Consic

a$)$ Write the transfer function for a setpoint change, $Y^{\text{'}}\frac{s}{Y_{sp}^{\text{'}}}(s),$ given that $G_{C1}=K_1,G_{C2}=\frac{K_1}{{}_{\text{I}}}s$ and $G_{C3}=K_1{}_{D}s.$

b$)$ Write the transfer function for a setpoint change, $Y^{\text{'}}\frac{s}{Y_{sp}^{\text{'}}}(s),$ given that $G_{C1}=K_1,G_{C2}=K_1$ and $G_{C3}=K_1(1+{}_{D}s)(1+\frac{1}{{}_{\text{I}}}s).$

c$)$ Consider the following values: ${}_1=2,{}_{\text{I}}=1,{}_D=5,K_1=3,K=2.$ Write the expressions and plot $($not sketch!$)$ the response for both system a$)$ and system b$)$ to a step change of $Y_{sp}^{\text{'}}=S(t).$