Assignment 3 Solutions 2009

1 2 12 y s b 2a 2 1 21 u s s a 1s a22 a 2a21 2 1 1

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Unformatted text preview: = u - u s . 1 1 2 2 2 1 1 1 2 2 2 This results in the fo llowing linear equat ions: dx1 = a 1 x1 + a 2 x + b 1 1 1 1 2 1 u dt dx 2 = a x + a x + b u 21 1 22 2 22 2 dt If we take the Laplace transform of both equations, we have: sx1 ( s = a 1 x1 ( s + a 2 x ( s + b 1 1 ( s ) 1 ) 1 2 ) 1 u ) sx ( s = a 1 x1 ( s + a 2 x ( s + b 2 u ( s 2 ) 2 ) 2 2 ) 2 2 ) Note that y1 ( s = x ( ); y ( ) = x ( s . Rearranging, and making the subst itutions, ) 1s 2s 2) a b y1 ( s = 12 y ( s + 11 u ( s ) 2 ) 1 ) s - a 1 s - a 1 1 1 a 1 b 2 2 2 y 2 ( s = ) y1 ( s + ) u ( s 2 ) s - a 2 s - a 2 2 2 and making the substitutions, é a ù a b 2 b 2 y1 ( s = 12 ê 21 y1 ( s + ) ) u ( s ú + 11 u ( s 2 ) 1 ) s - a 1 ë s - a 2 s - a 2 1 2 2 û s - a 1 1 y 2 ( s = ) ù a 1 é a 2 b 1 b 2 2 1 1 2 ) 1 ) 2 ) ê s - a y 2 ( s + s - a u ( s ú + s - a u ( s s - a 2 ë 2 11 11 û 22 Collect ing the terms, we end up with the fo llowing transfer funct ions, y1 ( s ) b 1 ( s - a 2 ) 1 2 = u ( s ( s - a 1 )( s - a 2 ) - a 2 a 1 1 ) 1 2 12 y ( s ) b 2 a 2 1 21 = u ( s ( s - a 1 )( s - a 22 ) - a 2 a 21 ) 2 1 1 y 2 ( s ) b 1 21 1 a = u ( s ( s - a 1 )( s - a 2 ) - a 2 a 1 1 ) 1 2 12 y 2 ( s ) b 2 ( s - a 1 ) 2 1 = u ( s ( s - a 1 )( s - a 2 ) - a 2 a 1 ) 2 1 2 12 Here are the two files needed to solve the problem in MATLA...
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