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

TL;DR

This paper proves that the backward reachable sets of robust controlled invariant sets for discrete-time linear systems with disturbances converge exponentially to the maximal RCIS, with conditions verifiable via linear programming.

Contribution

It introduces a simple condition ensuring exponential convergence of backward reachable sets to the maximal RCIS, generalizing existing conditions for disturbance-free systems.

Findings

Backward reachable sets converge exponentially to the maximal RCIS.

The convergence condition can be checked via linear programming.

The condition generalizes previous results for systems without disturbances.

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).