Introd Control Helmy

7/24/2019 Introd Control Helmy

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Instructors: Dr. Helmy El-Zoghby

Third Power

201

Princi!les o" #utom$tic %ontrol

&ystems

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'e"erences.

Te(t boo)s $nd re"erences

1-#utom$tic %ontrol &ystems*+,en$min %. uo / . oln$r$ghi

2-odern %ontrol Engineering*+ $tsuhi)o3g$t$

-eedb$c) %ontrol o" Dyn$mic &ystems*+

ene . r$n)lin et al

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o$ls o" this course

4nderst$nd the b$sic conce!ts o" $utom$ticcontrol

now how to conduct design $nd $n$lysiso"

line$r control system with the "ollowing

techni5ues: $them$tic$l modelingtechni5ues

Time-dom$in $n$lysis techni5ues

'oot-locus$n$lysis techni5ues

re5uency-dom$in$n$lysis techni5ues

4se $tl$bto design $nd $n$ly6e control systems

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%h$!ter 1

Introduction to %ontrol &ystems

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3utline o" ch$!ter.1

What is a control system?A brief history of control

Basic components of a control

system

Open-loopcontrol vs. closed-loop

control

Classification of control systems

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7h$t is $ control system8

In gener$l $ control system is $ system th$t is used to re$li6e

$ desired out!ut or obecti9e.

%ontrol systems $re e9erywhere

They $!!e$r in our homes+ in c$rs+ in industry+ in scienti"ic

l$bs+ $nd in hos!it$l

Princi!les o" control h$9e $n im!$ct on $ll "ields $s

engineering+ economics+ biology $nd medicine

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# brie" history o" control

Two o" the e$rliest e($m!les

1-7$ter cloc) ;2


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ly-b$ll go9ernor ;>$mes 7$tt+1


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,irth o" m$them$tic$l control theory

. ,. #iry ;1AB0=the "irst one to discuss inst$bilityin $ "eedb$c) control system

the "irst to $n$ly6e such $ system using di""erenti$l e5u$tions

>. %. $(well ;1A?A=

the "irst system$tic study o" the st$bility o" "eedb$c) control

E. >. 'outh ;1A


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,irth o" cl$ssic$l control design method

H. y5uist ;1@2=de9elo!ed $ rel$ti9ely sim!le !rocedure to determine st$bility "rom $

gr$!hic$l !lot o" the loo!-"re5uency res!onse.

H. 7. ,ode ;1@B="re5uency-res!onse method

7. '. E9$ns ;1@BA=root-locus method

7ith the $bo9e methods+ we c$n design control systems th$t

$re st$ble+ $cce!t$ble but not o!tim$l in $ny me$ning"ul sense.

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# brie" history o" controlDe9elo!ment o" modern control design

C$te 1@0s: designing o!tim$l systems in some me$ning"ul sense

1@?0s: digit$l com!uters hel! time-dom$in $n$lysis o" com!le( systems+ modern control

theory h$s been de9elo!ed to co!e with the incre$sed com!le(ity o" modern !l$nts

1@?0sF1@A0s: o!tim$l control o" both deterministic $nd stoch$stic systemsG $d$!ti9e

control $nd le$rning control

1@A0sF!resent: robust control+ H-in" control

'ecent $!!lic$tions o" modern control theory include such non-engineering systems $s

biologic$l+ biomedic$l+ economic $nd socioeconomic systems

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,$sic com!onents o" $ control system

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,$sic conce!ts o" $ control system

Plant

1.Pl$nt: $ !hysic$l obect to be controlledsuch $s $ mech$nic$l de9ice+ $ he$ting

"urn$ce+ $ chemic$l re$ctor or $ s!$cecr$"t.

ControlledControlledvariablevariable

2.%ontrolled 9$ri$ble: the 9$ri$blecontrolled by #utom$tic %ontrol &ystem +

gener$lly re"ers to the system out!ut.

