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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 Fall '08
 Hjortso,M
 pH, Alternat ive Linear, ive Linear model, transfer funct ions

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