# lect12 - SYSTEMS DEFINED BY DIFFERENTIAL EQUATIONS The...

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
[ ] ) 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 13

lect12 - SYSTEMS DEFINED BY DIFFERENTIAL EQUATIONS The...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online