Transient and Steady State Response - Control Systems Engineering
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Apr 26, 2024
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About This Presentation
. Two crucial aspects of this behavior are transient and steady-state responses. These concepts encapsulate how a system behaves over time, from the moment an input is applied to when the system settles into a stable state. The transient response of a system characterizes its behavior during the ini...
Slide Content
Transient and Steady
state Response
state Response
System Response
•Aftertheengineerobtainsamathematicalrepresentation(model)ofa
subsystem,thesubsystemisanalyzedforitstransientandsteady-state
responsestoseeifthesecharacteristicsyieldthedesiredbehavior.
•Inanalyzinganddesigningcontrolsystems,wemusthaveabasisof
comparisonofperformanceofvariouscontrolsystems.Thisbasis
maybesetupbyspecifyingparticulartestinputsignalsandby
comparingtheresponsesofvarioussystemstotheseinputsignals.
Withthesetestsignals,mathematicalandexperimentalanalysesof
controlsystemscanbecarriedouteasily,sincethesignalsarevery
simplefunctionsoftime.
•Aftertheengineerobtainsamathematicalrepresentation(model)ofa
subsystem,thesubsystemisanalyzedforitstransientandsteady-state
responsestoseeifthesecharacteristicsyieldthedesiredbehavior.
•Inanalyzinganddesigningcontrolsystems,wemusthaveabasisof
comparisonofperformanceofvariouscontrolsystems.Thisbasis
maybesetupbyspecifyingparticulartestinputsignalsandby
comparingtheresponsesofvarioussystemstotheseinputsignals.
Withthesetestsignals,mathematicalandexperimentalanalysesof
controlsystemscanbecarriedouteasily,sincethesignalsarevery
simplefunctionsoftime.
Standard test input signals
The u(t)shows that the response is zero until t = 0
The u(t)shows that the response is zero until t = 0
Standard test input signals
•An impulse; is infinite at t = 0 and zero else where, area under the
unit impulse is 1. An approximation of this type of waveform is used
to place initial energy into a system so that the response due to that
initial energy is only the transient response of a system.
•Astepinput;representsaconstantcommand,suchasposition,
velocity,oracceleration.Typicallythestepinputcommandisofthe
sameformastheoutput.Forexample,ifthesystem'soutputis
position,thestepinputrepresentsadesiredposition,andtheoutput
representstheactualposition.Thedesignerusesstepinputsbecause
boththetransientresponseandthesteady-stateresponseareclearly
visibleandcanbeevaluated.
•An impulse; is infinite at t = 0 and zero else where, area under the
unit impulse is 1. An approximation of this type of waveform is used
to place initial energy into a system so that the response due to that
initial energy is only the transient response of a system.
•Astepinput;representsaconstantcommand,suchasposition,
velocity,oracceleration.Typicallythestepinputcommandisofthe
sameformastheoutput.Forexample,ifthesystem'soutputis
position,thestepinputrepresentsadesiredposition,andtheoutput
representstheactualposition.Thedesignerusesstepinputsbecause
boththetransientresponseandthesteady-stateresponseareclearly
visibleandcanbeevaluated.
Standard test input signals
•Theramp-inputrepresentsalinearlyincreasingcommand.For
example,ifthesystem'soutputisposition,theinputramprepresentsa
linearlyincreasingposition,suchasthatfoundwhentrackinga
satellitemovingacrosstheskywithaconstantspeed.Theresponseto
aninputramptestsignalyieldsadditionalinformationaboutthe
steady-stateerror.
•Parabolic inputs; is also used to evaluate a system's steady state error.
•Sinusoidal inputs-can also be used to test a physical system to arrive
at a mathematical model.
•Theramp-inputrepresentsalinearlyincreasingcommand.For
example,ifthesystem'soutputisposition,theinputramprepresentsa
linearlyincreasingposition,suchasthatfoundwhentrackinga
satellitemovingacrosstheskywithaconstantspeed.Theresponseto
aninputramptestsignalyieldsadditionalinformationaboutthe
steady-stateerror.
