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Control Systems (Prof. Casella) Final Exam – June 26th, 2015 Answers sheets Question 1 Consider the control system shown in the figure. Define the control sensitivity transfer function and explain why is it important to assess the system performance. Finally, explain how to approximate its frequency response assuming that the loop transfer function satisfies the pre-conditions of Bode's criterion. The sensitivity function is defined as Q(s) = C(s)/(1+C(s)G(s)). Disregarding the sign, it is the transfer function between each of the inputs y°, d, and n, and the manipulated variable u, so it allows to understand how sensitive the latter is to all the disturbances acting on the system. It is possible to approximate Q(jw)  1/G(jw) for w << wc, and Q(jw)  C(jw) for w >> wc Question 2 Draw the block diagram of a 2-degrees-of-freedom controller using set-point pre-filtering. Briefly explain the design criteria for the controllers. The feed-back controller C(s) must be designed to meet the specifications regarding disturbance rejection (both from d and from n). The pre-filter F(s) must then be designed so that has the required dynamic behaviour. d C(s) y° - G(s) y n u d H(s) C(s)y° - G(s) yF(s) v° n Y (s) Y °( s)=F ( s) C (s) G( s) 1+ C ( s)G (s) Question 3 Consider the following control system (time constants are given in seconds), where C(s) is a real PID controller with Kp = 50, Ti = 100, Td = 20, N = 3: 3.1 Compute the steady-state errors corresponding to unit step changes of the set point y° and of the disturbance d. 3.2 Compute the crossover frequency and the phase margin of the control system. Then, plot an approximated diagram of the step response of the controlled variable y to a step change of the set point y°. Considering the asymptotic magnitude plot, wc = 0.075;…