Page 254 - Distributed model predictive control for plant-wide systems

228 Distributed Model Predictive Control for Plant-Wide Systems

s
∑
̃ s−l
+ A (k + l − 1|k), s = 0, 1, 2, … , N − 1
i ̂ x̃ i
l=1
s
∑
‖ p ‖
s−l ‖x (k + l|k) − ̂ x (k + l|k)‖
i
l=1 ‖ i ‖2

, s = 1, 2, … , N − 1
≤ √
2 mm 2
‖ p ‖
i
‖x (k + N |k) − ̂ x (k + N |k)‖ ≤ √
‖ i ‖P i
2 m
‖ p ‖ ‖ f ‖
‖x (k + s|k)‖ ≤ ‖x (k + s|k)‖
i
‖ i ‖P i ‖ ‖P i

+ √ , s = 1, 2, … , N
N m
p
u (k + s − 1|k)∈ U , s = 1, 2, … , N
i i
p
x (k + N |k)∈Ω ( ∕2)
i
i
It should be noted that the consistency constraints in inputs (10.13) do not appear here. This
is because there is no input, except for u (⋅), that appears in the predictive model. This result
i
is consistent with the linear version of what is presented in Ref. [51], which provides a local
performance index based DMPC for continuous nonlinear systems.
10.5.2 Cooperative DMPC
In this coordination strategy, each subsystem-based MPC calculates the optimal u (k)tomin-
i
imize the cost function of the entire system. The predictive model of each subsystem-based
MPC includes the state evolutions of all subsystems, and can be deduced as follows according
to (10.1):
s
∑
p s s−l p
x (k + s|k)= L A + A u (k + l − 1|k)
j,i
j,i i
l=1
(10.52)
s
∑ s−l
+ A ̂ u (k + l − 1|k), s = 1, 2, … , N, j ∈ P
j,i
l=1
(⋅) is nonexistent, m = m − 1, and P = P. Consequently, the optimization problem of
1 i
Also, x̃ i
the DMPC, where the global cost is minimized at each subsystem-based MPC, can be deduced
from the N-DMPC as follows:
∑ ‖ p 2
‖
J (k)= J (k)= ‖x (k + N |k)‖
i
i
j∈V ‖ j,i ‖P j
( )
N−1
∑ ∑ ‖ p 2 ‖ p 2
‖
‖
+ ‖x (k + s|k)‖ + ‖u (k + s − 1|k)‖
‖ j,i ‖Q j ‖ i ‖R i
s=1 j∈V