•Parabolic inputs; is also used to evaluate a system's steady state error.
•Sinusoidal inputs-can also be used to test a physical system to arrive
at a mathematical model.
Whichofthesetypicalinputsignalstouseforanalyzingsystem
characteristicsmaybedeterminedbytheformoftheinputthatthe
systemwillbesubjectedtomostfrequentlyundernormaloperation.
Iftheinputstoacontrolsystemaregraduallychangingfunctionsof
time,thenarampfunctionoftimemaybeagoodtestsignal.Similarly,
ifasystemissubjectedtosuddendisturbances,astepfunctionoftime
maybeagoodtestsignalandforasystemsubjectedtoshockinputs,an
impulsefunctionmaybebest.
Onceacontrolsystemisdesignedonthebasisoftestsignals,the
performanceofthesysteminresponsetoactualinputsisgenerally
satisfactory
Standard test input signals
characteristicsmaybedeterminedbytheformoftheinputthatthe
systemwillbesubjectedtomostfrequentlyundernormaloperation.
Iftheinputstoacontrolsystemaregraduallychangingfunctionsof
time,thenarampfunctionoftimemaybeagoodtestsignal.Similarly,
ifasystemissubjectedtosuddendisturbances,astepfunctionoftime
maybeagoodtestsignalandforasystemsubjectedtoshockinputs,an
impulsefunctionmaybebest.
Onceacontrolsystemisdesignedonthebasisoftestsignals,the
performanceofthesysteminresponsetoactualinputsisgenerally
satisfactory
Standard test input signals
•The output response of a system is the sum of two responses:
•the forced response also called steady-state response or particular solution and
•the natural response or homogeneous solution.
•Inthischapterwewilldiscusstheresponseoffirstandsecondorder
system.Theorderreferstotheorderoftheequivalentdifferential
equationrepresentingthesystem,theorderofthedenominatorofthe
transferfunctionaftercancellationofcommonfactorsinthe
numeratororthenumberofsimultaneousfirst-orderequations
requiredforthestate-spacerepresentation.
System Response
•the forced response also called steady-state response or particular solution and
•the natural response or homogeneous solution.
•Inthischapterwewilldiscusstheresponseoffirstandsecondorder
system.Theorderreferstotheorderoftheequivalentdifferential
equationrepresentingthesystem,theorderofthedenominatorofthe
transferfunctionaftercancellationofcommonfactorsinthe
numeratororthenumberofsimultaneousfirst-orderequations
requiredforthestate-spacerepresentation.
System Response
Poles and Zeros of a First-Order System
The use of poles and zeros and their relationship to the time response of
a system is a technique used to derive the desired result by inspection.
•Poles of a Transfer Function
The poles of a transfer function are the values of the Laplace transform
variable, s, that cause the transfer function to become infinite.In other
words they are the roots of denominator of the transfer function.
•Zeros of a Transfer Function
The zeros of a transfer function are the values of the Laplace transform
variable, s,that cause the transfer function to become zero.In other
words they are the roots of the numerator of the transfer function.
The use of poles and zeros and their relationship to the time response of
a system is a technique used to derive the desired result by inspection.
•Poles of a Transfer Function
The poles of a transfer function are the values of the Laplace transform
variable, s, that cause the transfer function to become infinite.In other
words they are the roots of denominator of the transfer function.
•Zeros of a Transfer Function
The zeros of a transfer function are the values of the Laplace transform
variable, s,that cause the transfer function to become zero.In other
words they are the roots of the numerator of the transfer function.
Poles and Zeros of a First-Order System
•Given the transfer function, G(s) with a unit step input;
•a pole exists at s= -5 and zero exists at -2
•unit step response of the system can be found as follows;
•Thus;
•Given the transfer function, G(s) with a unit step input;
•a pole exists at s= -5 and zero exists at -2
•unit step response of the system can be found as follows;
•Thus;
Poles and Zeros of a First-Order System
Pole-zero plot of the system
Poles and zeros of the input and system system
Pole-zero plot of the system
Poles and zeros of the input and system system
Poles and Zeros of a First-Order System
Conclusion;
•Apoleoftheinputfunctiongeneratestheformoftheforcedresponse
(thatis,thepoleattheorigingeneratedastepfunctionattheoutput).
