Transfer Function

Question

Hi Guys I am trying to evaluate the following transfer function below can anyone verify if this is correct

$$e_i = -d_{r,i} + d_i $$ $$d_i = q_{i-1}-q_i$$ $$d_{r,i} =H_iq_i$$ $$Therefore, e_i = -Hiq_i + q_{i-1} - q_i$$ $$q_i = e_iPDG_i$$ $$q_i = PDG_i(-H_iq_i+q_{i-1}-q_i)$$ $$q_i = -PDG_iH_iq_i+PDG_iq_{i-1}-PDG_iq_i$$ $$\frac{q_i}{q_{i-1}} = \frac{PDG}{1+HPDG+PDG}$$

Im thinking this is incorrect but I'm hoping someone can verify this for me?

Answer 1

Shouldn't the error equation have $-H_iq_i$ instead of $H_iq_i$? $$e_i=−d_{r,i}+d_i$$ $$d_i=q_{i−1}−q_i$$ $$−d_{r,i}=−H_iq_i$$ Therefore (modified equation): $$e_i=-H_iq_i+q_{i−1}−q_i$$ $$q_i=e_iPDG_i$$ $$q_i=PDG_i(-H_iq_i+q_{i−1}−q_i)$$ $$q_i=-PDG_iH_iq_i+PDG_iq_{i−1}−PDG_iq_i$$ $$\frac{q_i}{q_{i−1}}=\frac{PDG_i}{1+H_iPDG_i+PDG_i}$$