Q$3.$CERCL Consultants Ltd$.$ has asked two of its engineers to design a PD

controller for a plant with the following transfer function:

$G_p(s)=\frac{1}{(s+1)(s+3)}$

a$)$ What is the general form of the transfer function of a PD controller?

b$)$ What are the overall transfer function and characteristic equation for the

servo problem of the closed$-$loop system in terms of controller gain and

derivative time constant for the servo problem? Assume $G_v=G_m=1$

The engineers suggest two controllers with derivative time constants of $0\mathrm{.}2$ and $0\mathrm{.}5.$

Now we want to choose and implement the final design in the plant. The design

requirements include $1)$ fast response, $2)$ stable over a wide range of controller gain,

and $3)$ preferably less oscillatory response.

c$)$ Describe an engineering approach to quantitatively or qualitatively assess

the performance of the controllers and briefly how to use it in practice for the

given problem.

d$)$ The engineers have the Root$-$Locus diagrams for the two design scenarios as

shown below. Which of the two controllers is better and why? Use the design

requirements in your arguments.

At the time of implementation, it was found that the transfer function of the plant

was wrongly estimated, and the actual transfer function is:

$G(s)=\frac{1}{s(s+1)(s+3)}$

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