•Apoleofthetransferfunctiongeneratestheformofthenatural
response.
•Apoleontherealaxisgeneratesanexponentialresponseoftheform
�
−??????�
where-??????isthepolelocationontherealaxis.Thus,thefartherto
theleftapoleisonthenegativerealaxis,thefastertheexponential
transientresponsewilldecaytozero.
•Thezerosandpolesgeneratetheamplitudesforboththeforcedand
naturalresponses.
Conclusion;
•Apoleoftheinputfunctiongeneratestheformoftheforcedresponse
(thatis,thepoleattheorigingeneratedastepfunctionattheoutput).
•Apoleofthetransferfunctiongeneratestheformofthenatural
response.
•Apoleontherealaxisgeneratesanexponentialresponseoftheform
�
−??????�
where-??????isthepolelocationontherealaxis.Thus,thefartherto
theleftapoleisonthenegativerealaxis,thefastertheexponential
transientresponsewilldecaytozero.
•Thezerosandpolesgeneratetheamplitudesforboththeforcedand
naturalresponses.
Unit step Response of first order system
•A first-order system without zeros
can be described by the transfer
function;
??????
���=
1
��+1
•If the input is a unit step, where,
��=
1
�
,the Laplace transform of
the step response is;
��=�(�)??????�=
1
�(��+1)
Taking the inverse transform, the step
response is given by;
��=�
��+�
??????�=�−�
−
�
�
•A first-order system without zeros
can be described by the transfer
function;
??????
���=
1
��+1
•If the input is a unit step, where,
��=
1
�
,the Laplace transform of
the step response is;
��=�(�)??????�=
1
�(��+1)
Taking the inverse transform, the step
response is given by;
��=�
��+�
??????�=�−�
−
�
�
Unit step Response of first order system
Equation
��=1−�
−
??????
??????
states that initially the output ��is zero and finally it becomes unity.
One important characteristic of such an exponential response curve ��
is that at �=�the value of ��is 0.632,or the response ��has
reached 63.2%of its total change. That is,
��=1−�
−1
=0.632
Equation
��=1−�
−
??????
??????
states that initially the output ��is zero and finally it becomes unity.
One important characteristic of such an exponential response curve ��
is that at �=�the value of ��is 0.632,or the response ��has
reached 63.2%of its total change. That is,
��=1−�
−1
=0.632
Performance specification of first order system
for unit step input
Time constant �
�:
Is the time for�
−
�
�to decay to 37%of
its initial value. Or, the time constant is
the time it takes for the step response to
rise to 63%of its final value.
At �=�,�
−
�
�=�
−1
=0.37
Or�−�
−
�
�=0.63
the smaller the time constant �, the faster
the system response. Another important
characteristic of the exponential response
curve is that the slope of the tangent line
at �=0is
1
??????,since
We can say pole is located at
reciprocal of time constant
.
The output would reach the final value at �=�if
it maintained its initial speed of response
for unit step input
Time constant �
�:
Is the time for�
−
�
�to decay to 37%of
its initial value. Or, the time constant is
the time it takes for the step response to
rise to 63%of its final value.
At �=�,�
−
�
�=�
−1
=0.37
Or�−�
−
�
�=0.63
the smaller the time constant �, the faster
the system response. Another important
characteristic of the exponential response
curve is that the slope of the tangent line
at �=0is
1
??????,since
We can say pole is located at
reciprocal of time constant
.
The output would reach the final value at �=�if
it maintained its initial speed of response
Performance specification of first order
system for unit step input
Rise Time,�
�
•is defined as the time for the
waveform to go from 0.1 to 0.9
of its final value.
�
�=�
??????�%−�
��%
�
�=�.���−�.���=�.��
system for unit step input
Rise Time,�
�
•is defined as the time for the
waveform to go from 0.1 to 0.9
of its final value.
