Page 50 - Distributed model predictive control for plant-wide systems

24 Distributed Model Predictive Control for Plant-Wide Systems

where
y (k + P|k − 1)= y (k + P − 1|k − 1) (2.13)
i,0
0
By summarizing the above deductions, at each time k > 0, ̃ y (k|k) can be calculated by
0
m
∑
̃
̃ y (k|k)= ̃ y (k|k − 1)+ A Δu (k − 1)+ f (k) (2.14)
i,0 i,0 ij,1 j i i
j=1
where
[ ] T
A ij,1 = s ij,2 s ij,3 ··· s ij,P+1
[ ] T
f = f f ··· f
̃
i i1 i2 iP
And Y (k|k) can be calculated as
0
Y (k|k)= Y (k|k − 1)+ A ΔU(k|k − 1)+ FΥ(k) (2.15)
̃
̃
0
0
1
where
[ T ] T
Y (k|k − 1)= ̃ y (k|k − 1) ̃ y (k|k − 1) T ··· ̃ y (k|k − 1) T
0 1,0 2,0 n,0
[ ] T
̃ y (k|k − 1)= y (k + 1|k − 1) y (k + 2|k − 1) · · · y (k + P|k − 1)
i,0
i,0
i,0
i,0
[ ] T
ΔU(k − 1)= Δu (k − 1) Δu (k − 1) ··· Δu (k − 1)
m
1
2
⎡A 11,1 A 12,1 ··· A 1m,1⎤
⎢A A ··· A ⎥
̃
A = 21,1 22,1 2m,1
1 ⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣A r1,1 A r2,1 ··· A nm,1 ⎦
⎡ f ̃ 1 0 ··· 0 ⎤
⎢ 0 ̃ f ··· 0 ⎥
̃
F = 2
⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣0 0 ··· ̃ f ⎦
n
[ ] T
Υ(k)= (k) (k) ··· (k)
1 2 n
2.2.5 DMC with Constraint
In the real application of DMC, the constraints on actuator slew rates, actuator ranges, and
constraints on the controlled variables usually exist. In the following, we discuss how to handle
the constraint in DMC.
1. Output constraint: y i,min ≤ y (k + l|k) ≤ y i,max , l = 1, 2, … , P.
i
At each time instant k, the output prediction is (2.7). Hence, we can let the optimization
problem satisfies the following constraint:
Y min ≤ Y (k|k)+ AΔU(k|k) ≤ Y max (2.16)
̃
0