Model Predictive Control

98 6 Multiparametric Programming: a Geometric Approach
with
and
⎡
⎢
G = ⎢
⎣
where K is given by
H =
1.00 −1.00 1.00
−1.00 1.00 −1.00
0.00 0.00 −0.30 ⎥
−1.00 0.00 −1.00 ⎥
0.00 0.00 0.30 ⎥
1.00 0.00 1.00 ⎥
−0.30 0.00 −0.60 ⎥
0.00 −1.00 0.00 ⎥
0.30 0.00 0.60 ⎥
0.00 1.00 0.00 ⎥
−1.00 0.00 0.00 ⎥
1.00 0.00 0.00 ⎥
0.00 0.00 0.00 ⎥
0.00 0.00 −1.00 ⎥
0.00 0.00 0.00 ⎦
0.00 0.00 1.00
0.00 0.00 −1.00
0.00 0.00 1.00
8.18 −3.00 5.36
−3.00 5.00 −3.00 , F =
5.36 −3.00 10.90
0.60 0.00 1.80
5.54 −3.00 8.44
⎤
⎡
⎢
, E = ⎢
⎣
0.00 −1.00
0.00 1.00
1.00 0.60
0.00 1.00
−1.00 −0.60
0.00 −1.00
1.00 0.90
0.00 0.00
−1.00 −0.90
0.00 0.00
0.00 0.00
0.00 0.00
1.00 0.30
0.00 1.00
−1.00 −0.30
0.00 −1.00
0.00 0.00
0.00 0.00
−8 ≤ x1 ≤ 8
−8 ≤ x2 ≤ 8.
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ , W = ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
0.50
0.50
8.00
8.00
8.00
8.00
8.00
8.00
8.00
8.00
0.50
0.50
8.00
8.00
8.00
8.00
0.50
0.50
⎤
⎥
⎦
(6.60)
(6.61)
(6.62)
A solution to the mp-QP problem is shown in Figure 6.9 and the constraints which
are active in each associated critical region are reported in Table 6.3. Figure 6.9 and
Table 6.3 report only full dimensional critical regions.
Critical Region Active Constraints
CR1 {}
CR2 {1}
CR3 {2}
CR4 {11}
CR5 {12}
CR6 {17}
CR7 {18}
CR8 {1,11}
CR9 {1,18}
CR10 {2,12}
CR11 {2,17}
CR12 {11,17}
CR13 {12,18}
CR14 {1,11,17}
CR15 {1,11,18}
CR16 {2,12,17}
CR17 {2,12,18}
Table 6.3 Example 6.6: Critical regions and corresponding set of active
constraints.
Since A1 is empty in CR1 from equations (6.46) and (6.47) we can conclude
that the facets of CR1 are facets of primal feasibility and therefore do not belong
to CR1. In general, as discussed in Remark 6.7 critical regions are open on facets