Page 51 - Distributed model predictive control for plant-wide systems

Model Predictive Control 25

where
[ T T T ] T nP
Y min = ̃ y 1,min ̃ y 2,min ··· ̃ y n,min ∈ R
[ ] T P
T
̃ y = y y ··· y ∈ R
i,min i,min i,min i,min
[ T T T ] T nP
Y = ̃ y ̃ y ··· ̃ y ∈ R
max 1,max 2,max n,max
[ ] T P
T
̃ y = y i,max y i,max ··· y i,max ∈ R
i,max
2. Input increment constraint: Δu ≤ Δu (k + l|k) ≤ u . The concatenated form can
j,min j j,max
be expressed as
ΔU ≤ ΔU(k|k) ≤ ΔU (2.17)
min max
where
[ T T T ] T mM
ΔU min = Δ̃ u 1,min Δ̃ u 2,min ··· Δ̃ u m,min ∈ ℝ
[ ] T M
T
Δ̃ u = Δu Δu ··· Δu ∈ ℝ
j,min j,min j,min j,min
[ T T T ] T mM
ΔU max = Δ̃ u 1,max Δ̃ u 2,max ··· Δ̃ u m,max ∈ ℝ
[ ] T M
T
Δ̃ u = Δu Δu ··· Δu ∈ ℝ
j,max j,max j,max j,max
3. Input magnitude constraint u j,min ≤ u (k + l|k) ≤ u j,max . The optimization problem should
j
satisfy the following constraint:
U ≤ BΔU(k|k)+ ̃ u(k − 1) ≤ U (2.18)
min max
where
[ T T T ] T mM
U min = ̃ u 1,min ̃ u 2,min ··· ̃ u m,min ∈ ℝ
T [ ] T M
̃ u = u u ··· u ∈ ℝ
j,min j,min j,min j,min
[ T T T ] T mM
U max = ̃ u 1,max ̃ u 2,max ··· ̃ u m,max ∈ ℝ
T [ ] T M
̃ u = u u ··· u ∈ ℝ
j,max j,max j,max j,max
B = block − diag{B , … , B }(m blocks)
0
0
⎡1 0 ··· 0⎤
⎢1 1 ⋱ ⋮⎥ M×M
B = ⎢ ⋮ ⋱ ⋱ 0 ⎥ ∈ ℝ
0
⎢ ⎥
⎣1 ··· 1 1⎦
[ T T T ] T
̃ u(k − 1)= ̃ u (k − 1) ̃ u (k − 1) ··· ̃ u (k − 1)
m
1
2
[ ] T M
̃ u (k − 1)= u (k − 1) u (k − 1) · · · u (k − 1) ∈ ℝ
j j j j
Equations (2.16)–(2.18) can be written in a uniform form as
CΔU(k|k) ≤ b (2.19)