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

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 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 → ∞. 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.

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

“…Extending the optimal active sets for horizon N − 1 to those for N formally corresponds to a backward dynamic programming step [12]. It is an obvious question to ask whether also the geometric approaches (see Sect.…”

“…The essential idea is as follows. An optimal active set of the constrained LQR problem with horizon N always contains an optimal active set of the same problem with horizon N − 1 [12,Prop. 1].…”

Simulation studies on regional predictive control

“…Closed-loop optimal sequences of affine laws [14,15] With this approach all polytopes (and their feedback laws) that contain a state of the closed-loop trajectory can be computed from the solution of a QP at the current state, i.e., a single point x ∈ X f . If the terminal constraints are inactive at the current state x, then the solution of the QP for the state x does not only provide a single feedback law but the entire sequence of optimal feedback laws and their polytopes of validity along the closed-loop trajectory.…”

Accelerating MPC by online detection of state space sets with common optimal feedback laws

“…We do not treat numerical methods such as tailored optimization algorithms here, but exploit the piecewise-affine structure of the solution [1,23,22]. We stress that we never calculate explicit control laws, but the present paper belongs to a group of works [3,18,17,4,10] that exploit the affine structure, or the corresponding structure of the set of active sets [8,5,9,20,19], to accelerate online MPC.…”