Signal flow graph of G1–G3, H1–H3 block diagram

In the signal flow graph method, each signal becomes a node and each block a branch; then Mason's gain formula gives the transfer function.

Signal flow graph

Nodes: $x_1$ = output of $S_1$, $x_2$ = output of $G_1$, $x_3$ = output of $S_2$, $x_4$ = output of $S_3$ (node A), $x_5$ = node B, $x_6$ = output of $S_4$, $C$.

                         H1
                   +--------------+
                   |              v
R -1-> x1 -G1-> x2 -1-> x3 -1-> x4 -G2-> x5 -1-> x6 -G3-> C
       ^                ^       ^         |                |
       |                +-(-H2)-+---------+                |
       |                        +-------(-H3)--------------+
       +--------------------(-1)---------------------------+

Branches: $x_4\to x_6$ ($H_1$), $x_5\to x_3$ ($-H_2$), $C\to x_4$ ($-H_3$), $C\to x_1$ ($-1$); all others as labelled.

Forward paths

• $P_1 = G_1G_2G_3$ (through $G_2$)
• $P_2 = G_1H_1G_3$ (through the $H_1$ branch)

Individual loops

• $L_1 = -G_2H_2$ ($x_3, x_4, x_5, x_3$)
• $L_2 = -G_2G_3H_3$ ($x_4, x_5, x_6, C, x_4$)
• $L_3 = -G_3H_1H_3$ ($x_4, x_6, C, x_4$)
• $L_4 = -G_1G_2G_3$ ($x_1 \dots C, x_1$)
• $L_5 = -G_1G_3H_1$ ($x_1, x_2, x_3, x_4, x_6, C, x_1$)

Non-touching loops

$L_1$ touches $L_2$ and $L_3$ at $x_4$, and all others pass through $x_3$/$x_4$. So there are no non-touching pairs.

$$\Delta = 1 + G_2H_2 + G_2G_3H_3 + G_3H_1H_3 + G_1G_2G_3 + G_1G_3H_1$$

Cofactors

Both forward paths pass through $x_3$, $x_4$ and $C$, so they touch every loop: $\Delta_1 = \Delta_2 = 1$.

Mason's gain formula

$$T = \frac{P_1\Delta_1 + P_2\Delta_2}{\Delta}$$

Answer:

$$\frac{C(s)}{R(s)} = \frac{G_1G_3(G_2 + H_1)}{1 + G_2H_2 + G_2G_3H_3 + G_3H_1H_3 + G_1G_2G_3 + G_1G_3H_1}$$