ExpectedExpectedvaluevalue

.E(!ected 9$lue : the desired 9$lueo" controlled 9$ri$ble b$sed on re5uirement+

o"ten it is used $s the re"erence in!ut

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Disturbance


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,loc) di$gr$m o" $ control system

Controller Actuator Plant

Sensor

-

rExpected

value

e

Error

Disturbance

Controlledvariable

n y

com!$rison com!onent

;com!$rison !oint= :

its out!ut e5u$ls the

$lgebr$ic sum o" $ll in!utsign$ls.

*: !lusG -*: minus

le$d-out !oint:

Here+ the sign$l is

tr$ns"erred $long two

se!$r$te routes.

The ,loc) re!resentsthe "unction $nd n$me o" its

corres!onding mode+ we donJt

need to dr$w det$iled structure+

$nd the line guides "or the tr$ns"er route.

u

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3!en-loo! control systems

3!en-loo! control systems: those systems in which the out!ut h$s no e""ect on the

control $ction.

The out!ut is neither me$sured nor "ed b$c) "or com!$rison with the in!ut.

or e$ch re"erence in!ut+ there corres!onds $ "i(ed o!er$ting conditionsG the $ccur$cy

o" the system de!ends on calibration.

In the !resence o" disturbances,$n o!en-loo! system will not !er"orm the desired t$s).

CONTROLLER PLANT

Controlsignal

Systemoutput

Systeminput

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3!en-loo! control systems E($m!les

Washin !achine

Tra""ic sinals

ote th$t $ny control systems

th$t o!er$tes on $ time b$sis $re

o!en-loo!.

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%omments on 3!en-loo! control systems

&im!le construction $nd e$se o"m$inten$nce.

Cess e(!ensi9e th$n $ closed-loo!system.o st$bility !roblem.'ec$libr$tion is necess$ry "rom

time to time.

&ensiti9e to disturb$nces+ so less$ccur$te.

Good

Bad

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7hen should we $!!ly o!en-loo! control8

The rel$tionshi! between the in!ut $nd out!ut isexactly known.

There $re neither intern$l nor e(tern$l

disturb$nces.

e$suring the out!ut !recisely is very hard or

economically infeasible.

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%losed-loo! control systems

%losed-loo! control systems $re o"ten re"erred to $s "eedb$c) control systems.

The ide$ o" "eedb$c):

%om!$re the actual outputwith the expected value.

T$)e $ctions b$sed on the difference (error).

This seemingly sim!le ide$ is tremendously !ower"ul.

eedb$c) is a key idea in the disci!line o" control.

CONTROLLER PLANT

Controlsignal

SystemoutputExpeted!alue Error

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%losed-loo! control systems

In !r$ctice+ "eedb$c) control system $nd

closed-loo! control system $re used

interch$nge$bly

%losed-loo! control $lw$ys im!lies the use o"

"eedb$c) control $ction in order to reduce

system error

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%losed Coo! E($m!le : "lush toilet

Le!er"aterTan#

$loat

Piston0h ( )h t

( )q t

lever

Plant#$nput#Output#Expected value#

Sensor#Controller#Actuator#

0h

( )h t

0h

PlantController Atuator

Sensor

%ater tan&%ater "lo%%ater level

"loatleverpiston

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3ther e($m!les o" "eedb$c)

The hum$n body is highly $d9$nced "eedb$c) controlsystem.

-,ody tem!er$ture $nd blood !ressure $re )e!tconst$nt by me$ns o" !hysiologic$l "eedb$c).

eedb$c) m$)esthe hum$n body rel$ti9ely insensiti9eto e(tern$l disturb$nce.Thus we c$n sur9i9e in $ ch$ngingen9ironment.

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%omments on "eedb$c) control ;closed

loo!=

$in $d9$nt$ges o" "eedb$c):

reduce disturb$nce e""ects

m$)e system insensiti9e to 9$ri$tions

st$bili6e $n unst$ble system

cre$te well-de"inedrel$tionshi! between out!ut

$nd re"erence

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%omments on "eedb$c) control

Dr$wb$c)s o" "eedb$c):

ca!seinstability if not !sed properly

co!ple noise from sensors into the dynamics of a

system

increase the overall comple"ity of a system

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3!en-loo! 9s. closed-loo!

