Page 250 - Distributed model predictive control for plant-wide systems

224 Distributed Model Predictive Control for Plant-Wide Systems

f
First, for any s = 1, 2, … , N − 1, u (k + l − 1|k)= ̂ u (k + l − 1|k). Thus, constraint (10.13)
i i
is satisfied.
Also,
s
∑
‖ f ‖
s−l ‖x (k + l|k) − ̂ x (k + l|k)‖
i
i
‖ ‖2
l=1
s
1 ∑ ‖ f ‖
≤ s−l ‖x (k + l|k) − ̂ x (k + l|k)‖ (10.39)
i
i
min (P ) ‖ ‖P i
i l=1
√
s m
1 ∑ 2
≤ s−l √
min (P) l=1 2 mm 2
Thus, when
√ s
m
2 ∑
s−l ≤ 1
(P)
min l=1
f
state x (k + s|k), s = 1, 2, … , N − 1, satisfy constraint (10.14).
i
Finally,
‖ f ‖
‖x (k + N |k) − ̂ x (k + N |k)‖
i
i
‖ ‖P i
‖ f ‖
= ‖x (k + N |k) − ̂ x (k + N |k)‖ (10.40)
i,i
i,i
‖ ‖P i

≤ √
2 m
which shows that constraint (10.15) is satisfied. This concludes the proof.
In what follows we establish that, at time k, if conditions (8.18) and (8.20) are satisfied, then
f
f
x (k + s|k) and u (k + s|k), s = 1, 2, … , N, are a feasible solution of Problem 8.1.
j,i i
n
Lemma 10.4 Suppose Assumptions 10.1–10.3 hold, x(k )∈ ℝ x , and conditions (10.30) and
0
(10.25) are satisfied. For any k ≥ 0, if Problem 10.1 has a solution at every update time l, l =
f
1, 2, … ,k − 1, then u (k + s|k)∈ U, for all s = 1, 2, … , N − 1.
i
f
Proof. Since Problem 10.1 has a feasible solution at l = 1, 2, … , k − 1, and u (k +
p i
s − 1|k)= u (k + s − 1|k − 1) for all s = 1, 2, … , N − 1, we only need to show that
i
f
u (k + N − 1|k)∈ U.
i
Since has been chosen to satisfy the conditions of Lemma 10.1, K x ∈ U for all i ∈ P
i i
f
f
when x ∈Ω( ). Consequently, a sufficient condition for u (k + N − 1|k)∈ U is that x (k +
i i
N − 1|k)∈Ω( ).
In view of Lemmas 10.2 and 10.3, using the triangle inequality, we have
‖ f ‖ ‖ f ‖
i
‖x (k + N − 1|k)‖ ≤ ‖x (k + N − 1|k) − ̂ x (k + N − 1|k)‖
i
i
‖ ‖P i ‖ ‖P i
+ ̂ x (k + N − 1|k) ‖ (10.41)
‖
‖ i
‖P i

≤ √ + √
2(q + 1) m 2 m