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Laplace Transform 1.Definition Chapter 4 Modelling and Analysis for Process Control

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2.Input signals

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(c) A unit impulse function (Dirac delta function)

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＊ Properties of the Laplace transform 1.Linearity 2.Differentiation theorem

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Zero initial values Proof:

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3.Integration theorem

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4.Translation theorem Proof:

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5.Final value theorem 6.Initial value theorem

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7.Complex translation theorem 8.Complex differentiation theorem

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Example 4.1 Solution:

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Example 4.2 (S1)

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(S2)

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＊ Laplace transform procedure for differential equations Steps:

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Exercises: a second-order differential equation (1) Laplace transform

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Algebraic rearrangement (2) Transfer function Zero initials

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(3) Laplace Inversion Where

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Inversion method: Partial fractions expansion (pp.931) (i) Fraction of denominator and

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(ii) Partial fractions where

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(iii) Inversion ＊ Repeated roots If r 1 =r 2, the expansion is carried out as

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where Inversion

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＊ Repeated roots for m times If the expansion is carried out as

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and

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and A 3 =2 as (a) case.

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The step response: Example 4.3

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(S1)

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(S3) Find coefficients s=0 Inversion

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Example 4.4 (S1) Laplace transformation

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(S2) Find coefficients s=0 s=1-j s=-1+j

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(S3) Inversion and using the identity

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Time delays: Consider Y(s)=Y 1 (s)e -st 0 and

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Example:

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Input function f(t)

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＊ Input-Output model and Transfer Function Ex.4.5 Adiabatic thermal process example

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S1. Energy balance

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S2. Under steady-state initial conditions and define deviation variable

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S3. Standard form where

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S4. Transfer function (Laplace Step change ( )

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＊ Non-adiabatic thermal process example S1. model S2. Under deviation variables, the standard form

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where

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S3. Laplace Transfer functions

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Ex. 4.6 Thermal process with transportation delay

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@ Dead time

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@ Transfer functions

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※ Transfer function (G(s)) Note: The transfer function defines the steady-state and dynamic characteristic, or total response, of a system described by a linear differential equation.

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＊ Important properties of G(s) 1.Physical systems, 2.Transforms of the derivation of input and output variables 3.Steady state responses

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＊ Steady-state gain ( ) Ex. Consider two isothermal CSTRs in series

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Ans.: (1)Steady-state gain: (2) Final value of the reactant concentration in the second reactor:

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※ Block diagr ams

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@ Block diagram for

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Example 4.7 Block diagram for

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＊ Rules for block diagram

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Example 4.8 Determine the transfer functions

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Solution: ◎◎◎◎

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Example Determine the transfer functions =？=？

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@ Reduced block

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Example 4.9 =？=？

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◎ Answer

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◎ Design steps for transfer function

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@ Review of complex number c=a+ib

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Polar notations

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※ Frequency response

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◎ Experimental determination of frequency response S1. Process (valve, model, sensor/transmitter)

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S2. Input signal S3. Output response where

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P1. Amplitude of output signal P2. Output signal ‘lags’ the input signal by θ. P3. Amplitude ratio (AR): AR=Y 0 /X 0 P4. Magnitude ratio (MR): MR=AR/K P5. Phase angle (θ): if θ is negative, it is a lag angle.

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Ex.4.7 A first-order transfer function G(s)=K/(τs+1) ＊ Consider a form of If the input is set as Then the output

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＊ Through inverse Laplace transformation, the output response is reduced as P1. P2. (p.69)

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Ex.4.8 Consider a first-order system S1.

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S2. Amplitude ratio and phase angle Ex.4.9 Consider a second-order system

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S1. s=iω to decide amplitude ratio ＃

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S2. Phase angle ＃ Ex.4.10 Consider a first-order lead transfer function G(s)=K(1+τs)

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Ex.4 Consider a pure dead time transfer function G(s) =e -t 0 s

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Ex.5 Consider an integrator G(s)=1/s G(i )=-(1/ )i

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＊ Expression of AR and θ for general OLTF

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※ Bode plot A common graphical representation of AR (MR) and θ functions. Bode plot consists: (1) log AR or (log MR) vs. log ω (2) θ vs. log ω * (3) 20 log AR (db) vs. log ω

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Ex. 5 Consider a first-order lag by Ex. 1 To show Bode plot. S1. MR 1 as ω 0 S2. As ω

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＃ ＊ Types of Bode plots (a)Gain element (b)First-order lag (c)Dead time (d)Second-order lag (e)First-order lead (f)Integrator

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＊ Process control for a chemical reactor

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Homework 2# (1)Q4.6 (2)Q4.10 (3)Q4.16 (4)Q4.18 ( ※ Difficulty)

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