Relaxing the optimality condition in receding horizon control

Receding horizon control is based on iteratively solving an open-loop finite horizon optimization problem. Despite its success in a variety of industrial applications, theoretical issues such as stability were not completely addressed until recently. It was shown in Jadbabaie et al. (1999) that by utilizing a suitable control Lyapunov function (CLF) as terminal cost, the stability of the receding horizon scheme can be guaranteed and the region of attraction of the receding horizon controller can be estimated. The key point in this approach, which made it different from others, was removal of additional stability constraints, hence making the optimizations much easier to solve. A requirement implied in the previous results was being able to solve the optimizations globally. In this paper, that assumption is removed and it is shown that the optimality can be replaced by an improvement property. Specifically, instead of requiring the trajectories to be optimal, it is required that a certain amount of decrease in the cost is obtained at each receding horizon iteration. It is further shown that there always exist a sequence of controls which guarantee the necessary decrease in the cost. A numerical example using the inverted pendulum is presented to illustrate this point.