On the Convergence of the Backward Reachable Sets of Robust Controlled Invariant Sets For Discrete-time Linear Systems

Abstract

This paper considers discrete-time linear systems with bounded additive disturbances, and studies the convergence properties of the backward reachable sets of robust controlled invariant sets (RCIS). Under a simple condition, we prove that the backward reachable sets of an RCIS are guaranteed to converge to the maximal RCIS in Hausdorff distance, with an exponential convergence rate. When all sets are represented by polytopes, this condition can be checked numerically via a linear program. We discuss how the developed condition generalizes the existing conditions in the literature for (controlled) invariant sets of systems without disturbances (or without control inputs).