A linearized model of a torque-controlled crane hoisting a load with a fixed rope length is where 0 = g/L; L= the rope length,

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

where ω₀ = √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?

X7(s) _ 1s + o P(s) = FT(s) mT s(s2 + aw)