hw6-fa09

# hw6-fa09 - CALIFORNIA INSTITUTE OF TECHNOLOGY Control and...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 101 D. G. MacMynowski Fall 2009 Problem Set #6 Issued: 9 Nov 09 Due: 16 Nov 09 Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading). 1. Plot the (open loop) Nyquist and Bode plots for the following systems and compute the gain and phase margin of each. You should annotate your plots to show the gain and phase margin computations. For the Nyquist plot, mark the branches corresponding to the following sections of the Nyquist “D” contour: negative imaginary axis, positive imaginary axis, semi- circle at infinity (the curved part of the “D”). (a) Disk drive read head positioning system, using a lead compensator (described in Chap- ter 11): P ( s ) = 1 s 3 + 10 s 2 + 3 s + 10 , C ( s ) = 1000 s + 1 s + 10 . (b) Second-order system with PD compensator: P ( s ) = 100 (100 s + 1)( s + 1) , C ( s ) = s + 10 . Note: you may find it easier to sketch the Nyquist plot from the Bode plot (taking some liberties with the scale) rather than relying on MATLAB. 2. In this problem we will design a PI controller for a cruise control system, building on the example shown in class. Use the following transfer function to represent the vehicle and engine dynamics: P ( s ) = Tba/m ( s + a )( s + c/m ) where b = 25 is the transmission gain, T = 200 is the conversion factor between the throttle input and steady state torque, a = 0 . 2 is the engine lag coefficient, m = 1000 kg is the mass of the car, and c = 50 N s/m is the viscous damping coefficient. (a) Consider a proportional controller for the car, u = k p ( r y ). Assuming a unity gain feedback controller, this gives C ( s ) = k p .

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