Numerical treatment of the bounded-control LQR problem by updating the final phase value

Abstract

A novel approach has been developed for approximately solving the constrained LQR problem, based on updating the final state and costate of an unrestricted related regular problem, and the switching times (when the control meets the constraints). The main result is the expression of a suboptimal control in feedback form by using some corresponding Riccati equation. The gradient method is applied to reduce the cost via explicit algebraic formula for its partial derivatives with respect to the hidden final state/costate of the related regular problem. The numerical method results efficient because it does not involve integrations of states or cost trajectories and reduces the dimension of the relevant unknown parameters. All the relevant objects are calculated from a few auxiliary matrices, which are computed only once and kept in memory. The scheme is here applied to the "cheapest stop of a train" case-study whose optimal solution is already known.