Signal flow graph and Mason's formula: G3 feedforward

Signal flow graph

Nodes: $R$, $e_1$ (after $S_1$), $A$ (after $G_1$), $e_2$ (after $S_2$), $B$ (after $G_2$), $C$. A dummy output node $C$ is drawn with unity gain.

       1      G1      1       G2       1
 R ───►e1 ───►A ────►e2 ────► B ─────►C ──1──► C
       ▲      │       ▲       │        │
       │      └───────┼──G3───┼───────►│
       │              │       │        │
       │              └──────(−H2)─────┘
       └────────(−H1)─────────┘

Branches: $R\to e_1$ (1), $e_1\to A$ ($G_1$), $A\to e_2$ (1), $e_2\to B$ ($G_2$), $B\to C$ (1), $A\to C$ ($G_3$), $B\to e_1$ ($-H_1$), $C\to e_2$ ($-H_2$).

Mason's gain formula

$$T = \frac{1}{\Delta}\sum_k P_k\Delta_k$$

Forward paths

• $P_1 = G_1G_2$ ($R\to e_1\to A\to e_2\to B\to C$)
• $P_2 = G_1G_3$ ($R\to e_1\to A\to C$)

Individual loops

• $L_1 = -G_1G_2H_1$ ($e_1\to A\to e_2\to B\to e_1$)
• $L_2 = -G_2H_2$ ($e_2\to B\to C\to e_2$)
• $L_3 = +G_1G_3G_2H_1H_2$ ($e_1\to A\to C\to e_2\to B\to e_1$; two negative branches give a positive gain)

Non-touching loops: none (all share node $e_2$ or $B$).

Determinant

$$\Delta = 1 - (L_1 + L_2 + L_3) = 1 + G_1G_2H_1 + G_2H_2 - G_1G_2G_3H_1H_2$$

Cofactors: both forward paths touch all loops, so $\Delta_1 = \Delta_2 = 1$.

Transfer function

$$\frac{C(s)}{R(s)} = \frac{G_1G_2 + G_1G_3}{1 + G_1G_2H_1 + G_2H_2 - G_1G_2G_3H_1H_2}$$

This agrees with block diagram reduction of the same figure.