Page 240 - Distributed model predictive control for plant-wide systems

214 Distributed Model Predictive Control for Plant-Wide Systems

s
∑ s−l
+ A i B ̂ u (k + l − 1|k)
i i
l=1
s
∑ s−l
+ A i A ̂ x̃ i (k + l − 1|k) (10.5)
̃
i
l=1
where
[ ]
A ii A
A = ii
i
A A
ii ii
[ ]
A ̃ ii
̃
A =
i
A
i ̃ i
[ ]
B ii
B =
i
Σ j∈P +i x j
n ×n u i
[ n ]
n x i ×Σ j∈P +i u j
B =
i
B
ii
[ ]
A = A A … A
ii ij 1 ij 2 ij m i
j 1 i
⎡ A ⎤
A j 2 i
⎢ ⎥
A = ⎢ ⎥
ii
⎢ M ⎥
⎢ ⎥
A
⎣ j m i i⎦
⎡ A A ··· A ⎤
j 1 j 1 j 1 j 2 j 1 j m i
⎢ ⎥
⎢ A A ··· A ⎥
A = j 2 j 1 j 2 j 2 j 1 j m i
ii ⎢ M M ⋱ M ⎥
⎢ ⎥
⎢ A A ··· A ⎥
j
j
j m i 1 j a j 1 j m i m i ⎦
⎣
B = block-diag(B , B , … , B )
ii j 1 j 2 j m i
{j , j , … , j }= P ̆ i
1 2 m i
p p
Given x (k|k)= x (k|k), x (k|k)= x (k|k), the presumed control sequence for subsystem
i i i,i i
S is given by
i
̂ u (k + s − 1|k)= ̂ u (k + s − 1|k) (10.6)
i i,i
and
{ p
u (k + s − 1|k − 1) , s = 1, 2, … , N − 1
̂ u (k + s − 1|k)= j p (10.7)
j,i
K x (k + N − 1|k − 1), s = N
j j,i