Finite constraint set receding horizon quadratic control

This contribution addresses the problem of discrete time receding horizon quadratic control for plants whose input is restricted to belong to a finite set. We also study the dynamics of the resulting closed-loop system. Based upon the geometry of the underlying quadratic programme, a finitely parametrized expression for the control law is derived, which makes use of vector quantizers. Alternatively, the control law can be formulated by means of a polyhedral partition of the state space, which is closely connected with the partition induced when considering saturation-like constraints. Exact analytic expressions for the partition can be developed, therefore avoiding the need for on-line optimization. The closed-loop system, comprising controller and plant, exhibits highly nonlinear dynamics, due to the finite set restriction. Asymptotic stability only holds for very special cases. In general, this notion is too strong. Nevertheless, ultimate boundedness of state trajectories is often achieved. Tools for determining positively invariant sets, hence ensuring ultimate boundedness, are presented.