  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.
