Where you previously designed a lead $($and lag$)$ compensator to meet stability margin and

error requirements, you will now follow a different $($graphical$)$ approach to design a similar

compensator, however this time to meet a desired damping ratio $$ and desired natural

frequency ${}_n$ for the closed loop system.

In this case, suppose you wish to control the focus of a laser to conduct interferometry

experiments, done through adjustment of the position of an optical lens. Knowing you desire

quite a high natural frequency ${}_n$ of $5ra\frac{d}{sec}$ to provide a high response rate needed for

real$-$time control, along with a moderate damping ratio $$ of $0\mathrm{.}5$ to limit overshoot, you have

set out to employ a lead compensator to achieve this. With the block diagram for this

controlled system illustrated in Figure $1,$ complete the following steps to design and evaluate

the performance of the controller:

Figure $1$