$$ model for the inverted pendulum on a cart shown in figure below $...$

$$

$...$ is described by the following set of equations

$(M+m)x^{}+ml{}^{}cos=b{x}^{}+ml{}^{}{?}^{2}sin+f$

$(J+m{l}^2){}^{}+ml{x}^{}cos=$mglsin$$

Assume $M=m=1Kg,J=1Kg{m}^2,l=1m$ and $b=1K\frac{g}{s}.$

$($a$)$ Obtain a four$-$dimensional nonlincar state$-$space representation with output $y=x,$ input

$($b$)$ An cquilibrium poinh of a system of the form $x^{}=f(x,u)$ is obtained by solving for $x$

that satisfies $f(x,0)=0.$ Determine equilibrium points of this system $($You can revise

the notes on cquilibrium points$).$

$($c$)$ Lincarize this system of equations around its around the trajectory about the equilibrium

point when $J(t)=0.$ Write it in sitate$-$space form.

$($d$)$ Find the Lransfer finction for the linear system