The primary objective of many control systems is to maintain the output variable at the desired or reference condition when the system is subjected to a disturbance [23]. A typical chemical reactor control scheme is shown in Figure P9.17. The disturbance is represented by $U(s)$, and the chemical process by $G_3$ and $G_4$. The controller is represented by $G_1$ and the valve by $G_2$. The feedback sensor is $H(s)$ and will be assumed to be equal to 1 . We will assume that $G_2, G_3$, and $G_4$ are all of the form
$$
G_i(s)=\frac{K_i}{1+\tau_i s},
$$
where $\tau_3=\tau_4=4 \mathrm{~s}$ and $K_3=K_4=0.1$. The valve constants are $K_2=20$ and $\tau_2=0.5 \mathrm{~s}$. We want to maintain a steady-state error less than $5 \%$ of the desired reference position.
(a) When $G_1(s)=K_1$, find the necessary gain to satisfy the error-constant requirement. For this condition, determine the expected overshoot to a step change in the reference signal $r(t)$.
(b) If the controller has a proportional term plus an integral term so that $G_1(s)=K_1(1+1 / s)$, determine a suitable gain to yield a system with an overshoot less than $30 \%$, but greater than $5 \%$. For parts (a) and (b), use the approximation of the damping ratio as a function of phase margin that yields $\zeta=0.01 \phi_{\mathrm{pm}}$. For these calculations, assume that $U(s)=0$.
(c) Estimate the settling time (with a $2 \%$ criterion) of the step response of the system for the controller of parts (a) and (b).
(d) The system is expected to be subjected to a step disturbance $U(s)=A / s$. For simplicity, assume that the desired reference is $r(t)=0$ when the system has settled. Determine the response of the system of part (b) to the disturbance.