On the Marginal Instability of Linear Switched Systems

Yacine Chitour (1) , Paolo Mason (1) , Mario Sigalotti (2, 3)

1 L2S - Laboratoire des signaux et systèmes
2 IECN - Institut Élie Cartan de Nancy
3 CORIDA - Robust control of infinite dimensional systems and applications

Stability properties for continuous-time linear switched systems are determined by the Lyapunov exponent associated with the system, which is the analogous of the joint spectral radius for the discrete-time case. This paper is concerned with the characterizations of stability properties when the Lyapunov exponent is zero. In this case it is well known that the system can be stable as well as unstable, even if it is never asymptotically stable nor it admits a trajectory blowing up exponentially fast. Our main result asserts that a switched system whose Lyapunov exponent is zero may be unstable only if a certain resonance condition is satisfied.

Mots clés: switched systems; polynomial instability

Domaines: Automatique / Robotique; Optimisation et contrôle [math.OC]

Yacine Chitour, Paolo Mason, Mario Sigalotti. On the Marginal Instability of Linear Switched Systems. 49th IEEE Conference on Decision and Control, Dec 2010, Atlanta, United States. 6 p., ⟨10.1109/cdc.2010.5717638⟩. ⟨hal-00522676⟩