Chapter 5.P2, Problem 6P

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.

The Nonlinear Feedback Control System. Using the relation (3) and (4), show that Eqs. (1) and (2) can be expressed in terms of $\varphi$ and $y$ as

$mL\varphi \text{'}\text{'}=-\gamma \varphi \text{'}-m{\Omega }_0^{2}L\cos ({\theta }_0)\sin ({\theta }_0+\varphi )+m{({\Omega }_0+y)}^{2}L\sin ({\theta }_0+\varphi )\cos ({\theta }_0+\varphi )$ (i)

And $Jy\text{'}=-\beta y-K\varphi ,$ (ii)

Where the negative feedback loop has been incorporated into the nonlinear system.

$\begin{array}{cc}mL\varphi \text{'}\text{'}=-\gamma \varphi -mg\sin \theta +m{\Omega }_{}^{2}L\cos \theta \sin \theta .& (1)\\ J\frac{d\Omega }{dt}=-\beta \Omega +\tau ,& (2)\\ g=L{\Omega }_0^2\cos {\theta }_0& (3)\\ {\tau }_0=\beta {\Omega }_0& (4)\end{array}$