Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones

This paper presents a new (geometrical) approach to the computation of polyhedral   (robustly) positively invariant (PI) sets for general (possibly discontinuous) nonlinear discrete-time systems possibly affected by disturbances. Given a ββ-contractive ellipsoidal set EE, the key idea is to construct a polyhedral set that lies between the ellipsoidal sets βEβE and EE. A proof that the resulting polyhedral set is contractive and thus, PI, is given, and a new algorithm is developed to construct the desired polyhedral set. The problem of computing polyhedral invariant sets is formulated as a number of quadratic programming (QP) problems. The number of QP problems is guaranteed to be finite and therefore, the algorithm has finite termination. An important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs.