Page 75 - Distributed model predictive control for plant-wide systems

Structure Model and System Decomposition 49

where
[ T T T ] T
x(k)= x (k) x (k)· · · x (k) ∈ ℝ n x
1 2 m
[ T T T ] T
u(k)= u (k) u (k)· · · u (k) ∈ U ⊂ ℝ n u
m
1 2
[ T T T ] T
y(k)= y (k) y (k)· · · y (k) ∈ ℝ n y
m
1 2
are, respectively, the concatenated state, control input, and output vectors of the overall system
S.Also, u(k)∈ U = U × U ×···× U . A, B, and C are constant matrices of appropriate
1 2 m
dimensions and are defined as follows:
T
⎡ A 11 A 12 ··· A 1m ⎤
⎢ ⎥
A A ··· A
⎢ 21 22 2m⎥
A =
⎢ ⎥
⋮ ⋮ ⋱ ⋮
⎢ ⎥
⎢ ⎥
⎣A m1 A m2 ··· A mm ⎦
T
⎡ B 11 B 12 ··· B 1m ⎤
⎢ ⎥
B B ··· B
⎢ 21 22 2m⎥
B =
⎢ ⎥
⋮ ⋮ ⋱ ⋮
⎢ ⎥
⎢ ⎥
⎣B B ··· B
m1 m2 mm ⎦
T
C C ··· C
⎡ 11 12 1m⎤
⎢ C C ··· C ⎥
C = ⎢ 21 22 2m ⎥
⋮ ⋮ ⋱ ⋮
⎢ ⎥
⎢ ⎥
⎣ ⎦
C C ··· C
m1 m2 mm
If there is only the state interacting term, we call the model the state interacted model and
it can be expressed as
∑
x (k + 1)= A x (k)+ B u (k)+ A x (k) (4.3)
i
ii i
ii i
ij j
j∈P +i
Similarly, if there is only the input interacting term of input, we call the model the input
interacted model and it can be expressed as
∑
x (k + 1)= A x (k)+ B u (k)+ B u (k) (4.4)
ij j
ii i
ii i
i
u
j∈P
i
In fact, the state interacted model can be transformed into the input interacted model by
selecting a suitable virtue states. Let us take the following system which is composed of two
state interacted subsystems as an example.
1. Model of subsystem S
1
[ (1) ] [ (1) ][ (1) ] [ (1) ]
x (k + 1) A 0 x (k) B
1 11 1 11
= + u (k) (4.5)
(2) (2) (2) (2) 1
x (k + 1) 0 A x (k) B
1 11 1 11