lecture-22222222222_transfer_function.pdf

CS Slide on Transfer Function

1. Control Systems (CS)
Dr. Imtiaz Hussain
Associate Professor
Mehran University of Engineering & Technology Jamshoro, Pakistan
email: [email protected]
URL :http://imtiazhussainkalwar.weebly.com/
Lecture-2
Transfer Function and stability of LTI systems
1

2. Transfer Function
•TransferFunctionistheratioofLaplacetransformofthe
outputtotheLaplacetransformoftheinput.
Consideringallinitialconditionstozero.
•WhereistheLaplaceoperator.
Plant
y(t)
u(t))()(
)()(
SYty
andSUtuIf






2

3. Transfer Function
•ThenthetransferfunctionG(S)oftheplantisgiven
as
G(S) Y(S)U(S))(
)(
)(
SU
SY
SG
3

4. Why Laplace Transform?
•ByuseofLaplacetransformwecanconvertmany
commonfunctionsintoalgebraicfunctionofcomplex
variables.
•Forexample
Or
•Wheresisacomplexvariable(complexfrequency)and
isgivenas22





s
tsin as
e
at


 1
 js
4

5. Laplace Transform of Derivatives
•Notonlycommonfunctioncanbeconvertedinto
simplealgebraicexpressionsbutcalculusoperations
canalsobeconvertedintoalgebraicexpressions.
•Forexample)()(
)(
0xSsX
dt
tdx
 dt
dx
xSXs
dt
txd )(
)()(
)( 0
0
2
2
2

5

6. Laplace Transform of Derivatives
•Ingeneral
•Where istheinitialconditionofthesystem.)()()(
)(
00
11 

nnn
n
n
xxsSXs
dt
txd
 )(0x
6

7. Example: RC Circuit
•If the capacitor is not already charged then
y(0)=0.
•uis the input voltage applied at t=0
•yis the capacitor voltage
7

8. Laplace Transform of Integrals)()( SX
s
dttx
1

•The time domain integral becomes division by
sin frequency domain.
8

9. Calculation of the Transfer Functiondt
tdx
B
dt
tdy
C
dt
txd
A
)()()(

2
2
•ConsiderthefollowingODEwherey(t)isinputofthesystemand
x(t)istheoutput.
•or
•TakingtheLaplacetransformoneithersides)(')(')('' tBxtCytAx  )]()([)]()([)](')()([ 0000
2
xssXByssYCxsxsXsA 
9

10. Calculation of the Transfer Function
•ConsideringInitialconditionstozeroinordertofindthetransfer
functionofthesystem
•Rearrangingtheaboveequation)]()([)]()([)](')()([ 0000
2
xssXByssYCxsxsXsA  )()()( sBsXsCsYsXAs 
2 )(])[(
)()()(
sCsYBsAssX
sCsYsBsXsXAs


2
2 BAs
C
BsAs
Cs
sY
sX




2
)(
)(
10

11. Example
1.FindoutthetransferfunctionoftheRCnetworkshowninfigure-1.
Assumethatthecapacitorisnotinitiallycharged.
Figure-1)()(''')()()('' tytydttytutu  336
2.u(t)andy(t)aretheinputandoutputrespectivelyofasystemdefinedby
followingODE.DeterminetheTransferFunction.Assumethereisnoany
energystoredinthesystem.
11

12. Transfer Function
•In general
•Wherexistheinputofthesystemandyistheoutputof
thesystem.
12

13. Transfer Function
•When order of the denominator polynomial is greater
than the numerator polynomial the transfer function is
said to be ‘proper’.
•Otherwise ‘improper’
13

14. Transfer Function
•Transfer function helps us to check
–Thestabilityofthesystem
–Timedomainandfrequencydomaincharacteristicsofthe
system
–Responseofthesystemforanygiveninput
14

15. Stability of Control System
•Thereareseveralmeaningsofstability,ingeneral
therearetwokindsofstabilitydefinitionsincontrol
systemstudy.
–AbsoluteStability
–RelativeStability
15

16. Stability of Control System
•Roots of denominator polynomial of a transfer
function are called ‘poles’.
•And the roots of numerator polynomials of a
transfer function are called ‘zeros’.
16

17. Stability of Control System
•Polesofthesystemarerepresentedby‘x’and
zerosofthesystemarerepresentedby‘o’.
•Systemorderisalwaysequaltonumberof
polesofthetransferfunction.
•Followingtransferfunctionrepresentsn
th
orderplant.
17

18. Stability of Control System
•Polesisalsodefinedas“itisthefrequencyatwhich
systembecomesinfinite”.Hencethenamepole
wherefieldisinfinite.
•Andzeroisthefrequencyatwhichsystembecomes
0.
18