%losed-loo! control3!en-loo! control

Si!ple structure' lo% cost

Lo% accuracy and

resistance todisturbance

Easy to reulate

Ability to correct error

Co!plex structure' hih cost

(ih accuracy andresistance o" disturbance

Selectin para!eter iscritical )!ay causestability proble!*

3!en-loo! %losed-loo! %om!osite control system11/02/1526


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T%in#ing time&

E($m!les o" o!en-loo!

control $nd closed-loo!

control systems 8

or e$ch system+ could you identi"y

the sensor+ $ctu$tor $nd controller8

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%l$ssi"ic$tion o" control systems

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the re"erence input)expected value* is aconstant value the controller %or&s to&eep the output aroundthe constant value

e++ constant,te!perature control'li-uid level control andconstant,pressurecontrol+

the re"erence input!ay be un&no%n orvaryin the controller %or&sto !a&e the outputtrac& the varyin

re"erence e++ auto!aticnaviation syste!s onboats and planes'satellite,trac&inantennas

the input chanesaccordin to aprora! the controller%or&s accordin to

prede"ined co!!ande++ nu!erical

control !achine

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superposition priniple applies desri'ed 'y lineardi((erential e)uation

desri'ed 'y nonlineardi((erential e)uation

# #( ) ( )f x y f x y= =

# # #( ) ( ) ( )f x x f x f x y y+ = + = +superposition priniple

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All t%e signals are (untionso( ontinuous time !aria'le

Signals are in t%e (orm o(eit%er a pulse train or adigital ode

e.g. digital control system

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T%e parameters o( a ontrolsystem are stationary *it%respet to time

System ontain elements t%atdri(t or !ary *it% time

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,$sic re5uirements "or control systems

&t$bility: re"er to the $bility o" $ system to reco9er e5uilibrium

Kuic)ness: re"er to the dur$tion o" tr$nsient !rocess be"ore the

control system to re$ch its e5uilibrium

#ccur$cy: re"er to the si6e o" ste$dy-st$te error when the

tr$nsient !rocess ends

;&te$dy-st$te errorLdesired out!ut M $ctu$l out!ut=

------------------------------- ED.lec.1

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The Laplace transform was developedby the Frenchmathematician by the same

name (1749-1827 and was widely adapted

to en!ineerin! problems in the last

cent"ry# $ts "tility lies in the ability to

convert differential e%"ations to al!ebraicforms that are more easily solved# The

notation has become very common in

certain areas as a form of en!ineerin!

&lan!"a!e' for dealin! with systems#

The C$!l$ce Tr$ns"orm


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&te!s in9ol9ed in using the C$!l$ce tr$ns"orm

$ ( )% ( )L f t F s=

$ ( )% ( )L F s f t =

0( ) ( ) stF s f t e dt

=


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Basic Theorems o !inearity

$ ( )% $ ( )% ( )L Kf t KL f t KF s= =

The !a"lace transorm o a "rod#ct is notthe "rod#ct

o the transorms.

# #

#

$ ( ) ( )% $ ( )% $ ( )%

( ) ( )

L f t f t L f t L f t

F s F s

+ = +

= +

# #$ ( ) ( )% ( ) ( )L f t f t F s F s


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Laplace transform of the basic si!nals

0( ) () stF s e dt

= 0

0

( ) 0

ste e

F ss s s

= = =

#nit ste" #nction


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!a"lace transorm o the e$"onential #nction

( )

0 0

( ) 0

0

( )

0( ) ( )

t st s t

s t

F s e e dt e dt

e e

s s

s

+

+

= =

= = + +

= +

( ) t

f t e =


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%ommon tr$ns"orm !$irs.


