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4. A lead-lag compensator has been designed for a second order mechanical system. The controller has transfer function K(8) = 2(s + 10)(8 +5) (s + 50)(s + 1) The controller is to be implemented in a digital microcontroller which is capable of sampling the output at a rate of 100 Hz. (a) Explain briefly the purpose of the phase-lead and phase-lag terms in the controller, and why they might be used for a particular design problem. (10 Marks) (b) We can use the Matched Pole-Zero method to find a suitable z-domain transfer func- tion for the discretised controller in the form (2) k(z z)(2 22) (z p)(z P2) Find equivalent locations in the z-domain, 21, 22, P1 and p2, corresponding to the two poles and two zeroes of K(s). (4 Marks) (c) Calculate the DC gain for K(s), and hence find an appropriate value for k such that the DC gains of K(+) and K(s) match. Hence show that K(2) is (2) 1.68722 -3.1312 + 1.452 22 1.5972 +0.6005 (6 marks) (d) Use () to derive a difference equation relating the controller output u(k) to the tracking error (controller input) e(k). (10 marks) 4. A lead-lag compensator has been designed for a second order mechanical system. The controller has transfer function K(8) = 2(s + 10)(8 +5) (s + 50)(s + 1) The controller is to be implemented in a digital microcontroller which is capable of sampling the output at a rate of 100 Hz. (a) Explain briefly the purpose of the phase-lead and phase-lag terms in the controller, and why they might be used for a particular design problem. (10 Marks) (b) We can use the Matched Pole-Zero method to find a suitable z-domain transfer func- tion for the discretised controller in the form (2) k(z z)(2 22) (z p)(z P2) Find equivalent locations in the z-domain, 21, 22, P1 and p2, corresponding to the two poles and two zeroes of K(s). (4 Marks) (c) Calculate the DC gain for K(s), and hence find an appropriate value for k such that the DC gains of K(+) and K(s) match. Hence show that K(2) is (2) 1.68722 -3.1312 + 1.452 22 1.5972 +0.6005 (6 marks) (d) Use () to derive a difference equation relating the controller output u(k) to the tracking error (controller input) e(k). (10 marks)