frequency response and control stability

8/20/2019 Frequency Response and Control Stability

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Frequency Response&

Control System Stability


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Frequency Response• In practice the performance of a control system is

more realistically and directly measured by its time-domain response characteristics due to theperformance of most control systems is judged based

on the time response due to certain signal.• his is in contrast to the analysis and design of

communication systems for !hich the frequencyresponse is more importance" since it is the case"

most of signals to be processed are either sinusoidalor composed of sinusoidal components.

School of #ngineering $%F# Slide '


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Frequency Response• In design problems" the difficulties lie in the fact that

there are no unified methods of arri(ing at a designedsystem gi(en the time-domain performancespecifications" such as ma)imum o(er-shoot" rise time"

delay time" setting time and so on.• *n the other hand" there is a !ealth of graphical

methods a(ailable in the frequency domain" allsuitable for the analysis and design of the linear

control system.

School of #ngineering $%F# Slide +


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efinition of frequency response• he characteristics of a system can be determined by

measuring the response of the system to sinusoidalinputs.

• he steady-state frequency $or harmonic response of

a system is defined as the (ariation !ith frequency ofthe steady-stat& amplitude and phase difference of theoutput" corresponding to a forced sinusoidal input offi)ed amplitude.

• It should be noted that the frequency response and thetransient response characteristics of a system bothdepend on the same factors and !hen one of theresponses is /no!n" the other can be determined.

School of #ngineering $%F# Slide 0


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efinition of frequency response• 1oth methods are used in flight test !or/ to determine

the dynamic stability characteristics of aircraft" though"in this particular case" the transient technique isgenerally employed.

• If the amplitude of the sinusoidal input is represented byθ i , and the amplitude of the corresponding sinusoidaloutput by θ o , then the relationship bet!een the outputand the input can be e)pressed in the form


φ

θ

θ i

i

o Ae−=


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efinition of frequency response• his equation states that the amplitude of the output is A times the

amplitude of the input $i.e. the system has an amplification factor orgain of A), but that the output lags behind the input by an angle ϕ$i.e. the system has a phase lag of ϕ.

• he input and output are both sinusoidal oscillations of frequency"say" ω rads3sec" and they can be represented by t!o (ectors oflength θ i and θ o , respecti(ely" rotating at a rate of ω rads3sec" !iththe output (ector θ o lagging behind the input (ector θ i by an angle ϕas sho!n in the (ector representation diagram.



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efinition of frequency response



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Frequency response

• he frequency response method may be less intuiti(ethan other methods.• 6o!e(er" it has certain ad(antages" especially in real-

life situations such as modelling transfer functions from

physical data.• he frequency response of a system can be (ie!ed t!odifferent !ays7 (ia the 1ode plot or (ia the 8yquistdiagram.

• 1oth methods display the same information9 thedifference lies in the !ay the information is presented.

Information echnology Ser(ices Slide :


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Frequency response

• o plot the frequency response" it is necessary tocreate a (ector of frequencies $(arying bet!een ;ero$C and infinity and compute the (alue of the systemtransfer function at those frequencies.

• If


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Frequency response

• Since >


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Frequency response

Information echnology Ser(ices Slide >+

•hen !e use a phasor to describe a sinusoidal signal

all !e need specify is its magnitude and phase angle.


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Frequency response• In order to clearly indicate !hen !e are tal/ing of the

magnitude of a sinusoidal signal !e often !rite IBI forthe magnitude of the sinusoidal signal represented bythe phasor and bold" non-italic" letters for the symbolsfor phasors" e.g. Y. hus" Y implies a phasor !ith

both magnitude and phase.• % comple) number ; A a D jb can be represented on

an %rgand diagram" i.e. a graph of imaginary partplotted against real part" by a line $as sho!n of length

I;l at an angle ϕ.

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Frequency response•

he magnitude l;l of a comple) number ; and its angleϕ is thus gi(en by7• e can describe a phasor used to represent a

sinusoidal quantity by a comple) number.

• If !e ha(e y A B sin ωt then this is described by aphasor $Figure a consisting of just a real number.• hus" a unit magnitude phasor !ith phase angle ?E is

represented by > D ?j.

