The Implicit Maximal Positively Invariant Set

This article introduces an alternative characterization, together with indirect computation and implicit representation, of the maximal positively invariant set for the nonlinear dynamics and closed constraints. The introduced alternative characterization, indirect computation, and implicit representation of the maximal positively invariant set are novel features that, when combined together, overcome successfully dimensionality related limitations of the standard explicit computation of the maximal positively invariant set. The proposed approach enables, via an adequate numerical implementation, the utilization of the maximal positively invariant set in arbitrary finite dimensions from a theoretical perspective, which, in practical terms, facilitates its use for large-scale problems. The approach is worked out in fine detail for the setting of the linear dynamics and the constraints expressed in terms of closed, convex subsets that contain the origin in the interior.