Y/R of block diagram with G4 and H1–H3

Mason's gain formula is used (the block diagram reduction gives the same result). Name the signals: $A$ = output of $S_1$, $B$ = output of $S_2$, $N$ = output of $S_3$ (node C in the figure), $Y$ = output.

Signal flow graph

                     G4
        +-----------------------------+
        |                             v
R -1--> A --G1--> B --G2--> N --G3--> Y
        ^         |^        |^        |
        +--(-H1)--+|        ||        |
                   +-(-H2)--+|        |
                             +-(-H3)--+

Branches: $B\to A$ ($-H_1$), $N\to B$ ($-H_2$), $Y\to N$ ($-H_3$), $A\to Y$ ($G_4$).

Forward paths

• $P_1 = G_1G_2G_3$ (A, B, N, Y)
• $P_2 = G_4$ (A, Y)

Individual loops

• $L_1 = -G_1H_1$ (A, B, A)
• $L_2 = -G_2H_2$ (B, N, B)
• $L_3 = -G_3H_3$ (N, Y, N)
• $L_4 = G_4(-H_3)(-H_2)(-H_1) = -G_4H_1H_2H_3$ (A, Y, N, B, A)

Non-touching loops

$L_1$ (A, B) and $L_3$ (N, Y) do not touch: $L_1L_3 = G_1G_3H_1H_3$. No three loops are mutually non-touching.

$$\begin{aligned} \Delta &= 1 - (L_1 + L_2 + L_3 + L_4) + L_1L_3 \\ &= 1 + G_1H_1 + G_2H_2 + G_3H_3 + G_4H_1H_2H_3 + G_1G_3H_1H_3 \end{aligned}$$

Cofactors

• $P_1$ touches all loops: $\Delta_1 = 1$.
• $P_2$ (A, Y) does not touch $L_2$ (B, N): $\Delta_2 = 1 - L_2 = 1 + G_2H_2$.

Transfer function

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

Answer:

$$\frac{Y(s)}{R(s)} = \frac{G_1G_2G_3 + G_4(1 + G_2H_2)}{1 + G_1H_1 + G_2H_2 + G_3H_3 + G_1G_3H_1H_3 + G_4H_1H_2H_3}$$