On Finitely Determined Minimal Robust Positively Invariant Sets

For linear, time invariant stable systems with additive state disturbances that are bounded by polytopic sets, we establish connections between the minimal robust positively invariant set (mRPI) and ultimate-bound invariant (UBI) sets. We first identify cases for which the mRPI set is finitely determined. We then apply those cases to address the dual problem of finding (i) the A matrix of an LTI system, (ii) a disturbance set and (iii) a projection matrix, for which a given UBI set is a projection of the mRPI set associated with those three elements. Finally, these results are combined to iteratively compute converging outer approximations of the mRPI set associated with a given system via a sequence of sets that are projections of finitely determined mRPI sets in lifted spaces.