Stability of a set of matrices: an application to hybrid systems

Abstract

Asymptotic stability and stabilizability of a set of matrices are defined and investigated. Asymptotic stability of a set of matrices requires that all infinite products of matrices from that set tend to zero. Asymptotic stabilizability of a set of matrices, however, requires that there is at least one such sequence in the set. The upper and lower spectral radius of a set are defined to aid in the analysis. Necessary and sufficient conditions for asymptotic stability and stabilizability are provided leading to some methods using Lyapunov theory. Finally hybrid system stability is considered when the continuous state part of the hybrid system is modeled as a linear discrete time system. It is shown that the concept of stability of matrix sets may be helpful in analysis and control design of such hybrid systems.