Page 206 - Distributed model predictive control for plant-wide systems

180 Distributed Model Predictive Control for Plant-Wide Systems

Subtracting (8.22) from (8.21), and from the definition of (8.16), we obtain the discrepancy
between the feasible state sequence and the presumed state sequence as
‖ f ‖
j,i
‖x (k + s|k) − ̂ x (k + s|k)‖
j,i
‖ ‖P j
s
‖ ∑ ‖
‖ s−l ( p ) ‖
i
= ‖ A ii A ij x (k + l − 1|k − 1) − ̂ x (k + l − 1|k − 1) ‖
i
‖ ‖
‖ l=1 ‖ P i
s
∑ ( p )
‖ s−l ‖
≤ ‖A ii A ij x (k + l − 1|k − 1) − ̂ x (k + l − 1|k − 1) ‖
i
i
‖ ‖P i
l=1
s
∑ ‖ p
≤ ‖ (8.23)
s−l ‖x (k + l − 1|k − 1) − ̂ x (k + l − 1|k − 1)‖
i
̃
‖ i
‖2
l=1
Let the subsystems, which respectively maximize the following functions, as S
g
s
∑
‖ p ‖
s−l ‖x (k − 1 + l|k − 1) − ̂ x (k − 1 + l|k − 1)‖ , i ∈ P
i
‖ i
l=1 ‖2
Then, the following equation can be deduced from (8.23):
‖ f ‖
‖x (k + s|k) − ̂ x (k + s|k)‖
j
‖ j ‖P i
s
√ ∑ ‖ p
‖
≤ m 1 s−l ‖x (k + l − 1|k − 1) − ̂ x (k + l − 1|k − 1)‖
g
g
‖ ‖2
l=1
p
Since x (l|k − 1) satisfy constraints (8.10) for all times l = 1, 2, … , k − 1, the following
i
equation can be deduced:
‖ f ‖
i
‖x (k + s|k) − ̂ x (k + s|k)‖
i
‖ ‖P i
(1 − )(1 − ) (1 − )
≤ +
√ √
2 m 2 m

= √ (8.24)
2 m
Thus, (8.19) holds for all s = 1, 2, … , N − 1.
When l = N, we can get
∑
f
f
x (k + N|k)= A x (k + N − 1|k)+ A ̂ x (k + N − 1|k) (8.25)
i d,i i ij j
j∈P +i
∑
̂ x (k + N|k)= A ̂ x (k + N − 1|k)+ A ̂ x (k + N − 1|k) (8.26)
i d,i i ij j
j∈P +i
From the subtraction of the two equations, then, the discrepancy between the feasible state
f
x (k + N|k) and the presumed state ̂ x (k + N|k) is
j,i
j,i
f
f
x (k + N|k)− ̂ x (k + N|k)= A (x (k + N − 1|k)− ̂ x (k + N − 1|k)) (8.27)
i i d,i i i
This completes the proof of (8.19).