The explicit linear quadratic regulator for constrained systems

• Published in: Automatica (Oxford) Vol. 38; no. 1; pp. 3 - 20
• Main Authors: Bemporad, Alberto, Morari, Manfred, Dua, Vivek, Pistikopoulos, Efstratios N.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 2002
• Subjects: Constraints; Linear quadratic regulators; Piecewise linear controllers; Predictive control; Piecewise linear controllers; Linear quadratic regulators; Constraints; Predictive control
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/S0005-1098(01)00174-1

We present a technique to compute the explicit state-feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints. We show that this closed form solution is piecewise linear and continuous. As a practical consequence of the result, constrained linear quadratic regulation becomes attractive also for systems with high sampling rates, as on-line quadratic programming solvers are no more required for the implementation. For discrete-time linear time invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly, the state feedback control law which minimizes a quadratic performance criterion. We show that the control law is piece-wise linear and continuous for both the finite horizon problem (model predictive control) and the usual infinite time measure (constrained linear quadratic regulation). Thus, the on-line control computation reduces to the simple evaluation of an explicitly defined piecewise linear function. By computing the inherent underlying controller structure, we also solve the equivalent of the Hamilton–Jacobi–Bellman equation for discrete-time linear constrained systems. Control based on on-line optimization has long been recognized as a superior alternative for constrained systems. The technique proposed in this paper is attractive for a wide range of practical problems where the computational complexity of on-line optimization is prohibitive. It also provides an insight into the structure underlying optimization-based controllers.