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hw8-fa09

# hw8-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 #8 Issued: 23 Nov 09 Due: 2 Dec 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 a second-order process of the form P ( s ) = k s 2 + 2 ζω 0 s + ω 2 0 k, ζ, ω 0 > 0 . In this problem we will explore various methods for designing a PID controller for the system. (a) (Eigenvalue assignment) Suppose that we want the closed loop dynamics of the system to have a characteristic polynomial given by p ( s ) = s 3 + a 1 s 2 + a 2 s + a 3 . Compute a formula for the controller parameters of a PID controller ( k p , k i and k d ) that gives the desired closed loop response. (b) (Eigenvalue assignment) Let the process parameters be given by k = 1, ζ = 0 . 5 and ω 0 = 2. Using the formulas from part (a), compute a feedback control law that places the closed loop poles of the system at λ = {- 1 , - 2 ± j } . Plot the step response and frequency response for the closed loop systems, and compute the gain and phase margins for your design. (c) Optional: (Ziegler–Nichols step response) Using the same process parameters as above, plot the step response for the system and use the Ziegler–Nichols rules to design PID controllers. Plot the step response and frequency response for each of your controller designs, and compute the gain and phase margins for each design.

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