Page 204 - Distributed model predictive control for plant-wide systems

178 Distributed Model Predictive Control for Plant-Wide Systems

Proof. Since Problem 8.1 has a solution at time k − 1, by construction (8.6), it has
‖ p ‖
‖ ̂ x (k + N − 1) |k ‖
‖ i ‖P j = ‖x (k + N − 1) |k − 1‖
‖ i
‖P i

≤ √
2 m
In addition, since
∑
p p
̂ x (k + N|k)= A x (k + N − 1|k − 1)+ A x (k + N − 1|k − 1)
i
di i
ij j
j∈P +i
∑
= A ̂ x (k + N − 1|k)+ A ̂ x (k + N − 1|k)
ij j
di i
j∈P +i
It has
‖ ‖
‖ ∑ ‖
‖
‖ ̂ x (k + N|k) ‖ = A ̂ x (k + N − 1|k) + A ̂ x (k + N − 1|k) ‖
‖ i ‖P i ‖ di i ij j ‖
‖ ‖
j∈P +i
‖ ‖P i
T
T
T
̂
Consider Assumption 8.2, A PA + A PA + A PA < Q∕2
o o o d d o
Thus if, then
‖ ̂ x (k + N|k) ‖ ‖ ‖
‖ i
‖ i ‖P i ≤ ̂ x (k + N − 1|k) ̂
‖Q∕2
√
−1 T ̂
̂
≤ (Q P ) Q P −1‖ ̂ x (k + N − 1|k) ‖
max i i i i ‖ i ‖P i

≤ (1 − ) √
2 m
This completes the proof.
Lemma 8.3 Suppose Assumptions 8.1–8.3 hold and x(k )∈ X, ∀k ≥ 0, if Problem 8.1 has a
0
solution at every update time l, l = 1, 2, … ,k − 1, then
‖ f ‖
i
i
‖x (k + s|k) − ̂ x (k + s|k)‖ ≤ √ (8.19)
2 m
‖ ‖P i
for all i ∈ P and all s = 1, 2, … , N, provided that (8.18) and the following parametric condi-
tion hold:
√ N−2
m 2 ∑
≤ 1 (8.20)
l
min (P) l=0
f
where is as defined in (8.16). Furthermore, the feasible control u (k + s|k) and the feasible
l
i
f
state x (k + s|k) satisfy constraints (8.10) and (8.11).
i