Page 294 - Distributed model predictive control for plant-wide systems

268 Distributed Model Predictive Control for Plant-Wide Systems

2
m ̇v = u −(m c + m c v + m c v )− k x
1 1 1 1 01 1 11 1 i 21 1 1 1
2
m ̇v = u −(m c + m c v + m c v )+ k i−1 i−1 − k x , i = 2, … , n − 1
x
i 0i
i i
i 1i i
i
i i
i 2i i
(12.6)
2
m ̇v = u −(m c + m c v + m c v )+ k n-1 n-1
x
n
n 2n n
n 1n n
n 0n
n n
̇ x = v − v i+1 , i = 1, … , n − 1
i
i
Assuming that the equilibrium state has a cruising speed
e
e
e
v = v = ··· = v = v
1 2 n r
(12.7)
e
e
e
̇ v = ̇v = ··· = ̇v = 0
1 2 n
e
The inputs u at the equilibrium point can be written as
i
e2
e
e
u = m c + m c v + m c v , i = 1, … , n (12.8)
i i 0i i 1i i i 2i i
e
e
e
x = x + x , v = v + v , u = u + u i (12.9)
i
i
i
i
i
i
i
i
Let
x =[ v , v , … , v , x , x , … , x ] T
1 2 n 1 2 n−1
u =[ u , u , … , u ] T (12.10)
1
n
2
We can get the following linearized equations by substituting (12.9) into (12.6).
̇ x = Ax + Bu
(12.11)
y = Cx
where
[ ]
A = A 11 A 12 , A =−diag(c + c v , … , c + c v ), A =
A A 11 11 21 r 1n 2n r 22 (n−1)×(n−1)
21 22
⎡ k 1 ⎤
− 0 ··· 0 0
m 1
⎢ ⎥
⎢ ⎥
k 1 k 2
⎢ − ··· 0 0 ⎥
⎢ m 2 m 2 ⎥
A = ⎢ ⋮ ⋮ ⋱ ⋮ ⋮ ⎥
12
⎢ k k ⎥
⎢ 0 ··· 0 n−2 − n−1 ⎥
⎢ m n−1 m n−1 ⎥
⎢ k n−1 ⎥
⎢ 0 ··· 0 0 ⎥
⎣ m n ⎦
⎡1 −1 0 ··· 0 0 ⎤
[ ]
⎢0 1 −1 ··· 0 0 ⎥ B 11
A 21 = ⎢ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⎥ , B = ,
⎢ ⎥ (n−1)×n
⎣0 0 0 ··· 1 −1⎦
( )
1 1 1 [ ]
B 11 = diag , , … , , C = I n×n (n−1)×(n−1)
m 1 m 2 m n