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=+= −

b

aba Z

12tanand

2

φ


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Frequency response• 6o!e(er" for y A B sin $ωt D ϕ !e ha(e a phasor $Figure b !hich has" in

general" both a real and imaginary part and so is represented by a D bj.

• If the phase θ is :?Ethen for y A sin $ωt D :?E A cos ωt the phasor $Figure

c has only an imaginary part. hus" such a unit magnitude phasor isrepresented by ? D >j.



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Frequency response• If !e ha(e a phasor of length > and phase angle 0E then it !ill ha(e a comple)

representation of > D ?j.• he same length phasor !ith a phase angle of :?E !ill ha(e a comple)representation of ? D >j9 rotation of a phasor anticloc/!ise by :?E correspondsto multiplication of the phasor by j since j$> D?j A ? D j >.

• If !e no! rotate this phasor by a further :?E" then as j$?D>j A ? D >j !e ha(ethe original phasor multiplied by j.

• %s this phasor is just the original phasor in the opposite direction" it is just theoriginal phasor multiplied by-> and so j A-> and hence j A G$->.



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Frequency response• Rotation of the original phasor through a total of '4?Ei.e. + ) :?E is equi(alent to multiplying the original phasor by jH A j$j A -j.

Example 1:

• hat magnitude and phase is gi(en the phasors described as $a +j" $b > D 'j



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Frequency response• $a he magnitude is + units and" since !e only ha(ean imaginary component" the phase is :?E.

• $b he magnitude is G$a D b A G$> D A '.' unitsand the phase is gi(en by tan ϕ A b3a A '3> and ϕ A

2+. E.

Information echnology Ser(ices Slide >:


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Frequency response• he frequency response of a system is the steady

state response of the system to a sinusoidal inputsignal.

• he steady state output is a sinusoidal signal of thesame frequency as the input signal differing only inamplitude and phase angle.

• In order to arri(e at the principle of the frequencyresponse function !e !ill consider a simple system!ith a sinusoidal input and steady-state sinusoidaloutput and recognise that our conclusions can beapplied in a more general !ay to all systems.

Information echnology Ser(ices Slide '?


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Frequency response• Consider a system !here the input x is related to the

output by y = kx.• If !e ha(e an input of a sinusoidal signal x = sin ωt then

the output is y = k sin ωt and so a sinusoidal signal !iththe same frequency but a different amplitude.

• hus" if !e represent the sinusoidal signals byphasors7

• 8o! consider a system !here the input x is related tothe output y by the differential equation7

Information echnology Ser(ices Slide '>

k or Input phas

sor Output pha=

X

Y

kx ydt

dy =+τ


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Frequency response• hus" since the frequency does not change !e can ta/e

x = sin ωt and y = sin ωt and so" since dy/dt = ω cos ωt =ω sin (ωt +90°" the equation can be !ritten as7

!ω sin (ωt +90°+ sin ωt =k sin ωt

• e can represent sinusoidal signals by phasors anddescribe them by comple) numbers.

• hus" the abo(e equation can be !ritten in terms ofphasors as7 jτω YD Y=k X

• 6ence"

• his is the definition of a frequency response function asthe output phasor di(ided by the input phasor .

Information echnology Ser(ices Slide ''

ωτ "k

or Input phas sor Output pha += 1X

Y


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Frequency response• e can compare this !ith the different equation !ritten

in the s-domain as7 ! sY(s) + Y(s)= k X(s)• %nd the resulting transfer function7

• he frequency response function equation is of the

same form as the transfer function if !e replace s by jω.• 6ence the frequency response function is denoted by


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Frequency responseExample 2:

• etermine the frequency response function for asystem ha(ing a transfer function of !(s) = "#($ + s).

• Replacing s by jω gi(es the frequency response

function of


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Frequency response for

first-order system• In general" a first-order system has a transfer functionof the form7

• here τ is the time constant of the system. he

frequency response function


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first-order system• he frequency response function has thus a realelement of >3$> D ωτ and an imaginary element of-ωτ/$> D ωτ.

• Since


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first-order system• he phase difference ϕ bet!een the output phasor andthe input phasor is gi(en by tan ϕ A b#a as% tan ϕ A - ωτ.

• he negati(e sign indicates that the output phasor lags

behind the input phasor by this angle. hus7 The gain and phase of a system &hen sub'ect to a

sinusoidal input is obtained by putting the frequencyresponse function in the form a + 'b and then the gain

is (a + b ) and the phase is .

