Feedback stabilisation of switched systems via iterative approximate eigenvector assignment

TL;DR

This paper introduces an iterative feedback design algorithm for stabilizing discrete-time switched systems, aiming to find feedback gains that ensure a common quadratic Lyapunov function and exponential stability, while preserving system structure.

Contribution

It provides a numerical implementation for single-input systems based on iterative approximate eigenvector assignment and identifies cases where the algorithm guarantees success.

Findings

Algorithm successfully stabilizes systems under certain conditions.

Feedback matrices approximate Lie-algebraic structure for CQLF.

Numerical examples demonstrate advantages and limitations.

Abstract

This paper presents and implements an iterative feedback design algorithm for stabilisation of discrete-time switched systems under arbitrary switching regimes. The algorithm seeks state feedback gains so that the closed-loop switching system admits a common quadratic Lyapunov function (CQLF) and hence is uniformly globally exponentially stable. Although the feedback design problem considered can be solved directly via linear matrix inequalities (LMIs), direct application of LMIs for feedback design does not provide information on closed-loop system structure. In contrast, the feedback matrices computed by the proposed algorithm assign closed-loop structure approximating that required to satisfy Lie-algebraic conditions that guarantee existence of a CQLF. The main contribution of the paper is to provide, for single-input systems, a numerical implementation of the algorithm based on iterative approximate common eigenvector assignment, and to establish cases where such algorithm is guaranteed to succeed. We include pseudocode and a few numerical examples to illustrate advantages and limitations of the proposed technique.