Stability and stabilisability of switched discrete-time systems based on structured Lyapunov functions

This study addresses the stability and the stabilisability problem for discrete-time switched systems under arbitrary switching, by employing structured Lyapunov functions. The main contributions are: (i) the development of new necessary and sufficient conditions for the stability problem in terms of linear matrix inequalities (LMIs) that can provide stability certificates requiring a smaller number of decision variables and LMI rows than existing approaches; (ii) a new LMI condition derived in terms of the modes of the switched system to deal with the stabilisability problem considering switching state-feedback gains. The stabilisation problem makes use of the structured Lyapunov function to provide less conservative results. Benchmark examples from the literature are presented to illustrate the efficacy of the proposed approach.