A simple unity feedback control system has a process transfer function
$$
\frac{Y(s)}{E(s)}=G(s)=\frac{K}{s} .
$$
The system input is a step function with an amplitude $A$. The initial condition of the system at time $t_0$ is $y\left(t_0\right)=Q$, where $y(t)$ is the output of the system. The performance index is defined as
$$
I=\int_0^{\infty} e^2(t) d t .
$$
(a) Show that $I=(A-Q)^2 /(2 K)$. (b) Determine the gain $K$ that will minimize the performance index $I$. Is this gain a practical value? (c) Select a practical value of gain and determine the resulting value of the performance index.