The explicit linear quadratic regulator for constrained systems

For discrete-time linear time invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly, the state feed back control law which minimizes a quadratic performance criterion. We show that the control law is piece-wise linear and continuous for both the fini te horizon problem (model predictive control) and the usual infinite time m easure (constrained linear quadratic regulation). Thus, the on-line control computation reduces to the simple evaluation of an explicitly defined piec ewise linear function. By computing the inherent underlying controller stru cture, we also solve the equivalent of the Hamilton-Jacobi-Bellman equation for discrete-time linear constrained systems. Control based on on-line opt imization has long been recognized as a superior alternative for constraine d systems, The technique proposed in this paper is attractive for a wide ra nge of practical problems where the computational complexity of on-line opt imization is prohibitive. It also provides an insight into the structure un derlying optimization-based controllers. (C) 2001 Elsevier Science Ltd. All rights reserved.