P$3.(60$ marks$)$

Consider a closed loop system comprising of system$1$ and system $2$ given in the configuration shown in Figure $1.$ The open loop response of system $1$ and system $2$ to step inputs are given in Figure $2$ and Figure $3$ respectively.

R

Figure $2$: Step response of open loop System $1$

Figure $3$ Step response of open loop system $2$

Given the open step responses of both systems, answer the following with your reasons:

a$.$ What is the order of open loop system $1?$

b$.$ What is the order of open loop system $2?$

c$.$ What is the type of open loop system $1?$

d$.$ What is the type of open loop system $2?$

e$.$ Are the complex poles in system $1?$

f$.$ Are the complex poles in system $2?$

Given the step response, highlight the time domain characteristics $($e$.$g$.$ settling time, rise time, overshoot etc.$)$ to find the transfer function for both system $1$ and system $2.$

Given the closed loop configuration in figure $1,$ find the closed loop transfer function.

Comment on the stability of the closed loop system using Routh Hurwitz criteria, root locus and bode plot.

Comment of the performance of the closed loop system?

a$.$ Is there is steady error in the system response to impulse, step and ramp response?

Is there a need of designing the controller$/$compensator$?$ Why?

What controller will you design to achieve desired stability and performance from the system for impulse, step and ramp inputs. $($Explain your design process and reasons$)$