Receding horizon control Lyapunov function approach to suboptimal regulation of nonlinear systems

The problem of rendering the origin an asymptotically stable equilibrium point of a nonlinear system while optimizing some measure of performance has been the object of much attention in the past few years. In contrast to the case of linear systems where several optimal synthesis techniques (such as H(infinity), H(2), and l(1)) are well established, their nonlinear counterparts are just starting to emerge. Moreover, in most cases these tools lead to partial differential equations that are difficult to solve. In this paper we propose a suboptimal regulator for nonlinear affine systems based upon the combination of receding horizon and control Lyapunov function techniques. The main result of the paper shows that this controller is nearly optimal provided that a certain finite horizon problem can be solved on-line. Additional results include 1) sufficient conditions guaranteeing closed-loop stability even in cases where there is not enough computational power available to solve this optimization on-line and 2) an analysis of the soboptimality level of the proposed method. These results are illustrated with two simple examples comparing the performance of the suboptimal controller against that achieved by some other popular nonlinear control methods.