Chapter 4 Modelling and Analysis for Process Control Laplace Transform Definition

Input signals

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

＊ Properties of the Laplace transform Linearity Differentiation theorem

Zero initial values Proof:

Integration theorem

Translation theorem Proof:

Final value theorem Initial value theorem

Complex translation theorem Complex differentiation theorem

Example 4.1 Solution:

Example 4.2 (S1)

(S2)

＊ Laplace transform procedure for differential equations Steps:

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

Algebraic rearrangement Zero initials (2) Transfer function

(3) Laplace Inversion Where

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

(ii) Partial fractions where

＊ Repeated roots (iii) Inversion If r1=r2, the expansion is carried out as

where Inversion

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

and

and A3=2 as (a) case.

The step response: Example 4.3

(S1)

(S3) Find coefficients s=0 Inversion

Example 4.4 (S1) Laplace transformation

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

(S3) Inversion and using the identity

Time delays: Consider Y(s)=Y1(s)e-st0 and

Example:

Input function f(t)

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

S1. Energy balance

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

S3. Standard form where

S4. Transfer function (Laplace form) @ Step change ( )

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

where

S3. Laplace form @ Transfer functions

Ex. 4.6 Thermal process with transportation delay

@ Dead time

@ Transfer functions

※ 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.

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

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

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

※ Block diagrams

@ Block diagram for

Example 4.7 Block diagram for

＊ Rules for block diagram

Example 4.8 Determine the transfer functions

Solution: ◎

Example 3-4.3 Determine the transfer functions =？

@ Reduced block

Example 4.9 =？

◎ Answer

◎ Design steps for transfer function

@ Review of complex number c=a+ib

Polar notations

※ Frequency response

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

S2. Input signal S3. Output response where

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.

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

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

Ex.4.8 Consider a first-order system

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

S1. s=iω to decide amplitude ratio ＃

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

Ex.4 Consider a pure dead time transfer function G(s) =e-t0s

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

＊ Expression of AR and θ for general OLTF

※ 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

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

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

＊ Process control for a chemical reactor

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