The Minimal Robust Positively Invariant Set for Linear Discrete Time Systems: Approximation Methods and Control Applications

Abstract

This paper considers the minimal robust positively invariant set for linear discrete time systems and its robust positively invariant approximations. Efficient approximating techniques proposed by Rakovic et al., (2005) are extended to degenerate cases when the disturbance set is not necessarily full-dimensional. Two methods for handling degenerate case are proposed and two novel families of robust positively invariant sets are characterized. The minimal robust positively invariant set can be approximated arbitrarily closely with appropriate members of these families. The presented results are exploited, under mild assumptions, to construct robust positively invariant sets for the case when the state is also uncertain and only its estimate, obtained by the standard Luenberger type observer, is known. A simple example illustrates the proposed methods