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219
Basic problems in stability and design of switched systems
 IEEE Control Systems Magazine
, 1999
"... By a switched system, we mean a hybrid dynamical system consisting of a family of continuoustime subsystems and a rule that orchestrates the switching between them. This article surveys recent developments in three basic problems regarding stability and design of switched systems. These problems ar ..."
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Cited by 379 (10 self)
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By a switched system, we mean a hybrid dynamical system consisting of a family of continuoustime subsystems and a rule that orchestrates the switching between them. This article surveys recent developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.
Perspectives and Results on the Stability and Stabilizability of Hybrid Systems
 PROCEEDINGS OF THE IEEE
, 2000
"... This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable ..."
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Cited by 202 (2 self)
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This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavor, this paper surveys the major results in the (Lyapunov) stability of finitedimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.
Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results
, 2005
"... During the last decade, there has been increasing interest in the stability analysis and switching control design for switched linear systems. This paper aims to briefly survey recent results in this field, focusing on stability analysis and switching stabilization problems. First, the stability an ..."
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Cited by 126 (10 self)
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During the last decade, there has been increasing interest in the stability analysis and switching control design for switched linear systems. This paper aims to briefly survey recent results in this field, focusing on stability analysis and switching stabilization problems. First, the stability analysis problem for switched linear systems is reviewed. We focus on the asymptotic stability analysis for switched linear systems under arbitrary switching, and highlight necessary and sufficient conditions for this problem. Secondly, the switching stabilization problem is studied, and a variety of switching stabilization methods found in the literature are outlined. One of the most elusive problems in the switched systems literature has been the switching stabilizability problem, that is under what condition it is possible to stabilize a switched system by properly designing switching control laws. Necessary and sufficient conditions for asymptotic stabilizability of switched linear systems are described.
Stability criteria for switched and hybrid systems
 SIAM Review
, 2007
"... The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, an ..."
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Cited by 114 (8 self)
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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NPhardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
Stable flocking of multiple inertial agents on balanced graphs
 Computer Science, The University of Newcastle
, 2006
"... and the optimum value of max[P (0)] was max[P (0)] = 00:40844 < 0 which indicates that this system has no robustly unobservable states. For the optimal value of given above, a plot of max[P (t)] as a function of t is shown in Fig. 6. ..."
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Cited by 36 (7 self)
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and the optimum value of max[P (0)] was max[P (0)] = 00:40844 < 0 which indicates that this system has no robustly unobservable states. For the optimal value of given above, a plot of max[P (t)] as a function of t is shown in Fig. 6.
Nonlinear normobservability notions and stability of switched systems
 IEEE TRANS. AUTOMAT. CONTR
, 2005
"... This paper proposes several definitions of “normobservability” for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunovli ..."
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Cited by 35 (2 self)
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This paper proposes several definitions of “normobservability” for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunovlike sufficient condition for normobservability is also obtained. As an application, we prove several variants of LaSalle’s stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.
Disturbance Attenuation Properties of TimeControlled Switched Systems
 ECC2001
, 2001
"... In this paper, we investigate the disturbance attenuation properties of timecontrolled switched systems consisting of several linear timeinvariant subsystems by using a dwell time approach incorporated with piecewise Lyapunov functions. First, we show that when all subsystems are Hurwitz stable and ..."
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Cited by 35 (7 self)
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In this paper, we investigate the disturbance attenuation properties of timecontrolled switched systems consisting of several linear timeinvariant subsystems by using a dwell time approach incorporated with piecewise Lyapunov functions. First, we show that when all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than # 0 , then the switched system can achieve any disturbance attenuation level larger than # 0 if the dwell time is chosen sufficiently large. This result is extended to the case where not all subsystems are Hurwitz stable, by showing that if the dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then a desirable disturbance attenuation level is guaranteed. Finally, a discussion is made on the case for which nonlinear normbounded perturbations exist in the subsystems.
Multiple Model Adaptive Control, Part 2: Switching
"... This paper addresses the problem of controlling a continuoustime linear system with large modeling errors. We employ an adaptive control algorithm consisting of a family of linear candidate controllers supervised by a highlevel switching logic. Methods for constructing such controller families ..."
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Cited by 35 (10 self)
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This paper addresses the problem of controlling a continuoustime linear system with large modeling errors. We employ an adaptive control algorithm consisting of a family of linear candidate controllers supervised by a highlevel switching logic. Methods for constructing such controller families have been discussed in the recent paper by the authors. The present paper concentrates on the switching task in a multiple model context. We describe and compare two different switching logics, and in each case study the behavior of the resulting closedloop hybrid system.
Overcoming the Limitations of Adaptive Control By Means of LogicBased Switching
, 2002
"... In this paper we describe a framework for deterministic adaptive control which involves logicbased switching among a family of candidate controllers. We compare it with more conventional adaptive control techniques that rely on continuous tuning, emphasizing how switching and logic can be used to o ..."
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Cited by 34 (8 self)
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In this paper we describe a framework for deterministic adaptive control which involves logicbased switching among a family of candidate controllers. We compare it with more conventional adaptive control techniques that rely on continuous tuning, emphasizing how switching and logic can be used to overcome some of the limitations of traditional adaptive control. The issues are discussed in a tutorial, nontechnical manner and illustrated with specific examples.
Stability Analysis of Switched Systems with Stable and Unstable Subsystems: An Average Dwell Time Approach
 PROCEEDINGS OF THE 2000 AMERICAN CONTROL CONFERENCE
, 2000
"... We study the stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems using an average dwell time approach. We show that if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small co ..."
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Cited by 33 (6 self)
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We study the stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems using an average dwell time approach. We show that if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then exponential stability of a desired degree is guaranteed. We also derive a piecewise Lyapunov function for the switched system subjected to the switching law and the average dwell time scheme under consideration, and we extend these results to the case for which nonlinear normbounded perturbations exist in the subsystems. We show that when the norms of the perturbations are small, we can modify the switching law appropriately to guarantee that the solutions of the switched system converge to the origin exponentially with large average dwell time.