Page 100 - Distributed model predictive control for plant-wide systems

74 Distributed Model Predictive Control for Plant-Wide Systems

the stacked state and output predictions for the controller C take the form
i
{
̂
X (k + 1, P|k)= S [A ̂x (k|k)+ B U (k, M|k)+ W (k, P|k − 1)]
̂
i i i i i i i
(5.17)
̂
Y (k + 1, P|k)= C X (k + 1, P|k)+ T V (k, P|k − 1)
̂
̂
i
i
i
i
i
Define that
⎡ A 0 ···
ii 0 ⎤
⎢ ⎥
S ≜ ⋮ ⋱ ⋮
c ⎢ ⎥
⎢ p−1 0⎥
⎣ A ii ··· A ii ⎦ (5.18)
I
[ ]
(P−1)n y i ×n y i (P−1)n y i
T ≜
i I
n y i ×(p−1)n y i n y i
S ≜ diag{S , … , S }
1 n
(5.19)
T ≜ diag{T , … , T }
n
1
[ ]
A ii
A ≜
i
Pn y i ×n y i
{ }
⎡diag B
M ii ⎤
··· B
⎢ ⎥
⎢ n u i n u i ii⎥ (5.20)
B ≜
⋮ ⋱ ⋮ ⋮
⎢ ⎥
⎢ ⎥
···
⎢ B ii ⎥
⎣ n u i n u i ⎦
C ≜ diag {C }
i P ii
A ≜ diag{A , … , A }
1 m
B ≜ diag{B , … , B } (5.21)
1
m
C ≜ diag{C , … , C }
1 m
Then, by substituting W (k, P|k − 1) and V (k, P|k − 1) with their explicit expressions (5.15),
i i
it results
⎧X (k + 1, P|k)= S [A ̂ x (k|k)+ B U (k, M|k)+ A X(k, P|k − 1)
̂
̃ ̂
i
i i
i
i
i
i
⎪
̃
+ B U(k − 1, M|k − 1)] (5.22)
⎨ i
⎪
̂
̃ ̂
⎩Y (k + 1, P|k)= C X (k + 1, P|k)+ T C X(k, P|k − 1)
̂
i i i i i
Thus, under Assumption 5.1, for each controller C , i = 1, … , m, the stacked decentralized
i
predictions of state and output at time k are expressed by Equation (5.22), and the complete
stacked decentralized predictions take the following form:
̃ ̂
⎧X(k + 1, P|k)= S[Âx(k|k)+ BU(k, M|k)+ AX(k, P|k − 1)
̂
⎪
+ BU(k − 1, M|k − 1)] (5.23)
̃
⎨
⎪
⎩Y(k + 1, P|k)= CX(k + 1, P|k)+ TCX(k, P|k − 1)
̃ ̂
̂
̂