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Spring2003Quiz2

Spring2003Quiz2 - ME 132 Spring 2003 Quiz 2#1#2#3#4#5 TOTAL...

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ME 132, Spring 2003, Quiz # 2 # 1 15 # 2 12 # 3 12 # 4 6 # 5 15 TOTAL 60 1. Three systems are described below by their ODE (input u , output y ). All are started with zero initial conditions at t = 0 - , namely y (0 - ) = ˙ y (0 - ) = ¨ y (0 - ) = 0. A unit-step input is applied at t = 0. In each case, determine: the “new” initial conditions at t = 0 + , namely y (0 + ) , ˙ y (0 + ) and ¨ y (0 + ); the final value of y , ie., lim t →∞ y ( t ). (a) y [3] ( t ) + 2 y [2] ( t ) + y [1] ( t ) + y ( t ) = 6 u [2] ( t ) - 3 u [1] ( t ) + 2 u ( t ) (b) y [3] ( t ) + 2 y [2] ( t ) + y [1] ( t ) + 3 y ( t ) = - 3 u [1] ( t ) + 2 u ( t ) (c) y [3] ( t ) + 2 y [2] ( t ) + 4 y [1] ( t ) + 5 y ( t ) = 2 u ( t ) 2. Assume G 1 , G 2 and H are transfer functions of linear systems. Compute the transfer function from R to Y in the figure below. G 1 G 2 H - ²fl - - ? ²fl 6 - - R Y - + - + 3. A closed-loop feedback system consisting of plant P and controller C is shown below. C P - - - - 6 h + - r ( t ) e ( t ) u ( t ) y ( t ) In this problem, it is known that the nominal closed-loop system is stable. The plots below are the magnitude and phase of the product ˆ P ( ) ˆ C ( ), given both in linear and log scales, depending on which is easier for you to read.
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