The Explicit Solution of Model Predictive Control via Multiparametric Quadratic Programming

Control based on on-line optimization, popularly known as model predictive control (MPC), has long been recognized as the winning alternative for constrained sys t ems . The main limitation of MPC is, however, its on-line computat ional complexity. 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 associated with MPC, and show that it is piecewise linear and continuous. The controller inherits all the stability and performance properties of MPC, but the online computation is reduced to a simple linear function evaluation instead of the expensive quadratic program. The new technique is expected to enlarge the scope of applicability of MPC to small-size/fast-sampling applications which cannot be covered satisfactorily with anti-windup schemes. 1 I n t r o d u c t i o n As we extend the class of system descriptions beyond the class of linear systems, linear systems with constraints are probably the most important class in practice. The most popular approaches for designing controllers for linear systems with constraints fall into two categories: anti-windup and model predictive control. Anti-windup schemes assume that a well functioning linear controller is available for small excursions from the nominal operating point. This controller is augmented by the anti-windup scheme in a somewhat ad hoc fashion to take care of situations when constraints are met. Kothare et al. [13] reviewed numerous apparently different anti-windup schemes and showed that they differ only in their choice of two static matrix parameters. Anti-windup schemes are widely used in practice because in most SISO situations they are simple to design and work adequately. Model Predictive Control (MPC) has become the accepted standard for complex constrained multivariable control problems in the process industries. Here at each sampling time, start ing at the current state, an open-loop optimal control problem is solved over a finite horizon. At the next t ime step the computat ion is repeated starting from the new state and over a shifted horizon, leading to a moving horizon policy. The solution relies on a linear dynamic model, respects all input and output constraints, and optimizes a quadratic performance index. The big drawback of MPC is the relatively formidable on-line computational effort which limits its applicability to relatively slow and/or small problems. In this paper we show how to move all the computations necessary for the implementation of MPC offline while preserving all its other characteristics. This should largely increase MPC's range of applicability to problems where anti-windup schemes and other ad hoc techniques dominated up to now. The paper is organized as follows. The basics of MPC are reviewed first to derive the quadratic program which needs to be solved to determine the optimal control action. We note that the quadratic program depends on the current state which appears linearly in the constraints, i.e., it is a multi-parametric quadratic program. Next we study the multi-parametric quadratic programming problem. We show that the optimal solution is a piecewise affine function of the state (confirming previous investigations on the form of MPC laws [16, 17, 7]), analyze its properties, and develop an efficient algorithm to solve it. The paper concludes with an example which illustrates the different features of the method. The results in this paper can be extended easily to 1norm and o¢-norm objective functions instead of the 2-norm employed in this paper. The resulting multiparametric linear program can be solved in a similar manner as suggested here [6], or as in [10]. For MPC of hybrid systems, an extension involving multiparametric mixed-integer linear programming, is also possible [3]. 2 M o d e l P r e d i c t i v e C o n t r o l Consider the problem of regulating to the origin the discrete-time linear time invariant system x(t+l) = Ax(t)+Bu(t) y(t) = Cx(t) (1) while fulfilling the constraints Ymin ~ y(t) < Ymax, Umin ~ u(t) < Umax (2) at all time instants t > 0. In (1)-(2), x(t) e Nn, u(t) 6 Nm, and y(t) 6 NP are the state, input, and output vector respectively, Ymin _( Ymax (Umin < Umax) 0-7803-5519-9100 $10.00 © 2000 A A C C 8 7 2 are bounds on outputs (inputs), and the pair (A, B) is stabilizable. Model Predictive Control (MPC) solves such a constrained regulation problem in the following way. Assume that a full measurement of the state x(t) is available at the current t ime t. Then, the optimization problem . rain U={ut,... ,ul..t-Nu--1} J(U, X.(t) ) = Xlt+N~ltPXt+Nylt