ME 475/591 Control Systems Final Exam Fall '99

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1 ME 475/591 Control Systems Final Exam Fall '99 Closed book closed notes portion of exam. Answer 5 of the 6 questions below (20 points total) 1) What is a phase margin? Under ideal circumstances, what does it tell you about the system s closed-loop transient response? 2) From the Bode plot perspective, explain how a lead compensator increases the speed of the transient response. 3) How does a pure time delay affect the magnitude component of the frequency response of a system? How does a pure time delay affect the phase angle component of the frequency response of a system?

2 4) What is the nonlinearity of saturation and how does it affect the performance of a control system? 5) Briefly outline the steps required to set PID gains using the Ziegler-Nichols method. 6) What major advantage does compensator/controller design by root locus have over frequency response design?

3 Policies for computer use during ME 475 Test You may use the following computer resources to work the problems on this test: 1. Matlab (including Simulink) including any " *.M " files you may have created for homework or projects 2. Maple including any " *.ms " files you may have created for homework or projects 3. Excel including any " *.xls " files you may have created for homework or projects You may NOT access the Internet, mail servers, or any other program during the test. You may NOT access any Word, Acrobat PDF, text, or other documents on your personal account or any other account on the computer. Your signature on the line below indicates that you understand these policies and have agreed to abide by them. Date - Tuesday, December 14, 1999

4 ME 475/591 Control Systems Final Exam Fall '99 Closed book exam - one sheet of formulas and notes allowed. 1) [20] For the open-loop transfer function given below, do the following: ( s + 1) G( s) = 2 ( s + 4)( s + 100)( s + 20s + 400) a) plot the linear approximation for the magnitude on the graph paper provided, b) determine the phase and gain margins for this system (assuming unity feedback), c) write analytical expressions for magnitude and phase angle, d) compute the actual (analytical) magnitude and phase angle at ω = 2, 10, and 75 rad/sec. 2) [20] Design a control system such that the resulting unity feedback, closed-loop system meets the following specifications: a) dominant complex roots with damping ratio ζ = 0.5 b) 2% settling time t s for a step input of < 1.0 seconds c) steady-state error for a ramp input < 2% of input magnitude open loop G( s) = 5000 s( s+ 5)( s+ 1000) You must show (and justify) all steps in your design process. Matlab and Simulink outputs must be used to confirm the results of your design. 3) [20] A unity feedback, closed loop control system is shown below. a) Find the value of the gain K for the system below that gives a steady-state error of 0.10 for a unit step input. b) Design a lead controller that approximately triples the phase margin at this value of K, i.e., Φ M,new = 3* Φ M,original R(s) + - controller K plant 100 ( s + 4)( s + 8)( s + 26) C(s) 4) [20] All graduate students must work this problem undergraduates do NOT. For the same system and same K value found in part (a) of the previous problem, design a lag controller that will reduce the steady-state error to 0.02 for a unit step input, while maintaining approximately the same original phase margin, Φ M,original.

5 5) [20] The frequency response (magnitude and phase angle) for a system is shown below. a) What would be your "linear approximations" to the magnitude data (show on the diagram) b) From (a), what would be a reasonable first "guess" of the transfer function, G(s)? (I am not looking for an exact answer, only an educated engineering estimate with your reasons clearly stated) c) Is there anything unusual about the frequency response shown below? What is the most likely cause? Magnitude, db Phase Angle, degrees -50 Magnitude Phase Angle E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 Frequency, rad/sec

6 Magnitude, db E E E E E E+04 Frequency, rad/sec

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