Robust sliding mode control of non-linear systems with delay: a design via polytopic formulation

This paper outlines the possibility to control non-linear uncertain systems with single, possibly time-varying state-delay. These systems are submitted to additive perturbations (possibly non-vanishing). The control is of sliding mode type and the sliding surface is designed to maximize the calculable set of delays which do not destabilize the relay–delay system. The way to deal with the non-linear parts of the model relies on a polytopic formulation, i.e. a decomposition into a non-linear combination of linear models, whose weighting coefficients satisfy a convex property. The conditions for the existence of the sliding regime are studied by using both a Lyapunov–Krasovskii functional and a Lyapunov–Razumikhin function. The optimization procedure for the choice of the sliding surface is expressed in terms of LMIs. An example illustrates the proposed method.