Spectral and Convex Uniform Exponential Stability Determination in a Class of Switched Linear Systems

This communication is concerned with uniform exponential stability, under arbitrary switching, in discrete-time switched linear systems having all except one of their modes represented by rank-one matrices. It is proved that a switched linear system, in this class, is uniformly exponentially stable if and only if a new switched linear system with rank-one modes (associated to the original switched system) is also uniformly exponentially stable. It is shown that uniform exponential stability, in this class, can always be determined either in terms of the feasibility of a set of linear inequalities or in terms of the spectral radii of a (finite) set of matrices.