that the system described by equation (5.4) is an uncertain linear system without input delay. 5.3 The Design of the Sliding Mode Controller In the absence of perturbations, from (5.4), the nominal system is given by i(t) = Az(t) + Bu(t) (5.5) In the following we will design the switching function parameters on the basis of system (5.5). The design consists of two phases. The first phase is the construction of the switching function, so that, the nominal system restricted to the switching surface is asymptotically stable. The second phase involves the development of a control law, which must satisfy a sufficient condition for the existence and reachability of a sliding mode within a finite time period. Design of the switching function Since system in (5.5) is a typical linear system, choose the switching function to be s(t) = Cz(t) (5.6) where C is an m x n matrix to be determined. The idea is to design the switching function so that the nominal system is asymptotically stable in the sliding mode [Woodham and Zinober, 1993]. Note that other types of switching function such as an integral switching function can also be used to facilitate desirable switching dynamics. Since the matrix B has full rank m, it is possible to reduce (n m) of the rows to zero, i.e., there exists a nonsingular matrix T such that TB=L (5.7) where B1 is an m x m nonsingular matrix. In fact, we remark that such a T has a simple form for rn = 1 and is given by (n1) (n) 1 The transformed states are given by w(t) = Tz(t) (5.8) and (5.5) becomes 4(t) = TAT1w(t) + TBu(t) (5.9) while the switching function becomes s(t) = CT'w(t) = Qw(t) We introduce the following partitions for later use. W(t)=W([] (m) Q= [ w(t)] (in) (nm (1) (5.10) Qg](m) )(in) (5.11) T=[T T2]n (nm) (m) TAT =All A12](nm) [A21 A22/ (M) (nn) (m) equation (5.9) can be rewritten in the form W(t) = A1 W(t) + A12w1(t) (5.1 la) ii'1(t) = A2lW(t) + A22wl(t) + Blu(t) (5.1 lb) Furthermore, (5.10) can be written in the form s(t) = QW(t) + QwI(t) (5.12) where Q is an m x (n m) matrix, and Q, is an m x m nonsingular matrix. Differentiating (5.12) and substituting (5.11) in to the dynamic equation yields W(t) = (AI A12Q,I)W(t)+ A12Qls(t) (5.13) = [(QAI + QIA21) (A12 + QIA22)QIlQ]W(t) + (QA12 + QiA22)Qlls(t) + QiBlu(t) (5.14) When s(t) # 0, equations (5.13) and (5.14) describe the reaching mode. On the other hand, when the system is in sliding mode, i.e., s(t) = 0 and (t) = 0, (5.13) yields W(t) = (A,, A12QIla)W(t) (5.15) wI(t) = QQW(t) (5.15a) Since the pair (A11, A12) is completely controllable (see the Appendix E for a proof), there exists an m x (n m) matrix K that satisfies )L(A1 A12K)=A (5.16) where V, represents the desired poles in the sliding mode. Since A11 and A12 are known, a set of linear equations can be solved to yield K. Comparing (5.15) and (5.16), we have QIK (5.17) Choosing Q, = I yields Q = [Q QI] = [K I] (5.18) Substituting (5.18) into (5.10) results in C=QT=[K Im]T (5.19) Therefore, the switching function has the form s(t) = Cz(t) = [K I]Tz(t) (5.20) It is instructive to recast the dynamics of the system in terms of our choice for C, i. e., K. By using (5.11), it is easy to show the following: W(t) = (A,1 A12K)W(t) + A12s(t) (5.21) = [K(Al1 A12K) + (A21 A22K)]W(t) + (KA12 + A22)s(t) + Biu(t) (5.22) w1(t) = s(t) KW(t) (5.23) We also remark that in the sliding mode, wI(t) = KW(t) and hence, the asymptotic stability of (5.15) ensures that lim w, (t) 4 0. Remarks: (1) The above also illustrates an interesting systemtheoretic point. We are seeking to control an ndimensional system via a mdimensional input. From (5.21) (5.23), it is obvious that the input explicitly affects only a mdimensional subset of this realization and the remaining (nm) dimensional subsystem is affected via the feedback of the states as determined by the system. This information is not new. In fact, by recasting the system in the controllability realization, this fact can be trivially seen. But, in contrast to the controllability canonical form, (5.21)(5.23) also guide in choosing a controller to exploit the fact that the matrix BI is of rank 'm'. More about this will follow in later sections. (2) Note that with the choice of C, (CB)1 = B1' and since BI is nonsingular, (CB)1 always exists. Design of the Control Law The control law chosen is a combination of the traditional state feedback and a time varying discontinuous control element u(t) = (CB)I[CAz(t) + kjs(t)j" sgn[s(t)] + d(t)sgn[s(t)]] (5.24) where the parameters are chosen such that k > 0, 0!