Use of a motor to control the position of a certain load having inertia, damping, and elasticity

Question:

Use of a motor to control the position of a certain load having inertia, damping, and elasticity gives the following plant transfer function. See Figure P11.23.
Gp (s) = ( (s) / V (s) = 0.5 / (s2 + s + 1) (s + 0.5)
Use the ultimate cycle method to compute the controller gains for P, PI, and PID control.
Plot and compare the unit-step responses for the three designs obtained in (a). If the PID response is unsatisfactory, tune the gains to improve the performance.
Figure P11.23