Diff Equation 12 2012 FourierSeries

8/12/2019 Diff Equation 12 2012 FourierSeries

1/115

S.l.dr.ing.mat. Alina BogoiDifferential Equations

POLITEHNICA University

of

Bucharest

Faculty of Aerospace

Engineering

CHAPTER 7

Laplace Transform(Cont)


2/115

Diff_Eq_12_2012/201

3

Laplace Transform Theorems

Theorem Description

Definition of Laplace Transform

Linear Property

Derivatives

Integrals

First Shifting Property

Second Shifting Property

==0

)()()}({ sFdttfetf stL

)()()}()({ sbGsaFtbgtaf +=+L

)0()0()()}({ )1(1)( = nnnn ffssFstf LL

)(1

})({0

sFs

df =

L

)()}({ asFtfeat =L

)()}()({ sFeatuatf as

=L


3/115

Diff_Eq_12_2012/201

3

3

Transfer functionBlack-box

systemInput

x(t)

Output

y(t)

1

1 01( ) ( ) ... ( ) ( ) ... ( )

n n m

n mn n m

d d dy t a y t a y t b x t b x t

dt dt dt

+ + + = + +

Assume all initial conditions are zero, we get the transfer

function(TF) of the system as

Consider an input-output system from x(t) to y(t) whose

dynamic model has the form:


4/115

Diff_Eq_12_2012/201

3

Connection between the t- and s-domains

The transfer function, G(s), is defined by:

)t(xb...)t(xdt

db)t(ya...)t(ydt

da)t(ydt

dmm

m

nn

n

n

n++=+++

01

1

1

)s(Xb...)s(Xsb)s(Ya...)s(Ysa)s(Ys mm

n

nn

++=+++

0

1

1

n

nn

m

m

a...sas

b...sb

)s(X

)s(Y)s(G

+++

++==

1

1

0

011 =+++

n

nn a...sas Characteristic Equation

Poles of G(s) Roots of the Characteristic Eq


5/115

Diff_Eq_12_2012/201

3

Transfer functionBlack-box

systemInput

x(t)

Output

y(t)

differential

equation

x(t) y(t)

transferfunction

X(s)Y(s)

G(s)


6/115

Diff_Eq_12_2012/201

3 6

Example 1. Find the transfer function of the RLC

1) Writing the differential equation of the system according to

physical law:

R L

C

u(t)u

c (t)

i(t)Input Output

2) Assuming all initial conditions are zero and applying Laplace

transform

3) Calculating the transfer function as( )G s

2( ) 1( )( ) 1

cU sG sU s LCs RCs

= =+ +

( ) ( ) ( ) ( )C C CLCu t RCu t u t u t+ + =&& &

2 ( ) ( ) ( ) ( )c c cLCs U s RCsU s U s U s+ + =

Solution:


7/115

Diff_Eq_12_2012/201

3

Properties of transfer function

The transfer function is defined only for alinear time-invariant system, not fornonlinear system.

All initial conditions of the system are setto zero.

The transfer function is independent of theinput of the system.

7


8/115

Diff_Eq_12_2012/201

3

88

Position of poles

and zeros

-a

j

i0

( )X ss a

=+

Transfer function

( ) atx t Ae=

Time-domain impulse

response

0


9/115

Diff_Eq_12_2012/201

3

99

1 1

2 2( )

( )

A s BX s

s a b

+=

+ +

Transfer function

( ) sin( )atx t Ae bt = +

Time-domain

impulse response

Position of poles and

zeros

-a

j

i

b

0

0


10/115

Diff_Eq_12_2012/201

3

1010

1 1

2 2( )

s BX s

s b

+=

+

Transfer function

( ) sin( )x t A bt = +

Time-domain

impulse response


zerosj

i

b

00


11/115

Diff_Eq_12_2012/201

3

1111

Position of poles

and zeros

a

j

i0

( )X ss a

=

Transfer function

( ) atx t Ae=

Time-domain impulse

response


12/115

Diff_Eq_12_2012/201

3

1212

1 1

2 2( )

( )

A s BX s

s a b

+=

+

Transfer function

( ) sin( )atx t Ae bt = +

Time-domain

dynamic response


zeros

-a

j

i

b

0

0


13/115

Diff_Eq_12_2012/201

3

1313

Summary of pole position & system dynamics


14/115


15/115

Diff_Eq_12_2012/201

3

Significance of poles

The nature and value of the poles determine whether

the system is stable or instable, and the type of

response

The nature and value of any pole is classified as a

function of its location in theplan defined by:

the Real part and Imaginary part of the pole

[or in other words the s-domain]

Im(s)Re(s)


16/115

Diff_Eq_12_2012/201

3

Stability Criterion vs Pole

Locations)(sIm

UnstableStable

)(sRe

The locations of poles ins-domain determinewhether the system is stable or unstable.


