Page 48 - Distributed model predictive control for plant-wide systems

22 Distributed Model Predictive Control for Plant-Wide Systems

Suppose all the inputs can be changed. Then considering the ith output and applying the
superposition principle yields
r
∑
̃ y (k|k)= ̃ y (k|k)+ A Δ̃ u (k|k) (2.6)
i i,0 ij j
j=1
where
̃ y (k|k)=[y (k + 1|k), y (k + 2|k), … , y (k + P|k)] T
i i i i
Considering all the inputs and outputs yields

̃
Y(k|k)= Y (k|k)+ AΔU(k|k) (2.7)
0
where
[ T ] T
Y(k|k)= ̃ y (k|k) ̃ y (k|k) T ··· ̃ y (k|k) T
1 2 r
[ T ] T
Y (k|k)= ̃ y (k|k) ̃ y (k|k) T ··· ̃ y (k|k) T
1,0
r,0
2,0
0
[ T ] T
ΔU(k|k)= Δ̃ u (k|k) Δ̃ u (k|k) T ··· Δ̃ u (k|k) T
1 2 m
⎡A 11 A 12 ··· A ⎤
1m
⎢A A ··· A ⎥
A = 21 22 2m
̃
⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣A A ···
r1 r2 A ⎦
rm
2.2.3 Optimization
Suppose the criterion for optimizing ΔU(k|k) is to minimize the following cost function:
2 2
J(k)= ‖E(k|k)‖ + ‖ΔU(k|k)‖ (2.8)
̃ W ̃ R
where W ≥ 0 and R ≥ 0 are symmetric matrices
̃
̃
[ T ] T
E(k|k)= ̃ e (k|k) ̃ e (k|k) T ··· ̃ e (k|k) T
1 2 n
[ ] T
̃ e (k|k)= e (k + 1|k) e (k + 2|k) · · · e (k + P|k)
i i i i
e (k + l|k)= y (k + l)− y (k + l|k)
i,r
i
i
̃ T ̃ ̃
Then, when A WA + R is nonsingular, minimization of (2.8) yields
̃
̃ T ̃ ̃
ΔU(k)=(A WA + R) A WE (k) (2.9)
̃ −1 ̃ T ̃
0
where
[ T T T ] T
E (k|k)= ̃ e (k|k) ̃ e (k|k) ··· ̃ e (k|k)
2,0
1,0
n,0
0
[ ] T
̃ e (k|k)= e (k + 1|k) e (k + 2|k) · · · e (k + P|k)
i,0
i,0
i,0
i,0
e (k + l|k)= y (k + l)− y (k + l|k)
i,0
i,r
i,0
y i,r (k + l) is the set-point value at the future time k + 1for the ith output; y i,0 (k + l|k)isthe
prediction on the ith output at future time k + 1, when the control moves for the time k and
future sampling instants are kept invariant.