Model Predictive Control

12.7 Example 263
We denote with X Mv
0 ⊆ X the set of states x0 for which the robust optimal control
problem (12.101)-(12.102) is feasible, i.e.,
X Mv
0 =
x0 ∈ R n : P Mv
0 (x0) = ∅
P Mv
0 (x0) = {M, v : xk ∈ X, uk ∈ U, k = 0, . . .,N − 1, xN ∈ Xf
∀ w a k ∈ Wa k = 0, . . .,N − 1, where xk+1 = Axk + Buk + Ew a k , uk = k−1
i=0 Mk,iwi + vi}
(12.103)
The following result has been proven in [170].
Theorem 12.6 Consider the control parameterizations (12.92), (12.98) and the
corresponding feasible sets X Lg
0 in (12.97) and X Mv
0 in (12.103). Then,
and P Mv
N (x0) is convex in M and v.
X Lg
0
= X Mv
0
Note that in general X Mv
0 and JMv ∗
0 (x0) are different from the corresponding
CROC-CL solutions X0 and J ∗ 0 (x0). In particular X Mv
0 ⊆ X0 and JMv ∗
0 (x0) ≥
J ∗ 0(x0).
The idea of the parametrization (12.98) appears in the work of Gartska & Wets
in 1974 in the context of stochastic optimization [105]. Recently, it re-appeared
in robust optimization work by Guslitzer and Ben-Tal [124, 41], and in the context
of robust MPC in the wok of van Hessem & Bosgra, Löfberg and Goulart &
Kerrigan [247, 170, 118].
12.7 Example
Example 12.8 Consider the problem of robustly regulating to the origin the system
x(t + 1) =
subject to the input constraints
and the state constraints
1 1
0 1
x(t) +
0
1
U = {u ∈ R : − 3 ≤ u ≤ 3}
u(t) + w a (t)
X = {x ∈ R 2 : − 10 ≤ x ≤ 10 k = 0, . . . , 3}
The two-dimensional disturbance w a is restricted to the set W a = {v : w a ∞ ≤
1.5}.
Next we compute the state feedback control law (12.104) obtained by solving the
CROC-CL (12.47)–(12.51) and the CROC-OL (12.41)–(12.45). We use the cost func-
tion
N−1
PxN ∞ +
(Qxk∞ + |Ruk|)
k=0