Q4. (a) Explain the significance of the transfer function: w s² + 23wns + w²/ in determining the nature of the time response of a system. Give examples. [24 marks] (b) The transfer function of the mechanical vibratory system shown in Figure Q4a is given in equation (4.1). Gp(s) X(s) 1 F(s) ms² + ẞs + k When a force of 2 N (step input) is applied to the system, the mass oscillates, as shown in Figure Q4b. Determine m, ẞ, and k of the system from the time response graph in Figure Q4b. x(t)4 -0.16 0.8 k- -f(t) m x(t) Figure Q4a (4.1) [32 marks] 0.704 time(s) Figure Q4b (c) A proportional controller Ge(s) is in cascade with the plant Gp(s) as shown in Figure Q4c. Assume m = 0.1C kg, ẞ = 0.4D N-s/m and k = 0.5E N/m. Ge(s) Gp(s) ✗(s) X(s) Kp ms² + ẞs + k Figure Q4c (i) select a controller gain Kp that will give the overall system a damping ratio of 0.7 (for a step input). [16 marks] (ii) if the proportional controller is replaced with a proportional-derivative (PD) controller of the form G(s) = KpKDs. Comment on the merit(s) of the PD controller over the proportional controller. Suggest suitable values for K, and KD that would give a damping ratio of 0.7 and an undamped natural frequency of 12.F rad/s. [12 marks] (iii) compare and comment, on the effect of the proportional and PD controllers, on the steady-state error of the compensated system. [16 marks]