CONSTRAINED STABILIZATION WITH AN ASSIGNED INITIAL CONDITION SET

The problem of stabilizing a linear discrete-time system with control constraints is considered. Necessary and sufficient conditions are giv en for the existence of a state feedback controller which drives the s tate to the origin asymptotically from every initial state in an assig ned compact polyhedral set. These conditions can be checked via linear programming. It is shown that when the problem has a solution, a poly hedral function can be formed which turns out to be a Lyapunov functio n if a proper nonlinear feedback controller is applied. Two procedures are presented for the construction of the Lyapunov function. The firs t is based on the property that the stabilizing feedback compensator e xists if and only if for every initial condition chosen on a vertex of the set there exists an open-loop control driving the state to its in terior. The second procedure is based on the construction of the contr ollability regions to the given polyhedral set; this procedure can als o be applied to systems with parameter uncertainties. The resulting co mpensator is obtained by solving on-line an optimization problem which can be efficiently implemented on a digital computer.