Approximation of The Constrained Joint Spectral Radius via Algebraic Lifting

TL;DR

This paper introduces a novel algebraic lifting method using semi-tensor products to approximate the constrained joint spectral radius of switched linear systems, simplifying the computation process.

Contribution

It establishes an equivalence between the constrained joint spectral radius and that of a lifted arbitrary system, enabling easier approximation with existing algorithms.

Findings

Equivalence between constrained and lifted system spectral radii.

Facilitates approximation using standard algorithms.

Provides a new algebraic framework for constrained system stability.

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.