Computing the Maximal Positive Invariant Set for the Constrained Zonotopic Case

The maximal positive invariant (MPI) set results from a finite set recurrence instantiated by the intersection of input and state bounds (e.g., the stage constraints of the linear model predictive control problem). When these constraints take the form of hyper-rectangles, zonotopes or constrained zonotopes, the resulting polyhedral MPI set may be succinctly described as a constrained zonotope, eliminating the need for explicit enumeration of its halfspaces. In this letter we discuss the various MPI computation algorithms (with both exact and sufficient stop conditions), recasted in the framework of constrained zonotopes. We analyze one of these variations over a dynamical system whose dimension can be arbitrarily increased in order to asses changes in computation time and storage requirements with respect to the polyhedral case (under the simplifying assumption of closed-loop invertibility of the state matrix).