_ a < 1.0 and d(t) = PC[Ix(t)l + E, with e > 0. When s(t) is small, a faster approach rate than the linear approach algorithm is realized. The state feedback is designed to cancel the effect of the state matrix A in the switching dynamics. Once the system is in the vicinity of the switching surface, the magnitude of the discontinuous control is linearly proportional to the norm of the states. Since the states x(t) and z(t) are available, realizing the above control law is straight forward. Using the partitions (5.11), it is also easy to verify that (5.24) is equivalent to u = (B,)1{[K(A11 A12K) + (A21 A22K)]W(t) + (KA12 + A22)s(t)} + (B,)'[kjs(t)jasgn[s(t)] + d(t)sgn[s(t)]] (5.25) Comparing (5.25) and (5.22), it is obvious that we exploit the fact B1 is nonsingular m x m matrix, and obtain u by equating (5.22) to zero. With the chosen controller, the switching dynamics (5.22) is of the form (t)= kls(t)l' sgn[s(t)] d(t)sgn[s(t)] + Cf(x(t),t) (5.26) Remarks: (i) From (5.26), it is obvious that the discontinuous control is used to overpower the effect of uncertainty. Since the only information we have about the uncertainty is the norm bound, we can not use any lower value of d(t) and be guaranteed to reach the sliding mode within a finite time. From (5.25), it is clear that there will always be chattering for this choice of parameters. (ii) For the nominal system d(t) can be chosen arbitrarily close to zero. (iii) The parameter a can be used in a time varying fashion to guarantee a faster rate of approach to the switching surface than the normal choice c = 1.0. For example, c = 1.0 for ls(t)l 1.0 and (x = 0.5 for ls(t)l = 1.0 would assure a faster rate of approach than the proportion control (i.e., a = 1.0, Vt > 0 ). 5.4 Analysis of the Perturbed System In this section, the robust stability of the uncertain system (5.1) is analyzed. First, the analysis is carried out for system (5.4), in order to give conditions for reaching the sliding mode and the asymptotic stability in the sliding mode. Since system (5.4) is related to system (5.1) via (5.3), conditions are given under which robust stability can be inferred for the original system (5.1). For an uncertain system to be robustly stable under sliding mode control, the following conditions have to be met: (i) The system should attain the sliding mode within a finite time; that is, the state trajectories should reach the switching surface and should stay on the switching surface for all time thereafter; and (ii) the system in the sliding mode should be asymptotically stable. Reaching Condition for Sliding Mode. Theorem 5.1: The uncertain delay system (5.4) with modeluncertainty and bounded measurement uncertainty given by (5.2) achieves the desired sliding mode within a finite time if the control law (5.25) is applied. Furthermore, it can be shown that the value of oc chosen 1, then the sliding mode is attained within t = tr, where tr < (2 ln[l +k T{s (o)s(O) J] (5.27) k 2s) and the upper bound on the norm of the switching surface is given by 8 s(t) k e k (5.28) Proof: We choose a positive definite Lyapunov function V(t) = 1 (ST(t)S(t)) (5.29) Our aim is to show that '(t) < 0 for the choice of parameters k and F. The development here is fairly standard. V(t) = sT (t)(t) utilizing (5.26) for (t) = s T (t)[ks(t)ja sgn[s(t)] d(t)sgn[s(t)] + Cf (x, t)] (5.30) From the available bounds on the uncertainty Cf (x,t) < [CIf(x,t)l 13 C[x (t)[ <_ d (5.31) Using (5.31) in (5.30) (t)_Is (t)[ ksT(t)sgn[s(t)]] d(t)[ST (t)sgn[s(t)]] + sT (t)Cf(x,t) (5.32) We observe that sT (t)sgn[s(t)] = ys1T(t)sgn[si (t)] 2 si(t)l[sgn[si (t)]2 = X si(t) = s(t)1 (5.33) Using (5.33) in (5.32) V (t) < kls(t)als(t)l1 Es(t)l (5.34) Since both the terms on the right hand side are negative, the system is guaranteed to attain the sliding mode in finite time. Furthermore, for the choice of a = 1.0, (5.34) becomes V(t) < kjs(t)12 EIS(t)I1 (5.35) since for a vector 1.12 <1.11' Using (5.5.29) V (t) kV(t) eV"12(t) (5.36) The above is a scalar difference equation, and from Appendix F, V < + e 2 (5.37) k I s(t)l< s(0)l+ e2 (5.38) [ kj k Equating the right hand side to zero we get the result. V Corollary: For the choice 0 < a < 1.0, a tighter result can be obtained. Following the same steps as in the proof of the above theorem (see Appendix F), it can be shown that for a control law (5.24) with d(t) = I3Cjjx(t)!, the sliding mode is achieved within t= tr, where tr 1 S T.5(1S( ) r and IS(t)l F (l a( 2 a)X(0)]0l kt] 0!