Mason gain formula with positive H2 loop

Each signal becomes a node and each block a branch; summer signs go into branch gains. Then Mason's gain formula is applied.

Signal flow graph

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

                         G4
               +------------------------+
               |                        v
R -1-> x1 -1-> x2 -G1-> x3 -1-> x4 -G2-> x5 -G3-> C -1-> C
       ^       ^                ^        |        |
       |       +-----(-H1)------+--------+        |
       |                        +-----(+H2)-------+
       +-------------------(-1)-------------------+

Branches: $x_3\to C$ ($G_4$), $x_5\to x_2$ ($-H_1$), $C\to x_4$ ($+H_2$), $C\to x_1$ ($-1$).

Forward paths

• $P_1 = G_1G_2G_3$
• $P_2 = G_1G_4$

Individual loops

• $L_1 = -G_1G_2H_1$ ($x_2, x_3, x_4, x_5, x_2$)
• $L_2 = +G_2G_3H_2$ ($x_4, x_5, C, x_4$)
• $L_3 = -G_1G_2G_3$ ($x_1, \dots, C, x_1$)
• $L_4 = -G_1G_4$ ($x_1, x_2, x_3, C, x_1$)
• $L_5 = G_1G_4(H_2)(G_2)(-H_1) = -G_1G_2G_4H_1H_2$ ($x_2, x_3, C, x_4, x_5, x_2$)

Non-touching loops

Every pair of loops shares at least one node (for example, $L_2$ and $L_4$ share $C$), so there are no non-touching loops.

$$\Delta = 1 + G_1G_2H_1 - G_2G_3H_2 + G_1G_2G_3 + G_1G_4 + G_1G_2G_4H_1H_2$$

Cofactors

$P_1$ touches all loops. $P_2$ passes through $x_2$, $x_3$ and $C$, and every loop contains at least one of these nodes. So $\Delta_1 = \Delta_2 = 1$.

Mason's gain formula

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

Answer:

$$\frac{C(s)}{R(s)} = \frac{G_1G_2G_3 + G_1G_4}{1 + G_1G_2H_1 - G_2G_3H_2 + G_1G_2G_3 + G_1G_4 + G_1G_2G_4H_1H_2}$$