Stability of a set of matrices with applications to automatic control

Abstract

The concepts of 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 and linear matrix inequalities. Finally some problems from different areas of control are considered including hybrid systems. It is shown that the theory of matrix sets is helpful in analysis and design of certain classes of control systems.