Page 247 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Inputs and Information Structure Constraints 221

Lemma 10.3 Suppose Assumptions 10.1–10.3 hold and x(k )∈ X, ∀k ≥ 0, if Problem 10.1
0
has a solution at every update time l, l = 1, 2, … ,k − 1, then
‖ f ‖
j,i
j,i
‖x (k + s|k) − ̂ x (k + s|k)‖ ≤ √ (10.29)
‖ ‖P j
2 m
for all j ∈ P , i ∈ P and all s = 1, 2, … , N, provided that (10.28) and the following parametric
i
condition hold:
√ N−2
m 2 ∑
≤ 1 (10.30)
l
min (P) l=0
f
where is as defined in (10.21). Furthermore, the feasible control u (k + s|k) and the feasible
l
i
f
state x (k + s|k) satisfy constraints (10.13)–(10.15).
j,i
Proof. We will prove (10.29) first. Since a solution exists at update time 1, 2, … , k − 1,
according to (10.5), (10.7), and (10.24), for any s = 1, 2, … , N − 1, the feasible state is
given by
[ ]
s x (k|k)
i
f
x (k + s|k)= L A
j,i j,i i
x (k|k)
i
s
∑ s−l
+ L A B ̂ u (k + l − 1|k)
j,i i i i
l=1
s (10.31)
∑ s−l
+ L A B ̂ u (k + l − 1|k)
j,i i i i
l=1
l
∑ s−l
+ L A ̃ (k + l − 1|k)
j,i i A ̂ x̃ i
i
l=1
and the presumed state is
p
̂ x (k + s|k)= ̂ x (k + s|k − 1)
j,i j,i
[ p ]
x (k|k − 1)
i
= L p
j,i x (k|k − 1)
i,i
s
∑ s−l p
+ L A B u (k + l − 1|k − 1)
j,i i i i
l=1 (10.32)
s
∑ s−l
+ L j,i A i B ̂ u (k + l − 1|k − 1)
i i
l=1
l
∑ s−l
+ L A ̃ (k + l − 1|k − 1)
i
j,i i A ̂ x̃ i
l=1
where, according to (10.5),
[ p ] [ p ]
x (k|k − 1) x (k − 1|k − 1)
i i p
p = A i p + B u (k − 1|k − 1)
x (k|k − 1) x (k − 1|k − 1) i i (10.33)
i,i i,i
̃
+ B ̂ u (k − 1|k − 1)+ A ̂ x (k − 1|k − 1)
i i i i
̂ x (k − 1|k − 1)= x (k − 1)
i
i