The efficient computation of polyhedral invariant sets for linear systems with polytopic uncertainty

In this paper the concept of maximal admissable set (MAS), introduced by Gilbert et al. (1991) for linear time-invariant systems, is extended to linear systems with polytopic uncertainty under linear state feedback. It is shown that by constructing a tree of state predictions using the vertices of the uncertainty polytope and by imposing state and input constraints on these predictions, a polyhedral robust invariant set can be constructed. The resulting set is proven to be the maximal admissable set. The number of constraints defining the invariant set is shown to be finite if the closed loop system is quadratically stable (i.e. has a quadratic Lyapunov function). An algorithm is also proposed that efficiently computes the polyhedral set without exhaustively exploring the entire prediction tree. This is achieved through the formulation of a more general invariance condition than that proposed in Gilbert et al. (1991) and by the removal of redundant constraints in intermediate steps. The efficiency and correctness of the algorithm is demonstrated by means of a numerical example.