# Calculating the transfer function of this block diagram

I'm not really sure what steps to take to solve the problem. I tried with G4 and G2 first but I'm not sure if that is the correct way.

• Shift the $G1$ to the right of the summing junction. This will make $H$ and $G4/G1$ parallel. Please add your partial results into the question using the edit.
– AJN
Commented May 14, 2022 at 13:47
• Label the outputs of the two summers on the left (the summer on the right outputs C(s)). The write down each summer's output and gather all terms involving C(s). Finally write C(s)/R(s) as the ratio of two expressions involving the remaining variables. Show us your work at each stage for more help. Commented May 14, 2022 at 19:51
• figure out what the signal is at each point. For example R-H*C at the point immediately left of G1. eventually you'll have an equation that can be solved for C in terms of the others
– Abel
Commented May 15, 2022 at 14:55

Assuming scalars (i.e. SISO systems): \begin{align} C&=E_1G_3+E_2G_2\\ E_1&=R-CH\\ E_2&=E_1G_1-CG_4=(R-CH)G_1-CG_4\\ \end{align} Hence \begin{align}C&=(R-CH)G_3+((R-CH)G_1-CG_4)G_2\\ &=RG_3-CHG_3+RG_1G_2-CHG_1G_2-CG_2G_4 \end{align} The overall transfer function is $$T(s)=\frac{C(s)}{R(s)}=\frac{G_1G_2+G_3}{HG_1G_2+HG_3+G_2G_4+1}$$