Minimal RPI sets computation for polytopic systems using the Bounded-real lemma and a new shrinking procedure

Abstract

This paper deals with the problem of computing the minimal robustly positive invariant-set s (mRPI sets) for discrete-time linear and polytopic systems. Here, the subsystems, which describe a polytopic system, are assumed to be stable in presence of bounded disturbances and, in addition, to share a common Lyapunov function. The proposed approach is based on the computation of “ellipsoidal” RPI sets which are obtained by using the Bounded-real lemma. After that, a RPI outer-approximation of the minimal “polyhedral” RPI set for the polytopic system is obtained by applying a novel shrinking process. A shrinking index, which depends on the number of iterations, is proposed as an indicator of the approximation error. Thus, the method provides mRPI set approximations that are always invariant sets at any step of the algorithm, and allows the a priori choice of a desired precision. Two numerical examples illustrate the application of a such method.