# Transfer functions of a closed loop system subjected to a disturbance

There is a system with two controllers in series having the transfer functions G1(s) and G2(s) which is separated by a summing point through which the disturbance D(s) enters the system. The output from the 2nd controller is fed back through another element having the transfer function H(s). The closed loop transfer function CD is given by the formula CD/D(s) = G2(s) / (1 + G1(s) * G2(s) * H(s)).

Can someone help me figure out how we get to this final formula.

I included the diagram too. It is from the book Modern Control Engineering by Katsuhiko Ogata.

Assume $$R(s) = 0$$. Now the left most summing junction can be removed from the diagram taking care to shift the -ve sign to the remaining summing junction.
Then redraw the diagram with $$G_2$$ in $$\text{forward}$$ path and both $$H(s)$$ and $$G_1(s)$$ in $$\text{feedback}$$ path (with negative feedback).
Now the closed loop transfer function is given by $$\frac{\text{forward}}{1 + \text{forward} \cdot \text{feedback}}$$