Model Predictive Control

12.5 State Feedback Solution, 1-Norm and ∞-Norm Case 257
Proof: Since xk = Akx0 + k−1 k=0 Ai [B(w p
k−1−i )uk−1−i + Ewa k−1−i ] is a linear
function of the disturbances w a {w a 0, . . .,w a N−1 }, and wp {w p
0
, . . .,wp
N−1
} for a
fixed input sequence and x0, the cost function in the maximization problem (12.41)
is convex and piecewise affine with respect to the optimization vectors w a and w p
and the parameters U0, x0. The constraints in (12.44) are linear in U0 and x0,
for any w a and w p . Therefore, by Lemma 9, problem (12.41)–(12.45) can be
solved by solving an mp-LP through the enumeration of all the vertices of the sets
W a × W a × . . . × W a and W p × W p × . . . × W p . The theorem follows from the
mp-LP properties described Theorem 6.4. ✷
Remark 12.6 In case of OL-CROC with additive disturbances only (w(t) = 0) the
number of constraints in (12.45) can be reduced as explained in Remark 12.5.
12.5.2 Recursive Approach: Closed-Loop Predictions
Theorem 12.3 There exists a state-feedback control law u ∗ (k) = fk(x(k)), fk :
Xk ⊆ R n → U ⊆ R m , solution of the CROC-CL (12.47)–(12.51) with cost (12.42)
and k = 0, . . . , N − 1 which is time-varying, continuous and piecewise affine on
polyhedra
fk(x) = F i k x + gi k if x ∈ CR i k , i = 1, . . .,Nr k (12.78)
where the polyhedral sets CRi k = {x ∈ Rn : Hi kx ≤ Ki k }, i = 1, . . .,Nr k are a
partition of the feasible polyhedron Xk. Moreover fi, i = 0, . . .,N − 1 can be found
by solving N mp-LPs.
Proof: Consider the first step j = N − 1 of dynamic programming applied to
the CROC-CL problem (12.47)–(12.49) with cost (12.42)
J ∗ N−1 (xN−1) min JN−1(xN−1, uN−1) (12.79)
uN −1
subj. to
⎧
⎨
⎩
JN−1(xN−1, uN−1) max
w a N −1 ∈Wa , w p
FxN−1 + GuN−1 ≤ f
A(w p
∀wa N−1 ∈ Wa , w p
N−1
N −1 ∈Wp
N−1 )xN−1 + B(w p
N−1 )uN−1 + Ewa N−1 ∈ Xf
∈ Wp
⎧
⎨
⎩
QxN−1p + RuN−1p+
+P(A(w p
N−1 )xN−1+
+B(w p
N−1 )uN−1 + Ew a N−1 )p
(12.80)
⎫
⎬
⎭ .
(12.81)
The cost function in the maximization problem (12.81) is piecewise affine and
, wp and the parameters
convex with respect to the optimization vector wa N−1 N−1
uN−1, xN−1. Moreover, the constraints in the minimization problem (12.80) are
linear in (uN−1, xN−1) for all vectors wa N−1 , wp
N−1 . Therefore, by Corollary 12.1,