A note on stability conditions for planar switched systems

Abstract

This article is concerned with the stability problem for the planar linear switched system , where the real matrices A ₁, A ₂ ∈ ℝ^2×2 are Hurwitz and u(·) : [0, ∞ [→ {0, 1} is a measurable function. We give coordinate-invariant necessary and sufficient conditions on A ₁ and A ₂ under which the system is asymptotically stable for arbitrary switching functions u(·). The new conditions unify those given in previous papers and are simpler to be verified since we reduced to study 4 cases instead of 20. Most of the cases are analysed in terms of the function .

Keywords:

• planar switched systems
• asymptotic stability
• quadratic Lyapunov functions

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Corrigendum and addendum to

Acknowledgements

M. Balde and U. Boscain were supported by a FABER grant of région Bourgogne. M. Balde would like to thank the Laboratoire des Signaux et Systémes (LSS–Supélec) for its kind hospitality during the writing of this article.

Notes

Notes

1. The stability conditions given in Boscain (2002) were not correct in the case called RC.2.2.B. See Mason et al. (2006) for the correction.

2. In Molchanov and Pyatnitskiĭ (1986a, b), Mason et al. (2006), Blanchini and Miani (1999), it is actually shown that the GUAS property is equivalent to the existence of a polynomial LF.