17/115

Diff_Eq_12_2012/201

3

Stability Criterion based on

the Pole Locations

A system is stable if a l l its poles havenegative real parts

(i.e., there are al l st r i ct l y inside the

left-side s-plane)

and unstable otherwise

Note:

This criterion is valid only if the system is

linear time-invariant

(i.e., has constant parameters).


18/115

Diff_Eq_12_2012/201

3

How to analyze and design acontrol system

18

Controller Actuator Plant

Sensor

-r

Expectedvalue

e

Error

Disturbance

Controlledvariable

n

yu

The first thing is to establish system model(mathematical model)


19/115

Diff_Eq_12_2012/201

3

Block diagramThe transfer function relationship

19

( ) ( ) ( )Y s G s U s=

can be graphically denoted through a block diagram.

G(s)U(s) Y(s)


20/115

Diff_Eq_12_2012/201

3

20

Equivalent transform of blockdiagram

1 Connection in series

G(s)U(s) Y(s)

( ) ?G s =

X(s)G1(s) G2(s)

U(s) Y(s)

1 2

( )

( ) ( ) ( )( )

Y s

G s G s G sU s= =


21/115

Diff_Eq_12_2012/201

3

21

2.Connection in parallel

G(s)U(s) Y(s)

1 2

( )( ) ( ) ( )

( )

Y sG s G s G s

U s= = +

U(s)

G2(s)

G1(s) Y1(s)

Y2(s)

+Y(S)

( ) ?G s =


22/115

Diff_Eq_12_2012/201

3

22

3. Negative feedback

M(s)

R(s) Y(s)

( ) ( ) ( )

( ) ( ) ( ) ( )

Y s U s G s

U s R s Y s H s

=

=

( )( )1 ( ) ( )

G sM sG s H s= +

[ ]( ) ( ) ( ) ( ) ( )Y s R s Y s H s G s=

Y(s)G(s)

H(s)

U(s)R(s)_

Transfer function of a negative feedback system:


23/115

Diff_Eq_12_2012/201

3

23

Laplace

transform

Fourier

transform

Three models

Differential equation Transfer function Frequency characteristic

Transfer

function

Differential

equation

Frequency

characteristic

Linear systemStudy

time-domain

responsestudy

frequency-domainresponse


24/115

S.l.dr.ing.mat. Alina

Bogoi

Differential

Equations

POLITEHNICA University

of

Bucharest

Faculty of Aerospace

Engineering

CHAPTER 8

The Fourier SeriesThe Fourier Transform


25/115

Diff_Eq_12_2012/201

3

Content Periodic Functions Fourier Series

Complex Form of the Fourier Series Impulse


26/115

Diff_Eq_12_2012/201

3

Odd and Even Functions A function f(x) is said to be even

if

A function f(x) is said to be odd

if

)()( xfxf =

)()( xfxf =


27/115

Diff_Eq_12_2012/201

3

Odd and Even Functions

)()( xfxf = )()( xfxf =

f(x)

x

f(x)

x

Even Function Odd Function


28/115

Diff_Eq_12_2012/201

3

Odd and Even Functions Property

The product of an even and anodd function is odd.

evenisxfifdxxfdxxfLL

L)(,)(2)(

0 =

oddisxfifdxxfL

L)(,0)( =


29/115

Diff_Eq_12_2012/201

3

Periodic Functions

A function ( )f t is periodic

if it is defined for all real tand if there is some positive number

(the lowest),T such that ( ) ( )t T f t + = .


30/115

Diff_Eq_12_2012/201

3

Example:

4cos

3cos)(

tttf +=

Find its period.


31/115

Diff_Eq_12_2012/201

3

Example:

4cos

3cos)(

tttf +=

Find its period.

)()( Ttftf += )(41cos)(

31cos

4cos

3cos TtTttt +++=+

Fact: )2cos(cos += m

= mT

23

= nT 24

= mT 6

= nT 8

= 24T smallest T


32/115

Diff_Eq_12_2012/201

3

Example:

tttf 21 coscos)( +=

Find its period.

)()( Ttftf += )(cos)(coscoscos 2121 TtTttt +++=+

= mT 21

= nT 22 n

m=

2

1

2

1

must be a

rational number


33/115

Diff_Eq_12_2012/201

3

Example:

tttf )10cos(10cos)( ++=

Is this function a periodicone?

+=

1010

2

1 not a rationalnumber


34/115

Diff_Eq_12_2012/201

3


35/115

Diff_Eq_12_2012/201

3

Rectangular Pulse


36/115

Diff_Eq_12_2012/201

3

Sawtooth wave


37/115

Diff_Eq_12_2012/201

3

Orthogonal Functions

Call a set of functions {k }orthogonal on an interval a < t

Documents

Diff Equation 12 2012 FourierSeries