Chapter 5.P2, Problem 4P

In the following problem, we ask the reader to some of the details left out of the above discussion, to analyse the closed-loop system for the stability properties, and to conduct a numerical simulation of the nonlinear system.

Show that the transfer function of the closed-loop system linearized about the operating point is

$H_k(s)=\frac{s^2+2\delta s+{\omega }_0^2}{(Js+\beta )(s^2+2\delta s+{\omega }_0^2)+K}.$