Information echnology Ser(ices Slide '4

)/(tan 1 ab−


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first-order system#)ample +7• etermine the magnitude and phase of the output from a

system !hen subject to a sinusoidal input of ' sin +t if it

has a transfer function of !(s) = $#(s + *).

Information echnology Ser(ices Slide '5


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first-order system

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second-order system• In general" a second-order system has a transfer function of theform7

Information echnology Ser(ices Slide +?

( )

+

−

=

+−=

++−=

∴

++=

nn

nn

n

nn

n

n

nn

n

"

" "

s s s%

ω

ω ζ

ω

ω

ζω ω ω ω

ω ω ζω ω ω ω

ω

ζ

ω ω ζω

ω

21

1

22)G(j

:Thus. j ys

re!"a#ing yotainedisfun#tionres!onse

fre$uen#yTheratio.dam!ingtheis

andfre$uen#ynatura"theis%here

2)(

2

22

2

22

2

22

2


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second-order system• Rationali;es the equation gi(es7

Information echnology Ser(ices Slide +>

222

222

2

2

2

2

21

1 )G(j

fa#tor&a ygain,

thei.e, !hasor,in!utof n that igger thais !hasorout!ut

of magnitude the !hasor,in!ut y thedi'ided !hasor

out!uttheis)G(jsin#eso,and jaformtheisThis

21

21

)G(jgi'es

21

21

21

1

+

−

=

+

+

−

−

−

=

−

−

−

−

×

+

−

nn

nn

nn

nn

nn

nn

"

"

"

"

"

"

ω

ω ζ

ω

ω ω

ω

ω

ω ζ

ω

ω

ω

ω ζ

ω

ω

ω

ω

ω ζ

ω

ω

ω

ω ζ

ω

ω

ω

ω ζ

ω

ω


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second-order system

Information echnology Ser(ices Slide +'

!hasor.in!utthe

"ags !hasorout!utthat theindi#atessignminusThe

1

2

tan: ygi'enisout!utthe

andin!ute et%een thdifferen#e !haseThen the

2

=

n

n

ω ω

ω

ω ζ

φ

φ


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Frequency iagrams• *ne of the most straightfor!ard methods of presenting

the necessary information is a graphical presentationof the (alues of % and plotted against the frequency =.

• his representation is unsatisfactory" and ho!e(er"and the same information can be plotted more usefullyin the form of log>? % against log>?= and K againstlog>?=.

• *ne ad(antage of these logarithmic plots is that theinformation corresponding to (ery lo! frequencies $afe! cycles per second and to (ery high frequencies$hundreds of thousands of cycles per second can allbe con(eniently presented on the one graph.

Information echnology Ser(ices Slide +0


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Frequency iagrams• In this logarithmic presentation" the (ertical scale of

log>? % $i.e. log>? θo3θi ) is usually calibrated in terms of aunit called a decibel.

• he gain in decibels is defined as7

• #)ample7>. Find the gain for θo3θi=>??

'. hen the gain=2? decible" !hat is the (alue of θo3θi.



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Frequency Response Locus

@lots


• %nother con(enient representation of the frequencyresponse characteristics of a system is obtained bycombining together the gain and phase characteristics

into a single cur(e" often called the 8yquist diagram ofthe system.

• he relationship bet!een θo and θi is gi(en by equation

• It can be represented by a (ector on !hat is /no!n asan %rgand iagram.


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8yquist iagram

• he ordinate corresponding to any point on thediagram are the real and imaginary parts" respecti(ely"of the particular function being represented.

• Since

• In %rgand diagram" the θo3θi relationship isrepresented by a (ector of the length proportional tothe gain %" at an angle K belo! the positi(e real a)is.

• It should be noted that" !hene(er the (ector lies in thelo!er half of the %rgand iagram" the output lagsbehind the input9 !hene(er it lies in the upper half ofthe diagram" the output leads the input.



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8yquist iagram

• %s the frequency of the input is (aried from A ? to A M" the (alues of % $i.e. the length of the (ectorand K $i.e. the angle of the (ector to the real a)is(ary and the end of the (ector s!eeps out a cur(e ora locus plot.

• his cur(e is the 8yquist diagram of the system.

