Page 188 - Distributed model predictive control for plant-wide systems

162 Distributed Model Predictive Control for Plant-Wide Systems

[ ] T
̃ T
̃ T
B = B ··· B
̃
1 n
L = diag{L , … , L }
1 n
{[ ]}
L = diag P I n x i n x i ×(n ⌢ −n x i )
i
x i
S = diag{S , … , S } (E.5)
1 n
Then, for each controller C , i = 1, … , n, by Lemma 2 and definitions (E.5), the stacked
i
distributed state prediction at time k are expressed by
⌢ ̂
X (k + 1, P|k)= L X (k + 1, P|k)
̂
i
i
i
= L S [A ̂x(k|k)+ B U (k, M|k)
i i i1 i i
+ A X(k, P|k − 1)+ B U(k − 1, M|k − 1)] (E.6)
̃
̃ ̂
i i
By definitions (E.5), the completed stacked distributed prediction can be expressed as
̂
(k + 1, P|k)= LS[Âx(k|k)+ B (k, M|k)
+ AX(k, P|k − 1)+ BU(k − 1, M|k − 1)] (E.7)
̃
̃ ̂
Substituting (E.3) and (E.2) into (E.7), the following complete version of the stacked dis-
tributed prediction can be deduced:
̂
(k + 1, P|k)= LS[Âx(k|k)+ B (k, M|k)
̃
̃
+ A (k, P|k − 1)+ B (k − 1, M|k − 1)] (E.8)
̂
̂
Considering that the local control action applied at time k − 1 is given by u (k − 1)=
i
U (k − 1, m|k − 1), the open-loop optimal sequence U (k, M|k) of controller C at time k can
i i i i
be expressed as
′
U (k, M|k)= U (k − 1, M|k − 1)+ ΔU (k, M|k).
i i i i i i
Then by Equations (7.35) and (7.37), the stacked open-loop optimal control sequence at
time k can be directly expressed as
d
′
U (k, M|k)= u (k − 1)+ K [Y (k + 1, P|k)− Z (k + 1, P|k)]
̂
i i i i i i i
d
′
= u (k − 1)+ K {Y (k + 1, P|k)
i i i i i
(1)
′
− S [B u (k − 1)+ A ̂x(k|k)+ A X(k, P|k − 1)
̃ ̂
i i i i i i
̃ ̂
̃
+ B U(k − 1, M|k − 1)] − T C X(k, P|k − 1)} (E.9)
i i i
Define that
′
′
′
= diag{ , … , }
1 n
= diag{ , … , }
1
n
S = diag{S , … , S }
1
n