< t:5 tr This is advantageous because, we avoid an extra discontinuous control element, 8. Asymptotic Stability in the Sliding Mode Note that when the system is in the sliding mode, the control law reduces to a linear state feedback. In this chapter, we follow a LyapunovRazhamikhin type approach to derive sufficient stability condition. The dynamic equation in sliding mode is given by w W)= (A1 A12K)W(t) + Tlf(x,t) (5.39) Since in the sliding mode, wl(t) = KW(t), it suffices to show the asymptotic stability of (5.39). The first task is to convert the uncertainty description in terms of W(t). z(t) = T[ W "(t) ] = T'[_ Wm(t) T1 Im(5.40) LW1(t). KwI ()K Next we seek an expression relating x(t) and z(t). From (5.5.3) Ix(t) Iz(t)l + h ea (th )Bdu(,r)dT (5.41) where u(t) = (CB)' CAz(t). Thus Jx(t)J: z(t)J+ max eA8 ] Bd (CB)CAfhhz(Jdr (5.42) The main hurdle at this point is to express z('r)j in terms of Jz(t)l. To obviate this difficulty, we take LyapunovRazhamikhin type approach. First of all we show that for the case of matched uncertainties, proving asymptotic stability is straight forward. Definition: Matching condition [Hung et al., 1993]. If the uncertainties satisfy f(x, t) = Bf(x, t) where f(x, t) e 91rex1, then the matching condition is satisfied. This means that the perturbation is in the image of B. Tf(x, t) = TBf(x, t) =[T2Bf(xt)] Hence, in the sliding mode, the system equation becomes W(t) = (AlI Al2QI1Q)(t) (5.43) Note that (5.43) is of the same form as the equation describing sliding mode for the nominal system (5.15). This implies that the uncertainties will have no effect on the system in the sliding mode. This is one of the major advantages of the sliding mode design. The state feedback is simply used to place the poles of the nominal system in the sliding mode. However, the assumption of matched uncertainties is a highly restrictive one, as for example in the m = 1 case, we are constraining the perturbation to be a scaled version of B. One should note that the situation represented in (5.43) is an ideal case. That is, if the controller is able to exactly cancel the perturbation, then the sliding mode is independent of the perturbation. This obviously suggests that if one could estimate and cancel the perturbation (or) the effect of the perturbation on the sliding mode, then the system in sliding mode can be made to be independent of the matching perturbations. These kinds of approaches are commonly called perturbation compensation and have been applied to linear state delay systems [Chan et al., 1996]. Even if mismatched uncertainties are present, perturbation compensation can be employed to reduce the magnitude of the required discontinuous control. In the mismatched case, it is not possible to completely cancel the perturbation and there will be a residual effect of the perturbations in the sliding mode and the state feedback should be such that the overall system is asymptotically stable. Theorem 5.2 gives a sufficient condition for the mismatched case. Before we state and prove Theorem 5.2, we explain about a difficulty that commonly arises in proving asymptotic stability in statedelayed systems and inputdelayed systems (as the control usually involves state feedback, the inputdelay problem has relevance and resemblance to statedelayed problem). There are two classical approaches in the literature for proving asymptotic stability for delayed systems. Over the years there have been many incremental improvements, but the message is that all methods are conservative. This is because the differential equations that determine the dynamics of the system have two difficult elements. One, the perturbations. Two, the delayed states. We know only the norm bound on the perturbation, and while using this sufficiency creeps in. The problem with delayed states is more interesting. In using LyapunovRazhamikhin type approaches, a restriction is imposed on the dynamics of the system. Consider a Lyapunov candidate V(t) = WT(t)pW(t) where P = PTP > 0. At this point, we introduce the usual Razhamikhin type restriction on the system. We restrict that V(O) It is important to understand, how restrictive the above assumption is. The reason for using such an