Practice Final Problem 8 solution

Practice Final Problem 8 solution - (Spring 1999)Find the...

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Name_________________________ Page 1 of 2 12/8/2005 (Spring 1999) Find the total response of the system 5 ( ) x x u t + = d , ( ) 0 2 x = - , for the input shown to the right. Note that ( ) 1 5 G s s = + and that the homogeneous solution is ( ) 5 t x t Ae - = . Next write out an analytical form of the input function, ( ) ( ) ( ) 10 5 1 5 3 s r r u t u u t u t = - - + - . Alternatively note that the following inputs occur, ( ) { } 10; 0 1,10 5 1 ;1 3,0; 3 u t t t t = ≤ ≤ - - ≤ ≤ . As the Laplace Transform of a unit step is 1/ s , for the first time region ( ) ( ) ( ) 1 1 10 10 5 5 a b x s G s u s s s s s = = = + + + i i . Solving for a and b by multiplying both sides of the equation by the denominator of the left hand side gives ( ) 1 5 a s bs = + + . Equating coefficients of like powers of s gives 1 5 a = and 0 a b = + . Thus, 1/5 a = and 1/5 b a = - = - . Therefore ( ) 10 1 1 5 5 x s s s = - + and ( ) ( ) 5 2 1 t p x t e - = - . The total
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Practice Final Problem 8 solution - (Spring 1999)Find the...

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