On topological entropy of switched linear systems with pairwise commuting matrices

Abstract

We study a notion of topological entropy for switched systems, formulated in terms of the minimal number of initial states needed to approximate all initial states within a finite precision. This paper focuses on the topological entropy of switched linear systems with pairwise commuting matrices. First, we prove there exists a simultaneous change of basis under which each of the matrices can be decomposed into a diagonal part and a nilpotent part, and all the diagonal and nilpotent parts are pairwise commuting. Then a formula for the topological entropy is established in terms of the component- wise averages of the eigenvalues, weighted by the active time of each mode, which indicates that the topological entropy is independent of the nilpotent parts above. We also present how the formula generalizes known results for the non-switched case and the case with simultaneously diagonalizable matrices, and construct more general but more conservative upper bounds for the entropy. A numerical example is provided to demonstrate properties of the formula and the upper and lower bounds for the topological entropy.