The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem

Mason, Oliver; Shorten, Robert

doi:10.13001/1081-3810.1144

Cited by 17 publications

(4 citation statements)

References 32 publications

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“…These functions are only required to satisfy the requirements of a traditional Lyapunov function within the nonnegative orthant and may lead to less conservative stability conditions for positive switched linear systems than can be obtained using traditional Lyapunov functions. Some initial results on common copositive Lyapunov function existence can be found in [61,108,107] …”

Section: Positive Switched Linear Systemsmentioning

confidence: 99%

Stability Criteria for Switched and Hybrid Systems

Shorten¹,

Wirth²,

Mason³

et al. 2007

SIAM Rev.

Self Cite

914

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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Section: Positive Switched Linear Systemsmentioning

confidence: 99%

Stability Criteria for Switched and Hybrid Systems

Shorten¹,

Wirth²,

Mason³

et al. 2007

SIAM Rev.

Self Cite

914

558

View full text Add to dashboard Cite

show abstract

“…This, together with (13), shows that for t ∈ {1, 2}, (12) holds. Suppose that (12) holds for any t ∈ {1, 2, .…”

Section: X(t) Y(t)mentioning

confidence: 57%

Stability Analysis of Positive Systems With Bounded Time-Varying Delays

Liu

Wang

2009

IEEE Trans. Circuits Syst. II

157

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This brief addresses stability of the discrete-time positive systems with bounded time-varying delays and establishes some necessary and sufficient conditions for asymptotic stability of such systems. It turns out that, for any bounded time-varying delays, the magnitude of the delays does not affect the stability of these systems. In other words, system stability is completely determined by the system matrices.

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“…The convex cones C 1 , C 2 are non-empty as the systemsÊ iẋi (t) =Â ixi (t) are asymptotically stable [32,115] and they comprise all stability-preserving solutions P i . Figure 5.3: Convex cones C 1 (green), C 2 (blue) and P 1 , P 2 (red).…”

Section: Academic Examplementioning

confidence: 99%