hw7-fa10 - CALIFORNIA INSTITUTE OF TECHNOLOGY Control and...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 101 D. G. MacMynowski Fall 2010 Problem Set #7 Issued: 15 Nov 10 Due: 22 Nov 10 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 problem of stabilizing the orientation of a flying insect, modeled as a rigid body with moment of inertia J = 0 . 41 and damping constant D = 1. 1 We assume there is a small delay τ = 0 . 01 seconds given by the neural circuitry that implements the control system. The resulting transfer function for the system is taken to be P ( s ) = 1 Js 2 + Ds e τs . (a) Suppose that we can measure the orientation of the insect relative to its environment and we wish to design a control law that that gives zero steady state error, less than 10% tracking error from 0 to 0.5 Hz and has a phase margin of at least 60 . Convert these specifications to appropriate bounds on the loop transfer function and sketch the resulting constraints on a Bode plot. (b) Using a lead compensator, design a controller that meets the specifications in part (a). Provide whatever plots are required to verify that the specification is met. You may use a Pad´ e approximation for the time delay, but make sure that it is a good approximation over a frequency range that includes your gain crossover frequency.
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