Page 154 - Distributed model predictive control for plant-wide systems

128 Distributed Model Predictive Control for Plant-Wide Systems

Table 7.1 Notations used in this chapter
Notations Explanations

̂ x (l|h), ̂ y (l|h) The predictions of x (l) and y (l) computed at time h,and l, h ∈ P, h < l
i
i
i
i
Δu (l|h), u (l|h) The input u (l) and the input increment Δu (l) computed by controller C at time
i
i
i
i
i
h, l, h ∈ P,and h < l
d
y (l|h) The set-point of y (l|h)
i i
⌢
x (k), y (k) The state and output vectors of the downstream neighbors of S .
⌢
i
i
i
[ ] T
T
T
T
x (k)= x (k) x (k) · · · x (k) and
⌢
i i i 1 i n
[ ] T
T
T
T
⌢
y (k)= y (k) y (k) · · · y (k) , n is the number of the downstream
i i i 1 i n
neighbors of S i
⌢
⌢
w (k), v (k) The interactions act on the state and output of downstream neighbors of S .See
i
i
i
(7.13) and (7.14)
⌢ ̂ x (l|h), ̂ y (l|h) The predictions of x (l) and y (l) computed at time h, l, h ∈ P and h < l
⌢
⌢
⌢
i i i i
⌢
⌢ ̂ w (l|h), ̂ v (l|h) The estimations of w (l) and v (l) computed at time h, l, h ∈ P and h < l
⌢
⌢
i i i i
⌢d ⌢
y (l|h) The set-point of y (l|h)
i i
U (l, p|h) A complete input vector,
i
[ ] T
T
T
T
U (l, p|h)= u (l |h) u (l + 1|h) · · · u (l + p|h) , p, l, h ∈ ℕ and
i i i i
h < l;
ΔU (l, p|h) Input increment sequence vector,
i
[ ] T
T
T
T
ΔU (l, p|h)= Δu (l |h) Δu (l + 1|h) · · · Δu (l + p|h) , h < l;
i
i
i
i
U(l, p|h) A complete stacked input vector,
[
T
T
U(l, p|h)= u (l |h) ··· u (l|h)
1 m ;
T
T
··· u (l + p|h) · · · u (l + p|h)] T
1 m
X (l, p|h) A stacked distributed state vector,
̂
i
[ T T T ] T
X (l, p|h)= ̂ x (l |h) ̂ x (l + 1|h) · · · ̂ x (l + p|h) ;
̂
i
i
i
i
̂
X(l, p|h) A complete stacked state vector,
[
T
T
X(l, p|h)= ̂ x (l |h) ··· ̂ x (l|h) ;
̂
m
1
T
T
··· ̂ x (l + p|h) · · · ̂ x (l + p|h)
1 m
⌢ ̂
X (l, p|h) A stacked state vector,
i
⌢ ̂ [ ⌢ T ⌢ T ⌢ T ] T
X (l, p|h)= ̂ x (l |h) ̂ x (l + 1|h) · · · ̂ x (l + p|h) , p, l, h ∈ P and
i
i
i
i
h < l
⌢ ̂
Y (l, p|h) A stacked output vector,
i
[ ] T
⌢ ̂ ⌢ T ⌢ T ⌢ T
Y (l, p|h)= ̂ y (l |h) ̂ y (l + 1|h) · · · ̂ y (l + p|h) , p, l, h ∈ P and
i i i i
h < l
⌢ d ⌢
̂
Y (l, p|h) The set-point of Y (l, p|h)
i
i
[ ] T
⌢
⌢ T
̂
̂ w (l + p|h) ,
W (l, p|h) A stacked interaction vector, ̂ w (l |h) ⌢ T · · · ⌢ T
̂ w (l + 1|h)
i i i i
p, l, h ∈ P and h < l
[ ] T
⌢ ̂ ⌢ T ⌢ T ⌢ T
V (l, P|h) A stacked interaction vector, ̂ v (l |h) ̂ v (l + 1|h) · · · ̂ v (l + p|h) ,
i i i i
p, l, h ∈ P and h < l
[ ] T
̂ T
X (l, p |h)
̂ =(l, p|h) A complete stacked state vector, ̂ (l, p|h)= ̂ T ··· X (l, p|h)
1 m
[ ] T
T
T
=(l, p|h) A complete stacked state vector, (l, p|h)= U (l, p |h) ··· U (l, p|h)
1 m