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Unformatted text preview: Written Exam Control Systems 1 (Et 3101) On Tuesday the 29th of October 2002 from 9:00 to 12:00 Read the following VERY carefully. The exam consists of 6 pages with 5 exercises. You are only allowed to use the BOOK and a CALCULATOR. NEVER talk with your neighbor. If the answers are not easily readable, the corresponding answer will be given 0 points. Therefore WRITE CLEARLY. Read every question well before answering. Write ALL your reasoning steps on paper. Write your name and student number clearly readable on EACH piece of paper. Good luck ! Technische Universiteit Delft Faculteit der Informatie Technologie en Systemen Vakgroep Regeltechniek Mekelweg 4 2628 CD Delft Question 1 (weight:1.5) Given: The differential equation for the van der Pol oscillator is given by y ( t ) + (1 y 2 ( t )) y ( t ) + y ( t ) = u ( t ) Asked: 1. Calculate the differential equation of the form x = f ( x, ) describing the dynamics of the system with x = bracketleftbig y y bracketrightbig T . 2. Linearize the system around an equilibrium x and u . 3. For what values of u do we obtain a stable linearized system ? Question 2 (weight:2) Given: The transfer function of a system G ( s ) is: G ( s ) = ( s + 1)( s + 3) ( s 2 + 2 s + 17)( s + 4) The system is put in a feedback loop with the proportional gain D ( s ) = K P as follows: D(s) + G(s) Y(s) R(s) S Asked: 1. To analyse the variation of the proportional gain K P , draw the root locus for the poles of the transfer function Y ( s ) R ( s ) for variations of K P from to . Compute all details like asymptotes, departure/arrival angles, etc. if they are applicable for this specific case. The four solutions of the equation that needs to be solved to determine the location of the multiple roots are 2 . 20 , 3 . 39 , 4 . 60 2 . 30 j . Give this equation. 2. For K P K 1 it holds that i . 8 , where i denotes the damping coefficient of the i th pole. Give the equation that needs to be solved to determine K 1 . Question 3 (weight:2.5) Given: The transfer function of a system G ( s ) is: G ( s ) = K ( s + 0 . 2) ( s + 0 . 5)( s + 1) 2 ( s + 2) The system is put in a feedback loop as in Question 2....
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 Spring '10
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