�
�=�
??????�%−�
��%
�
�=�.���−�.���=�.��
Performance specification of first order
system for unit step input
2%, Settling time, �
�
•Settling time is defined as the
time for the response to reach,
and stay within, 2% of its final
value.
�
�=��
system for unit step input
2%, Settling time, �
�
•Settling time is defined as the
time for the response to reach,
and stay within, 2% of its final
value.
�
�=��
Performance specification of first order
system for unit step input
Example [page 160, Norman Nise]
A system has a transfer function, ??????�=
50
�+50
.Find the time constant,
�
�settling time,�
�and rise time,�
�of it’s step response.
Ans:
�
�=0.02s
�
�=0.08�
�
�=0.044�
system for unit step input
Example [page 160, Norman Nise]
A system has a transfer function, ??????�=
50
�+50
.Find the time constant,
�
�settling time,�
�and rise time,�
�of it’s step response.
Ans:
�
�=0.02s
�
�=0.08�
�
�=0.044�
Unit-Ramp Response of First-Order Systems
Unit-Impulse Response of First-Order
Systems
Systems
Second Order System
Whereas varying a first-order system's parameter simply changes the speedof
the response, changes in the parameters of a second-order system can change
the formof the response. Two quantities i.e. natural frequency and damping
ratio, can be used to describe the characteristics of the second-order transient
response just as time constants describe the first-order system response.
Natural Frequency, ??????
??????
The natural frequency of a second-order system is the frequency of oscillation
of the system without damping.
Damping Ratio,�
Is a quantity that compares the exponential decay frequency of the envelope
to the natural frequency. This ratio is constant regardless of the time scale of
the response.
Hence, �=
��������????????????����??????����������
�??????���??????���������
=
??????
????????????
Whereas varying a first-order system's parameter simply changes the speedof
the response, changes in the parameters of a second-order system can change
the formof the response. Two quantities i.e. natural frequency and damping
ratio, can be used to describe the characteristics of the second-order transient
response just as time constants describe the first-order system response.
Natural Frequency, ??????
??????
The natural frequency of a second-order system is the frequency of oscillation
of the system without damping.
Damping Ratio,�
Is a quantity that compares the exponential decay frequency of the envelope
to the natural frequency. This ratio is constant regardless of the time scale of
the response.
Hence, �=
��������????????????����??????����������
�??????���??????���������
=
??????
????????????
Second Order System
General second order transfer function;
Consider the following servo system
Typeequationhere.
(a) Servo system;
(b) block diagram;
(c) simplified block
diagram.
The closed-loop transfer function of the system is
�(�)
??????(�)
=
�
��
2
+��+�
=
�
�
�
2
+
??????
�
�+
�
�
In the transient-response analysis, it is convenient to
write
where ??????is called the exponential decay frequency or
attenuation; ??????
??????, the undampednatural frequency; and ??????,
the damping ratio of the system.
General second order transfer function;
Consider the following servo system
Typeequationhere.
(a) Servo system;
(b) block diagram;
(c) simplified block
diagram.
The closed-loop transfer function of the system is
�(�)
??????(�)
=
�
��
2
+��+�
=
�
�
�
2
+
??????
�
�+
�
�
In the transient-response analysis, it is convenient to
write
where ??????is called the exponential decay frequency or
attenuation; ??????
??????, the undampednatural frequency; and ??????,
the damping ratio of the system.
Second Order System
By definition, the natural frequency, ??????
�, is the frequency of oscillation of this
system.
??????
�=
�
�and
�
�=??????
�
2
??????=
���������??????����??????����������
�??????���??????���������(�??????�/���)
=
??????
??????
�
∴??????����??????������������������������������;
??????�=
??????
??????
�
�
�
+�????????????
??????�+??????
??????
�
By definition, the natural frequency, ??????
�, is the frequency of oscillation of this
system.
??????
�=
�
�and
�
�=??????
�
2
??????=
���������??????����??????����������
�??????���??????���������(�??????�/���)
=
??????
??????
�
∴??????����??????������������������������������;
??????�=
??????
??????
�
�
�
+�????????????
??????�+??????
??????
