Page 138 - Distributed model predictive control for plant-wide systems

112 Distributed Model Predictive Control for Plant-Wide Systems

Remark 6.3 It should be noticed that the first two block rows of the dynamic matrix A N
included in Equation (6.27) depend on the elements of matrix A and the elements of matrix B,
while the third block row depends on A, B, C, Q, R , P, and M. The degree of freedom available
i
to the designer is based on the choices of Q, R , P, and M in the C-DMPC design phase, which
i
introduces significant modifications only on the third block row of A used for the stability test.
N
6.2.4.2 Performance Analysis
To explain the essential difference between the C-MPC and the centralized MPC, for each C ,
i
i = 1, … , m, the optimization problem of C-DMPC (6.6) is rewritten as
P M
∑ ∑
d
2
i
min ‖̂ y (k + l|k)− y (k + l|k)‖ + ‖Δu (k + l − 1|k)‖ 2
i
ΔU i (k,M|k) Q R i
l=1 l=1
i
̂ x (k + l + 1|k)
1
⎡ i ⎤ ⎡ ̂ x (k|k − 1) ⎤
⎢ ⋮ ⎥ ⎢ ⋮ ⎥
⎢ i (k + l + 1|k) ⎥ ⎢ ̂ x (k|k − 1) ⎥ l
̂ x
⎢ i−1 ⎥ l ⎢ i−1 ⎥ ∑
BU(k, l|k)
s.t. ̂ x (k + l + 1|k) = A x (k|k) + A s−1 ̃
i
i
⎢ ⎥ ⎢ ⎥
⎢̂ x i (k + l + 1|k)⎥ ⎢̂ x (k|k − 1)⎥ s=1
i+1 i+1
⋮
⎢ ⎥ ⎢ ⋮ ⎥
⎢ ⎥ ⎢ ⎥
i
⎣ ̂ x (k + l + 1|k) ⎦ ⎣ ̂ x (k|k − 1) ⎦
m
m
[ T T
U(k, M|k)= u (k|k − 1) … u (k|k − 1) u (k|k)
̃
1 i−1 i
T
u T (k|k − 1) ··· u (k|k − 1) · · ·
m
i+1
T
u (k + l|k − 1) … u T (k + l|k − 1) u (k + l|k)
1 i−1 i
T
u T (k + l|k − 1) ··· u (k + l|k − 1)] T
m
i+1
i
i
̂ y (k + l|k)= Ĉ x (k + l|k) (6.39)
The optimization problem of the centralized MPC can be written as
P M
∑ d 2 ∑ 2
min ‖̂ y(k + l|k)− y (k + l|k)‖ + ‖Δu (k + l − 1|k)‖
i
ΔU i (k,M|k) Q R i
l=1 l=1
1
1
⎡ ̂ x (k + l + 1 |k) ⎤ ⎡ x (k |k) ⎤
⋮ ⋮
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
̂ x i−1 (k + l + 1|k) x i−1 (k|k) l
⎢ ⎥ l ⎢ ⎥ ∑ s−1
s.t. ̂ x (k + l + 1|k) = A x (k|k) + A BU(k, l|k)
⎢ i ⎥ ⎢ i ⎥
⎢̂ x ⎢x s=1
i+1 (k + l + 1|k)⎥ i+1 (k|k)⎥
⎢ ⋮ ⎥ ⎢ ⋮ ⎥
⎢ ⎥ ⎢ ⎥
⎣ ̂ x (k + l + 1|k) ⎦ ⎣ x (k|k) ⎦
m m
[
T
U(k, M|k)= u (k |k) … u T (k|k) u (k|k)
̃
1 i−1 i
T
u T (k|k) ··· u (k|k) ···
i+1 m
T
u (k + l|k) … u T (k + l|k) u (k + l|k)
1 i−1 i
T
u T (k + l|k) ··· u (k + l|k)] T
i+1 m
̂ y(k + l|k)= Ĉ x(k + l|k) (6.40)