19. Stability of Control System
•Polesisalsodefinedas“itisthefrequencyatwhich
systembecomesinfinite”.
•Likeamagneticpoleorblackhole.
19

20. Relation b/w poles and zeros and frequency
response of the system
•Therelationshipbetweenpolesandzerosandthefrequency
responseofasystemcomesalivewiththis3Dpole-zeroplot.
20
Single pole system

21. Relation b/w poles and zeros and frequency
response of the system
•3Dpole-zeroplot
–Systemhas1‘zero’and2‘poles’.
21

22. Relation b/w poles and zeros and frequency
response of the system
22

23. Example
•ConsidertheTransferfunctioncalculatedinprevious
slides.
•Theonlypoleofthesystemis
23BAs
C
sY
sX
sG


)(
)(
)( 0BAs is polynomialr denominato the A
B
s

24. Examples
•Considerthefollowingtransferfunctions.
–Determine
•Whetherthetransferfunctionisproperorimproper
•Polesofthesystem
•zerosofthesystem
•Orderofthesystem
24)(
)(
2
3



ss
s
sG ))()((
)(
321 

sss
s
sG )(
)(
)(
10
3
2
2



ss
s
sG )(
)(
)(
10
1
2



ss
ss
sG
i)
ii)
iii) iv)

25. Stability of Control Systems
•Thepolesandzerosofthesystemareplottedins-plane
tocheckthestabilityofthesystem.
25
s-plane
LHP RHP j js Recall

26. Stability of Control Systems
•Ifallthepolesofthesystemlieinlefthalfplanethe
systemissaidtobeStable.
•Ifanyofthepoleslieinrighthalfplanethesystemissaid
tobeunstable.
•Ifpole(s)lieonimaginaryaxisthesystemissaidtobe
marginallystable.
26
s-plane
LHP RHP j

27. Stability of Control Systems
•Forexample
•Thentheonlypoleofthesystemlieat
271031 

 CandBA
BAs
C
sG if ,,)( 3pole
s-plane
LHP RHP j
X
-3

28. Examples
•Considerthefollowingtransferfunctions.
Determinewhetherthetransferfunctionisproperorimproper
CalculatethePolesandzerosofthesystem
Determinetheorderofthesystem
Drawthepole-zeromap
DeterminetheStabilityofthesystem
28)(
)(
2
3



ss
s
sG ))()((
)(
321 

sss
s
sG )(
)(
)(
10
3
2
2



ss
s
sG )(
)(
)(
10
1
2



ss
ss
sG
i)
ii)
iii) iv)

29. Another definition of Stability
•Thesystemissaidtobestableifforanybounded
inputtheoutputofthesystemisalsobounded
(BIBO).
•Thustheforanyboundedinputtheoutputeither
remainconstantordecreasewithtime.
29
u(t)
t
1
Unit Step Input
Plant
y(t)
t
Output
1
overshoot

30. Another definition of Stability
•Ifforanyboundedinputtheoutputisnot
boundedthesystemissaidtobeunstable.
30
u(t)
t
1
Unit Step Input
Plant
y(t)
t
Outputat
e

31. BIBO vsTransfer Function
•Forexample3
1
)(
)(
)(
1


ssU
sY
sG 3
1
)(
)(
)(
2


ssU
sY
sG -4 -2 0 2 4
-4
-3
-2
-1
0
1
2
3
4
Pole-Zero Map
Real Axis
Imaginary Axis -4 -2 0 2 4
-4
-3
-2
-1
0
1
2
3
4
Pole-Zero Map
Real Axis
Imaginary Axis
stable
unstable

32. BIBO vsTransfer Function
•Forexample3
1
)(
)(
)(
1


ssU
sY
sG 3
1
)(
)(
)(
2


ssU
sY
sG )()(
3
1
)(
)(
)(
3
11
1
1
tuety
ssU
sY
sG
t



  )()(
3
1
)(
)(
)(
3
11
2
1
tuety
ssU
sY
sG
t






33. BIBO vsTransfer Function
•Forexample)()(
3
tuety
t
 )()(
3
tuety
t
 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
exp(-3t)*u(t) 0 2 4 6 8 10
0
2
4
6
8
10
12
x 10
12
exp(3t)*u(t)

34. BIBO vsTransfer Function
•Wheneveroneormorethanonepolesarein
RHPthesolutionofdynamicequations
containsincreasingexponentialterms.
•Suchas.
•Thatmakestheresponseofthesystem
unboundedandhencetheoverallresponseof
thesystemisunstable.t
e
3

35. END OF LECTURE-2
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