Stability and stabilizability of discrete-time switched linear systems with state delay

The problems of stability and stabilizability of discrete-time switched linear systems whose subsystems are subject to state delays are investigated in this paper using linear matrix inequalities. The conditions, based on a Lyapunov-Krasovskii functional, assure the stability of the system for arbitrary switching functions, irrespective of the value of the time-delay. The switching function is assumed not known a priori but available in real time, the state as well as the delayed state vectors are supposed available for feedback, and the time-delay is considered unknown and unbounded. The proposed conditions can reduce the conservatism of the analysis and, thanks to extra variables, can be extended for synthesis purposes, providing switched gains for the state feedback control which encompass the fixed gains obtained from the standard quadratic stabilizability approach. Numerical examples illustrate the results.