�
Second Order System
•Thedynamicbehaviorofthesecond-ordersystemcanthenbe
describedintermsoftwoparameters??????and??????
??????.Asecond-order
systemcandisplaycharacteristicsmuchlikeafirst-ordersystemor
dependingoncomponentvalues,displaydampedorpureoscillations
foritstransientresponse.
•Thepolestellustheformoftheresponsewithoutthetedious
calculationoftheinverseLaplacetransform.
•Thedynamicbehaviorofthesecond-ordersystemcanthenbe
describedintermsoftwoparameters??????and??????
??????.Asecond-order
systemcandisplaycharacteristicsmuchlikeafirst-ordersystemor
dependingoncomponentvalues,displaydampedorpureoscillations
foritstransientresponse.
•Thepolestellustheformoftheresponsewithoutthetedious
calculationoftheinverseLaplacetransform.
Second order Unit-stepResponse
Critically damped response
•The system TF has two real and repeated polesat −??????
??????
•Has a damping ratio,�=1
•One term of the natural response is an exponential whose time constant is equal to the reciprocal of
the pole location. Another term is the product of time, t, and an exponential with time constant
equal to the reciprocal of the pole location.
��=�−�
−??????
??????�
−??????
??????��
−??????
??????�
,
•Is the fastest response without overshoot
poles on s-plane step response
Critically damped response
•The system TF has two real and repeated polesat −??????
??????
•Has a damping ratio,�=1
•One term of the natural response is an exponential whose time constant is equal to the reciprocal of
the pole location. Another term is the product of time, t, and an exponential with time constant
equal to the reciprocal of the pole location.
��=�−�
−??????
??????�
−??????
??????��
−??????
??????�
,
•Is the fastest response without overshoot
poles on s-plane step response
Second order Unit-stepResponse
Under-damped second order system unit step response
•Has two complex poles at −??????
�±�??????
�, where, ??????
�=??????
�1−�
2
•Has damping ratio,0<�<1
•The natural response is damped sinusoid with an exponential envelope whose time constant is
equal to the reciprocal of the pole's real part. The radian frequency of the sinusoid, the damped
frequency of oscillation, is equal to the imaginarypart of the poles,
��=�−
�
�−??????
�
�
−????????????
??????�
�????????????(??????
��+??????),Where??????-is phase angle
poles on s-plane step response
Under-damped second order system unit step response
•Has two complex poles at −??????
�±�??????
�, where, ??????
�=??????
�1−�
2
•Has damping ratio,0<�<1
•The natural response is damped sinusoid with an exponential envelope whose time constant is
equal to the reciprocal of the pole's real part. The radian frequency of the sinusoid, the damped
frequency of oscillation, is equal to the imaginarypart of the poles,
��=�−
�
�−??????
�
�
−????????????
??????�
�????????????(??????
��+??????),Where??????-is phase angle
poles on s-plane step response
Second order Unit-stepResponse
Under-damped Response
approaches a steady-state value via a transient
response
Thetransientresponseconsistsofan
exponentiallydecayingamplitudegeneratedby
therealpartofthesystempoletimesa
sinusoidalwaveformgeneratedbythe
imaginarypartofthesystempole.Thetime
constantoftheexponentialdecayisequaltothe
reciprocaloftherealpartofthesystempole.
Thevalueoftheimaginarypartistheactual
frequencyofthesinusoid,asdepictedinfigure
ontheright.Thissinusoidalfrequencyisgiven
thenamedampedfrequencyofoscillation,??????
�.
Finally,thesteady-stateresponse(unitstep)was
generatedbytheinputpolelocatedattheorigin.
UnderdampedResponse
Under-damped Response
approaches a steady-state value via a transient
response
Thetransientresponseconsistsofan
exponentiallydecayingamplitudegeneratedby
therealpartofthesystempoletimesa
sinusoidalwaveformgeneratedbythe
imaginarypartofthesystempole.Thetime
constantoftheexponentialdecayisequaltothe
reciprocaloftherealpartofthesystempole.
