832 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006 On the Stability of Constrained MPC Without Terminal Constraint D. Limon, T. Alamo, F. Salas, and E. F. Camacho Abstract—The usual way to guarantee stability of model predictive control (MPC) strategies is based on a terminal cost function and a terminal constraint region. This note analyzes the stability of MPC when the terminal constraint is removed. This is particularly interesting when the system is unconstrained on the state. In this case, the computational burden of the optimization problem does not have to be increased by introducing terminal state constraints due to stabilizing reasons. A region in which the terminal constraint can be removed from the optimization problem is characterized depending on some of the design parameters of MPC. This region is a domain of attraction of the MPC without terminal constraint. Based on this result, it is proved that weighting the terminal cost, this domain of attraction of the MPC controller without terminal constraint is enlarged reaching (practically) the same domain of attraction of the MPC with terminal constraint; moreover, a practical procedure to calculate the stabilizing weighting factor for a given initial state is shown. Finally, these results are extended to the case of suboptimal solutions and an asymptotically stabilizing suboptimal controller without terminal constraint is presented. Index Terms—Asymptotic stability, predictive control, suboptimal con- trol. I. INTRODUCTION AND PROBLEM STATEMENT Consider a system described by a nonlinear invariant discrete time model (1) where is the system state, is the current control vector and is the successor state. The system is subject to constraints on both states and control actions, and they are given by (2) where is a closed set and a compact set, both of them containing the origin. In what follows, and will denote the state and the control action applied to the system at sampling time . A sequence of control actions to be applied to the system at current state is denoted as where its dependence with may be omitted. The predicted state of the system at time , when the initial state is (at time 0) and the control sequence is applied, will be denoted as . It has been proved that this class of systems can be stabilized by a MPC control law ; this control law is obtained by solving a constrained optimization problem at each sampling time and applying it to the system in a receding horizon way. The finite horizon nominal MPC optimization problem with terminal cost and terminal constraint Manuscript received December 28, 2003; revised November 23, 2004 and January 23, 2006. Recommended by Associate Editor L. Magni. This work was supported by MCYT-Spain under Contracts DPI2004-07444 and DPI2005- 04568. Preliminary results of this note were presented at the American Control Conference, 2003. The authors are with the Departamento de Ingeniería de Sistemas y Au- tomática, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla. Spain (e-mail: limon@cartuja.us.es; alamo@cartuja.us.es; salas@cartuja.us.es; eduardo@cartuja.us.es). Digital Object Identifier 10.1109/TAC.2006.875014 is the standard way of formulating the MPC controller [1], and it will be denoted as general MPC. This optimization problem, denoted as , is given by where , is the stage cost, which is assumed to be a positive definite function on (i.e., there exists a function 1 such that [2]); is the terminal cost and is the terminal region. In what follows, denotes the op- timal solution to and , , denotes the optimal predicted trajectory. The set of states where the op- timization problem is feasible (and hence is defined) is denoted by . The terminal cost and the terminal constraint are usually chosen sat- isfying the following assumption. Assumption1: Let be a control Lyapunov function (CLF) and let be a set given by , with , such that and for all : (3) (4) where , are -functions. In [3], it is proved that if the terminal cost and terminal set satisfy assumption 1, the optimal cost of is a Lyapunov function and the model predictive control (MPC) control law stabilizes asymp- totically the system in . If the terminal constraint is removed from the optimization problem, then the optimal cost may not be a Lyapunov function for the system and, moreover, the feasibility may be lost. However, there are some predictive controllers with guaran- teed stability which do not consider an explicit terminal constraint, as in [4]–[7]. Notice that the removal of the terminal constraint may be interesting, for instance, if the system is not constrained on the states. In this case, the terminal constraint is the only one that depends on the predicted state of the system. So, the removal of this constraint makes the problem much easier to solve and the computational burden is re- duced, but at the expense of a reduction of the domain of attraction. In [5], stability is guaranteed by considering a quadratic terminal cost function and it is proved that, for any stabilizable initial state, there is a triple ( , , and ) such that the system is stabi- lized. Based on these results, stability of MPC with a CLF as terminal cost for a class of unconstrained nonlinear systems is analyzed in [6]. In [7], using a slightly modified Lypaunov function as terminal cost it is proved that the MPC without terminal constraint stabilizes the system asymptotically for any initial state where the terminal constraint is not active; that is, in . This note presents some novel results on this topic. Generalizing pre- vious results presented in [5] to the general MPC, a region where the terminal constraint is satisfied in the optimization problem is character- ized. This region is a domain of attraction of the MPC without terminal constraint. This characterization allows us to prove that this region can be enlarged by weighting the terminal cost. Furthermore, it is proved that a larger weighting factor implies a bigger domain of attraction. Thus, the proposed MPC, by means of weighting the terminal cost, can stabilize the system at any initial state such that , where 1 A function is a function if it is continuous, strictly increasing and 0018-9286/$20.00 © 2006 IEEE