04_Transfer_functions_E2009 - 1 28150 Introduction to...

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Unformatted text preview: 1 28150. Introduction to proces control 4. Transfer functions Krist V. Gernaey 21 September 2009 28150 Learning objectives • At the end of this lesson you should be able to: – Derive a transfer function for a given system – Linearize a non-linear model – Write a linear system of ODEs in state-space form 21 September 2009 28150 Outline • Definition • Properties of transfer functions • Example: variable volume tanks in series • Linearization of non-linear models • State-space models • Group work 21 September 2009 28150 Outline • Definition • Properties of transfer functions • Example: variable volume tanks in series • Linearization of non-linear models • State-space models • Group work 21 September 2009 28150 Laplace transform for solving linear ODEs 2 4 5 = ⋅ + ⋅ y dt dy ( ) 1 = y ( ) = y ( ) ( ) 4 5 2 + ⋅ ⋅ = s s s Y ( ) ( ) 4 5 2 5 + ⋅ ⋅ + ⋅ = s s s s Y ( ) t e t y ⋅- ⋅ + = 8 . 5 . 5 . ( ) t e t y ⋅- ⋅- = 8 . 5 . 5 . 21 September 2009 28150 Laplace transform vs. transfer function • Laplace transform technique: derivation procedure needs to be repeated for any change in the initial conditions or in the type of forcing function (input)! A clear disadvantage! • Transfer function (TF): – Algebraic expression for the dynamic relation between one input and one output – Independent of initial conditions or type of forcing function applied!!! – Contains information on the fundamental dynamic properties of a system, when written in standard form ) s ( Y ) t ( y system ) s ( U ) t ( u → → 2 21 September 2009 28150 Terminology ) s ( Y ) t ( y system ) s ( U ) t ( u → → u input forcing function “cause” y output response “effect” 21 September 2009 28150 Definition • Let G(s) denote the transfer function between an input, u, and an output, y. Then, by definition: where: ) ( ) ( ) ( s U s Y s G = [ ] [ ] ) ( L ) ( ) ( L ) ( t u s U t y s Y = = 21 September 2009 28150 Derivation of transfer functions from linear ODEs: Examples • Example 1: Constant volume tank, 1 st order reaction • Example 2: Constant volume, constant density blending tank 21 September 2009 28150 Constant volume tank, 1 st order reaction • Assume: – Tank with overflow (= constant volume tank) – Dissolved substance A disappears from the tank according to a first order reaction – Feed flow rate is constant • The following mass balance applies for A: ( ) A A Ai A c k V c c F dt dc V ⋅ ⋅-- ⋅ = ⋅ 21 September 2009 28150 Constant volume tank, 1 st order reaction • At steady-state, we have: • Subtracting the steady-state from the differential equation, we obtain deviation variables: • Rearranging to standard form gives: ( ) A A Ai c k V c c F ⋅ ⋅-- ⋅ = ( ) ( ) ( ) ( ) A A A A Ai Ai A A c c k V c c F c c F dt c c d V- ⋅ ⋅-- ⋅-- ⋅ =- ⋅ ( ) A Ai A c k V F c F dt c d V ′ ⋅ ⋅ +- ′ ⋅ = ′ ⋅ Ai A A c k V F F c dt c d k V F V ′ ⋅ ⋅ + = ′ + ′ ⋅ ⋅ + 21 September 2009...
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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.

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04_Transfer_functions_E2009 - 1 28150 Introduction to...

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