# ps08 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #8 Posted: Friday, Nov. 2, ’07 Due: Nov. 9, ’07 1. Consider the complex number s 1 = 1+ j and its geometrical representation on the complex s –plane shown below. Do not use Matlab until you reach question (d) of this problem. ω j s 1 σ -1 1 0 a) Using geometrical arguments and calculations on the s –plane, compute the phase of the complex number ( s 1 + 2)( s 1 + 0). Use complex number algebra to verify your geometrical computation. b) Using geometrical arguments and calculations on the s –plane, compute the value of the real number K such that K | s 1 +2 || s 1 +0 | =1 . Use complex number algebra to verify your geometrical computation. c) Based on the answers to the previous two questions, do you expect s 1 to be a closed–loop pole of a feedback system which has two open–loop poles located at 0, 2 and no open–loop zeros? Justify your answer. If yes, what is the value of the feedback gain that would yield a closed–loop pole at location s 1 ? d) Verify numerically, using Matlab , your answers to all the above questions. Include corroborating plots in your argument. 1

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2. Consider a feedback system with open–loop poles at 0, 1, 2 and no open–loop zeros. The geometrical representation of this system’s open–loop poles on the complex s –plane is shown below. Do not use Matlab until you reach question (f) of this problem. j ω 2 σ -2 -1 0 2 a) Using geometrical arguments and calculations on the s –plane, show that ± j 2 belongs to the root locus of this system. b) Using geometrical arguments and calculations on the s –plane, compute the feedback gain K that would be required to drive two closed–loop poles of this system to the
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ps08 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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