$1.$ Using $=0$ at the operating equilibrium, and the small angle approximation sin$()$

around the operating equilibrium, obtain the linearized version of Equation $(1)$ and the

Laplace transform of this equation. Note that $$ is the input and $$ is the output.

$2.$ Can this transfer function be stabilized with a simple proportional controller? Why$/$Why

not?

Figure $1.$ Balance beam system

Figure $2.$ Ball $-$Beam model definition

$3.$ Write a MATLAB script to define all the constants in Table $1$ and create the transfer

function for the Laplace transformed linearized system.

$4.$ Plot the root locus of the linearized system, using the rlocus command.

$5.$ Plot its step response using step command.