System Structure Modeling Technology#
Concept and Significance of Structural Analysis#
Concept: Structural analysis is a process of realizing and interpreting system structure modeling.
Significance: Structural analysis is the essential content of system analysis and the foundation of system optimization analysis, design, and management.
Main Content:
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Understanding of system objectives and functions;
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Selection of system components;
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Analysis of the connections and hierarchical relationships between elements;
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Determination and interpretation of the overall system structure.
Basic Expression Methods of System Structure#
There are multiple ways to express system structure, and they can generally be converted between each other. In the following introduction, we will provide two examples.
Set Expression of System Structure#
Assuming that a system A contains five elements, $ S_1 $ to $ S_5 $, the binary relations between them can be expressed as a set:
$ R_b = {(S_1, S_2), (S_1, S_5), (S_2, S_3), (S_3, S_4), (S_5, S_2), (S_5, S_3), (S_5, S_4)} $
Similarly, for system B, which contains six elements, $ S_1 $ to $ S_6 $, the set expression of the binary relations between them is:
$ R_b = {(S_1, S_3), (S_1, S_5), (S_2, S_4), (S_4, S_2), (S_4, S_6), (S_5, S_1), (S_5, S_2)} $
Directed Graph Expression of System Structure#
Based on the description, we can create a directed graph as follows:
Matrix Expression of System Structure#
Adjacency Matrix#
The adjacency matrix $ (A) $ represents the basic binary relations or direct connections between system elements. If there is a binary relation between two elements, "1" is filled in the corresponding position; otherwise, "0" is filled. With the set expression $ (S, R_b) $ or the directed graph $ (D) $ that expresses the system structure, the adjacency matrix $ (A) $ can be easily obtained. The adjacency matrices for system A and B are as follows:
It is obvious that the number of "1" in matrix $ A $ is the same as the number of element pairs in $ R_b $ and the number of directed arcs in graph $ D $.
In the adjacency matrix, if a column is completely filled with 0, the corresponding element is a input element of the system. If a row is completely filled with 0, the corresponding element is an output element of the system.
Reachability Matrix#
If there is a transitive binary relation between element $ S_i $ and $ S_j $, or if there is a directed path from node i to j in the directed graph, it is said that $ S_i $ can reach $ S_j $. The reachability matrix $ (M) $ expresses the transitive binary relations or the reachability between any two nodes in the directed graph.
The elements in matrices $ A $ and $ M $ are either "1" or "0" and follow the rules of Boolean algebra. By operating on the adjacency matrix $ A $, the reachability matrix $ M $ can be obtained. Assuming the maximum path length or number of transmissions without loops is $ r $, and $ I $ is the identity matrix of the same order as $ A $, the calculation formula for the reachability matrix is $ M=(A+I)^r $.
The reachability matrices for system A and B are as follows:
Reduced Matrix#
Based on the replaceable rows of strongly connected elements, in the existing reachability matrix $ M $, a group of elements with strong connectivity can be treated as one element, and one representative element can be retained while deleting the rest to obtain the reduced matrix $ M' $ of the reachability matrix $ M $. The reduced matrices for system A and B are as follows:
Establishing Hierarchical Structure Models#
To establish a hierarchical structure model that reflects the hierarchical relationships between elements in a system, we can start with the reachability matrix $ M $.
When using the normative method, we generally need to go through four stages: local partitioning, level partitioning, skeleton matrix extraction, and multi-level hierarchical directed graph drawing.
When using the practical method, we need to go through three steps: establishing the reduced matrix, hierarchical processing, and multi-level hierarchical directed graph drawing.
Since this part involves many concepts and is relatively abstract, we choose to explain it through solving practical problems.
Given the reachability matrix of the system shown below, establish its hierarchical structure model.
Due to the incomplete support for $ \LaTeX $ on this platform, the demonstration will be conducted in the form of images, and the relevant documents will be open-sourced later.
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