Information echnology Ser(ices Slide +:


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Response of First-*rder

Systems to Sinusoidal Inputs

Information echnology Ser(ices Slide ?

• hen a sinusoidal input is fed to a first-order system"a sinusoidal output is produced.

• %t lo! frequencies the signals are transmitted!ith little change of amplitude $unless the system hasa steady-state gain represented by N, in !hich casethe amplitude is appro)imately N times the input orphase.

• %t the higher frequencies the amplitude is reduced andthe output lags behind the input and as follo! = M"

% ?" K :?E.


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Response of Second-*rder

Systems to Sinusoidal Inputs• he response of a second-order system to sinusoidaloscillations of (arying frequency is sho!n in thefrequency response cur(es.

Information echnology Ser(ices Slide >


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Systems to Sinusoidal Inputs• he gain and phase lag of the system depend on boththe frequency of the input and the relati(e damping of thesystem.

• For lo! (alues of damping" as the input frequency isincreased the gain increases up to a ma)imum at afrequency just less than the natural frequency =n of thesystem.

• his pea/ amplification frequency is gi(en by =nG$>- 'O.• For (alues of OP I3 G' no pea/ e)ists in the amplification

cur(e and at the higher (alues of O" the amplificationdecreases continuously !ith increase of frequency.

Information echnology Ser(ices Slide '


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Systems to Sinusoidal Inputs• he corresponding phase lag increases !ithincreasing frequency" being higher for the higher(alues of O up to A =n" and smaller at the higher

(alues of P =n.• Irrespecti(e of the amount of damping ip the system"the phase difference is al!ays -:?E !hen A =n$!hen the applied frequency is equal to the natural

frequency of the system.• hen A ? the gain of the system is N $i.e. thesteady-state gain and the phase is ;ero.

Information echnology Ser(ices Slide +


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Systems to Sinusoidal Inputs• %s the frequency increases the gain and phase (ary"until at A =n, A -:?E and the gain A l3'O.

• ith further increase in " the phase increases from

-:?E to - >5? E" and the gain decreases steadily until"!hen KA ->5?E" the gain is ;ero.

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Stability Characteristics of

Closed-loop Control Systems• Ideally" the output of a control system should follo!

e)actly any (ariations of the input signal.

• In practice this is not possible" but" it is desirable thatthe system should respond quic/ly" that steady-stateerrors should be eliminated" and that the systemshould be a stable one.

• he (arious errors in(ol(ed depend upon the type ofsystem and the type of input being considered.

Information echnology Ser(ices Slide 0


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Closed-loop Control Systems• Qarious feedbac/ signals can be used to impro(e theperformance of a particular system" e.g. signalsproportional to the error" to the deri(ati(e of the output

$J(elocity feedbac/" to the deri(ati(e of the error" tothe integral of the output" to the integral of the error"etc." or any desired combination of these.

• % compromise is al!ays necessary in the design of any

system9 a high gain produces good response" but maylead to instability" !hereas a large amount of dampingimpro(es the stability but increases the response timeand the magnitude of the steady-state errors.



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Closed-loop Control Systems



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Stability Criteria

• he closed-loop transfer function is gi(en by equation

• !here < is the for!ard-path transfer function and !- the

loop transfer function.• From this equation it can be seen that" !hen thedenominator is ;ero" the output !ould be theoreticallyinfinite and the system !ould be unstable.

• hus the condition of instability can be stated in the form i.e. the system is unstable !hen the loop gainbecomes equal to -> $i.e. !hen the loop gain has anamplitude of unity and a phase lag of >5?E.


%&

%

i

o

+=1θ

θ


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8yquist Criterion

• he critical can be represented on the 8yqist diagramby the $->" ? point and if the locus of the loop transferfunction passes through" or to the left of this point" thecorresponding closed-loop system is unstable one.

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8yquist Criterion

• ore generally" the 8yquist criterion can be stated bysaying that the locus" !hen tra(elled in the directionfrom = A ? to = A M" must pass the $->" ? point insuch a !ay that the point lies to the left of the locus.

• he 8yquist diagram of the loop transfer function canalso be used to determine the amount of stability that!ould be possessed by the corresponding closed loopsystem.

