Consider a pendulum as shown in Fig. $1.$ We assume that the mass $m$ is concentrated at pendulum end,

with length as $l.$ Gravity should be considered. The pendulum is driven by a torque input $T$ at the base,

and the base rotation joint is subject to rotational damping $b.$ The equation of motion of this pendulum is

$m{l}^2{}^{}+b{}^{}+$mglsin$=T$

where T is input and pendulum angle $$ is output.

Figure $1$: Simple pendulum

$($a$)2$ points. Define the $2$ state variables of the system. The input is $u=T.$ Put the equations of

motion in nonlinear state$-$space form, $x^{}=f(x,u).$

$($b$)4$ points. Consider the initial angle of pendulum is ${}_0$ and initial torque $T_0=0$ for a passive system.

A small toque input $T$ is at the base joint, leading to perturbation angle of $.$ Linearize the nonlinear

state$-$space model about ${}_0$ and $T_0$ to obtain the linear state$-$space models for ${}_0=0$ and ${}_0=.$