Thevalueoftheimaginarypartistheactual
frequencyofthesinusoid,asdepictedinfigure
ontheright.Thissinusoidalfrequencyisgiven
thenamedampedfrequencyofoscillation,??????
�.
Finally,thesteady-stateresponse(unitstep)was
generatedbytheinputpolelocatedattheorigin.
UnderdampedResponse
Second order Unit-stepResponse
Underdamped second order system unit step response
C�=
????????????
2
�(�
2
+2??????????????????�+????????????
2
)
=
�
�
+
��+�
�
2
+2??????????????????�+????????????
2
,??????���=1,�=−1,�=−2�??????
�
Hence; ��=
1
�
−
(�+2??????????????????)
�
2
+2??????????????????+????????????
2
Which can be written as;
C�=
1
�
−
�+????????????
??????+
??????
1−??????
2
??????
??????1−??????
2
�+????????????
??????
2
+??????
??????
2
1−??????
2
=
1
�
−
�+????????????
??????
�+????????????
??????
2
+??????
??????
2
1−??????
2
−
??????
1−??????
2
??????
??????1−??????
2
�+????????????
??????
2
+??????
??????
2
1−??????
2
Underdamped second order system unit step response
C�=
????????????
2
�(�
2
+2??????????????????�+????????????
2
)
=
�
�
+
��+�
�
2
+2??????????????????�+????????????
2
,??????���=1,�=−1,�=−2�??????
�
Hence; ��=
1
�
−
(�+2??????????????????)
�
2
+2??????????????????+????????????
2
Which can be written as;
C�=
1
�
−
�+????????????
??????+
??????
1−??????
2
??????
??????1−??????
2
�+????????????
??????
2
+??????
??????
2
1−??????
2
=
1
�
−
�+????????????
??????
�+????????????
??????
2
+??????
??????
2
1−??????
2
−
??????
1−??????
2
??????
??????1−??????
2
�+????????????
??????
2
+??????
??????
2
1−??????
2
Second order Unit-stepResponse
Under-damped second order system unit step response
Substituting ??????
�=??????
�1−�
2
and taking inverse Laplace transform;
and
��=1−�
−????????????
??????�
(cos??????
�t+
??????
�−??????
�
sin??????
�t),
��=�−
�
�−??????
�
�
−??????????????????�
�????????????(??????
��+??????), Whereφ=�??????�
−1
(
1−??????
2
??????
)
Under-damped second order system unit step response
Substituting ??????
�=??????
�1−�
2
and taking inverse Laplace transform;
and
��=1−�
−????????????
??????�
(cos??????
�t+
??????
�−??????
�
sin??????
�t),
��=�−
�
�−??????
�
�
−??????????????????�
�????????????(??????
��+??????), Whereφ=�??????�
−1
(
1−??????
2
??????
)
Second-order underdampedunit-step
responses for different damping ratio values
The lower the value of�, the more
oscillatory the response will be.
��=�−
�
�−??????
�
�
−????????????
??????�
�????????????(??????
��+??????),
.
responses for different damping ratio values
The lower the value of�, the more
oscillatory the response will be.
��=�−
�
�−??????
�
�
−????????????
??????�
�????????????(??????
��+??????),
.
Second order Unit-stepResponse
Undamped response
•Has two imaginary poles at ±�??????
??????
•Has a damping ratio,�=0
•Natural response is undamped sinusoid with radian frequency equal to the
imaginary part of the poles, the absence of a real part in the pole pair
corresponds to an exponential that does not decay.
??????�=??????�????????????(??????
??????�+??????),is naturalresponse , where A-is constant
poles on s-plane step response
Undamped response
•Has two imaginary poles at ±�??????
??????
•Has a damping ratio,�=0
•Natural response is undamped sinusoid with radian frequency equal to the
imaginary part of the poles, the absence of a real part in the pole pair
corresponds to an exponential that does not decay.
??????�=??????�????????????(??????
??????�+??????),is naturalresponse , where A-is constant
poles on s-plane step response
Second order Unit-stepResponse
Overdamped Response
•Has real and distinct poles at ,−??????
1??????��−??????
