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**Chapter 4 Modelling and Analysis for Process Control**

Laplace Transform Definition

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Input signals

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

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**＊ Properties of the Laplace transform**

Linearity Differentiation theorem

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Zero initial values Proof:

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Integration theorem

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Translation theorem Proof:

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Final value theorem Initial value theorem

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**Complex translation theorem**

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**

Zero initials (2) Transfer function

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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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**＊ Repeated roots (iii) Inversion**

If r1=r2, 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 A3=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)=Y1(s)e-st0 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 form)**

@ 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 form @ 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)**

Physical systems, Transforms of the derivation of input and output variables Steady state responses

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

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

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※ Block diagrams

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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 3-4.3 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=Y0/X0 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**

P2. (p.69)

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**Ex.4.8 Consider a first-order system**

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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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**G(s)=K(1+τs) 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-t0s

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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. MR1 as ω 0 S2. As ω

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＃ ＊ Types of Bode plots Gain element First-order lag Dead time Second-order lag First-order lead Integrator

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**＊ Process control for a chemical reactor**

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

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