Stability conditions for linear discrete‐time switched systems in block companion form

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.