Model Predictive Control

216 11 Receding Horizon Control
A different tuning of the controller is required when such polyhedral regions appear
and the overall performance is not satisfactory. The second issue is the presence of
multiple solutions, that might arise from the degeneracy of the dual problem (6.14).
Multiple optimizers are undesirable, as they might lead to a fast switching between
the different optimal control moves when the optimization program (10.75) is solved
on-line, unless interior-point methods are used. The mp-LP solvers [101, 55] can
detect critical regions of degeneracy and partition them into sub-regions where a
unique optimizer is defined. Example 11.5 illustrates a RHC law where multiple
optimizers and idle control occur.
Example 11.5 Consider the double integrator of Example 11.4, with N = 1, Q =
1 0
0 1
, R = 1, P = Q subject to the input constraints
and the state constraints
X = {x :
The associated mp-LP problem is
min
ε1,ε2,u0
ε1 + ε2
U = {u : − 1 ≤ u ≤ 1} (11.42)
−10
−10
subj. to
⎡
−1 0
⎤
1
⎡
⎢ −1
⎢ 0
⎢ 0
⎢ 0
⎢ 0
⎢ 0
⎢ 0
⎢ 0
⎢ 0
⎣
0
0
−1
−1
−1
−1
0
0
0
0
0
−1 ⎥ ⎢
⎥ ⎢
0 ⎥ ⎢
⎥ ⎢
−1 ⎥ ⎢
⎥
0 ⎥ ⎡ ⎤ ⎢
⎥ ε1 ⎢
1 ⎥ ⎣ ε2 ⎦ ⎢
≤ ⎢
1 ⎥ ⎢
⎥ u0 ⎢
0 ⎥ ⎢
⎥ ⎢
−1 ⎥ ⎢
0
⎥ ⎢
⎥ ⎢
1
⎦ ⎣
0 0 −1
≤ x ≤
0
0
0
0
0
0
10
10
10
10
1
1
10
10
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ + ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
} (11.43)
0 0
0 0
1 1
0 1
−1 −1
0 −1
0 −1
−1 −1
0 1
1 1
0 0
0 0
⎤
⎥ x(t)
⎥
⎦
The solution of (11.44) gives rise to idle control and multiple optimizers.
(11.44)
In fact, the corresponding polyhedral partition of the state-space is depicted in
Fig. 11.8. The RHC law is
⎧
degenerate if
(Region #1)
0
⎪⎨
if
u =
degenerate if
⎪⎩
0 if
−1.00 −2.00
1.00 0.00
1.00 1.00
−1.00 −1.00
x ≤
0.00 1.00
1.00 0.00
1.00 2.00
−1.00 −1.00
x ≤
−1.00 0.00
1.00 2.00
1.00 1.00 x ≤
−1.00 −1.00
0.00 −1.00
−1.00 −2.00
−1.00 0.00 x ≤
1.00 1.00
0.00
0.00
10.00
10.00
11.00
0.00
0.00
10.00
0.00
0.00
10.00
10.00
11.00
0.00
0.00
10.00
(Region #2)
(Region #3)
(Region #4)