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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 theory for hybrid dynamical systems
 IEEE Transactions on Automatic Control
, 1998
"... Abstract — Hybrid systems which are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system (e.g., continuoustime dynamics, discretetime dynamics, jump phenomena, switching and logic commands, and the like) are of great current interest. In the present ..."
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Cited by 90 (8 self)
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Abstract — Hybrid systems which are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system (e.g., continuoustime dynamics, discretetime dynamics, jump phenomena, switching and logic commands, and the like) are of great current interest. In the present paper we first formulate a model for hybrid dynamical systems which covers a very large class of systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of an invariant set (e.g., equilibrium) for hybrid dynamical systems and we define several types of (Lyapunovlike) stability concepts for an invariant set. We then establish sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability). To demonstrate the applicability of the developed theory, we present specific examples of hybrid dynamical systems and we conduct a stability analysis of some of these examples (a class of sampleddata feedback control systems with a nonlinear (continuoustime) plant and a linear (discretetime) controller, and a class of systems with impulse effects). Index Terms — Asymptotic stability, boundedness, dynamical system, equilibrium, exponential stability, hybrid, hybrid dynamical
Stability And Robustness For Hybrid Systems
, 1996
"... Stability and robustness issues for hybrid systems are considered in this paper. General stability results that are extensions of classical Lyapunov theory have recently been formulated. However, these results are in general not straightforward to apply due to the following reasons. First, a search ..."
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Cited by 59 (6 self)
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Stability and robustness issues for hybrid systems are considered in this paper. General stability results that are extensions of classical Lyapunov theory have recently been formulated. However, these results are in general not straightforward to apply due to the following reasons. First, a search for multiple Lyapunov functions must be performed. However, existing theory does not unveil how to find such functions. Secondly, if the most general stability result is applied, knowledge about the continuous trajectory is required, at least at some time instants. Because of these drawbacks stronger conditions for stability are suggested, in which case it is shown that the search for Lyapunov functions can be formulated as a linear matrix inequality (LMI) problem for hybrid systems consisting of linear subsystems. Additionally, it is shown how robustness properties can be achieved when the Lyapunov functions are given. Specifically, it is described how to determine permitted switch regions ...
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 ...
A notion of passivity for hybrid systems
"... We propose a notion of passivity for hybrid systems. Our work is motivated by problems in haptics and teleoperation where several computer controlled mechanical systems are connected through a communication channel. To account for time delays and to better react to user actions it is desirable to de ..."
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Cited by 17 (0 self)
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We propose a notion of passivity for hybrid systems. Our work is motivated by problems in haptics and teleoperation where several computer controlled mechanical systems are connected through a communication channel. To account for time delays and to better react to user actions it is desirable to design controllers that can switch between different operating modes. Each of the interacting systems can be therefore naturally modeled as a hybrid system. A traditional passivity definition requires that a storage function exists that is common to all operating modes. We show that stability of the system can be guaranteed even if different storage function is found for each of the modes, provided appropriate conditions are satisfied when the system switches.
Towards a Stability Theory of General Hybrid Dynamical Systems
 Automatica
, 1998
"... In recent work we proposed a general model for hybrid dynamical systems whose states are dened on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possib ..."
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Cited by 16 (6 self)
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In recent work we proposed a general model for hybrid dynamical systems whose states are dened on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but dened on real time R + = [0; 1). The motions of this class of systems are in general discontinuous. This class of systems may be nite or innite dimensional. For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specic classes of hybrid dynamical systems. In the present paper we continue the...
Design of switching controllers for systems with changing dynamics
 Proc. Conf. on Decision and Control
, 1998
"... We present a framework for designing stable control schemes for systems with changing dynamics (SCD). Such systems form a subset of hybrid systems; their stabilization is therefore a problem in hybrid control. It is often difficult or even impossible to design a single controller that would stabiliz ..."
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Cited by 15 (5 self)
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We present a framework for designing stable control schemes for systems with changing dynamics (SCD). Such systems form a subset of hybrid systems; their stabilization is therefore a problem in hybrid control. It is often difficult or even impossible to design a single controller that would stabilize a SCD. An appealing alternative are switching control schemes, where a different controller is employed in each dynamic regime and the stability of the overall system is ensured through an appropriate switching scheme. We formulate a set of sufficient conditions for the stability of a switching control scheme. We show that by imposing a hierarchy among the controllers, sufficient conditions can be formulated in a form suitable for the controller design. The hierarchy is formally defined through a partial order. Our methodology is applied to stabilization of a twowheel mobile robot of the Hilare type, where the wheels are allowed to slip.
Stabilization of systems with changing dynamics
 in Hybrid Systems: Computation and Control, LNCS 1386
, 1998
"... Abstract. We present a framework for designing stable control schemes for systems whose dynamic equations change as they evolve on the state space. It is usually difficult or even impossible to design a single controller that would stabilize such a system. An appealing alternative are switching cont ..."
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Cited by 11 (1 self)
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Abstract. We present a framework for designing stable control schemes for systems whose dynamic equations change as they evolve on the state space. It is usually difficult or even impossible to design a single controller that would stabilize such a system. An appealing alternative are switching control schemes, where a different controller is employed on each of the regions defined by different dynamic characteristics and the stability of the overall system is ensured through appropriate switching scheme. We derive sufficient conditions for the stability of a switching control scheme in a form that can be used for controller design. An important feature of the proposed framework is that although the overall hierarchy can be very complicated, the stability depends only on the immediate relation of each controller to its neighbors. This makes the application of our results particularly straight forward. The methodology is applied to stabilization of a shimmying wheel, where changes in the dynamics are due to switches between sliding and rolling. 1
Stabilization of systems with changing dynamics by means of switching
, 1998
"... We present a framework for designing stable control schemes for systems whose dynamics change. The idea is to develop a controller for each of the regions defined by different dynamic characteristics and design a switching scheme that guarantees the stability of the overall system. We derive suffic ..."
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Cited by 11 (0 self)
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We present a framework for designing stable control schemes for systems whose dynamics change. The idea is to develop a controller for each of the regions defined by different dynamic characteristics and design a switching scheme that guarantees the stability of the overall system. We derive sufficient conditions for the stability of the switching scheme for systems evolving on a sequence of embedded manifolds. An important feature of the proposed framework is that if the conditions are satisfied by pairs of controllers adjacent in the hierarchy, the overall system will be stable. This makes the application of our results particularly straight forward. The methodology is applied to stabilization of a shimmying wheel, where changes in the dynamic behavior are due to switches between sliding and rolling.