2
•Has a damping ratio,�>1
•Natural response is two exponentials with time constants equal to the reciprocal of the
pole locations;
��=�−��
−??????��
+��
−??????��
, where �and �??????�������??????���
poles on s-plane step response
Overdamped Response
•Has real and distinct poles at ,−??????
1??????��−??????
2
•Has a damping ratio,�>1
•Natural response is two exponentials with time constants equal to the reciprocal of the
pole locations;
��=�−��
−??????��
+��
−??????��
, where �and �??????�������??????���
poles on s-plane step response
Second order system with zero
??????�=
(�+�)
??????
�
2
�
�
2
+2�??????
��+??????
�
2
Which can be written as;
??????�=
????????????
2
�
2
+2????????????
??????�+??????
??????
2
+
�
�
(
????????????
2
�
2
+2????????????
??????�+??????
??????
2
)
The response will be,
�
��=��+
1
�
�
��
��
Theeffectofzeroiscontributepronounced
earlypeakingeffecttotheresponse.Thecloser
thezerotothedominantpole,themore
pronouncedthepeakingphenomenon.
Consider a 2
nd
order system with
poles �
12=−1±�2.83
??????�=
(�+�)
??????
�
2
�
�
2
+2�??????
��+??????
�
2
Which can be written as;
??????�=
????????????
2
�
2
+2????????????
??????�+??????
??????
2
+
�
�
(
????????????
2
�
2
+2????????????
??????�+??????
??????
2
)
The response will be,
�
��=��+
1
�
�
��
��
Theeffectofzeroiscontributepronounced
earlypeakingeffecttotheresponse.Thecloser
thezerotothedominantpole,themore
pronouncedthepeakingphenomenon.
Consider a 2
nd
order system with
poles �
12=−1±�2.83
Performance specification of unit step
Second order under damped Response
Other parameters associated with the
underdamped response are rise time,
peak time, percent overshoot and
settling time.
•Rise time, �
�;The time required for
the waveform to go from 0 to 100% of
the final value.
�
�=
??????−??????
??????�
??????−���??????��??????�
•Peak time, �
??????; The time required to
reach the first,or maximum, peak.
�
??????=
??????
??????
??????�−�
�
Second order under damped Response
Other parameters associated with the
underdamped response are rise time,
peak time, percent overshoot and
settling time.
•Rise time, �
�;The time required for
the waveform to go from 0 to 100% of
the final value.
�
�=
??????−??????
??????�
??????−���??????��??????�
•Peak time, �
??????; The time required to
reach the first,or maximum, peak.
�
??????=
??????
??????
??????�−�
�
Performance specification of unit step
Second order under damped Response
•Percent overshoot, %OS. The
amount that the waveform
overshoots the steady state,or final,
value at the peak time expressed as
a percentage of the steady state
value.
%??????�=�
−(
????????????
�−??????
�
)
���
and
�=
−????????????(
%??????�
���
)
??????
�
+????????????
�
(
%??????�
���
)
Second order under damped Response
•Percent overshoot, %OS. The
amount that the waveform
overshoots the steady state,or final,
value at the peak time expressed as
a percentage of the steady state
value.
%??????�=�
−(
????????????
�−??????
�
)
���
and
�=
−????????????(
%??????�
���
)
??????
�
+????????????
�
(
%??????�
���
)
Performance specification of unit step
Second order under damped Response
•�%, Settling time �
�. The time
required for the transient's
damped oscillations to reach and
stay within �%of the steady-
state value.
�
�=
�
�??????
??????
Second order under damped Response
•�%, Settling time �
�. The time
required for the transient's
damped oscillations to reach and
stay within �%of the steady-
state value.
�
�=
�
�??????
??????
Exercise
Given the transfer function;
??????�=
100
�
2
+15�+100
1. Find the unit step response
2. What is the nature of the response?
3. Find �
�,�
�, �
�??????��%��and
Given the transfer function;
??????�=
100
�
2
+15�+100
1. Find the unit step response
2. What is the nature of the response?
3. Find �
�,�
�, �
�??????��%��and
Exercise-Answer
Answer:��=�+��
−7.5�
sin6.61�+??????
