Uniform exponential stability characterizations in switched linear systems with rank-one modes

Abstract

Novel characterizations for uniform exponential stability, under arbitrary switching, in discrete-time switched linear systems, whose modes are described by rank-one matrices, are reported and proved in the present paper. It is shown that the uniform exponential stability, of a switched linear system in this class, can be determined by means of the spectral radii of a given finite set of matrices associated to the system. It is also proved that the uniform exponential stability, in this class of switched linear systems, can be characterized in terms of the feasibility of a set of linear matrix inequalities (LMIs) associated to the given system (and equivalently, it can also be characterized in terms of the feasibility of a set of linear inequalities associated to the system). A constructive converse Lyapunov theorem, for uniformly exponentially stable switched linear systems in this class, is also included in the last characterization. It establishes that a uniformly exponentially stable switched linear system (in this class) always has a convex common Lyapunov function that is represented by as many quadratic functionals as modes composing the switched system. The reported results show that, for this class of switched linear systems, the uniform exponential stability can be determined in a computationally efficient manner. An example illustrates the reported results.