assumption is that it allows one to bound the norm of the delayed states of the system in terms of the norm of the present states of the system. Obviously any condition based on this will only be sufficient. Since the following condition holds for any positivedefinite matrix P = pT > 0, and W = 91nm 1min(P)WTW _! jTpW _< ',min(P)WTW it follows that where a 4 )min(P) X'min (P) ' The above condition is not very restrictive as it is true for some (51 > 85 > 1, for all stable systems whose state norm continuously decreases (and it is easy to show that for unstable systems, the above is true for any positive 8 ). The actual values of 8 and a (and hence P) quantify the decay rate of the system. This condition, however could be violated for a short durations, for example, when a delayed disturbance hits the system (i.e, the norm of the states could increase). However, the controller will act against the disturbance and restore the performance. Hence, temporary violation of this constraint is not fatal, and will not affect the stability conclusion. Theorem 5.2: The system (5.4) in sliding mode with the mismatched uncertainties (5.2) is asymptotically stable if the following condition is satisfied Tr Ifl 1+ha max e 16 Bd(CBY1CA ]} <'min(Q) (544) K L h ij 211PI1TI l where a =ImxjP) and P > 0 is the solution of min (P) P(A _A12K) + (Aj1 _A12K)T p = _Q, Q >0 (5.45) Proof. Under the similarity transformation (5.8) the system (5.4) transforms to (5.11). If the uncertainties are present, using (5.11) equation (5.1 la)(5.1 lb) adopts the form W(t) = A1 W(t) + A12Wl(t) + Tf(x,t) (5.46) i1(t) = A21W(t) + A22Wl(t) + Btu(t) + T2f (x,t) (5.47) Consider a Lyapunov candidate V(t) = WT(t)PW(t) (5.48) where P = pT > p = pT > ,P (n)(n Taking the derivative of (5.48) V,(t) = WT(t)[P(A11 A12K) + (Al1 A12K)Tp]wT (t) + 2wT (t)PTf (x,t)(5.49) At this point, we make the usual Razhamikhin type restriction on the system. We restrict that V(o)!<_ N(t), t h<__0 ___t, V(5>1 (5.50) it follows that, Ii(o)1 (! 4 m (P)l (t)I (5.51) Amin (P) Denoting a = x} using this in (5.42) Ix(t)l< T1[:] {l+h8a[ max eAO Rd(CB)) (5.52) Therefore, If (Xt) < P T1[Im] {i + haa[_max eAO Bd(CB)1CA ]}iv(t)I (5.53) We combine all the terms to get f3l = / Tl[Inm] {1 + amax eAO1 Bd(CB)IcA if(,q P wtj(5.54) hence V(t) = WT(t)[P(A11 A12K) + (A, I A12K)T P]w(t) + 2/, 11PJ II W(t)12(5.55) Choose P > 0, Q > 0, such that P(A11 A12K) + (AI A12K)TP = Q <w T(t)QW(t) + 2/ llPjTI (t)12 (5.56) V7(t) < [Amn(Q) + 2P,111l] W(t)12 (5.57) Hence a sufficient condition for asymptotic stability in the sliding mode is given by P 1in(Q) 211PJJIJTalI i.e., T 1[IMlI+h'a[ max e A 1nCd(cA)ca] < min(Q) (5.58) K il L heO 211PI11T1 Note that if the above expression is true for 8 = 1 then there exists a 3 > 1, sufficiently small that it will still hold true. Therefore, a sufficient condition for stability is given by the above expression, with 8 = 1. V Remarks: (i) In the sliding mode, the derivative of the Lyapunov function is given V (t) < [,Imin (Q) + 2pa, 1P[T1 2 JjW(t)j2 <0. Denoting g := (/min (Q) + 231 11PIIT1 [), we can obtain <(t) __ I /Iin () 1(5.59) This explains why we can get only asymptotic stability guarantee, that is, we can only state that the norm of the states will reach zero as time goes to infinity. In the next section we will elaborate more about this issue. (ii) For a given system description, once the poles of the closedloop nominal system are chosen, everything except P and hence Q is known. Hence P is an optimization parameter. We remark that the best choice is to make sure that the condition number of P is minimized while satisfying (5.44)(5.45). For large systems, this constrained optimization problem can be cast as an convex LMI problem and easily solved through available commercial software [Niculescu et al., 1997]. Theorem 5.3. If the uncertain system (5.4) is asymptotically stable, it implies the asymptotic stability of system (5.1). Proof. From the transformation (5.3), we have x(t)<lz(t)j+ h max eAO]. Bd. ut (5.60) From Theorem 5.2, the transformed system (5.4) is asymptotically stable. Hence, lim Jz(t)l  0. Since u(t) = (CB) CAz(t), in the sliding mode, lim [u(t)l  0 From f>t400 (5.60) the result follows. V Implementation Issues Due to inherent limitations such as the switching delay, system inertia, etc., the sliding motion does not take place in any real system. Instead, the trajectories chatter in a