Modern Control System Theory and Design, 2nd Edition

2.16.  APPLICATION OF MASON’S THEOREM AND THE SIGNAL-FLOW GRAPH TO MULTIPLE-FEEBACK SYSTEMS

It is important at this point to differentiate between signal-flow graphs and block diagrams. Basically, the signal-flow graph represents a detailed picture of a system’s topological structure, whereas the block diagram focuses on the transfer functions that comprise the various elements of the system. The signal-flow graph is useful in analyzing multiple-loop feedback systems and in determining the effect of a particular element or parameter in an overall feedback system. Essentially both present the same information in different ways, and Mason’s theorem can be applied to both. However, Mason’s theorem is conventionally used with the signal-flow graph, because the topology is more clearly depicted by the signal-flow graph.

Let us next analyze various control-system configurations, depicted by both their block diagrams and signal-flow graphs, using Mason’s theorem.

Example 1. The signal-flow graph corresponding to the block diagram shown in Figure 2.10 is illustrated in Figure 2.15:

Applying Mason’s theorem to this feedback control system results in the following:

  Δ = 1 − [−G(s)H(s)] = 1 + G(s)H(s)

G_A = G(s)

Δ_A = 1

Therefore,

which agrees with Eq. (2.121) that was derived from block-diagram relationships.

Example 2. Let us consider the feedback control system illustrated in Figure 2.16 ...