Page 205 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 179

Proof. We will prove (8.19) first. Since a solution exists at update time 1, 2, … , k − 1,
according to (8.4), (8.5), and (8.17), for any s = 1, 2, … , N − 1, the feasible state is given by
f
l
f
x (k + l|k)= A x (k|k)
i ii i
l
∑
f
+ A l−h B u (k + l|k)
ii ii i
h=1
l
∑ ∑ l−h
+ A A ̂ x (k + h − 1|k)
ii ij j
j∈P +i h=1
(
l
= A l A x (k − 1|k − 1)
ii ii i
)
∑
+ B u (k − 1|k − 1) + A x (k − 1|k − 1)
ii i
ij j
j∈P +i
l
∑
+ A l−h B ̂ u (k + l|k)
ii ii i
h=1
l
∑ ∑ l−h P
+ A A x (k + h − 1|k − 1) (8.21)
ii ij j
j∈P +i h=1
and
l
̂ x(k + l|k)= A x (k|k − 1)
ii i
l
∑ l−h
+ A B u (k + l|k − 1)
ii ii i
h=1
l
∑ ∑ l−h
+ A A ̂ x (k + h − 1|k − 1)
ii ij j
j∈P +i h=1
(
l
= A l A x (k − 1|k − 1)
ii ii i
)
∑
+ B u (k − 1|k − 1) + A ̂ x (k − 1|k − 1)
ij j
ii i
j∈P +i
l
∑ l−h
+ A B ̂ u (k + l|k)
ii ii i
h=1
l
∑ ∑ l−h
+ A A ̂ x (k + h − 1|k − 1) (8.22)
ii ij j
j∈P +i h=1