Page 81 - Distributed model predictive control for plant-wide systems

Structure Model and System Decomposition 55

operating rules, the above equation can be simply rewritten as
R =(I ∪ A ) N−1 (4.12)
a
In dynamic systems, there are three basic facts: system state, inputs, and outputs. To find out
the interactions between subsystems, to find out whether the states are able to be accessed by
the input, to find out whether the states of the system impact the output, the system dynamic
structure analysis is very important. If every state x can be accessed by at least one input u ,
i j
then we say that this system is input accessible. If every state x can be accessed to at least
i
one output y , we say that this system is output accessible. The input accessibility and output
l
accessibility reflect whether the system states can be adjusted by inputs and whether all the
system states can be reflected by outputs.
The accessible matrix (4.12) can be rewritten in the following bock form:
x u y
x ⎡R xx R xu R ⎤
xy
N−1
R =(I ∪ A ) = (4.13)
a
u ⎢ R ux R uu R uy ⎥
y ⎢ ⎣R yx R yu R ⎦ ⎥
yy
Obviously, the input accessibility and output accessibility can be described by the structure
matrices R and R , respectively. Consider that
xy
ux
( T ) N−1 T
⎡ N−2 T ⎤
A + I (A + I) C
⎢ ⎥
R =(I ∪ A ) N−1 = ⎢ T T T T T ⎥ (4.14)
a ⎢ B (A + I) N−2 I B (A + I) N−3 C ⎥
⎢ ⎥
⎢ I ⎥
⎣ ⎦
and notice that the interaction relationship can at most go through n units in R, just like u →
j
x → x → ··· → x and x → x → ··· → x → y , where n is the dimension of state x.
i 1 i 2 i n i 1 i 2 i n k
We have the input accessibility matrix
T T n−1
R = B (A + I) ,
ux
and the output accessibility matrix
T n−1 T
R =(A + I) C .
xy
If there are no columns where all elements are zeros in R , this system is input accessible.
ux
If there are no rows where all elements are zeros in R , this system is output accessible.
xy
Example 4.1 Consider the following system:

0 0 ∗ ∗ 0
⎡ ⎤
⎢0 0 ∗ 0 ∗⎥
A = 0 0 ∗ ∗ 0 ⎥ (4.15)
⎢
⎢ ⎥
0 0 ∗ ∗ 0
⎢ ⎥
⎣0 0 ∗ 0 0⎦