Stabilization of polytopic delay difference inclusions: Time-varying control Lyapunov functions

This paper deals with stabilization of polytopic delay difference inclusions (DDIs) via the Razumikhin approach. As not every DDI that is globally exponentially stable (GES) admits a standard Lyapunov-Razumikhin function (LRF), time-varying Lyapunov functions are proposed as a tool for stability analysis and stabilizing controller synthesis for polytopic DDIs. It is shown that any polytopic DDI that is GES admits a quadratic time-varying Lyapunov-Krasovskii function and, under an additional mild assumption, it also admits a quadratic time-varying LRF. These results are further exploited to derive a stabilizing receding horizon control scheme that does not suffer from an exponential increase in complexity when the size of the delay increases. The proposed control scheme requires solving on-line a low-dimensional semi-definite program.