Using Subsets Sequence to Approach the Maximal Terminal Region for MPC

Due to the ability to handle control and state constraints, MPC has become quite popular recently. In order to guarantee the stability of MPC, a terminal constraint and a terminal cost are added to the on-line optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function [1, 9]. As we know, the domain of attraction of MPC can be enlarged by increasing the prediction horizon, but it is at the expense of a greater computational burden. In [2], a prediction horizon larger than the control horizon was considered and the domain of attraction was enlarged. On the other hand, the domain of attraction can be enlarged by enlarging the terminal region. In [3], an ellipsoidal set included in the stabilizable region of using linear feedback controller served as the terminal region. In [4], a polytopic set was adopted. In [5], a saturated local control law was used to enlarge the terminal region. In [6], SVM was employed to estimate the stabilizable region of using linear feedback controller and the estimated stabilizable region was used as the terminal region. The method in [6] enlarged the terminal region dramatically. In [7], it was proved that, for the MPC without terminal constraint, the terminal region can be enlarged by weighting the terminal cost. In [8], the enlargement of the domain of attraction was obtained by employing a contractive terminal constraint. In [9], the domain of attraction was enlarged by the inclusion of an appropriate set of slacked terminal constraints into the control problem. In this paper, the domain of attraction is enlarged by enlarging the terminal region. A novel method is proposed to achive a large terminal region. First, the sufficient conditions to guarantee the stability of MPC are presented and the maximal terminal region satisfying these conditions is defined. Then, given the terminal cost and an initial subset of the maximal terminal region, a subsets sequence is obtained by using one-step set expansion iteratively. It is proved that, when the iteration time goes to infinity, this subsets sequence will converge to the maximal terminal region. Finally, the subsets in this sequence are separated from the state space one by one by exploiting SVM classifier (see [10,11] for details of SVM).