On the structure of the set of active sets in constrained linear quadratic regulation

TL;DR

This paper explores the structure of active sets in constrained linear quadratic regulation, revealing how they evolve with horizon extension and identifying invariant sets, which simplifies analysis and computation in model predictive control.

Contribution

It demonstrates that active sets for increasing horizons can be constructed iteratively from previous sets, and identifies horizon-invariant polytopes and special active sets without solving complex optimization problems.

Findings

Active sets for horizon N+1 extend those for horizon N by adding stage constraints.

Invariant polytopes and affine pieces can be detected without solving optimization problems.

A subset of active sets generates a positive invariant region around the origin.

Abstract

The constrained linear quadratic regulation problem is solved by a continuous piecewise affine function on a set of state space polytopes. It is an obvious question whether this solution can be built up iteratively by increasing the horizon, i.e., by extending the classical backward dynamic programming solution for the unconstrained case to the constrained case. Unfortunately, however, the piecewise affine solution for horizon N is in general not contained in the piecewise affine law for horizon N + 1. We show that a backward dynamic programming does, in contrast, result in a useful structure for the set of the active sets that defines the solution. Essentially, every active set for the problem with horizon N + 1 results from extending an active set for horizon N , if the constraints are ordered stage by stage. Consequently, the set for horizon N + 1 can be found by only considering the constraints of the additional stage. Furthermore, it is easy to detect which polytopes and affine pieces are invariant to increasing the horizon, and therefore persist in the limit N to infinity. Several other aspects of the structure of the set of active sets become evident if the active sets are represented by bit tuples. There exists, for example, a subset of special active sets that generates a positive invariant and persistent (i.e., horizon invariant) set around the origin. It is very simple to identify these special active sets, and the positive invariant and persistent region can be found without solving optimal control or auxiliary optimization problems. The paper briefly discusses the use of these results in model predictive control. Some opportunities for uses in computational methods are also briefly summarized.