On the Stability of Switched Positive Linear Systems

In this paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of "linear" stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results.

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“…, A N } are the system matrices of the constituent systems, which are Metzler matrices. See [21,7] for more details on systems of this type.…”

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confidence: 99%

On linear co-positive Lyapunov functions for sets of linear positive systems

2009

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“…Moreover, there can exist system classes for which the existence of a CQLF for any pair of systems in a finite family implies the existence of a CQLF for the entire family. This was shown to be the case for the class of second order positive systems in Gurvits et al (2004). This fact provides further motivation for the study of the problem of CQLF existence for pairs of systems.…”

Section: Introductionmentioning

confidence: 65%

Some results on quadratic stability of switched systems with interval uncertainty

Zeheb

Mason

Solmaz

et al. 2007

International Journal of Control

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“…Recently, Gurvits et al [13] proved that this conjecture is in general false (see also [12]). However, they also showed that it does hold when n = 2 (even when the number of matrices is arbitrary).…”

Section: Conjecture 2 If (1) Is a Pls Then Assumption 1 Provides A Sumentioning

confidence: 99%

Stability Analysis of Positive Linear Switched Systems: A Variational Approach

Margaliot

Branicky

2009

IFAC Proceedings Volumes

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We consider the stability analysis of switched systems under arbitrary switching laws. A powerful approach for addressing this problem is based on studying the "most unstable" switching law (MUSL). If the switched system is stable for the MUSL, it is stable for any switching law. The MUSL can be defined as the solution to a certain optimal control problem. The main analysis tool is then the celebrated maximum principle (MP). This provides an implicit characterization of the MUSL in terms of the adjoint vector and switching function defined in the MP. In this paper, we consider the optimal control problem defining the MUSL for the particular case of positive linear switched systems. We show that in this case the adjoint vector must be positive for all time. We demonstrate two consequents of this property. First, there exist regions in the state-space for which the MUSL is known explicitly. Second, if certain Lie-brackets are componentwise positive matrices then the number of switchings in the MUSL is bounded by a fixed number for any final time. We describe several applications of our results to the stability analysis of positive switched linear systems under arbitrary switching laws.

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On the Stability of Switched Positive Linear Systems

Cited by 316 publications

References 15 publications

On linear co-positive Lyapunov functions for sets of linear positive systems

On linear co-positive Lyapunov functions for sets of linear positive systems

Some results on quadratic stability of switched systems with interval uncertainty

Stability Analysis of Positive Linear Switched Systems: A Variational Approach

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