Feedback rectifiable pairs and stabilization of switched linear systems

TL;DR

This paper develops a feedback design method for switched linear systems to ensure stability by aligning eigenstructures of subsystems through a constructive feedback rectification approach.

Contribution

It introduces necessary and sufficient conditions for feedback rectifiability and provides an algorithm for stabilizing feedback design for switched systems.

Findings

Conditions for feedback rectifiability of matrix pairs

Constructive algorithm for stabilizing feedback design

Examples demonstrating the effectiveness of the method

Abstract

We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share the same eigenstructure. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. In particular the proposed algorithm provides sets of eigenvalues and corresponding eigenvectors for the closed-loop subsystems that guarantee stability for arbitrary switching. Several examples illustrate the characteristics of the problem considered and the application of the proposed design procedure.