Stability conditions for linear discrete‐time switched systems in block companion form. Issue 19 (9th February 2021)

• Record Type: Journal Article
• Authors: De Iuliis, Vittorio; D'Innocenzo, Alessandro; Germani, Alfredo; Manes, Costanzo
• Is Part Of: IET control theory & applications. Volume 14:Issue 19(2020)
• Page Start: 3107
• Page End: 3115
• Publication Date: 2021-02-09
• Subjects: time‐varying systems -- linear systems -- delays -- Lyapunov methods -- stability -- discrete time systems -- matrix algebra; stability conditions -- linear discrete‐time switched systems -- block companion form -- switched models whose dynamic matrices -- theoretical applicative problems -- ARX models -- state‐space form -- delay‐independent stability -- discrete‐time systems -- time‐varying delays -- arbitrary switching stability -- block companion discrete‐time switched systems -- block‐row -- nonnegative matrices -- simple necessary stability condition -- sufficient stability condition -- linear common Lyapunov function; Control theory -- Periodicals; Automatic control -- Periodicals; 629.8312
• DOI: 10.1049/iet-cta.2020.0754
• Languages: English
• ISSNs: 1751-8644

Abstract

Abstract : Switched models whose dynamic matrices are in block companion form arise in theoretical and applicative problems such as representing switched ARX models in state‐space form for control purposes. Inspired by some insightful results on the delay‐independent stability of discrete‐time systems with time‐varying delays, in this work, the authors study the arbitrary switching stability for some classes of block companion discrete‐time switched systems. They start from the special case in which the first block‐row is made of permutations of non‐negative matrices, deriving a simple necessary and sufficient stability condition under arbitrary switching. The condition is computationally less demanding than the sufficient‐only existence of a linear common Lyapunov function. Then, both non‐negativity and combinatorial assumptions are dropped, at the expense of introducing conservatism. Some implications on the computation of the joint spectral radius for the aforementioned families of matrices are illustrated.