Outer Approximations of The Minimal Disturbance Invariant Set

This paper is concerned with outer approximations of the minimal disturbance invariant set (MDIS) of a discrete-time linear system with an additive set-bounded disturbance. The k-step disturbance reachable sets (Minkowski partial sums) are inner approximations of MDIS that converge to MDIS. Enlarged by a suitable scaling, they lead to outer approximations of MDIS. Two families of approximations, each based on partial sums, are considered: one minimizes the scalings of the partial sums and is not disturbance invariant, the other is generated by maximal disturbance invariant subsets of scaled partial sums. Theoretical properties of the families are proved and interrelated. Algorithmic questions, including error bounds for the approximations, are addressed. The results are illustrated by computational data from several examples.