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40

!a"lace Transorms o %ommon

nctions

&ame f(t) F(s)

'mp!lse

tep

amp

*"ponential

ine

s

#

s

as

##

s+

)( =tf

ttf =)(

atetf =)(

)sin()( ttf =

>

==

00

0)(

t

ttf


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%ommonly #sed #nctions

1

2

N

1

1

1=;

1=;

n

n

st

s

nt

s

t

ase

s

tu

t

##

##

##

##

sincos)cos(

cossin)sin(

)(

)cos(

)()sin(

ws

wswt

ws

wswt

was

aswte

was

wwte

at

at

+

+

+

++

++

+

++


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'$am"le.1

#( ) + sin ,tv t e t =

# #

# #

,( ) $ ( )% +

( #) (,)

#0 #0

, , - , #0

V s L v t s

s s s s

= = + +

= =+ + + + +


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'$am"le .2

,( ) +cos # tp t t e= +

# #

#

( ) $ ( )% +

(#) ,

+

, ,

sP s L p t

s s

s

s s

= = + + +

= ++ +

I C l T " b Id ti"i ti


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In9erse C$!l$ce Tr$ns"orms by Identi"ic$tion

hen a differential e%"ation is solved by Laplace

transforms) the sol"tion is obtained as a f"nction ofthe variable s# The inverse transform m"st be

formed in order to determine the time response# The

simplest forms are those that can be reco!ni*ed

within the tables and a few of those will now beconsidered#


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'$am"le .3 (etermine the in)erse transorm o

the #nction *elo+.

#

+ # /( )

.F s

s s s= + +

+

.( ) + # /

tf t t e= + +


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the #nction *elo+.

#

#00( )

00V s

s=

+

# #

0( ) #0(0)

V ss

= +

( ) #0sin0v t t=


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the #nction *elo+.

#

/ ,( )

- .

sV s

s s

+=

+ +hen the denominator contains a %"adratic) chec+ the roots# $f

they are real) a partial fraction e,pansion will be re%"ired# $f they

are comple,) the table may be "sed# $n this case) the roots are

#. #s i=

#

# # #

#

# #

- .

- (.) . (.)

- 1 ,

( .) (#)

s s

s s

s s

s

+ +

= + + +

= + + +

= + +


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'$am"le .5 %ontin#ation.

# # # #

# # # #

/( ) , #,( )

( ) (#) ( ) (#)

/( ) 0(#)

( ) (#) ( ) (#)

sV s

s s

s

s s

+ = ++ + + +

+= + + + +

( ) / cos# 0 sin #

t tv t e t e t

=


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rdinary .ifferential /%"ations Transforms

( )( )

( )

N sF s

D s=

0( ) ...

n n

n nN s a s a s a s a

= + + + +

0( ) ...m m

m mD s b s b s b s b

= + + + +

#

( )( )( )( )....( )m m

N sF sb s p s p s p=


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The roots o D,s- are calledpoles and they may *e

classiied in o#r +ays.

1# 0eal poles of first order# 2# omple, poles of first order (incl"din! p"rely

ima!inary poles

# 0eal poles of m"ltiple order

4# omple, poles of m"ltiple order (incl"din!p"rely ima!inary poles


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'$am"le .6 (etermine in)erse transorm o

#nction *elo+.

#-( )( )( #) #

A AsF ss s s s

+= = ++ + + +

]

- -( ) ( ) +

# #

s

s

sA s F s

s

=

=

+ += + = = =+ +

#

- -( )

# ( )( #)

s sF s

s s s s

+ += =

+ + + +


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]# ##

- # -( #) ( ) ,

# ss

sA s F s

s=

=

+ += + = = = + +

+ ,( )

#F s

s s=

+ +

#( ) + ,

t tf t e e

=


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'$am"le .7 (etermine e$"onential "ortion o

in)erse transorm o #nction *elo+.

#

+0( ) (+0)(#)#+

( #)( # +) ()(,)s

sA

s s s=

+= = =+ + +

# ##

+0( ) (+0)()0

( )( # +) ( )(+)s

sA

s s s =

+= = =

+ + +

#

+0( .)( )

( )( #)( # +)

sF s

s s s s

+=

+ + + +

#( )

#

A AF s

s s= +

+ +

#

( ) #+ 0

t tf t e e

=

' l 8 i d th i


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'$am"le .8 ind the in)erse

transorm

# #

+0( .) #+ 0

( )( #)( # +) # # +

s As B

s s s s s s s s

+ += +

+ + + + + + + +

+0(.) #+ 0

()(#)(+) # +

B

= +#+B=

+0(,) #+ 0

(#)(.)(/) # . /

A B+= + +A =

#

#+ 0 + #+( )

