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Unformatted text preview: 1 28150. Introduction to proces control 3. Laplace transforms Krist V. Gernaey 07 September 2009 28150 Learning objectives At the end of this lesson you should be able to: Solve linear differential equations using Laplace transforms Note: exercises on Laplace transforms will be given next week! 07 September 2009 28150 Outline Definition Solution of linear differential equations by Laplace transform techniques Partial fraction expansion Useful Laplace transform properties Exercise 3.5 07 September 2009 28150 Outline Definition Solution of linear differential equations by Laplace transform techniques Partial fraction expansion Useful Laplace transform properties Exercise 3.5 07 September 2009 28150 What is the Laplace transform? A standard notation in dynamics and control A mathematical tool that allows to convert rather complicated mathematical problems (solving ODEs to find the dynamic trajectory of the output variables for a change in the input variables) to rather simple algebraic operations for a system of linear ODEs Advantageous for block diagram analysis of systems 07 September 2009 28150 Definition The Laplace transform of a function f(t) is defined as: Where: F(s) = symbol for the Laplace transform of f(t) s = a complex independent variable f(t) = a function of time L = an operator, defined by the integral Consequently: the inverse Laplace transform (L-1 ) will operate on the function F(s), and convert it to f(t) f(t) is in that case not defined for t < 0!!! ( ) ( ) [ ] ( ) - = = dt e t f t f L s F st 2 07 September 2009 28150 Superposition principle The Laplace transform of a sum of functions is the sum of the individual Laplace transforms Multiplicative constants can be factored out of the Laplace operator Or: In this course: We need to be able to use the Laplace transform to obtain solutions for linear ODEs (see Table 3.1 in the textbook) Deeper understanding of the mathematical aspects of the Laplace transform is out of the scope of this course!!! ( ) ( ) ( ) ( ) ( ) s Y b s X a t y b t x a L + = + 07 September 2009 28150 Examples A constant A unit step function- Def.: ( ) [ ] s a s a e s a dt e a a L t s t s = -- =- = = - - Time 1 ( ) < = 1 t t t S 1 ( ) ( ) s t S L 1 = 07 September 2009 28150 Examples An exponential function Note: The Laplace transform for b < 0: unbounded for s < -b Real part of s must be restricted to be larger than b for the integral to be finite!!! (will be the case for the problems we consider) ( ) ( ) ( ) [ ] b s e s b dt e dt e e e L t s b t s b t s t b t b + =- + = = = +- +- - - - 1 1 07 September 2009 28150 Examples The first derivative of a function It will often be assumed that f(0) = 0 (the system is in steady state at t = 0):...
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This note was uploaded on 11/20/2009 for the course CHME DTU-abroad taught by Professor Rafiqulgani during the Fall '09 term at Rensselaer Polytechnic Institute.
- Fall '09