MECH466 L3 ODESolution

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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 3

Lecture 3

Solution to

Solution to ODEsODEsvia Laplace transformvia Laplace transform

About laboratory

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

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.

Lab groups can be modified during this week. Lab groups can be modified during this week.

Please check it regularly.

Please check it regularly.

Jan. 19 (Mon):Jan. 19 (Mon):Lab starts. See the files Lab starts. See the files MECH466_LabInfo_0809.pdf MECH466_LabInfo_0809.pdf MECH466_LabSchedule_0809.pdf MECH466_LabSchedule_0809.pdf

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems

•

•electricalelectrical

•

•mechanicalmechanical

•

•electromechanicalelectromechanical

Linearization

Linearization Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient

•

•Steady stateSteady state

Frequency response

•

•Bode plotBode plot

Stability

•

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

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

PID & Lead

PID & Lead--laglag

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Laplace transform (review)

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

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

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

f(t

f(t))

t t 0

0 F(sF(s))

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Laplace transform table (review)

Inverse Laplace Inverse Laplace Transform Transform

(

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

Advantages of s

-

domain (review)

We can transform an ordinary differential We can transform an ordinary differential equation into an algebraic equation which is equation into an algebraic equation which is easy to solve.

easy to solve. (This lecture)(This lecture)

It is easy to analyze and design interconnected It is easy to analyze and design interconnected (series, feedback etc.) systems. (Throughout the (series, feedback etc.) systems. (Throughout the course)

course)

Frequency domain information of signals can be Frequency domain information of signals can be dealt with. (Lectures for frequency responses: dealt with. (Lectures for frequency responses: after midterm)

after midterm)

An advantage of Laplace transform

We can transform an ordinary differential We can transform an ordinary differential equation (ODE) into an algebraic equation (AE). equation (ODE) into an algebraic equation (AE).

ODE

ODE AEAE

Partial fraction

expansion

Solution to ODE

t

t--domaindomain ss--domaindomain

1 1

2 2

3 3

Example 1 (distinct roots)

ODE with initial conditions (ICs) ODE with initial conditions (ICs)

1.

1. Laplace transformLaplace transform

distinct roots

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Properties of Laplace transform

Differentiation (review)

t

t--domaindomain

s

s--domaindomain

2.

2. Partial fraction expansionPartial fraction expansion

Multiply both sides by s & let s go to zero: Multiply both sides by s & let s go to zero:

Similarly, Similarly,

unknowns

unknowns

Example 1 (cont

’

d)

3.

3. Inverse Laplace transformInverse Laplace transform

Example 1 (cont

’

d)

If we are interested in only the final value of

If we are interested in only the final value of y(ty(t), apply ), apply Final Value Theorem:

Final Value Theorem:

Example 2 (repeated roots)

ODE with initial conditions (ICs) ODE with initial conditions (ICs)

1.

Repeated roots

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2.

To obtain A: To obtain A:

To obtain B: To obtain B:

unknowns

unknowns

Example 2 (cont

’

d)

2.

To obtain C: To obtain C:

unknowns

unknowns

Example 2 (cont

’

d)

Take derivative

Take derivative

Let s go to

Let s go to --2.2.

3.

Example 2 (cont

’

d)

If we are interested in only the final value of

If we are interested in only the final value of y(ty(t), apply ), apply Final Value Theorem:

Final Value Theorem:

Properties of Laplace transform

8.

Frequency shift theorem (review)

Ex. Ex. Proof. Proof.

t

t--domaindomain

s

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Example 3 (complex roots)

ODE with zero initial conditions (ICs) ODE with zero initial conditions (ICs)

1.

Complex roots

2.

To obtain A, B & C: To obtain A, B & C:

unknowns

unknowns

Example 3 (cont

’

d)

3.

Example 3 (cont

’

d)

Example: Newton

’

s law

We want to know the trajectory of

We want to know the trajectory of x(tx(t). By Laplace transform,). By Laplace transform,

M

(Total response)

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Ex: Mechanical accelerometer

Taken from Taken from DorfDorf& Bishop book& Bishop book

Ex: Accelerometer (cont

’

d)

We would like to know how y(tWe would like to know how y(t) moves) moveswhen unit when unit step

step f(tf(t) is applied with zero ICs.) is applied with zero ICs.

By Newton’By Newton’s laws law

Ex: Mechanical accelerometer (cont

Ex: Mechanical accelerometer (cont’

’d)

d)

Suppose that b/M=3, k/M=2 and Ms=1.Suppose that b/M=3, k/M=2 and Ms=1.

Partial fraction expansionPartial fraction expansion

Inverse Laplace transformInverse Laplace transform

-0.4 -0.3 -0.2 -0.1 0

A

m

pl

it

ude

Summary & Exercises

Solution procedure to Solution procedure to ODEsODEs 1.

1. Laplace transformLaplace transform 2.

2. Partial fraction expansionPartial fraction expansion 3.

Next, modeling of physical systems using Next, modeling of physical systems using Laplace transform

Laplace transform

ExercisesExercises

Read Sections 2Read Sections 2--2 up to page 42.2 up to page 42.