Improved active set dynamic programming for solving linear-quadratic optimal control problems

The solution of a constrained linear-quadratic optimal control problem is a continuous piecewise-affine function. This solution can be determined by constructing all active sets that define the affine pieces. It has recently been proposed to combine the construction of all active sets with dynamic programming (DP). In this type of approach, DP is not applied to find the optimal input signal sequence that solves the optimal control problem, but DP is used to build up the set of all active sets for increasing horizon length. The existing approach for constructing the set of active sets with DP outperforms classical active set enumeration methods for long horizons, but they are less efficient for short horizons. We show that a simple bound on the candidate active set cardinality, which also is used in active set enumeration, renders the DP approach competitive also for short horizons.