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Structure Model and System Decomposition 53

loops in the strong interacted structure, which means all the elements of a , a , … , a are
i 1 i 2 i 2 i 3 i p i 1
“*” in the structure matrix. This structure cannot transform into a diagonal form or triangular
form. In this case, the system model cannot be completely divided.
In many large-scale systems, the structure models can be directly deduced by the physical
composition of systems or the product processing. However, the decomposability of system
may not be obtained directly in this structure model. In this case, we can transform the system
model into a triangular or diagonal form to divide the system. If some subsystems consist of
many units after partitioned by the triangular form, we can continue to, if possible, transform
the subsystem to the diagonal form for further decomposition.

4.3.3 Input–Output Accessibility

By the structure model, we are not only able to analyze the static characteristics of a system
for providing the important information for system decomposition but also able to analyze
the dynamic characteristics of a system. In this section, we induce the concept of structure
accessibility and general rank of the structure matrix, which are very important for discussing
the structure controllability of a system.
Consider the following LTI system:
{
x (k + 1) = Ax(k)+ Bu(k)
(4.9)
y(k + 1)= Cx(k)

where A, B, and C are coefficient matrices. There is a structure description (A, B, C) cor-
responding to the mathematical model (A, B, C), where zero in A, B, C expresses the “0”
element in A, B, and C and “*” expresses nonzero elements in A, B, and C. It can be seen
that when two mathematical models (A , B , C ) and (A , B , C ) are not equal to each other,
1 1 1 2 2 2
their structure model may be the same and expressed as (A, B, C). Here, we think that these
two systems are structure equivalence.
Above, we have induced the neighbor matrix to describe the relationship of each subsystem.
For dynamic system (4.9), the inputs, states, and outputs are the basic elements of the system;
thus, we can express the neighboring connection matrix as

x u y
T T
x ⎡ A C ⎤
A = ⎢ ⎥
a u ⎢ T ⎥
⎢ B ⎥
y ⎢ ⎥
⎣ ⎦
⎡x ⎡A B ⎤ ⎡ x
a,k+1 ⎤ a,k ⎤
⎢ u ⎥ = ⎥ ⎢ u ⎥
⎢
a,k+1 a,k
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ y a,k ⎦ ⎣ C ⎦ ⎣ y a,k−1⎦
⎡ A B ⎤
A = ⎥
⎢
d
⎢ ⎥
⎣C ⎦