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Article

A Matrix Approach for Analyzing Signal Flow Graph

1
Department of Mechanical Engineering, Lunghwa University of Science and Technology, Taoyuan City 333326, Taiwan
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Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan
*
Author to whom correspondence should be addressed.
Information 2020, 11(12), 562; https://doi.org/10.3390/info11120562
Received: 19 October 2020 / Revised: 25 November 2020 / Accepted: 27 November 2020 / Published: 30 November 2020
(This article belongs to the Special Issue Selected Papers from IIKII 2020 Conferences)
Mason’s gain formula can grow factorially because of growth in the enumeration of paths in a directed graph. Each of the (n − 2)! permutation of the intermediate vertices includes a path between input and output nodes. This paper presents a novel method for analyzing the loop gain of a signal flow graph based on the transform matrix approach. This approach only requires matrix determinant operations to determine the transfer function with complexity O(n3) in the worst case, therefore rendering it more efficient than Mason’s gain formula. We derive the transfer function of the signal flow graph to the ratio of different cofactor matrices of the augmented matrix. By using the cofactor expansion, we then obtain a correspondence between the topological operation of deleting a vertex from a signal flow graph and the algebraic operation of eliminating a variable from the set of equations. A set of loops sharing the same backward edges, referred to as a loop group, is used to simplify the loop enumeration. Two examples of feedback networks demonstrate the intuitive approach to obtain the transfer function for both numerical and computer-aided symbolic analysis, which yields the same results as Mason’s gain formula. The transfer matrix offers an excellent physical insight, because it enables visualization of the signal flow. View Full-Text
Keywords: signal flow graph; transfer function; Mason’s graph; linear system signal flow graph; transfer function; Mason’s graph; linear system
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MDPI and ACS Style

Jeng, S.-L.; Roy, R.; Chieng, W.-H. A Matrix Approach for Analyzing Signal Flow Graph. Information 2020, 11, 562. https://doi.org/10.3390/info11120562

AMA Style

Jeng S-L, Roy R, Chieng W-H. A Matrix Approach for Analyzing Signal Flow Graph. Information. 2020; 11(12):562. https://doi.org/10.3390/info11120562

Chicago/Turabian Style

Jeng, Shyr-Long, Rohit Roy, and Wei-Hua Chieng. 2020. "A Matrix Approach for Analyzing Signal Flow Graph" Information 11, no. 12: 562. https://doi.org/10.3390/info11120562

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