Abstract
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.
Highlights
A signal flow graph set up directly after inspecting a physical system without first formulating the associated equations is one of the most common tools for representing a complicated linear control system
This paper proposes a systematic method called the transfer matrix method [19] to determine the transfer function of a system, and it presents a physical insight into the transfer matrix
The determinant of the cofactor matrix is equivalent to the loop group gain associated with the backward edges
Summary
A signal flow graph set up directly after inspecting a physical system without first formulating the associated equations is one of the most common tools for representing a complicated linear control system. It offers a visual structure upon which causal relationships among several variables can be compared. Mason’s gain formula [11,12], or Mason’s rule, is a systematic method for obtaining the transfer function of a signal flow graph between input and output nodes, especially for complex and high-dimensional systems. As an alternative to Coates’ gain formula for solving the system, a solution can be obtained by considering the eigenvector of the transfer matrix for a signal flow graph.
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