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hw5-fa09

# hw5-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 #5 Issued: 2 Nov 09 Due: 9 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. Consider the block diagram for the following second-order system - - + + 1 s 1 s c a b r y z (a) Compute the transfer function H yr between the input r and the output y . (b) Show that the following state space system has the same transfer function, with the appropriate choice of parameters: d dt x 1 x 2 = 0 1 - a 2 - a 1 x 1 x 2 + 0 1 r y = b 2 b 1 x 1 x 2 + dr Give the values of a i , b i , and d that correspond to the transfer function you computed in (a). (c) Compute the transfer function H zr between the input r and the output z . (Hint: It is not H zr = 1.) 2. Consider the following simplified equations of motion for a cruise control system (these are a linearization of the equations from Section 3.1 in ˚ Astr¨ om and Murray): m dv dt = - cv + + F hill , where m = 1000 kg is the mass of the vehicle, c = 50 N s/m is the viscous damping coefficient, b = 25 is the conversion factor between engine torque and the force applied to the vehicle.

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