ECE3085-ControlSystems-Exam1_Spring08_solution

# ECE3085-ControlSystems-Exam1_Spring08_solution - ECE 3085...

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ECE 3085 Spring 2008 Exam 1 Closed-Book, Closed-Notes, 4 Problems. Name: 1. Complete the table below, by identifying the response generated by each of the given systems. System Dynamic Equations Initial Condition(s) Response # ˙ y ( t ) + 3 y ( t ) = 0 y (0) = 1 6 ¨ y ( t ) + 2 ˙ y ( t ) + 10 y ( t ) = 0 y (0) = 1 , ˙ y (0) = 0 2 ¨ y ( t ) - 0 . 2 ˙ y ( t ) + 9 y ( t ) = 0 y (0) = 1 , ˙ y (0) = 0 8 ¨ y ( t ) + ˙ y ( t ) + 0 . 25 y ( t ) = 0 y (0) = 1 , ˙ y (0) = 0 . 5 1 0 5 10 0 0.5 1 1.5 t y(t) response #1 0 5 10 -0.5 0 0.5 1 t y(t) response #2 0 5 10 1 1.5 2 2.5 3 t y(t) response #3 0 5 10 0 0.5 1 t y(t) response #4 0 5 10 0 0.5 1 t y(t) response #5 0 5 10 0 0.5 1 t y(t) response #6 0 5 10 -1 -0.5 0 0.5 1 t y(t) response #7 0 5 10 -4 -2 0 2 4 t y(t) response #8 0 5 10 0 0.5 1 t y(t) response #9 modes at s = - 3 y ( t ) = αe - 3 t exponential decay with τ = 1 3 # 6 modes at s = - 1 ± j 3 y ( t ) = αe - t cos(3 t + θ ) oscillating decay with τ = 1 # 2 modes at s = 0 . 1 ± j 2 . 998 y ( t ) = αe 0 . 1 t cos(2 . 998 t + θ ) oscillating growth with τ = 10 # 8 modes at s = - 0 . 5 , - 0 . 5 y ( t ) = ( α + βt ) e - 0 . 5 t decay with initial slope = ˙ y (0) > 0 # 1

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2. Determine the transfer function of the following system: x ( k + 1) = 1 2 3 4 x ( k
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