Root Locus

For the unity feedback system

shown, sketch the root locus as a

function of $k$ and

A$.$ Discuss the stability of the closed

loop system as a function of the

gain $k.$

B$.$ Find the value of the gain $k$ that

makes the closed$-$loop system

marginally stable.

State Space Control

For the system whose transfer function is given in Question $1$:

A$.$ Represent the system in any state$-$space form you like

B$.$ Check the controllability

C$.$ Check the observability

D$.$ Design a state feedback controller $($not integral control$)$ to yield a $10\%$ overshoot and

$0\mathrm{.}5$ second settling time to a step input

Nyquist Plot and Stability

For the system shown in Question $1$:

A$.$ Plot the Nyquist diagram

B$.$ Based on the Nyquist diagram, discuss the stability of the closed$-$loop system

C$.$ Find the value of gain $k$ that makes the system marginally stable