MECH466 L2 LaplaceTransform

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

University of British Columbia Lecture 2

Lecture 2

Laplace transform

About laboratory

Each Lab group consists of about 4 students.Each Lab group consists of about 4 students.

By Jan. 9 (Fri):By Jan. 9 (Fri):Register both the course and the lab.Register both the course and the lab.

By Jan. 12 (Mon)By Jan. 12 (Mon): The instructor will decide the : The instructor will decide the groups, and post tentative groups on Vista.

groups, and post tentative groups on Vista.

By Jan. 13 (Tue):By Jan. 13 (Tue):Contact the instructor in case of Contact the instructor in case of inconvenience.

inconvenience.

By Jan. 16 (Fri):By Jan. 16 (Fri):Lab groups will be finalized and Lab groups will be finalized and

posted on Vista.

Jan. 19 (Mon):Jan. 19 (Mon):Lab starts. See the schedule Lab starts. See the schedule MECH466_LabSchedule_0809.pdf MECH466_LabSchedule_0809.pdf

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems Models for systems •

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response Time response •

•TransientTransient •

•Steady stateSteady state

Frequency response Frequency response •

•Bode plotBode plot

Stability Stability •

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead PID & Lead--laglag

Design examples Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Laplace transform

One of most important math tools in the course!One of most important math tools in the course!

Definition: For a function Definition: For a function f(tf(t) () (f(tf(t)=0 for t<0),)=0 for t<0),

We denote Laplace transform of We denote Laplace transform of f(tf(t) by ) by F(sF(s).).

f(t

f(t))

t t 0

0 F(sF(s))

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Advantages of s

-

domain

We can transform an ordinary differential We can transform an ordinary differential

equation into an algebraic equation which is

easy to solve.

easy to solve. (Next lecture)(Next lecture)

It is easy to analyze and design interconnected It is easy to analyze and design interconnected

(series, feedback etc.) systems.

(series, feedback etc.) systems. (Throughout the (Throughout the course)

course)

Frequency domain information of signals can be Frequency domain information of signals can be

dealt with.

dealt with. (Lectures for frequency responses: (Lectures for frequency responses:

after midterm)

Examples of Laplace transform

Unit step functionUnit step function

Unit ramp functionUnit ramp function

f(t f(t))

t t 0

0 1 1

f(t f(t))

t t 0

0

(Memorize this!)

(Integration by parts)

Integration by parts

FormulaFormula

Why?

Ex. of Laplace transform (cont

’

d)

Unit impulse functionUnit impulse function

Exponential functionExponential function

f(t f(t))

t t 0

0

f(t f(t))

t t 0

0 1 1

(Memorize this!)

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Examples of Laplace transform (cont

’

d)

Sine functionSine function

Cosine functionCosine function

(Memorize these!)

Remark:

Remark:Instead of computing Laplace Instead of computing Laplace

transform for each function, and/or

memorizing complicated Laplace transform,

use the

use the Laplace transform table Laplace transform table !!

Laplace transform table

(Table 2.1 of the textbook)

Inverse Laplace Inverse Laplace Transform Transform

(

(u(tu(t) is often omitted.)) is often omitted.)

Properties of Laplace transform

1.

Linearity

Ex.

Proof.

2.Time delay

Ex.

Proof.

f(t

f(t))

0

0 TT

f(t

f(t--T)T)

t

t--domaindomain

s

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Properties of Laplace transform

3.

Differentiation

Ex.

Proof.

t

t--domaindomain

s

s--domaindomain

4.

Integration

Proof.

t

t--domaindomain

s

s--domaindomain

5.

Final value theorem

Ex.

if all the poles of

if all the poles of sF(ssF(s) are in ) are in the left half plane

the left half plane(LHP)(LHP)

Poles of

Poles of sF(ssF(s) are in LHP) are in LHP, so final value , so final value thmthmapplies.applies.

Ex.

Some poles of

Some poles of sF(ssF(s) are not in LHP) are not in LHP, so final value , so final value

thmdoes NOT apply.

Properties of Laplace transform

6.

Initial value theorem

Ex.

Remark: In this theorem, it does not matter if

pole location is in LHS or not.

if the limits exist.

Ex.

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Properties of Laplace transform

7.

Convolution

IMPORTANT REMARK

Convolution

8.

Frequency shift theorem

Ex.

Proof.

t

t--domaindomain

s

s--domaindomain

Exercise 1

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Exercise 3

Summary & Exercises

Laplace transform (Important math tool!)Laplace transform (Important math tool!)

DefinitionDefinition

Laplace transform tableLaplace transform table

Detailed Detailed LaplaceLaplacetransform table is posted on Vista.transform table is posted on Vista.

Properties of Laplace transformProperties of Laplace transform

Next Next

Solution to Solution to ODEsODEsvia Laplace transformvia Laplace transform

ExercisesExercises

Read Sections 2Read Sections 2--1,21,2--2 up to page 34.2 up to page 34.

By using the definition of the By using the definition of the LaplaceLaplacetransform, solve transform, solve

Problems 1 & 2 in page 94.