Computation of stable convex combinations of a set of matrices through LMIs with application in H∞ control of switched linear systems

Abstract

This paper proposes an LMI-based algorithm to compute a stable (Hurwitz) convex combination of a set of (unstable) matrices as primary contribution. Using stability analysis conditions based on Finsler’s lemma and a simplicial partitioning of the unit simplex, a semidecidable procedure is provided. As immediate application, the problem of $H_{\infty }$ state- and output-feedback control of continuous-time switched linear systems is addressed. Exploring the connection between stable convex combination and the Lyapunov–Metzler inequalities, the synthesis conditions are also formulated in terms of an LMI-based algorithm, which presents advantages when compared with the existing techniques for switching control. The technical novelty in this context is a new linearization strategy to deal with the product between matrix and scalar variables. Numerical experiments illustrate the quality of the achievements.