  Model predictive control (MPC) samples a generally unknown and complicated
feedback law point by point. The solution for the current state $x$ contains,
however, more information than only the optimal signal $u$ for this particular
state. In fact, it provides an optimal affine feedback law $x\rightarrow u(x)$
on a polytope $\Pi \subset \mathbb{R}^n$, i.e., on a full-dimensional state
space set. It is an obvious idea to reuse this affine feedback law as long as
possible. Reusing it on its polytope $\Pi$ is too conservative, however,
because any $\Pi$ is a state space set with a common affine law $x\rightarrow
(u_0^\prime (x), \dots, u_{N-1}^\prime (x))^\prime\in\mathbb{R}^{Nm}$ for the
entire horizon $N$. We show a simple criterion exists for identifying the
polytopes that have a common $x\rightarrow u_0(x)$, but may differ with respect
to $u_1(x), \dots, u_{N-1}(x)$. Because this criterion is too computationally
expensive for an online use, we introduce a simple heuristics for the fast
construction of a subset of the polytopes of interest. Computational examples
show (i) a considerable fraction of QPs can be avoided (10% to 40%) and (ii)
the heuristics results in a reduction very close to the maximum one that could
be achieved if the explicit solution was available. We stress the proposed
approach is intended for use in online MPC and it does not require the explicit
solution.
