ModelingLecture - Modeling ME 279 Automatic Control of Dynamic Systems Dr Robert G Landers Laplace Transform 2 Laplace Transform – mathematical

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Modeling ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers Laplace Transform 2 Laplace Transform – mathematical transformation that transforms a function from the time domain (t) to the Laplace domain (s). This transformation converts a differential equation into an algebraic equation. The Laplace transform is defined as Modeling Dr. Robert G. Landers – is an infinitesimal amount of time before time zero. Note the Laplace transform is only defined for integrals that converge. ( 29 ( 29 st F s f t e dt- ∞- = ∫ Example 1 3 Determine the Laplace transforms of f(t) = A and f(t) = e –at . Modeling Dr. Robert G. Landers Example 1 4 ( 29 ( 29 ( 29 ( 29 1 s a t s a t at at st e e F s e e dt e dt s a s a--- ∞ ∞ ∞- +- +--- = = = = = - + + ∫ ∫ L [ ] ( 29 st st Ae A A F s Ae dt s s-- ∞ ∞-- = = = =- ∫ L In Matlab syms a t % declares a and t to be symbols laplace(t^0) % returns 1/s laplace(exp(-a*t)) % returns 1/(s+a) Modeling Dr. Robert G. Landers Laplace Transform Properties 5 ( 29 ( 29 ( 29 ( 29 st st af t af t e dt a f t e dt aF s-- ∞ ∞-- = = = ∫ ∫ L Multiplication by a constant ( 29 ( 29 ( 29 ( 29 f t g t F s G s + = + L Superposition Modeling Dr. Robert G. Landers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 st st st f t g t f t g t e dt f t e dt g t e dt F s G s--- ∞- ∞ ∞-- + = + = + = + ∫ ∫ ∫ L Proof Laplace Transform Properties 6 ( 29 ( 29 ( 29 ( 29 f t g t F s G s ≠ L Multiplication of functions ( 29 ( 29 ( 29 ( 29 ( 29 s a t at at st f t e f t e e dt f t e dt F s a-- ∞ ∞- +--- = = = + ∫ ∫ L Multiplication by an exponential in general Modeling Dr. Robert G. Landers Proof Example ( 29 ( 29 at f t e F s a- = + L ( 29 ( 29 2 2 sin at t e s a ϖ ϖ ϖ- = + + L Laplace Transform Properties 7 ( 29 ( 29 d F s tf t ds = - L Multiplication by time Modeling Dr. Robert G. Landers ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 sin sin d d s t t t ds ds s s ϖ ϖ ϖ ϖ ϖ ϖ = - = - = ÷ + + L L ( 29 ( 29 ( 29 ( 29 st st d F s d f t e dt tf t e dt tf t ds ds-- ∞ ∞-- = = - = - ∫ ∫ L Proof Example Laplace Transform Properties 8 ( 29 ( 29 ( 29 st df f t e dt sF s f dt- ∞-- = =- ÷ ∫ & L Differentiation with respect to time ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 st st t st st t u e du se dt v f t a b f t e f t f t se dt f sF s----- ∞ =∞--- = = → = - = = = ∞ =-- =- + ∫ & L Proof, use integration by parts ( 29 ( 29 ( 29 ( 29 1 1 n n n n n d f s F s s f f dt---- =--- L L In general Modeling Dr. Robert G. Landers Laplace Transform Properties 9 ( 29 ( 29 ( 29 1 t t st f d f d e dt F s s λ λ λ λ--- ∞- = =...
View Full Document

This note was uploaded on 02/01/2012 for the course MECH ENG 279 taught by Professor Landers during the Spring '11 term at Missouri S&T.

Page1 / 80

ModelingLecture - Modeling ME 279 Automatic Control of Dynamic Systems Dr Robert G Landers Laplace Transform 2 Laplace Transform – mathematical

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online