$1.)(40$ pts$)$ Please find the transfer function $\frac{(s)}{U(s)}$ for pendulum and $\frac{x(s)}{U(s)}$ for the cart in the inverted pendulum system shown in Ogata $($page $69-70).$ Hint: Perform Laplace Transform for both Eq$.(3-16)$ and Eq$.(3-17)$

$I{}^{}=Vl{}_{d^2}Hl$

$V-mg=m\frac{d^2}{d{t}^2}(l*1)=0$

$H=m\frac{d^2}{d{t}^2}(x+l)=m(x^{}+l{}^{})$

$I{}^{}=Vl-Hl$

$I{}^{}=mgl-m(x^{}+l{}^{})l$

$I{}^{}=mgl-ml{x}^{}-m{l}^2{}^{}$

$u-H=M\frac{d^{2}x}{d{t}^2}=M{x}^{}$

$u-m(x^{}+l{}^{})=M{x}^{}$

$R_{z1}+R_{z2}-V-Mg=0$

$u-H=M\frac{d^{2}x}{d{t}^2}=M{x}^{}$