Stabilization of polytopic delay difference inclusions via the Razumikhin approach

Polytopic delay difference inclusions (DDIs) have received increasing attention recently, mostly due to their ability to model a wide variety of relevant processes, including networked control systems. One of the fundamental problems for DDIs that poses a non-trivial challenge is stabilization. This paper embraces the Razumikhin approach and provides several solutions to the stabilization problem as follows. Firstly, a method to synthesize a control Lyapunov–Razumikhin function (cLRF) is presented for unconstrained DDIs. Secondly, for constrained DDIs, a receding horizon controller based on the cLRF for the unconstrained system is proposed, along with a closed-loop stability analysis. Thirdly, it is shown that a tractable implementation of the developed control algorithm can be attained even for large delays, by means of an on-line Minkowski set addition. An advantageous feature of the developed methodology is that all the synthesis algorithms can be formulated as a low complexity semi-definite programming problem for quadratic cLRF candidates. A comparison with alternative synthesis methods demonstrates the advances provided by the developed theory.