Page 298 - Distributed model predictive control for plant-wide systems

272 Distributed Model Predictive Control for Plant-Wide Systems

12.3.5 Optimization Problem

According to the N-DMPC introduced in Chapter 7, the optimization problem of each
subsystem-based MPC can be written as per the following quadratic programming (QP):

T
T
min J (k)= U HU + f U i
i
i
i
{ e e
u min − u ≤ U ≤ u max − u
i
⌢ ⌢
x min ≤ Ax (k) + BU ≤ x max
i
i
where
H = B QB + R
̃ T ̃ ̃
̃
i i
Head coach:
̃ T
̃
̃
̃
f =−B × Q ×(Y − A Z (k − 1)− A Z (k − 1)) + ̃u e
i r ii ni ii−1 ni−1
Middle coach:
̃ T
̃
f =−B × Q ×(Y − A Z (k − 1)− A Z (k − 1)− A Z (k − 1)) + ̃u e
̃
̃
̃
i r ii ni ii+1 ni+1 ii−1 ni−1
Tail coach:
̃ T
̃
̃
̃
f =−B × Q ×(Y − A Z (k − 1)− A Z (k − 1)) + ̃u e
i r ii ni ii+1 ni+1
C A ii ⎡ C A ii+1 ⎤
⎡ ̂ ̂ ⎤ ̂ ̂
i
i
⎢ C A ⎢ ̂ ̂ ̂ ̂ ̂
̂ ̂ 2 ⎥
A = ⎢ i ii⎥ ̃ = C A A ii+1 C A ii+1 ⎥
ii
i
i
, A
̃
ii ii+1 ⎢ ⋮ ⋮ ⋮ ⋮ ⎥
⎢ ⋮ ⎥
⎢ ⎥
C A
A
̂ ̂ P−2 ̂
̂ ̂
C A
⎢ ̂ ̂ P⎥ ⎣ ̂ ̂ P−1 ̂ C A A ··· C A ⎦
⎣ i ii⎦ i ii ii+1 i ii ii+1 i ii+1
̂ ̂
C A
⎡ i ii+1 ⎤
C A A
̂ ̂
i
ii
i
A = ⎢ ̂ ̂ ̂ ii+1 C A ii+1 ⎥ ,
̃
ii+1 ⎢ ⋮ ⋮ ⋮ ⋮ ⎥
⎢ ⎥
C A
A
⎣ ̂ ̂ P−1 ̂ C A A ··· C A ⎦
̂ ̂ P−2 ̂
̂ ̂
i ii ii+1 i ii ii+1 i ii+1
C B
̂ ̂
⎡ i i ⎤
C A B
⎢ ̂ ̂ ̂ C B ⎥
̂ ̂
i ii i i i
B = ⎢ ⋮ ⋮ ⋮ ⋮ ⎥
̃
i
⎢
P−M+1 ⎥
⎢ ∑ ⎥
̂ ̂ P−1 ̂
⎢C A ii B i ··· ··· C A ii B ⎥
̂ ̂ i−1 ̂
i
i
i
⎣ ⎦
i=1
12.4 Simulation Results
We take the half of CRH2 EMUs as the simulation model as shown in Figure 12.6.