Asymptotic stability of linear switched systems: Observability approach and convergence rate

This paper mainly deals with switched linear systems defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function. Its aim is to give sufficient conditions for such a system to be GUAS and to study its convergence rate. We show that this property of being GUAS is equivalent to the uniform observability on [0, +∞) of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system. Then we focus our attention on the convergence rate of the solutions of linear switched systems. For that purpose we consider GUAS systems on the one hand and systems asymptotically stable only for inputs with dwell-time on the other one.