# # +

sF s

s s s s

= +

+ + + +


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# #

+ #+( )

# +

sF s

s s

=

+ +

# # # ## + # + ( ) (#)s s s s s+ + = + + + = + +

# # # # # # #

+ #+ +( ) +(#)( )

( ) (#) ( ) (#) ( ) (#)

s sF s

s s s

+ = = +

+ + + + + +

#

#

( ) ( ) ( )

#+ 0 + cos # + sin #t t t t

f t f t f t

e e e t e t = +

=


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econdrder eal oles

ss#me that F,s- contains a denominator actor o the

orm ,s-2. The e$"ansion +ill tae the orm sho+n

*elo+.

#

#( ) ( )

( )

C CF s R s

s s = + +

+ +

#

( ) ( ) sC s F s

=

= +

# #( ) ( )t t tf t C te C e C t C e

= + = +


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'$am"le.9 (etermine in)erse transorm o

#nction *elo+.

#

-0( )

( #)F s

s s=

+

#

# #

-0( )

( #) ( #) ( #)

C CAF s

s s s s s

= = + +

+ + +

] # #00

-0 -0( ) +

( #) (0 #)ss

A sF ss

=

=

= = = =+ +

#

##

-0 -0( #) ( ) .0

#s sC s F s

s= =

= + = = =


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58


#

# #

-0 + .0( )

( #) ( #) #

CF s

s s s s s= = +

+ + +

## #-0 + .0()( #) ( #) ( #)

C= ++ + +# +C =

# #

-0 + .0 +( )

( #) ( #) #F s

s s s s s= =

+ + +

# # #( ) + .0 + + + ( # )

t t tf t te e e t

= = +


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!a"lace Transorm "erations


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60

4sing $tl$b with C$!l$ce tr$ns"orm:

E($m!le 4se $tl$b to "ind the in9erse tr$ns"orm o"

)/-

#

)(.(

)-()(

+++

+=

sss

sssF

syms s t

ilaplace(s*(s+6)/((s+3)*(s!+6*s+"#)))

ans $

%exp(%3*t)+!*exp(%3*t)*cos(3*t)


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61

Initi$l O$lue Theorem

I" the "unction ";t= $nd its "irst deri9$ti9e $re C$!l$ce tr$ns"orm$ble $nd ";t=

H$s the C$!l$ce tr$ns"orm ;s=+ $nd the e(ists+ then=;lim ss&

0=0;=;lim=;lim

=

tsftfss&

The utility o" this theorem lies in not h$9ing to t$)e the in9erse o" ;s=in order to "ind out the initi$l condition in the time dom$in. This is

!$rticul$rly use"ul in circuits $nd systems.

s

'nitial alue heorem


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62

E($m!le:

i9enG 22D=1;

=2;=;

=

s

ss&

ind ";0=

1=2?;2

2lim

2D12

2lim

D=1;

=2;lim=;lim=0;

2222

222

2

2

22

=

=

=

=

sssss

ssss

ss

ss

s

ssss&f

ss s

s


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63

in$l O$lue Theorem

I" the "unction ";t= $nd its "irst deri9$ti9e $re C$!l$ce tr$ns"orm$ble $nd ";t=

h$s the C$!l$ce tr$ns"orm ;s=+ $nd the e(ists+ then=;lim ss&

s

=;=;lim=;lim ftfss&

0s t

#g$in+ the utility o" this theorem lies in not h$9ing to t$)e the in9erse

o" ;s= in order to "ind out the "in$l 9$lue o" ";t= in the time dom$in.

This is !$rticul$rly use"ul in circuits $nd systems.

&inal alue

heorem


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E($m!le:

i9en:

ttes&notesss& t cos=;

=2;=2;=; 2122

22

=

=

ind =;f

. 0

=2;

=2;lim=;lim=;

22

22

=

=

s

ssss&f

0s0s

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Introd Control Helmy