Page 156 - Distributed model predictive control for plant-wide systems

130 Distributed Model Predictive Control for Plant-Wide Systems

For a simplifying reason, redefine J (k) as
i
P M
∑ d 2 ∑ 2
‖y (k + l |k) − y (k + l|k)‖ ⌢ +
J (k)= ‖ ⌢ ̂ ‖ ‖ Δu (k + l − 1 |k) ‖ (7.8)
i i i ‖ i ‖R i
‖ ‖Q i
l=1 l=1
⌢
where Q = diag(Q , Q , … , Q ).
i i i 1 i b
The optimization index J (k) considers not only the performance of subsystem S but
i
i
also that of the downstream neighbors of S . The impact of the control decision of S to
i i
S ∈ P are fully considered in the neighborhood optimization, and, therefore, the global
j −i
performance improvement is guaranteed. It should be noticed that the global performance
may further be improved by using the optimization objective (7.5) in each subsystem, but
it requires a high-quality and complicated network communication and introduces more
complex computation.
7.2.2.2 Prediction Model
Since the state evolution of S ∈ P is affected by u (k), to improve the predictive precision,
j −i i
subsystem S , its downstream neighbors should be considered as one relatively large inte-
i
gral subsystem when predicting the future states of S and its downstream neighbors. Assume
i
that the number of downstream neighbors of S is n; then the state evolution model of the
i
downstream neighbors of S can be easily deduced by (5.1) and expressed as
i
{ ⌢ ⌢
⌢
⌢
⌢
x (k + 1) = A x (k)+ B u (k)+ w (k)
i i
i
i
i i
⌢ ⌢ ⌢ (7.9)
⌢
y (k)= C x (k)+ v (k)
i
i
i i
where
A ii A ··· A
⎡ ii 1 ii n ⎤
[ ]
⌢ ⌢ (1) ⌢ (2) ⎢A i 1 i A ··· A ⎥
A = = i 1 i 1 i n i n (7.10)
i A i A i ⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣A A ··· A
i n i i n i 1 i n i n ⎦
B
⎡ ii ⎤
⌢ ⎢B ⎥
i 1 i
B = (7.11)
i ⎢ ⋮ ⎥
⎢ ⎥
⎣B ⎦
i n i
C ii C ··· C
⎡ ii 1 ii m ⎤
⌢ ⎢C i 1 i C ··· C ⎥
C = i 1 i 1 i 1 i m (7.12)
i ⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣C C ··· C
i m i i m i 1 i m i m ⎦
∑
B u (k) +
⎡ ij j ⎤
⎢ j∈P +i , j≠i ⎥
∑ ∑
B u (k)+ A x (k)
⎢ ⎥
i 1 j j
i 1 j j
⎢ ⎥
⌢
w (k)= ⎢ j∈P +i 1 , j≠i j∈P +i 1 , j∉j∈P −i , ⎥ (7.13)
i
⋮
⎢ ⎥
∑ ∑
⎢ ⎥
B u (k)+ A x (k)
⎢ i n j j i n j j ⎥
, j≠i
⎣j∈P +i n j∈P +i n , j∉j∈P −i , ⎦