Gain scheduled control for discrete-time systems depending on bounded rate parameters

In this paper we consider a linear, discrete-time system depending multi-affinely on uncertain, real time-varying parameters. A new sufficient condition for the stability of this class of systems, in terms of a feasibility problem involving linear matrix inequalities (LMIs), is obtained under the hypothesis that a bound on the rate of variation of the parameters is known. This condition, obtained by the aid of parameter dependent Lyapunov functions, obviously turns out to be less restrictive than that one obtained via the classical quadratic stability (QS) approach, which guarantees stability in presence of arbitrary time-varying parameters. An important point is that the methodology proposed in this paper may result in being less conservative than the classical QS approach even in the absence of an explicit bound on the parameters rate of variation. Concerning the synthesis context, the design of a gain scheduled compensator based on the above approach is also proposed. It is shown that, if a suitable LMI problem is feasible, the solution of such problem allows to design an output feedback gain scheduled dynamic compensator in a controller-observer form stabilizing the class of systems which is dealt with. The stability conditions are then extended to take into account L2 performance requirements. Some numerical examples are carried out to show the effectiveness and to investigate the computational burden required by the proposed approach.