Chapter 6, Problem 48P

A linearized model of a torque-controlled crane hoisting a load with a fixed rope length is

$P\left(s\right)=\frac{X_T\left(s\right)}{F_T\left(s\right)}=\frac{1}{m_T}\frac{s^2+{\omega }_0^2}{s^2\left(s^2+a{\omega }_0^2\right)}$

where ${\omega }_0=\sqrt{\frac{g}{L}}$ , L = the rope length, m_T= the mass of the car, a = the combined rope and car mass, f_T= the force input applied to the car, and x_T= the resulting rope displacement (Marttinen, 1990). If the system is controlled in a feedback configuration by placing it in a loop as shown in Figure P6.11, with $K>0$ , where will the closed-loop poles be located?