Page 207 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 181

f
In what follows we will prove that the feasible state x (k + s|k) satisfies the constraints (8.10)
i
and (8.11) when (8.19) holds.
f
When l = 1, 2, … , N − 1, substitute x (k + l|k) in the constraint (8.10) with considering
i
(8.20), we can get
s
∑
‖ f ‖
‖x (k + l|k) − ̂ x (k + l|k)‖
s−l i i
‖ ‖2
l=1
s
1 ∑ ‖ f
‖
≤ s−l ‖x (k + l|k) − ̂ x (k + l|k)‖
i
i
(P ) ‖ ‖P i
min i l=1
s √
1 ∑ m 2
≤ s−l √ (8.28)
(P) 2 mm
min l=1 2
Thus, when
√ s
m 2 ∑
s−l ≤ 1
min (P) l=1
f
the state x (k + s|k), s = 1, 2, … , N − 1, satisfies the constraint (8.10).
i
f
Finally, when l = N, x (k + N|k) satisfies the constraint (8.11).
i
‖ f ‖
i
‖x (k + N|k) − ̂ x (k + N|k)‖ ≤ √ (8.29)
i
2 m
‖ ‖P i
which shows that the constraint (8.11) 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 8.4 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 u (k + s|k)∈ U, for all s = 1, 2, … ,N − 1.
i
f
Proof. Since Problem 8.1 has a feasible solution at l = 1, 2, … , k − 1, and u (k + s − 1|k)=
p f i
u (k + s − 1|k − 1) for all s = 1, 2, … , N − 1, we only need to show that u (k + N − 1|k)∈ U.
i i
Since has been chosen to satisfy the conditions of Lemma 8.1, 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 8.2 and 8.3, 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 P i i
‖ ‖P i
+ ̂ x (k + N − 1|k) ‖
‖
‖ i ‖P i

≤ √ + √
2(q + 1) m 2 m

≤ √ (8.30)
m
f
that is, x (k + N|k)∈Ω ( ). This completes the proof.
i
i