A dynamic programming approach to solving constrained linear-quadratic optimal control problems

TL;DR

This paper introduces a dynamic programming algorithm that efficiently constructs optimal active sets for constrained linear-quadratic control problems across different horizons, enabling easier detection of infinite-horizon solutions.

Contribution

The paper presents a novel recursive method to build optimal active sets for successive horizons, simplifying the solution process for constrained LQ control problems.

Findings

Efficient recursive construction of active sets for different horizons.

Easy detection of finite-horizon solutions matching infinite-horizon solutions.

Reduced computational effort through stagewise treatment.

Abstract

The solution of a constrained linear-quadratic regulator problem is determined by the set of its optimal active sets. We propose an algorithm that constructs this set of active sets for a desired horizon N from that for horizon N-1. While it is not obvious how to extend the optimal feedback law itself for horizon N-1 to horizon N, a simple relation between the optimal active sets for two successive horizon lengths exists. Specifically, every optimal active set for horizon N is a superset of an optimal active set for horizon N-1 if the constraints are ordered stage by stage. The stagewise treatment results in a favorable computational effort. In addition, it is easy to detect the solution of the current horizon is equal to the infinite-horizon solution, if such a finite horizon exists, with the proposed algorithm.