Page 118 - Distributed model predictive control for plant-wide systems

92 Distributed Model Predictive Control for Plant-Wide Systems

The Nash optimal solution of the whole system in this case is
−1
dis
Δu (k)=(I − T D T ) D [ (k)− ̃ y (k)] (5.67)
M r E c 1 P0
Using the matrix decomposition technique, it gives
− [ ] −
(I − T D T ) = (I − D ) +(I + D − T D T )
r
r
c
E
E
E
E
c
− − [ − − ] − −
= (I − D ) − (I − D ) (I − D ) + (I + D − T D T ) (I − D ) (5.68)
E
c
E
E
E
E
r
E
In general, (I − D ) − 1 and (I + D − 2T D T ) − 1 exist; therefore, the above equation holds.
E E r E c
Substitute (5.68) into (5.67) to obtain
[ −1 −1 ] −1 ∗
−1
∗
dis
Δu (k)= 2Δu (k)− 2(I − D ) (I − D ) +(I + D − 2T D T ) Δu (k)
E
E
c
E
r
E
M M M
(5.69)
∗
= SΔu (k)
M
with [ ] −
S = I − (I − D ) − (I − D ) − + (I + D − T D T ) −
r
E
E
E
E
c
[ ]
−1
∗
From Δu (k)= (I-D ) D (k)− ̃ y (k) , it has
P0
E
M
∗
−1
(k)− ̃ y (k)= D (I − D )Δu (k)
P0 E M
Then it gives
∗ ∗ 2 ∗ 2
J = ‖ (k)− ̃ y (k)− AΔu (k)‖ + ‖Δu (k)‖ R
P0
Q
M
M
−1 ∗ ∗ 2 ∗ 2
= ‖D (I − D )Δu (k)− AΔu (k)‖ + ‖Δu (k)‖ (5.70)
E
1 M M Q M R
∗ 2
= ‖Δu (k)‖
M F
−1
T
−1
with F =[D (I − D )− A] Q[D (I − D )− A]+ R
E
E
1 1
Let
⎡A 11 ⎤
A = ⎢ ⋱ ⎥
0
⎢ ⎥
⎣ 0 A mm⎦
Then the prediction model of the whole distributed system under the mixed failure can be
written as
dis dis dis
y = ̃ y (k)+(A + T GT )Δu (k)= ̃ y (k)+ A u (k) (5.71)
PM P0 0 r c M P0 M
with
A = A + T GT
r
0
Substituting (5.69) and (5.71) into (5.53), we derive
∗
2
∗
J dis = ‖ (k)− ̃ y (k)− A S u (k)‖ + ‖S u (k)‖ 2 R
P0
M
M
Q
∗ ∗ 2
= ‖ (k)− ̃ y (k)− A u (k)+(A − A S)Δu (k)‖ Q
P0
M
M
(5.72)
∗ ∗ 2
+ ‖Δu (k)+(S − I)Δu (k)‖
M M R
∗ ∗ 2
= J + ‖Δu (k)‖
M W