1. $Y_2 = G_1 Y_1 - G_2 Y_2$
2. $Y_3 = G_3 Y_2 + G_4 Y_3$
3. $Y_4 = G_5 Y_1 + G_6 Y_3$

Where Y4 is the output , Obtain the overall transfer function by using Mason's Gain formula.

Mumbai University > Electronics Engineering > Sem 4 > Linear Control Systems

Topic : Models for Control System

Difficulty : High

Marks : 10M

i) $Y_2$ = $G_1 Y_1 - G_2 Y_4$

ii) $Y_3$ = $G_3 Y_2 - G_4 Y_3$

iii) $Y_4$ = $G_{\sigma} Y_1 - G_{\sigma} Y_3$

Now add all the SFG to obtain TF

step I => Forward path,

$P_1 = G_1,G_3,G_6$

$P_2 = G_5$

step II => Total number of loops,

$L_1 = G_4$

$L_2 = - G_3 G_6 G_2$

Step III => Total number of two non-touching loops,

=> There are no two-non touching loops

Step IV => Total number of three non-touching loops,

=> There are no three-non touching loops

Step V => Find out value of $\Delta$,

$\Delta$ = 1 - ($L_1 + L_2$)

$\Delta$ = 1 - $G_4 + G_3 G_6 G_2$

Step VI => Find out value of $\Delta_1$ and $\Delta_2$,

All loops touch the forward path,

$\Delta_1$ = 1

Loop $G_4$ does not touch the forward path $P_2$,

$\Delta_2$ = 1 - $G_4$

=> $\frac{C(s)}{K(s)} = \frac{1}{\Delta}\Sigma P_K(\Delta)_K$

$\frac{C(s)}{R(s)} = \frac{P_1(\Delta)_1 + P_2(\Delta)_2}{(\Delta)}$

$\frac{C(s)}{R(s)} = \frac{G1.G3.G6 + G5 (1 - G4)}{1- G4 + G3.G6.G2}$