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ECE3085-ControlSystems-Exam3_Spring08_solution

ECE3085-ControlSystems-Exam3_Spring08_solution - ECE 3085...

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ECE 3085 Spring 2008 Exam 3 Closed-Book, Closed-Notes, 4 Problems. Name: 1. A control system has the following structure, where L ( s ) is a ratio of two polynomials; L ( s ) has more poles than zeros, and all its poles are located in the open left half plane. Let H ( s ) denote the overall transfer function satisfying Y ( s ) = H ( s ) R ( s ). L y r + - (a) The loop transfer function L ( s ) is characterized in the diagram to the left and the overall transfer function H ( s ) is characterized in the diagram to the right; the labeling of these diagrams establishes a unique correspondence between them. Precisely mark the diagram to the right with an asterisk corresponding to the asterisk found on the diagram to the left. -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Re{L(j ω )} Im{L(j ω )} -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Re{H(j ω )} Im{H(j ω )} L ( ) = - j H ( ) = L ( ) 1 + L ( ) = - j 1 - j = - j 1 - j · 1 + j 1 + j = 1 - j 2 (b) Prove that this system is internally stable. The given plot of L ( ) is at least a portion of the Nyquist plot of L ( s ). Since L ( s ) is strictly proper, the point L ( ) = 0 corresponds to ω = ±∞ . Since the coefficients of L ( s ) are real, the point L ( ) = 1 corresponds to ω = 0. As further confirmation that the entire axis has been accounted for, note that the given plot is a closed curve symmetric about the real axis. Finally, since
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