{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ModelingLecture - ME 279 Automatic Control of Dynamic...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Modeling ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers
Background image of page 1

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

View Full Document Right Arrow Icon
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 0 is an infinitesimal amount of time before time zero. Note the Laplace transform is only defined for integrals that converge. ( 29 ( 29 0 st F s f t e dt - - =
Background image of page 2
Example 1 3 Determine the Laplace transforms of f(t) = A and f(t) = e –at . Modeling Dr. Robert G. Landers
Background image of page 3

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

View Full Document Right Arrow Icon
Example 1 4 ( 29 ( 29 ( 29 ( 29 0 0 0 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 0 0 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
Background image of page 4
Laplace Transform Properties 5 ( 29 ( 29 ( 29 ( 29 0 0 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 0 0 0 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
Background image of page 5

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

View Full Document Right Arrow Icon
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 0 0 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
Background image of page 6
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 0 0 st st d F s d f t e dt tf t e dt tf t ds ds - - - - = = - = - L Proof Example
Background image of page 7

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

View Full Document Right Arrow Icon
Laplace Transform Properties 8 ( 29 ( 29 ( 29 0 0 st df f t e dt sF s f dt - - - = = - ÷ & L Differentiation with respect to time ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 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 0 0 n n n n n d f s F s s f f dt - - - - = - - - L L In general Modeling Dr. Robert G. Landers
Background image of page 8
Laplace Transform Properties 9 ( 29 ( 29 ( 29 0 0 0 1 t t st f d f d e dt F s s λ λ λ λ - - - - = = ∫ ∫ L Integration from 0 to t ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 0 0 0 0 0 0 0 1 1 0 0 t st t t t st st t st e u f d du f t dt v a b s e e f d f d f t dt s s e f t dt F s s s λ λ λ λ λ λ - - - - - - - - =∞ - - = - = = = = = ∞ - = - - - = - + = L Proof, use integration by parts Modeling Dr. Robert G. Landers
Background image of page 9

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

View Full Document Right Arrow Icon
Real Distinct Roots 10 ( 29 2 5 5 4 s F s s
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}