Approximation of The Constrained Joint Spectral Radius via Algebraic Lifting

Abstract

This paper studies the constrained switching (linear) system which is a discrete-time switched linear system whose switching sequences are constrained by a deterministic finite automaton. The stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate. Using the semi-tensor product of matrices, the matrix-form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system. Then the constrained joint/generalized spectral radius of a constrained switching system is proved to be equal to the joint/generalized spectral radius of its lifted arbitrary switching system which can be approximated by off-the-shelf algorithms.