Absolute characteristic exponent of a class of linear nonstatinoary systems of differential equations

Barabanov, Nikita

doi:10.1007/bf00969859

Cited by 60 publications

(107 citation statements)

References 2 publications

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“…The existence of a quadratic common Lyapunov function for a family of linear systems whose matrices can be simultaneously put into the upper-triangular form has been pointed out before (see, e.g., [8,15] and related earlier work in [1]). It is important to recognize, however, that while it is a nontrivial matter to find a basis in which all matrices take the triangular form or even decide whether such a basis exists, the Lie-algebraic condition given by Theorem 2 is formulated in terms of the original data and can always be checked in a finite number of steps if P is a finite set.…”

Section: Lemmamentioning

confidence: 83%

“…The problem of finding conditions that guarantee asymptotic stability of (1) for an arbitrary switching signal σ has recently attracted a considerable amount of attention-see the work reported in [2,3,8,9,10,11,14,15] and the references therein. Some of the aforementioned results suggest that certain properties of the Lie algebra {A p : p ∈ P} LA generated by the matrices A p may be of relevance to the question of stability of (1). In particular, it is well known and easy to show that if these matrices commute pairwise, i.e., the Lie bracket [A p , A q ] := A p A q − A q A p equals zero for all p, q ∈ P, and if P is a finite set, then the system (1) is asymptotically stable for any switching signal σ.…”

Section: Introductionmentioning

confidence: 99%

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Stability of switched systems: a Lie-algebraic condition

Liberzon

Hespanha

Morse

1999

Systems & Control Letters

612

324

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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.

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Section: Lemmamentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Stability of switched systems: a Lie-algebraic condition

Liberzon

Hespanha

Morse

1999

Systems & Control Letters

612

324

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“…It is known that the related concepts of attractivity and asymptotic stability together are equivalent to exponential stability for switched linear systems [38,9,37]. For a switched linear system the exponential growth rate κ defined in (10) is negative if and only if (11) is satisfied for some β > 0, that is if and only if the origin is uniformly exponentially stable.…”

Section: Definitionmentioning

confidence: 99%

Stability Criteria for Switched and Hybrid Systems

Shorten¹,

Wirth²,

Mason³

et al. 2007

SIAM Rev.

912

558

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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 NP-hardness 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.

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“…Some results, relating marginal stability of (1) to the existence of limit cycles and periodic trajectories can be found in [2,4], while some general observations about marginal stability and instability can be found in [15]. It has to be noted that a qualitative study of the properties of the trajectories in the case ρ(A) = 0 leads to similar properties for all values of ρ, since, as observed in [2], ρ(A ′ ) = 0, where A ′ is the set {A − ρ(A)Id | A ∈ A} with Id denoting the n × n identity matrix. In this paper, we introduce the concept of resonance, which is rather natural for reducible switched systems with ρ(A) = 0.…”

mentioning

confidence: 99%

On the marginal instability of linear switched systems

Chitour

Mason

Sigalotti

2012

Systems & Control Letters

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Stability properties for continuous-time linear switched systems are at first determined by the (largest) Lyapunov exponent associated with the system, which is the analogous of the joint spectral radius for the discrete-time case. The purpose of this paper is to provide a characterization of marginally unstable systems, i.e., systems for which the Lyapunov exponent is equal to zero and there exists an unbounded trajectory, and to analyze the asymptotic behavior of their trajectories. Our main contribution consists in pointing out a resonance phenomenon associated with marginal instability. In the course of our study, we derive an upper bound of the state at time t, which is polynomial in t and whose degree is computed from the resonance structure of the system. We also derive analogous results for discrete-time linear switched systems. IntroductionWe consider linear switched systems of the forṁwhere x ∈ R n and the switching law A(·) is an arbitrary measurable function taking values on a compact and convex set A of n × n matrices. In the following, a switched system of the form (1) will be often identified with the corresponding set of matrices A. This paper is concerned with stability issues for (1), where the stability properties are assumed to be uniform with respect to the switching law A(·). * Y. Chitour and P. Mason are with

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Absolute characteristic exponent of a class of linear nonstatinoary systems of differential equations

Cited by 60 publications

References 2 publications

Stability of switched systems: a Lie-algebraic condition

Stability of switched systems: a Lie-algebraic condition

Stability Criteria for Switched and Hybrid Systems

On the marginal instability of linear switched systems

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