Page 248 - Distributed model predictive control for plant-wide systems

222 Distributed Model Predictive Control for Plant-Wide Systems

p p p
Consider that x (k − 2|k − 2)= x (k − 2), x (k − 2|k − 2)= x (k − 2), and u (k − 2|k −
i i i i i
2)= u (k − 2), the above equation can be rewritten as
i
[ p ]
x (k|k − 1) [ x (k|k) ]
i i p
p = − B u (k − 1|k − 1)
x (k|k + 1) x (k|k) i i (10.34)
i,i i
+ B ̂ u (k − 1|k − 1)
i i
Subtracting (10.32) from (10.31), and substituting (10.34) into the resulting equation, we
obtain the discrepancy between the feasible state sequence and the presumed state sequence as
‖ f ‖
‖x (k + s|k) − ̂ x (k + s|k)‖
j,i
j,i
‖ ‖P j
s
‖ ∑ s−l ( )‖
‖ p ‖
j,i
≤ ‖ L A i B i u (k + l − 1|k − 1) − ̂ u (k + l − 1|k − 1) ‖
i
‖ i ‖
‖ l=0 ‖ P j
s
‖ ∑ s−l ( )‖
‖ ̃ p ‖
+ ‖ L A i A i x (k + l − 1|k − 1) − ̂ x̃ i (k + l − 1|k − 1) ‖
j,i
̃ i
‖ ‖
‖ l=1 ‖ P j
s
∑ ‖ s−l ( p ) ‖
≤ ‖ L A i B i u (k + l − 1|k − 1) − ̂ u (k + l − 1|k − 1) ‖ (10.35)
‖ j,i
i
‖
l=0 ‖ i ‖P j
s
∑ ‖ s−l ( p ) ‖
+ ‖ L A i A i x (k + l − 1|k − 1) − ̂ x̃ i (k + l − 1|k − 1) ‖
̃
‖ j,i
̃ i
‖
l=1 ‖ ‖P j
s
∑ ‖ p
≤ ‖
s−l ‖u (k + l − 1|k − 1) − ̂ u (k + l − 1|k − 1)‖
i
‖ i
l=0 ‖2
s
∑ ‖ p
+ ‖
s−l ‖x (k + l − 1|k − 1) − ̂ x̃ i (k + l − 1|k − 1)‖
‖ i
̃
‖2
l=1
Let the subsystems, which respectively maximize the following two functions, be S and S
g h
s
∑
‖ p ‖
‖u (k − 1 + l|k − 1) − ̂ u (k − 1 + l|k − 1)‖ , i ∈ P,
s−l ‖ i i ‖2
l=0
s
∑
‖ p ‖
‖x (k − 1 + l|k − 1) − ̂ x (k − 1 + l|k − 1)‖ , i ∈ P
s−l ‖ i i ‖2
l=1
Then, the following equation can be deduced from (10.35):
‖ f ‖
‖x (k + s|k) − ̂ x (k + s|k)‖
j,i
j,i
‖ ‖P j
s
1 ∑ ‖ p
≤ m 2 ‖
g
g
1 s−l ‖u (k + l − 1|k − 1) − ̂ u (k + l − 1|k − 1)‖
‖2
‖
l=1
s
1 ∑ ‖ p
+ m 2 ‖
h
2 s−l ‖x (k + l − 1|k − 1) − ̂ x (k + l − 1|k − 1)‖
‖ h
‖2
l=1