Page 226 - Distributed model predictive control for plant-wide systems

200 Distributed Model Predictive Control for Plant-Wide Systems

Since the performance index of S , ∀i ∈ P, with the optimal solution of u (⋅|k) must be not
i
i
f
more than the performance index of S with the feasible solution of u (⋅|k). Thus it has
i
i
V − V
k,i k−1,i
≤ −‖̂ x (k − 1|k − 1, i)‖ − u (k − 1|k − 1) ‖
‖
Q ‖ i ‖R i
N−2 ( )
∑ ‖ f ‖ f
‖
‖
+ ‖x (k + l|k)‖ + ‖u (k + l|k)‖
i
‖ ‖Q ‖ ‖R i
l=0
( )
‖ f ‖ ‖ f ‖
+ ‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖
i
‖ ‖Q ‖ ‖R i
‖ f ‖
+ ‖x (k + N|k)‖
‖ ‖P
N−2 ( )
∑
− ‖̂ x (k + l|k − 1, i)‖ + ̂ u (k + l|k − 1) ‖
‖
Q ‖ i ‖R i
l=0
( )
− ‖̂ x (k + N − 1|k − 1, i)‖ P
assuming x(k)∈ X∖Ω( ), that is ‖̂ x (k − 1|k − 1, i)‖ ≥ . Considering that
Q
‖u (k − 1| k − 1)‖ > 0 and substituting Equation (9.18) into Equation (9.31) yields
i R
′
V − V k−1,i ≤ − e + (N − 1) e
k,i
‖ f ‖ ‖ f ‖
+ ‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖
i
‖ ‖Q ‖ ‖R
‖ f ‖
+ ‖x (k + N|k)‖ − ‖̂ x (k + N − 1|k − 1, i)‖ P
‖ ‖P
In the above equation, consider the terms from the third term to the fifth term, it has
( ) 2
1 ‖ f ‖ ‖ f ‖ ‖ f ‖ 2
‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖ + ‖x (k + N|k)‖
i
2 ‖ ‖Q ‖ ‖R i ‖ ‖P
2 2 2
‖ f ‖ ‖ f ‖ ‖ f ‖
≤ ‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖ + ‖x (k + N|k)‖
i
‖ ‖Q ‖ ‖R i ‖ ‖P
2 2 2
‖ f ‖ ‖ f ‖ ‖ f ‖
≤ ‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖ + ‖x (k + N|k)‖
‖ ‖Q ‖ ‖R ‖ ‖P
T
f
T
̂
Notice that ‖ f f ‖ 2 = A x (k + N − 1|k) ‖ 2 and Q = Q + K RK, A PA −
x x (k + N|k)
‖
‖ ‖P ‖ c ‖P c c
̂
P =−Q, it has
2 2 2
‖ f ‖ ‖ f ‖ ‖ f ‖
‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖ + ‖x (k + N|k)‖
‖ ‖Q ‖ ‖R ‖ ‖P
2 2
‖ f ‖ ‖ f ‖
c
≤ ‖x (k + N − 1|k)‖ + ‖A x (k + N − 1|k)‖
‖ ‖ ̂ Q ‖ ‖P
2
‖ f ‖
= ‖x (k + N − 1|k)‖
‖ ‖P