Page 224 - Distributed model predictive control for plant-wide systems

198 Distributed Model Predictive Control for Plant-Wide Systems

f
In addition, from definition (9.13), it has u (k + l − 1|k)− u (k + l − 1|k − 1)= 0. Thus
i i
f
u (k + l − 1|k) satisfied the constraint (9.8) when l = 1, 2, … , N − 1, concluding the proof.
i
In what follows we establish that, at time k, if the condition (9.18) is satisfied, then
f
u (k + l − 1|k), l = 1, 2, … , N, is a feasible solution of Problem 9.1.
i
Lemma 9.3 Suppose Assumptions 9.1 and 9.2 hold, x(k )∈ X, and the conditions (9.12)
0
and (9.18) are satisfied. For any k ≥ 0 , if Problem 9.1 has a solution at every update time t,
f
t = 0, … ,k − 1, then u (k + l − 1|k)∈ U for all l = 1, 2, … , N, and for any i ∈ P.
i
f
Proof. Since Problem 9.1 has a feasible solution at k − 1, and u (k + l − 1|k)= u (k + l −
i
i
f
1|k − 1) for all l ∈ {1, … , N − 1}, it only needs to be shown that u (k + N − 1|k) is in U.
i
Since is chosen to satisfy the conditions of Lemma 9.1, K x ∈ U for all i ∈
i
f
P, when x ∈Ω( ). Consequently, a sufficient condition for u (k + N − 1|k) is that
i
f
x (k + N − 1| k) ∈Ω( ).
In view of Lemma 9.2 and ≤ 0.5, using the triangle inequality, it has
‖ f ‖
‖x (k + N − 1|k)‖
‖ ‖P
‖ f ‖
≤ ‖x (k + N − 1|k) − ̂ x(k + N − 1|k − 1)‖
‖ ‖P
+ ‖̂ x (k + N − 1|k − 1)‖
P
≤ +
≤
f
that is, x (k + N − 1| k) ∈Ω( ) for any i ∈ P. This concludes the proof.
Lemma 9.4 Suppose Assumptions 9.1 and 9.2 hold, x(k )∈ X, and the conditions (9.12)
0
and (9.18) are satisfied. For any k ≥ 0, if Problem 9.1 has a solution at every update time,
t = 0, … ,k − 1, then the terminal-state constraint is satisfied, for any i ∈ P.
Proof. Since there is a solution for Problem 9.1 at updates t = 1, … , k − 1, Lemmas 9.2–9.4
can be invoked. According to Lemma 9.2, conditions (9.12) and (9.18), and using the triangle
inequality, it has
‖ f ‖
‖x (k + N|k)‖
‖ ‖P
‖ f ‖
≤ ‖x (k + N|k) − ̂ x(k + N|k − 1, i)‖
‖ ‖P
+ ‖̂ x (k + N|k − 1, i)‖ P
≤ (1 − ) +(1 − )

≤
for each i ∈ P. This shows that the terminal-state constraint is satisfied.