Model Predictive Control

110 6 Multiparametric Programming: a Geometric Approach
Recursion
For all the region CRi not excluded from the MILP’s subproblem (6.80)-(6.84) the
algorithm continues to iterate between the mp-LP (6.78) with ¯z Nbi+1
= z di
∗ and the di
MILP (6.80)-(6.84). The algorithm terminates when all the MILPs (6.80)-(6.84)
are infeasible.
Note that the algorithm generates a partition of the state space. Some parameter
x could belong to the boundary of several regions. Differently from the LP and
QP case, the value function may be discontinuous and therefore such a case has to
be treated carefully. If a point x belong to different critical regions, the expressions
of the value function associated with such regions have to be compared in order
to assign to x the right optimizer. Such a procedure can be avoided by keeping
track of which facet belongs to a certain critical region and which not. Moreover,
if the value functions associated with the regions containing the same parameter x
coincide this may imply the presence of multiple optimizers.
6.4.3 Theoretical Results
The following properties of J ∗ (x) and Z ∗ (x) easily follow from the algorithm described
above.
Theorem 6.10 Consider the mp-MILP (6.75). Then, the set K ∗ is the union of
a finite number of (possibly open) polyhedra and the value function J ∗ is piecewise
affine on polyhedra. If the optimizer z ∗ (x) is unique for all x ∈ K ∗ , then the
optimizer functions z ∗ c : K ∗ → R sc and z ∗ d : K∗ → {0, 1} sd are piecewise affine and
piecewise constant, respectively, on polyhedra. Otherwise, it is always possible to
define a piecewise affine optimizer function z ∗ (x) ∈ Z ∗ (x) for all x ∈ K ∗ .
Note that differently from the mp-LP case, the set K ∗ can be non-convex and
even disconnected.