Page 55 - Distributed model predictive control for plant-wide systems

Model Predictive Control 29

where Δu(k + l|k) = 0, when l ≥ M. Define that
[ T ] T
Y(k)= y(k + 1|k) y(k + 2|k) T ··· y(k + P|k) T
[ T T T ] T
X(k)= x(k + 1|k) x(k + 2|k) ··· x(k + P|k)
[ T ] T
U(k)= u(k|k) u(k + 1|k) T ··· u(k + M − 1|k) T
[ T T T ] T
W(k)= d(k|k) d(k + 1|k) ··· d(k + P − 1|k)
And
T
⎡ A ⎤
2
⎢A ⎥
H =
⎢ ⋮ ⎥
⎢ ⎥
P
⎣A ⎦
B ···
⎡ ⎤
⎢ (A + I) B B ··· ⎥
⋮ ⋮ ··· ⋮
⎢ ⎥
⎢ ⎥
M M−1
⎢∑ i−1 ∑ i−1 ⎥
A B A B ··· B
⎢ ⎥
⎢ i=1 i=1 ⎥
G = ⎢ ⎥
M+1 M
⎢∑ i−1 ∑ i−1 ⎥
A B A B ···
⎢ (A + I)B ⎥
⎢ i=1 i=1 ⎥
⋮ ⋮ ⋮
⎢ ··· ⎥
⎢ ⎥
P P−1 P−M+1
∑ ∑ ∑
⎢ i−1 i−1 i−1 ⎥
A B A B ··· A B
⎢ ⎥
⎣ i=1 i=1 i=1 ⎦
B
⎡ ⎤
⎢ (A + I) B ⎥
⋮
⎢ ⎥
⎢ ⎥
M
⎢∑ i−1 ⎥
A
⎢ B⎥
⎢ i=1 ⎥
F = ⎢ M+1 ⎥
⎢∑ i−1 ⎥
A
⎢ B⎥
⎢ i=1 ⎥
⎢ ⋮ ⎥
P
⎢ ⎥
∑
⎢ i−1 ⎥
A B
⎢ ⎥
⎣ i=1 ⎦
E ···
⎡ ⎤
⎢ AE E ··· ⎥
V =
⎢ ⋮ ⋮ ··· ⋮ ⎥
⎢ P−1 P−2 ⎥
⎣A E A E ··· E⎦
⎡C ··· ⎤
⎢ C ⋱
T = ⋮⎥
⎢ ⋮ ⋱ ⋱ ⎥
⎣ ··· C⎦
⎥
⎢