Page 158 - Distributed model predictive control for plant-wide systems

132 Distributed Model Predictive Control for Plant-Wide Systems

The only information that each C , i = 1, … , m, needs is the future behavior of S ∈ P and
i
j
i
S ∈ P . Similarly, C broadcasts the future behavior of S to the controller of S ∈ P and that
g j i i j i
of S ∈ P .
g j
7.2.3 Closed-Form Solution
The main result of this subsection is the computation of the closed-form solution to the
ND-MPC proposed. For this purpose, expressions of the interaction prediction and the state
prediction are provided first. Define that

(1)
̃ A = diag
i P
⎧ ⎫
⎪ ⎪
⎪⎡ A i,1 ··· A i,i−1 n x i ×n x i A i,i+1 ··· A i,i 1 −1 n x i ×n x i1 A i,i 1 +1 ··· A i,i n −1 n x i ×n x in A i,i n +1 ··· A i,m ⎤ ⎪
⎪⎢ ⎥ ⎪
⎪⎢A i 1 ,1 ··· A i 1 ,i−1 A i 1 ,i+1 ··· A i 1 ,i 1 −1 A i 1 ,i 1 +1 ··· A i 1 ,i n −1 A i 1 ,i n +1 ··· A i 1 ,m ⎥ ⎪
× ⎨⎢ n x i1 ×n x i n x i1 ×n x i1 n x i1 ×n x in ⎥ ⎬
⎪⎢ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⎥ ⎪
⎪⎢ A ··· A A ··· A A ··· A A ··· A ⎥ ⎪
⎪ ⎣ i n ,1 i n ,i−1 n x in ×n x i i n ,i+1 i n ,i 1 −1 n x in ×n x i1 i n ,i 1 +1 i n ,i n −1 n x im ×n x in i m ,i n +1 i m ,m⎦ ⎪
⎪⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎪
⎩ ⎭
n columns
(7.18)
⎛
··· A ··· A ···
⎜
⎜⎡ n x i ×n x 1 n x i ×n x i1−1 i,i 1 n x i ×n x i1+1 n x i ×n x in−1 i,i n n x i ×n x in+1 n x i ×n x m ⎤
⎜⎢ ··· A ··· A ··· ⎥
(2)
̃ A = diag ⎜⎢ n x i1 ×n x1 n x i1 ×n x i1−1 i 1 ,i 1 n x i1 ×n x i1+1 n x i1 ×n x in−1 i 1 ,i n n x i1 ×n x in+1 n x i ×n x m ⎥ ,
i
⎜⎢ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⎥
⎜ ⎣ n x in ×n x 1 ··· n x i1 ×n x i1−1 A i n ,i 1 n x i1 ×n x i1+1 ··· n x i1 ×n x in−1 A i n ,i n n x i1 ×n x in+1 ··· n x i ×n x m ⎥ ⎦
⎜⎢
⎜⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
⎝
n columns
⎞
⎟
⎧ ⎫
⎟
⎪ ⎪
⎟
⎪
diag P−1 ⎨ m ⎪ ⎟ (7.19)
⎬
∑
∑
⎟
⎪ n × n ⎪
x l ⎟
⎪ l=1 x l ⎪
⎩ l∈P +i ⎭ ⎟
⎟
⎠
⎡ B i,1 ··· B i,i−1 B i,i+1 ··· B i,m ⎤
⎧ ⎫
⎪ n x i ×n u i ⎪
⎢
⎪ B i 1 ,1 ··· B i 1 ,i−1 B i 1 ,i+1 ··· B i 1 ,m ⎥ ⎪
B = diag ⎢ n x i1 ×n u i ⎥ (7.20)
̃ ̃
i P ⎨ ⎢ ⋮ ··· ⋮ ⋮ ⋮ ··· ⎬
⎪ ⋮ ⎥ ⎪
⎪ B ··· B i m ,i−1 n x in ×n u i B i n ,i+1 ··· B i n ,m⎦ ⎥ ⎪
⎢
⎣ i n ,1
⎩ ⎭
⎧ ⎫
⎪ ⎪
⎪⎡ C ··· C C ··· C C ··· C C ··· C ⎤ ⎪
i,1 i,i−1 n y i ×n x i i,i+1 i,i 1 −1 n y i ×n x i1 i,i 1 +1 i,i n −1 n y i ×n x in i,i n +1 i,m
⎪⎢ ⎥ ⎪
⎪⎢C i 1 ,1 ··· C i 1 ,i−1 n y i1 ×n x i C i 1 ,i+1 ··· C i 1 ,i 1 −1 n y i1 ×n x i1 C i 1 ,i 1 +1 ··· C i 1 ,i n −1 n y i1 ×n x in C i 1 ,i n +1 ··· C i 1 ,m ⎥ ⎪
̃ C = diag ⎨⎢
i P ⎥ ⎬
⎪⎢ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⋮ ··· ⋮ ⎥ ⎪
⎪⎢ C ··· C C ··· C C ··· C C ··· C ⎥ ⎪
⎪ ⎣ i n ,1 i n ,i−1 n y in ×n x i i n ,i+1 i n ,i 1 −1 n y in ×n x i1 i n ,i 1 +1 i n ,i n −1 n y in ×n x in i n ,i n +1 i n ,m⎦ ⎪
⎪⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎪
⎩ n columns ⎭
(7.21)
where A , B , and C i, j are the zero blocks of congruent dimensions if S ∉ P +h (S ∈ P ).
−i
i, j
j
i, j
h