neighborhood of the switching surface. The chattering phenomenon is undesirable because it may adversely affect the actuating mechanism and excite the highfrequency unmodeled plant dynamics, leading to instability. The discussion about chattering and the ways to alleviate this problem, as outlined in Chapter 4, applies to the present case of stabilizing systems with inputdelay as well. 5.5 Some Open Issues in Sliding Mode Literature Although the sliding mode control approaches have been used for a long time for a variety of systems, there are nevertheless some open issues. We raise the following issues of interest: (1) Is it possible to guarantee that the norm of the states continuously decrease during the reaching phase?. If so, what is the nontrivial upper bound, on the norm of the states? (2) Is it possible to get an upper bound the norm of the states when the system is in the reaching phase (other than of course, the trivial estimate Jx(0)[) ? (3) We are able to prove that the sliding mode will be attained within a finite time. At the same time, the best we can prove for the system in the sliding mode is asymptotic stability. We also seek to explain why this is so. We also remark that these questions apply to all sliding mode control schemes (for linear uncertain, Uncertain linear systems with state delay and so on..) and not just for the inputdelay case that we consider here. Note that the usual approach in the literature for sliding mode systems is to prove the reaching condition within a finite time (as we did via Theorem 5.1), and then prove asymptotic stability for the reduced system in the sliding mode (as in Theorem 5.2). However, there are problems with this approach. For example, one does not have an estimate of the norm of the states, when the system reaches the sliding mode. Hence, we do not know the proper initial condition for the scalar Lyapunov differential equation given by (5.59). One only hopes that it is less than the norm corresponding to the initial condition. Even assuming so, it is still highly conservative. What we need is a way of estimating a tight upperbound on the norm of the states, during the reaching phase. If we obtain such an expression, then we will be able to answer questions (Ql)(Q2). Before we seek to get an analytical expression to the above questions, it is instructive to ponder over the significance of Theorem 5.1. What (5.27)(5.28) imply is that the norm of a linear combination of the states continuously decreases and attains zero within a finite time. But, a linear combination of two timedependent functions (here states of the system) may be decreasing towards zero, while the individual elements may be diverging in opposite direction, making their sum (or difference) approach zero. Hence, it seems like that we do not even have a guarantee that the norm of the states of the system will continuously decrease, when the norm of the switching function continuously decreases and attains the sliding mode. In other words, the central question here is, can one obtain sliding mode when the system is unstable. However, upon reflection, one simple fact emerges. Consider the linear uncertain system (5.4) and bounded perturbation (5.2). The solution will be driven by the interaction of the control and perturbations and the rate at which the states converge (or) diverge will depend on the state transition matrix, and whether or not it is stable enough to overcome the perturbations. Ultimately the form of the solution will contain complex exponentials, apart from other terms. Since, the perturbation is bound by the norm of the states, an unique solution does exist. It is not our intention to derive it. In fact, it is impossible to derive it, unless the form of the perturbation is exactly known. Now, Theorem 5.1 implies that the norm of a linear combination of the complex exponential terms is continuously decreasing. If the system is unstable, it is the real part of the complex exponentials that will dominate the system dynamics. In that case, the resulting linear combination of the complex exponentials will diverge. Hence, it is not possible for js(t)j to continuously decrease and attain zero within a finite time, if the system is unstable. This however, does not necessarily mean that the norm of the states will continuously decrease. For example, for a system with state delay, when the delayed states first come