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homework2_solution.pdf - Solutions to Homework 2 ECE 460...

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Solutions to Homework 2 ECE 460: Automatic Control Due: Wednesday, January 20, at 2:10pm Note 1. Your solutions should be neat and legible; always show your work. Problem 1. (20 Points) Find the Laplace transform of the following time functions using Table 2.1 and Table 2.2 on the textbook. (a) g ( t ) = e - αt cos( ωt - φ ) Solution: g ( t ) = e - αt (cos( ωt ) cos( φ ) + sin( ωt ) sin( φ )) = cos( φ ) e - αt cos( ωt ) + sin( φ ) e - αt sin( ωt ) G ( s ) = cos( φ ) s + α ( s + α ) 2 + ω 2 + sin( φ ) ω ( s + α ) 2 + ω 2 = cos φ ( s + α ) + sin( φ ) ω ( s + α ) 2 + ω 2 = cos φ s + α + tan( φ ) ω ( s + α ) 2 + ω 2 (b) f ( t ) = 2 e - 2 t cos(10 t ) - e - 2 t sin(10 t ) Solution: F ( s ) = 2( s + 2) ( s + 2) 2 + 10 2 - 10 ( s + 2) 2 + 10 2 = 2 s - 6 ( s + 2) 2 + 10 2 Problem 2. (20 Points) Find the time function, y ( t ), corresponding to each of the following Laplace transforms: Hint: Use Table 2.1 and Table 2.2 on the textbook. 1
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(a) Y ( s ) = 11 s ( s 2 + 6 s + 11) Solution: Using the partial fraction decomposition, we have Y ( s ) = 11 s 2 + 6 s + 11 1 s = 1 s - s + 6 s 2 + 6 s + 11 We can write it in Y ( s ) = 1 s - s + 3 + 3 ( s + 3) 2 + 2 = 1 s - s + 3 ( s + 3) 2 + 2 - 3 2 2 ( s + 3) 2 + 2 Taking the inverse Laplace transform, we have y ( t ) = u ( t ) - e - 3 t u ( t ) cos( 2 t ) - 3 2 e - 3 t u ( t ) sin( 2 t ) = u ( t ) " 1 - r 11 2 e - 3 t r 2 11 cos(
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