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

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