Modern Control System Theory and Design, 2nd Edition

2.14.  SIGNAL-FLOW GRAPHS AND MASON’S THEOREM

Signal-flow graphs and Mason’s theorem [7, 8] enable the control engineer to determine the response of a complicated linear, multiloop system to any input much more rapidly than do block-diagram reduction techniques.

A signal-flow graph is a topological representation of a set of linear equations having the form

The equation expresses each of the n variables in terms of the others and themselves. A signal-flow graph represents a set of equations of this type by means of branches and nodes. A node is assigned to each variable of interest in the system. For example, node i represents variable y_i. Branch gains are used to relate the different variables. For example, branch gain a_ij relates variable y_i to y_j, where the branch originates at node i and terminates at node j. Consider the following set of linear equations

The signal-flow graph which represents this set of equations is shown in Figure 2.12. Here y₁ can be interpreted as the input ...