Finding Matrix Sequences with a High Asymptotic Growth Rate for Linear Constrained Switching Systems

TL;DR

This paper presents a novel algorithm to generate matrix sequences with high asymptotic growth rates for constrained switching systems, leveraging lifted systems and sum-of-squares optimization.

Contribution

It introduces a new method using lifted systems and sum-of-squares optimization to approximate the constrained joint spectral radius more effectively.

Findings

Proposed algorithms outperform existing methods in numerical tests.

The method effectively approximates the constrained joint spectral radius.

Lifted system approach enables applying arbitrary switching system algorithms.

Abstract

Linear constrained switching systems are linear switched systems whose switching sequences are constrained by a deterministic finite automaton. This work investigates how to generate a sequence of matrices with an asymptotic growth rate close to the constrained joint spectral radius (CJSR) for constrained switching systems, based on our previous result that reveals the equivalence of a constrained switching system and a lifted arbitrary switching system. By using the dual solution of a sum-of-squares optimization program, an algorithm is designed for the lifted arbitrary switching system to produce a sequence of matrices with an asymptotic growth rate that is close to the CJSR of the original constrained switching system. It is also shown that a type of existing algorithms designed for arbitrary switching systems can be applied to the lifted system such that the desired sequence of matrices can be generated for the constrained switching system. Several numerical examples are provided to illustrate the better performance of the proposed algorithms compared with existing ones.