Page 208 - Distributed model predictive control for plant-wide systems

182 Distributed Model Predictive Control for Plant-Wide Systems

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Lemma 8.5 Suppose Assumptions 8.1–8.3 hold, x(k )∈ ℝ x , and conditions (8.18) and
0
(8.20) are satisfied. For any k ≥ 0, if Problem 8.1 has a solution at every update time l,
f
l = 1, 2, … ,k − 1, then x (k + N|k)∈Ω( ∕2), ∀i ∈ P.
i
Proof. In view of Lemmas 8.2 and 8.3, using the triangle inequality, we have
‖ f ‖ ‖ f ‖
i
‖x (k + N|k)‖ ≤ ‖x (k + N|k) − ̂ x (k + N|k)‖
i
i
‖ ‖P i ‖ ‖P i
+ ̂ x (k + N|k) ‖
‖
‖ i
‖P i
(1 − )
+ (8.31)
≤ √ √ = √
2 m 2 m 2 m
for each j ∈ P , i ∈ P. This shows that the terminal state constraint is satisfied. This completes
i
the proof.
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Theorem 8.1 Suppose Assumptions 8.1–8.3 hold, x(k )∈ ℝ x and constraints (8.10), (8.11),
0
f
f
and (8.13) are satisfied at k . Then, for every i ∈ P, the control u (⋅|k) and state x (⋅|k),
0
i
j,i
respectively, defined by (8.17) and (8.4), are a feasible solution of Problem 8.1 at every
update k.
Proof. We will prove the theorem by induction.
p f
First, consider the case of k = 1. The state sequence x (⋅|1)= x (⋅|1) trivially satisfies the
j,i j,i
dynamic equation (8.4), the stability constraint (8.12), and the consistency constraints (8.10)
and (8.11).
Observe that
p f
̂ x (1|1)= x (1|0)= x (1|1)= x (1), i ∈ P,
i i i i
and that
p
f
x (1 + s|1)= x (1 + s|0)
i i
s = 1, 2, … , N − 1
f
Thus, x (N|1)∈Ω ( ∕2). By the invariance of Ω( ) under the terminal controller and the
i i
conditions in Lemma 8.1, it follows that the terminal state and control constraints are also
satisfied. This completes the proof of the case of k = 1.
p f
Now suppose u (⋅|l)= u (⋅|l) is a feasible solution for l = 1, 2, … , k − 1. We will show that
i i
f
u (⋅|k) is a feasible solution at update k.
i
f
As before, the consistency constraint (8.10) is trivially satisfied, and x (⋅|k) is the cor-
i
responding state sequence that satisfies the dynamic equation. Since there is a solution for
Problem 8.1 at updates l = 1, 2, … , k − 1, Lemmas 8.2–8.5 can be invoked. Lemma 8.4 guar-
antees control constraint feasibility. Lemma 8.5 shows that the terminal state constraint is
satisfied and the proof of Theorem 8.1 is completed.