Page 132 - Distributed model predictive control for plant-wide systems

106 Distributed Model Predictive Control for Plant-Wide Systems

6.2.2.1 Performance Index
Since the optimal control decision of S affects, or even destroys, the performance of other
i
subsystems, the performance of other subsystems should be considered in finding the optimal
of the solution of S . To improve the global performance of the whole closed-loop system, the
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following so-called global performance index is adopted in each C , i = 1, … , m:
i
P M
∑ i d 2 ∑ 2
J (k)= ‖̂ y (k + l|k)− y (k + l|k)‖ + (6.4)
i
i Q ‖Δu (k + l − 1|k)‖ R i
l=1 l=1
where Q = diag{Q , Q , … , Q }. It should be noticed that Δu (k + l − 1|k) is excluded in the
1 2 m j
performance index, since it is independent of the future inputs sequence of S .
j
6.2.2.2 Prediction Model

Since the state evolution of other subsystems is affected by u (k) after one or several control
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periods, to improve the prediction precision, this influence is considered in C when predicting
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the future states of all subsystems. In addition, due to the unit delay introduced by the net-
work, the information of other subsystems is available only after one sampling time interval.
Therefore, in C , the states and outputs of all subsystems in l-step ahead are predicted by
i
l
l ′
i
⎧ ̂ x (k + l + 1 |k) = A L x(k)+ A L ̂ x(k|k − 1)
i
i
l l
⎪
∑ ∑ ∑
+ A B u (k + l|k)+ A B u (k + l|k − 1)
⎪ s−1 s−1
i i
j j
⎨ (6.5)
s=1 j∈{1,…,m} s=1
⎪
j≠i
⎪ i i
⎩̂ y (k + l + 1|k)= Ĉ x (k + l + 1|k)
where
⎡
⎤
⎢
⎥
L = i−1 I m ⎥
⎢
i
⎢
∑ n xi ∑ ⎥
⎢ n xi × n
n xi × n xj xj⎥
⎢ j=i+1 ⎦
j=1
⎣
⎧ ⎫
⎪ ⎪
⎪
′
L = diag I , , I m ⎪
⎨ i−1
i n xi ∑ ⎬
∑
n xj xj
⎪ n ⎪
⎪ ⎪
⎩ j=1 j=i+1 ⎭
[ T T T ] T
B = B 1i B 2i ··· B mi
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Remark 6.1 It should be noticed that the input of this neighborhood model is still the input of
S , and the inputs and states of other subsystem are regarded as disturbances. The estimations
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of future states and outputs of all subsystems (except S ) are only used in the controller C , and
i i
these estimations are different from that estimated by the controller C itself.
i