(a) Derive the transfer function H ( s ) H ( s ) for the general second-order...

(a) Derive the transfer function $H\left(s\right)$ for the general second-order system given by

$$\ddot{y}+2\xi {\omega }_n\dot{y}+{\omega }_n^{2}y={\omega }_n^{2}u$$

where $y\left(t\right)$ is the output, $u\left(t\right)$ is the input, and all initial conditions are assumed to be zero.

(b) Find the natural response of the second-order system.

(c) When is this system stable?

(d) Explain how the unit step response can be obtained from the natural response in (b).