Inner-Outer Approximation of Robust Control Invariant Sets

This work proposes an approach to replace the use of a robust control invariant set by a pair of simpler sets that provide an inner and an outer approximation of the former. In the proposed approach, the outer set plays the role of the target region and the inner set is such that the trajectories that start inside it can be kept inside the outer set and be driven back to the inner set within a finite-time horizon. We show that the existence of these two sets implies the existence of a robust control invariant set between both regions. We also provide results that allow finding an inner set from a given target outer set and we show a way of using both sets in Model Predictive Control schemes such that the target region is never abandoned in spite of the fact that nor that region neither the inner set are invariant. We also illustrate the ideas with an example in which the inner and outer sets are very simple notwithstanding that any robust invariant set is not convex