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Question 4 (30 marks) The block diagram of the control system for a second order mechanical positioning system is shown in Figure Q4. D(s) R(s) Y(3) k(1+2+3) (32+48+1) 1 Figure Q4: Closed-loop System where R(8) represents the desired position, D(8) represents the disturbance, Y (5) is the actual position and the gain k > 0. a) Explain the purpose of each term in a typical PID controller. (5 marks) b) Assume the disturbance, D(s), is zero. Write down the transfer function relating Y(s) to R(s) and show the characteristic equation is given by 6+(4+ k)s? + (1 + k)s + 5% = 0 (10 marks) C) Use the Routh-Hurwitz criteria to find the range of values of k for which the closed-loop system is stable (10 marks) d) Compute the steady state output response to a unit step input on Ris). (5 marks)