Chapter 5.P2, Problem 7P

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.

Simulations. Consider the following parameter values expressed in SI units.

$\begin{array}{cccc}m=12,& L=\frac{1}{2},& g=9.8,& \gamma =0.001,\\ J=400,& \beta =20,& {\theta }_0=\pi /6& \end{array}$

With ${\Omega }_0$ determine by the relation (3).

a) Using the above parameter values, construct a root locus plot of the poles of $H_K(s)$ as $K$ varies over an interval containing $K_c$. Verify that a pair of poles cross from the left half plane to the right half plane as $K$ increases through $K_c$.

b) Conduct computer simulations of the system (i),(ii) in Problem 6 using the above parameter values. Do this various values of $K$ less than $K_c$ and greater than $K_c$ while experimenting different values of $y(0)\ne 0$ to represent departure from the equilibrium operating points. For simplicity, you may always assume that $\varphi (0)-\varphi \text{'}(0)=0$. Plot graphs of the function $\varphi (t)$ and $y(t)$ generated from simulations. Note that if $\varphi (t)$ wanders outside the interval $[-{\theta }_0,\pi -{\theta }_0]$, the results are nonphysical. (why?) Are the results for your simulations consistent with your theoretical predictions? Discuss the result of your computer experiments addressing such issues as whether the feedback control strategy actually works, good choices for $K$, stability and instability, and the ”hunting” phenomenon discussed above.

$g=L{\Omega }_0^2\cos {\theta }_0$ … (3)