Graph-Based Minimum Dwell Time and Average Dwell Time Computations for Discrete-Time Switched Linear Systems

TL;DR

This paper introduces a graph-based approach to compute minimum and average dwell times ensuring stability in discrete-time switched linear systems, including those with defective matrices, by leveraging cycle ratios and means.

Contribution

It presents a novel graph-theoretic method for dwell time computation that handles defective matrices using Jordan decomposition and scaling algorithms for improved estimates.

Findings

Effective computation of dwell times via maximum cycle ratio and mean.

Application of Jordan decomposition to defective matrices.

Scaling algorithms improve dwell time estimates.

Abstract

Discrete-time switched linear systems where switchings are governed by a digraph are considered. The minimum (or average) dwell time that guarantees the asymptotic stability can be computed by calculating the maximum cycle ratio (or maximum cycle mean) of a doubly weighted digraph where weights depend on the eigenvalues and eigenvectors of subsystem matrices. The graph-based method is applied to systems with defective subsystem matrices using Jordan decomposition. In the case of bimodal switched systems scaling algorithms that minimizes the condition number can be used to give a better minimum (or average) dwell time estimates.