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Unformatted text preview: The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic ControlFall 2009 Solutions to Problem Set Seven C. Caramanis November 4, 2009. 1. We denote the unit step function by 1 ( t ). f ( t ) = f 1 ( t ) * f 2 ( t ) where f 1 ( t ) = sin( t ) 1 ( t ) and f 2 ( t ) = cos( t ) 1 ( t ). This gives L{ f ( t ) } = L{ f 1 ( t ) * f 2 ( t ) } = L{ f 1 ( t ) }L{ f 2 ( t ) } = L{ sin( t ) 1 ( t ) }L{ cos( t ) 1 ( t ) } = 1 s 2 + 1 s s 2 + 1 = s ( s 2 + 1) 2 R t f ( ) d = f ( t ) * 1 ( t ) and L{ 1 ( t ) } = 1 s gives L ( h ) = 1 s 1 s 1 s L{ f ( t ) } = 1 s 2 ( s 2 + 1) 2 L{ f ( t ) t } = R s F ( ) d and L{ sin( t ) } = 1 s 2 +1 gives L{ sin( t ) t } = Z s d 2 + 1 = [tan 1 ( )] s = tan 1 ( ) tan 1 ( s ) = 2 tan 1 ( s ) 2. H ( s ) is expanded as H ( s ) = A s + 1 + Bs + C s 2 + 4 = ( A + B ) s 2 + ( B + C ) s + (4 A + C ) ( s + 1)( s 2 + 4) 2( s + 2) ( s + 1)( s 2 + 4) which gives...
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