Transfer Function Of A Multiple-loop System

Transfer function of a multiple-loop system

Fig: 1

A multiple-loop feedback system is shown in Figure 1 in block diagram form. There is no need to redraw the diagram in signal-flow graph form, and so we shall proceed as usual by using Mason's signal-flow gain formula, Equation

 There is one forward path P_x = G₁G₂G₃G₄. The feedback loops are

All the loops have common nodes and therefore are all touching. Furthermore, the path P₁ touches all the loops, so Δ₁ = 1. Thus, the closed-loop transfer function is

Fig: 2 Signal flow graph of a multiple-loop system