Signal flow graph and Mason's formula: G1–G4, H1–H3

A signal flow graph has a node for each signal and a directed branch (with gain) for each block. Summing points become nodes; a minus sign at a summer goes into the branch gain.

Signal flow graph

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

                  G4
        +--------------------------+
        |                          v
R --1--> x1 --1--> x2 --G1--> x3 --G2--> x4 --G3--> C --1--> C
         ^          ^          ^          |         |
         |          +---(-H2)--+----------+         |
         |                     +--------(-H3)-------+
         +-----------------(-H1)--------------------+

Branches: $R\to x_1$ (1), $x_1\to x_2$ (1), $x_2\to x_3$ ($G_1$), $x_3\to x_4$ ($G_2$), $x_4\to C$ ($G_3$), $x_2\to C$ ($G_4$), $x_4\to x_2$ ($-H_2$), $C\to x_3$ ($-H_3$), $C\to x_1$ ($-H_1$).

Mason's gain formula

$$T = \frac{C}{R} = \frac{\sum_k P_k\Delta_k}{\Delta}$$

Forward paths

• $P_1 = G_1G_2G_3$ (R, x1, x2, x3, x4, C)
• $P_2 = G_4$ (R, x1, x2, C)

Individual loops

• $L_1 = -G_1G_2H_2$ (x2, x3, x4, x2)
• $L_2 = -G_2G_3H_3$ (x3, x4, C, x3)
• $L_3 = -G_1G_2G_3H_1$ (x1, x2, x3, x4, C, x1)
• $L_4 = -G_4H_1$ (x1, x2, C, x1)
• $L_5 = (G_4)(-H_3)(G_2)(-H_2) = +G_2G_4H_2H_3$ (x2, C, x3, x4, x2)

Non-touching loops: every pair of loops shares at least one node (all pass through x2/x3 or C), so there are none.

$$\Delta = 1 - (L_1+L_2+L_3+L_4+L_5) = 1 + G_1G_2H_2 + G_2G_3H_3 + G_1G_2G_3H_1 + G_4H_1 - G_2G_4H_2H_3$$

Cofactors: $P_1$ and $P_2$ both touch all loops (they pass through x2 and C), so $\Delta_1 = \Delta_2 = 1$.

Answer:

$$\frac{C(s)}{R(s)} = \frac{G_1G_2G_3 + G_4}{1 + G_1G_2H_2 + G_2G_3H_3 + G_1G_2G_3H_1 + G_4H_1 - G_2G_4H_2H_3}$$