Find the transfer function of the signal flow graph shown below.

A forward path is a path from the input node ($x_1$) to the output node ($x_8$) that does not traverse any node more than once.

• Path 1 ($P_1$): $x_1 \to x_2 \to x_3 \to x_4 \to x_5 \to x_6 \to x_7 \to x_8$

  • The gain is the product of the branch gains along this path: $P_1 = (1)(G_1)(G_2)(G_3)(G_4)(G_5)(1) = G_1 G_2 G_3 G_4 G_5$

• Path 2 ($P_2$): $x_1 \to x_2 \to x_3 \to x_4 \to x_5 \to x_7 \to x_8$ (via $G_6$)

  • The gain is the product of the branch gains along this path: $P_2 = (1)(G_1)(G_2)(G_3)(G_6)(1) = G_1 G_2 G_3 G_6$

A loop is a path that starts and ends at the same node, traversing no other node more than once.

• Loop 1 ($L_1$): $x_2 \to x_3 \to x_2$

  • $L_1 = G_1 \cdot (-H_1) = -G_1 H_1$

• Loop 2 ($L_2$): $x_3 \to x_4 \to x_3$

  • $L_2 = G_2 \cdot (-H_2) = -G_2 H_2$

• Loop 3 ($L_3$): $x_4 \to x_5 \to x_4$

  • $L_3 = G_3 \cdot (-H_3) = -G_3 H_3$

• Loop 4 ($L_4$): $x_2 \to x_3 \to x_4 \to x_5 \to x_6 \to x_2$

  • $L_4 = G_1 \cdot G_2 \cdot G_3 \cdot G_4 \cdot (-H_4) = -G_1 G_2 G_3 G_4 H_4$

Non-touching loops are loops that do not share any common nodes.

• Pairs of non-touching loops:

  • $L_1$ (nodes $x_2, x_3$) and $L_3$ (nodes $x_4, x_5$) are non-touching.
    • $L_1 L_3 = (-G_1 H_1)(-G_3 H_3) = G_1 G_3 H_1 H_3$
  • All other pairs of loops ($L_1 L_2$, $L_1 L_4$, $L_2 L_3$, $L_2 L_4$, $L_3 L_4$) share common nodes and are therefore touching.

• Triplets or higher order non-touching loops:

  • Since there is only one pair of non-touching loops, there are no triplets or higher order sets of non-touching loops.

$$\Delta = 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + \dots$$ $$\Delta = 1 - (L_1 + L_2 + L_3 + L_4) + (L_1 L_3)$$ $$\Delta = 1 - (-G_1 H_1 - G_2 H_2 - G_3 H_3 - G_1 G_2 G_3 G_4 H_4) + (G_1 G_3 H_1 H_3)$$ $$\Delta = 1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3$$

$\Delta_k$ is the determinant of the graph when all loops touching the $k$-th forward path are removed.

• For $P_1 = G_1 G_2 G_3 G_4 G_5$:

  • $P_1$ passes through nodes $x_2, x_3, x_4, x_5, x_6, x_7$.
  • All loops ($L_1, L_2, L_3, L_4$) touch $P_1$.
  • Therefore, $\Delta_1 = 1$.

• For $P_2 = G_1 G_2 G_3 G_6$:

  • $P_2$ passes through nodes $x_2, x_3, x_4, x_5, x_7$.
  • All loops ($L_1, L_2, L_3, L_4$) touch $P_2$.
  • Therefore, $\Delta_2 = 1$.

$$T = \frac{P_1 \Delta_1 + P_2 \Delta_2}{\Delta}$$ $$T = \frac{(G_1 G_2 G_3 G_4 G_5)(1) + (G_1 G_2 G_3 G_6)(1)}{1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3}$$ $$T = \frac{G_1 G_2 G_3 G_4 G_5 + G_1 G_2 G_3 G_6}{1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3}$$

We can factor out $G_1 G_2 G_3$ from the numerator:

$$\boxed{T = \frac{G_1 G_2 G_3 (G_4 G_5 + G_6)}{1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3}}$$