lect12 - SYSTEMS DEFINED BY DIFFERENTIAL EQUATIONS The...

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SYSTEMS DEFINED BY DIFFERENTIAL EQUATIONS The general constant coefficient linear differential equation will be written as ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( 2 ) 1 ( 1 ) ( 0 ) 2 ( 2 ) 1 ( 1 ) ( t x b t x b t x b t x b t y a t y a t y a t y m m m m n n n n + + + + = + + + + - - - - (12.1) in which the coefficients are assumed known, as are n and m . THE UNILATERAL LAPLACE TRANSFORM We can analyze sytems described by differential equations by employing the unilateral laplace transform, defined by - - = 0 ) ( ) ( dt e t f s F t s . (12.2) The only modification here, with respect to the bilateral laplace transform is that the integral ranges only from zero-minus to infinity. Therefore, the unilateral laplace transform contains information only about the signal from zero to infinity. Although the signal need not be zero for t < 0, the unilateral laplace transform contains no information about its values for t < 0. The lower limit is taken to be - 0 so that if the function contains an impulse at the origin, the effect of that impulse will be included in the laplace transform.
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The main theorem used in applying the unilateral laplace transform to system analysis is the derivative theorem. Theorem: (Derivative) ) 0 ( ) ( ) ( - - f s F s t f . Proof: Integration by parts yields - - - - - - + = 0 0 0 ) ( ) ( ) ( ) ( dt e t f s e t f dt e t f t f t s t s t s or, ) 0 ( ) ( ) ( ) 0 ( ) ( - - - = + - f s F s s F s f t f . We have used the fact that, for Re{s} sufficiently large, 0 ) ( - t s e t f as t . Q.E.D. This theorem can be extended to higher order derivatives. For example, ) 0 ( ) 0 ( ) ( ) ( 2 - - - - f f s s F s t f Proof: By applying the theorem above to ) ( t f we have
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[ ] ) 0 ( )} ( { ) ( ) ( - - = f t f L s t f t f where )} ( { t f L is the laplace transform of ) ( t f . Now substituting for )} ( { t f L by using again the theorem above yields ( 29 ) 0 ( ) 0 ( ) ( ) 0 ( ) 0 ( ) ( ) ( 2 - - - - - - = - - f f s s F s f f s F s s t f . Q.E.D. By extending this process, we can easily find the general formula below ) 0 ( ) 0 ( ) 0 ( ) ( ) ( ) 1 ( 2 1 ) ( - - - - - - - - - - n n n n n f f s f s s F s t f (12.3) Example 12.1 Consider a system defined by the differential equation below ) ( 4 ) ( 2 ) ( 3 ) ( 4 ) ( t x t x t y t y t y - = + + where the values ) 0 ( - y and ) 0 ( - y are known, and the input is ) ( ) ( t U t x = , with laplace transform s s X 1 ) ( = . Taking the unilateral laplace transform of both sides of this
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This note was uploaded on 10/18/2009 for the course EE ee332 taught by Professor Ee during the Spring '07 term at Istanbul Technical University.

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lect12 - SYSTEMS DEFINED BY DIFFERENTIAL EQUATIONS The...

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