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Unformatted text preview: 304304A Signals and Systems II Thursday, April 18, 2008 FinalExam9:AM1/8 304304A Signals and Systems II Final Exam Date: Thursday, April 18, 2008 Time: 9:00 AM Department of Electrical and Computer Engineering McGill University Examiner: Prof. Richard Rose 3981749 CoExaminer: Prof. Jan Bajcsy 3987462 This is a closedbook examination Two hand written crib sheets (2sided) are permitted Facultyapproved calculators only Attempt all problems (total of 100 points) Show all your work and write your answers in the exam booklet provided. Clearly indicate your answer by circling or drawing a box around it Draw sketches to aid your thinking Grade Summary Problem # Points Score 1 15 2 15 3 20 4 15 5 20 6 15 Total 100 304304A Signals and Systems II Thursday, April 18, 2008 FinalExam9:AM2/8 Problem 1 (15 points) Consider the closed loop control system with open loop transfer function: 22(10)( )100(1)K sG sss+=+The bode plot for ( )( )K s H sfor 1k=and =is shown below: a)Identify all gain and phase crossover frequencies from the above magnitude and phase plots. Explain your answer. b)What is the phase margin for the above system? Explain your answer. c)What is the smallest value of k, if any, that will bring the closed loop system to the verge of instability? Explain your answer. ( )K s( )H skje304304A Signals and Systems II Thursday, April 18, 2008 FinalExam9:AM3/8 Problem 2 (15 points) State Space Representations:You are given the continuoustime LTI system shown in the following figure: a)Find the state space representation of the system with state variables1( )x tand2( )xt. b)Determine the range of values of for which the system will be BIBO stable. . 1s1s( )u t2( )xt1( )x t2( )xt1( )x t( )y t2421304304A Signals and Systems II Thursday, April 18, 2008 FinalExam9:AM4/8 Problem 3 (20 points) Stability Criteria for Closed Loop Systems:Consider the open loop transfer function 21( ) ( )(1)G s H ss ss=++which is part of the following closed loop system: a)Write the expression for the closed loop transfer function ( )( )/ ( )T sY sR s=b)Write the poles of the open loop transfer function ( )( )G s H sc)Plot the root loci fork>. Provide arrows to indicate the direction along the root loci corresponding to increasing values of k. d)Determine the range of values of k(if any) for which the closed loop system is stable. (Hint: The RouthHurwitz criterion may be helpful here) Explain your answer. ControllerPlant( )r t( )u t( )y t( )kG s( )H s( )e t304304A Signals and Systems II Thursday, April 18, 2008 FinalExam9:AM5/8 Problem 4 (15 points) zTransform:You are given five discrete time sequences labelled i) through v). The grey areas of the zplanes drawn below represent regions where the ztransform of the appropriate sequence converges. For each of the discrete time sequences, identify which of the regions in a) through f) is the closest approximation to the ROC for that sequence....
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 Winter '09
 FRANCISCODGALIANA

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