hw1_soln - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 486: Control Systems Homework 1 Solutions Spring 2011 Problem 1 (i) Compute by hand the Laplace transforms F i = L ( f i ) with, f 1 ( t ) = sin( t ) (recall Eulers formula) f 2 ( t ) = e - t f 3 ( t ) = sin( t ) + e - t (ii) Use the Final Value Theorem to compute lim t →∞ f i ( t ) in each case based on the transform F i ( s ). In which cases is the theorem valid? Solutions: (i) L{ sin( t ) } = 0 (sin( t )) e - st dt = 0 ( e jt - e - jt 2 j ) e - st dt = 1 2 j 0 ± e ( j - s ) t - e - ( j + s ) t ² dt = 1 s 2 + 1 L{ e - t } = 0 ³ e - t ´ e - st dt = 0 e - ( s +1) t dt = 1 s + 1 By the Linearity of the Laplace Transform, L{ sin( t ) + e - t } = L{ sin( t ) } + L{ e - t } = 1 s 2 + 1 + 1 s + 1 (ii) lim t →∞ f 1 ( t ) = lim s 0 sF 1 ( s ) = lim s 0 s s 2 + 1 = 0 . (Not valid) lim t →∞ f 2 ( t ) = lim s 0 sF 2 ( s ) = lim s 0 s s + 1 = 0 . (Valid) lim t →∞ f
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hw1_soln - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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