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ModelingLecture

# ModelingLecture - ME 279 Automatic Control of Dynamic...

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Modeling ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers

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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 - - =
Example 1 3 Determine the Laplace transforms of f(t) = A and f(t) = e –at . Modeling Dr. Robert G. Landers

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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
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

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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
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

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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
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

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Real Distinct Roots 10 ( 29 2 5 5 4 s F s s
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