Linearization of dynamic model $(20\%)$

$M\frac{d^{2}y(t)}{d{t}^2}=Mg-\frac{i^2(t)}{y^2(t)}$

$e(t)=Ri(t)+L\frac{di(t)}{dt}$

$e(t)=$ input voltage

$y(t)=$ ball position

$i(l)=$ winding current

$R=$ winding resistance

$L=$ winding inductance

$M=$ mass $of$ ball

$g=$ gravitational acceleration

$x_1=y$

Magnetic$-$ball$-$suspension system

$x_2=d\frac{y}{d}t$

Equivalent state: $Mg=f(y_0,i_0)$

$x_3=$i Determine the linearized dynamic equation for

the Magnetic$-$ball$-$suspension system and

$u=e$

Obtain its state space form.