Model Predictive Control

104 6 Multiparametric Programming: a Geometric Approach
one could simply generate all the possible combinations of active sets. However, in
many problems only a few active constraints sets generate full-dimensional critical
regions inside the region of interest K. Therefore, the goal is to design an active
set generator algorithm which computes only the active sets Ai with the associated
full-dimensional critical regions covering only K ∗ .
Next an implementation of a mp-QP algorithm is described. See Section 6.6 for
a literature review on alternative approaches to the solution of mp-QPs.
In order to start solving the mp-QP problem, we need an initial vector x0
inside the polyhedral set K ∗ of feasible parameters. A possible choiche for x0 is the
Chebychev center (see Section 3.4.5) of K ∗ :
maxx,¯z,ǫ ǫ
subj. to Tix + ǫTi2 ≤ Ni, i = 1, . . .,nT
G¯z − Sx ≤ W
(6.73)
where nT is the number of rows Ti of the matrix T defining the set K in (6.2).
If ǫ ≤ 0, then the QP problem (6.37) is infeasible for all x in the interior of K.
Otherwise, we set x = x0 and solve the QP problem (6.37), in order to obtain the
corresponding optimal solution z∗ 0. Such a solution is unique, because H ≻ 0. The
value of z∗ 0 defines the following optimal partition
A(x0) {j ∈ J : Gjz ∗ 0 − Sjx0 − Wj = 0}
NA(x0) {j ∈ J : Gjz ∗ 0 − Sjx0 − Wj < 0}
(6.74)
and consequently the critical region CRA(x0). Once the critical region CRA(x0) has been defined, the rest of the space Rrest = K\CRA(x0) has to be explored and
new critical regions generated. An approach for generating a polyhedral partition
{R1, . . .,Rnrest} of the rest of the space Rrest is described in Theorem 3.1 in Section
3.4.7. Theorem 3.1 provides a way of partitioning the non-convex set K \CR0
into polyhedral subsets Ri. For each Ri, a new vector xi is determined by solving
the LP (6.73), and, correspondingly, an optimum z ∗ i
, a set of active constraints Ai,
and a critical region CRi. The procedure proposed in Theorem 3.1 for partitioning
the set of parameters allows one to recursively explore the parameter space. Such
an iterative procedure terminates after a finite time, as the number of possible
combinations of active constraints decreases with each iteration. Two following
main elements need to be considered:
1. As for the mp-LP algorithm, the partitioning in Theorem 3.1 defines new
polyhedral regions Rk to be explored that are not related to the critical regions
which still need to be determined. This may split some of the critical
regions, due to the artificial cuts induced by Theorem 3.1. Post-processing
can be used to join cut critical regions [38]. As an example, in Figure 6.8 the
critical region CR {3,7} is discovered twice, one part during the exploration of
R1 and the second part during the exploration of R2. Although algorithms
exist for convexity recognition and computation of the union of polyhedra,
the post-processing operation is computationally expensive. Therefore, it is
more efficient not to intersect the critical region obtained by (6.27) with halfspaces
generated by Theorem 3.1, which is only used to drive the exploration