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8yquist Criterion

• his stability is e)pressed in terms of(a) the gain margin, !hich is the amount by !hich thegain differs from unity !hen KA >5?E"(b) the phase margin, !hich is the amount by !hich

the phase angle differs from >5?E !hen the gain isunity.

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Routh Criterion• %nother criterion !hich can be used to determine

!hether a particular system is stable or not is theRouth criterion.

• his criterion is based on the signs of the coefficientsof the characteristic equation of the system and thesign of the Routhian iscriminant and" for stability" thesigns must all be positi(e.

• %lthough it indicates !hether a system is a stable oran unstable one" it does not" unfortunately" gi(e ane)act measure of the amount of stability possessed bythe system.

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Routh Criterion• he characteristic equation of a system is obtained from

the e)pression for the closed loop transfer functionby equating the denominator $i.e. I D !-) to ;ero.

• he solution of the equation$> D !-)θo A ?

• %nd represents the transient $i.e. free response of thesystem and this is indicati(e of the o(erall stability of thesystem.

• In order that this equation should ha(e no solutions

in(ol(ing real roots $i.e. no di(ergent motions" thecoefficients of the equation must satisfy the Routhcriterion.

Information echnology Ser(ices Slide 0+


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Root-Locus @lots


• he characteristic equation of the closed-loop system isgi(en by > D !- A ?.

• %s the transfer function can be represented in its mostgeneral form as /(p) !here is the steady- state gain and/(p) represents the frequency-dependent part of the gain.

• Replacing the loop gain !- in >DD /(p)=0.


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Root-Locus @lots


• he frequency-dependent part of the gain usually consists of anumerator and a denominator" both of !hich are" in general"functions of p. hus >D /(p) =0 can be !ritten as7

• he starting point of the root locus plot can be considered to bethe condition of ;ero steady-state loop gain $i.e. ;ero feedbac/.If A ?" the equation

can only be satisfied if F$p A ?.

1)(

)( (*) +

)(

)(1

−==+ '

' )

'

' )

*

+

*

+

1)(

)( −=

'

' )

*


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Root-Locus @lots


• he (alues of p !hich satisfy this equation represent the basic

dynamic stability characteristics of the system9 they are usuallyreferred to as the poles of the root locus plot.

• hen feedbac/ is introduced the (alues of the roots of thecharacteristic equation are changed and as the (alue of isincreased the roots s!eep out the root locus plot. he ends of the

root locus plot correspond to A,.• In order to satisfy" the equation " F8$p A ? !hich is the ends

of the root locus plot correspond to the solution of the equation !iththe numerator of the loop transfer function equated to ;ero.

1)()( −= ' ' ) *


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Root-Locus @lots


• hese points are referred to as the 1eros of the root locus plot.

• he system is a stable one if the roots lie in the left hand halfof the %rgand diagram $i.e. if the real parts of the roots ha(enegati(e (alues" and any points in the right hand half indicateinstability.

• he root locus plot technique is particularly useful forrepresenting the effects on the dynamic characteristics of anaircraft of mo(ing the C.< $i.e. changing the static stability" orof changing the gain in an autostabili;ation system.


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ethods of Impro(ing the

Stability of a System

Information echnology Ser(ices Slide 0:

• he stability characteristics of a system can beimpro(ed by modifying the basic characteristics of thesystem" by the addition of e)tra feedbac/ signals" orby incorporating additional electronic circuits.

• he stability can be impro(ed by reducing the gain ofthe system" but this has the disad(antage of reducingthe performance as !ell.

• he damping can be increased by using J(elocityfeedbac/" i.e. feedbac/ of a signal proportional to thederi(ati(e of the output.


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Stability of a System

Information echnology Ser(ices Slide 2?

• his has the disad(antage of producing a lag in somesystems" and this method can be impro(ed by usingJtransient (elocity feedbac/ so that a feedbac/ signalis produced only !hen the output is changing.

• It is desirable to ha(e a high gain in the system" aslong as it does not result in instability. o achie(e this itis possible to add electrical circuits !hich eitherincrease the gain at lo! frequencies lea(ing thehigh frequency gain unchanged" or reduce the gain athigh frequencies lea(ing the lo! frequency gainunchanged.


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Stability of a System• hese phase-lag and phase-ad(ance circuits aremade up of different combinations of resistances andcapacitances and their effects are illustrated as belo!.

frequency response and control stability

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