into action, there could be a temporary increase in the norm. Alternately, there could be a nonlinear uncertainty that satisfies all our constraints and still has a spike leading to an increase in the norm of the states. Of course, the controller will seek to react to this situation and bring the system under control. Therefore, although mathematically not precise, it is nevertheless acceptable to assume in general, that the norm of the states decrease in the reaching phase. This conclusion is important to assess the behavior of the state norm in the reaching phase. For example, without such a guarantee, it is impossible to bound the discontinuous control signal d(t). From (5.24), d(t)= /31Cjx(t)j + E. Hence, the value of the discontinuous control is linked to the norm of the states. In proving asymptotic stability in the sliding mode, we obviate this difficulty, as sgn(s(t)) = 0. With the above conclusion, however, one could make a confident assumption on the expected value of d(t), much in the same vein of the Razhamikhin type assumption, that seeks to restrict the norm of the states. The following lemma is needed for the ensuing developments. Lemma 5.1 Consider the following scalar differential equation k(t)< ax(t)+(bexp(ct/2)+e exp(82t/2)d)x12(t) (5.61) with the initial condition x(O) and the following constraints: (1) The constants (a,b,c,d,e) > 0 and a # c and a # 52 (2) x>0, Vt >0 and Vx>0, Vt >0 Then, x+ bexp t) + exp t exp t a (a c) (5.62) e [(s 2 ( xpa t)] d +2)exp ex t i +(a 382)[ 2) (_ 2 a Proof: See Appendix G V Remark: In (5.62) the first three terms are positive, while the third term is a negative constant. Furthermore, the first term continuously decreases from the initial value, while the second and terms always have a maxima. As a parallel we remark that the second and third terms in (5.62) look like the response of an under damped second order system to unit impulse input. Upper Bound on the Norm of the States During the Reaching Phase In this section, the aim is to find an upper bound on the norm of the states, for the stable closedloop system. The analysis for the nominal system is straightforward and is presented in the sequel. However for the perturbed system, the analysis is quite involved. In the reaching phase, the dynamic equations are given by W(t) = (A,1 Al2K)W(t) + Tlf(x, t) + Al2st) (5.63) wi(t) = s(t) KW(t) (5.64) In complete parallel to the proof of Theorem 5.2, consider a Lyapunov candidate V(t) = WT(t)pW(t) (5.65) where P = PT > 0, p = PT > 0, p = 9 (nm)x(nin) and impose the restriction V(O)_8V(t) th<_O<_t,V8>1 (5.66a) It is straightforward to show from (5.66a) that /IZmin (P ) ITF(t)[ 56b Choose P > 0, Q > 0, such that 3(A11 A12K) + (A,1 A12K)Tp= Q (5.67) Taking the derivative of (5.65) yields V(t) WT(t)[P(A A12K) + (A,1 A12K)TP]wT(t) +2w (t)PTf(x,t) + 2wT(t)PA12s(t) Converting the Uncertainty in Terms of W(t) Again, the challenge is to convert the uncertainty description in terms of W(t). Since f(x(t),t)l < Ix(t)l, we run into a problem, since the stated aim is to find an upper bound on Ix(t)l In the ensuing development, we will pinpoint this difficulty and propose an additional assumption on the perturbation that will enable us to obviate this hurdle.. From Theorem 5.1, we have s(t) [S(O)I+ e 2 (5.69) I 1t~ ks0 l  k For simplicity, we denote /[=s(0)+ 2] d qe =k, d Hence s(t) < be 2 da. In the reaching mode, z(t) = T'w(t) = T W =) T [ Wt Lw = K(t). + T (t) Z(t) = T' 4, W(t) + T1 0"M]Lst 1' n 1 (q + where T2 is a matrix of dimension n x m, partitioned out of T. Since we used the transformation (5.3), from (5.42) Ix(t) < z(t)l + th a(thBU()d where the controller is given by u(t) = (CB)[CAz(t) + kjs(t)jsgn[s(t)] + d(t)sgn[s(t)]] since Isgn[s(t)] < m, we take an upper bound u(Ol (cn)'CAz(t) + m (&)1 [, kd + d(t)] Ix(t)lIz(t)j+I max e A] Bd(CB)1CA Jj hl(T)Id + Mmaxe eAO8 ] Bdjjj(Cffh] fth &kb2 0+ d()jdT Lh<_O<_O I Substituting for lz(t)l in terms of IW(t)l and [s(t)j, and using Razhamikhin type assumption for the delayed states, via (5.66b), results in Ix(01_< T1[ IW(t)+11T21 be d + h~a Tl[Inml max eA0 Bd(CB)ICA W(t)I LKoJ hO _o + mx aoe Bd(CB)1CA fth be2 d dz (5.70a) + m max eA1] JBd1 (CB jk)1 b 2 kd+d r)1dr kh<_5<_O5 "Jth kbI d+ (c ' In the above equation, the only unknown quantity is d(t). According to our control law, d(t) = P13 Icllx(t)l + E, and our stated aim is to find an upper bound on the norm of the 