The transfer function of the system below is

Solution

We are asked to find the transfer function C(s)/R(s) of the given feedback control system.
Identify the structure ●       Forward path: G(s) ●       Feedback path: H(s) ●       Summing point: adds R(s) and  + B(s)  (where B(s) = output through feedback). So,  E(s) = R(s) + B(s)  But B(s) = H(s)C(s) Express output  C(s) = G(s)E(s)  = G(s)[R(s) + H(s)C(s)] Rearrange  C(s) = G(s)R(s) + G(s)H(s)C(s) Bring terms with C(s) together: C(s) – G(s)H(s)C(s) = G(s)R(s) C(s)[1 – G(s)H(s)] = G(s)R(s) Transfer function  C(s)/R(s) = G(s) / [1 – G(s)H(s)]