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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 204 (9 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.
Stability of switched systems: a Liealgebraic condition
 Systems Control Lett
, 1999
"... We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family ..."
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Cited by 53 (9 self)
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We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.
LogicBased Switching Algorithms in Control
, 1998
"... This thesis deals with the use of logicbased switching in the control of imprecisely modeled nonlinear systems. Each control system considered consists of a continuoustime dynamical process to be controlled, a family of candidate controllers, and an eventdriven switching logic. The need for switc ..."
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Cited by 39 (23 self)
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This thesis deals with the use of logicbased switching in the control of imprecisely modeled nonlinear systems. Each control system considered consists of a continuoustime dynamical process to be controlled, a family of candidate controllers, and an eventdriven switching logic. The need for switching arises when no single candidate controller is capable, by itself, of guaranteeing good performance when connected with a poorly modeled process. In this thesis we develop provably correct switching strategies capable of determining in realtime which candidate controller should be put in feedback with a process so as to achieve a desired closedloop performance. The resulting closedloop systems are hybrid in the sense that in each case, continuous dynamics interact with eventdriven logic. In the process of designing these switching algorithms, we develop several tools for the analysis and synthesis o...
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 35 (4 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.
Stabilization of SecondOrder LTI Switched Systems
, 1999
"... In this paper, the problem of asymptotically stabilizing switched systems consisting of secondorder LTI subsystems is studied and solved. In particular, necessary and suÆcient stabilizability conditions are derived and a design procedure to construct stabilizing switching laws is introduced. Switch ..."
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Cited by 23 (5 self)
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In this paper, the problem of asymptotically stabilizing switched systems consisting of secondorder LTI subsystems is studied and solved. In particular, necessary and suÆcient stabilizability conditions are derived and a design procedure to construct stabilizing switching laws is introduced. Switching is needed for the stabilization of a switched system if none of its subsystems is stable. Switched systems consisting of subsystems with unstable foci are studied rst and stabilizing conic switching control laws for such systems are introduced. In particular, necessary and suÆcient conditions for asymptotic stabilizability are derived for such systems. This result is then extended to switched systems with unstable nodes and saddle points. If a switched system is asymptotically stabilizable, then using the conic switching approach introduced earlier, asymptotically stabilizing switching control laws can be obtained. Furthermore, the conic switching laws derived in the paper are shown to ...
Benchmark problems in stability and design of switched systems
 IEEE Control Systems Magazine
, 1999
"... systems ..."
InputOutput Gains of Switched Linear Systems
 in Open Problems in Mathematical Systems Theory and Control
, 1998
"... oe in S, the preceding defines a timevarying linear system of the form \Sigma oe \Delta = ( x = M oe x + D oe u y = H oe x ) where u is an integrable input signal taking values in IR nu . Thus if x(0) \Delta = 0, then y = Y oe (u), where Y oe is the inputoutput mapping u 7\Gamma! Z t ..."
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Cited by 5 (4 self)
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oe in S, the preceding defines a timevarying linear system of the form \Sigma oe \Delta = ( x = M oe x + D oe u y = H oe x ) where u is an integrable input signal taking values in IR nu . Thus if x(0) \Delta = 0, then y = Y oe (u), where Y oe is the inputoutput mapping u 7\Gamma! Z t 0 H oe(t) \Phi oe (t; ø)D oe(ø ) dø; and \Phi oe is the state transition matrix of M oe . Let prime denote transpose and, for any integr
On the Reachability of a Class of SecondOrder Switched Systems
, 1999
"... In this paper, the reachability problem for a class of secondorder LTI switched systems is solved. The reachability problem is rst explored for switched systems consisting of two subsystems and switching control laws are proposed that can drive the system state from an initial point to a target poi ..."
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Cited by 4 (1 self)
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In this paper, the reachability problem for a class of secondorder LTI switched systems is solved. The reachability problem is rst explored for switched systems consisting of two subsystems and switching control laws are proposed that can drive the system state from an initial point to a target point via nitely many switches. The method is then extended to the case of several subsystems. The robustness issue is also addressed briey and the relationship between stabilizability and reachability is discussed. 1 Introduction A switched system is a system that consists of several subsystems and a switching law that speci es which subsystem dynamics will be followed by the system trajectory at each instant of time. Switchings may be timedriven (i.e., a switching happens at specic time instants) or eventdriven (i.e., a switching happens when some internal or external event takes place). Recently, there has been increasing interest in the stability analysis and design of such systems ...