Problem $2.$ Cable Lift $[4$ points$]$

Figure P$2\mathrm{.}9.$

$($from KGS$)$

The above diagram is an idealized model of a cable lift $(m_2)$ that is actuated through a slightly flexible cable $($stiffness $k_2)$ by a motor with mass $m_1$ and viscous damping $b$ that is feedback controlled to track a desired position $x_1$ with a virtual stiffness of $k_1.($For your convenience, the motor, virtual spring, damping, etc. are all defined as translational elements, but almost certainly are implemented with rotational elements like a drum, a gear$-$motor, angle controller, etc.$)$ The pulley can be idealized as massless and frictionless. Please write the equations of motion for state variables $x_2,v_2,x_3,$ and $v_3,$ with $x_1$ acting as an input. $[2$ points$]$

And please develop expressions for ${}_2$ and ${}_3,$ the displaced references resulting from the action of gravity. $[2$ points$]$

${}_2=$

${}_3=$

$x_2^{}=0x_2+\frac{1}{0}{v}_2+0x_3+0v_3+0x_1,$

$v_2^{}=\frac{k_2}{m_1}-\frac{k_1}{m_1}{x}_2+\frac{-b}{m_1}{v}_2+\frac{-k_2}{m_1}{x}_3+0,v_3+\frac{k_1}{m_1}{x}_1,$

$x_3^{}=0,x_2+0,1_2+0,x_3+,*1_3+0x_1,$

$v_3^{}=\frac{k_2}{m_2}{x}_2+$dots,$v_2+-\frac{k_2}{m_2}{x}_3+,v_3+0,x_1.$