??????
�=10,�=0.75,�
�≈0.22,�
�,=0.475,
�
�=0.533,%��=2.84
The response is underdamped
Answer:��=�+��
−7.5�
sin6.61�+??????
??????
�=10,�=0.75,�
�≈0.22,�
�,=0.475,
�
�=0.533,%��=2.84
The response is underdamped
Exercise 2
Consider the system shown in figure below, where �=0.6and ??????
�=
5�??????�/���.
Obtain the closed loop TF, rise time, peak time, percent overshoot, and
2% settling time, when the unity feedback system is subjected to a unit-
step input.
Plot the step response in MATLAB
Consider the system shown in figure below, where �=0.6and ??????
�=
5�??????�/���.
Obtain the closed loop TF, rise time, peak time, percent overshoot, and
2% settling time, when the unity feedback system is subjected to a unit-
step input.
Plot the step response in MATLAB
Steady State Error
Imperfectionsinthesystemcomponents,suchasstaticfriction,backlash,
andamplifierdrift,aswellasagingordeterioration,willcauseerrorsat
steadystate.Inthissection,however,weshallnotdiscusserrorsdueto
imperfectionsinthesystemcomponents.Rather,weshallinvestigatea
typeofsteady-stateerrorthatiscausedbytheincapabilityofasystemto
followparticulartypesofinputs.
Controlsystemsmaybeclassifiedaccordingtotheirabilitytofollow
stepinputs,rampinputs,parabolicinputs,andsoon.Thisisareasonable
classificationscheme,becauseactualinputsmayfrequentlybe
consideredcombinationsofsuchinputs.Themagnitudesofthesteady-
stateerrorsduetotheseindividualinputsareindicativeofthegoodness
ofthesystem.
Imperfectionsinthesystemcomponents,suchasstaticfriction,backlash,
andamplifierdrift,aswellasagingordeterioration,willcauseerrorsat
steadystate.Inthissection,however,weshallnotdiscusserrorsdueto
imperfectionsinthesystemcomponents.Rather,weshallinvestigatea
typeofsteady-stateerrorthatiscausedbytheincapabilityofasystemto
followparticulartypesofinputs.
Controlsystemsmaybeclassifiedaccordingtotheirabilitytofollow
stepinputs,rampinputs,parabolicinputs,andsoon.Thisisareasonable
classificationscheme,becauseactualinputsmayfrequentlybe
consideredcombinationsofsuchinputs.Themagnitudesofthesteady-
stateerrorsduetotheseindividualinputsareindicativeofthegoodness
ofthesystem.
Steady State Error
Consider the unity-feedback control system with the following open-
looptransfer function G(s):
Itinvolvestheterm�
??????
inthedenominator,representingapoleof
multiplicity�attheorigin.Thepresentclassificationschemeisbasedon
thenumberofintegrationsindicatedbytheopen-looptransferfunction.
Asystemiscalledtype0,type1,type2,,,ifN=0,N=1,N=2,,,
respectively.Notethatthisclassificationisdifferentfromthatofthe
orderofasystem.Asthetypenumberisincreased,accuracyis
improved.However,increasingthetypenumberaggravatesthestability
problem.Acompromisebetweensteady-stateaccuracyandrelative
stabilityisalwaysnecessary.
Consider the unity-feedback control system with the following open-
looptransfer function G(s):
Itinvolvestheterm�
??????
inthedenominator,representingapoleof
multiplicity�attheorigin.Thepresentclassificationschemeisbasedon
thenumberofintegrationsindicatedbytheopen-looptransferfunction.
Asystemiscalledtype0,type1,type2,,,ifN=0,N=1,N=2,,,
respectively.Notethatthisclassificationisdifferentfromthatofthe
orderofasystem.Asthetypenumberisincreased,accuracyis
improved.However,increasingthetypenumberaggravatesthestability
problem.Acompromisebetweensteady-stateaccuracyandrelative
stabilityisalwaysnecessary.
Steady State Error
For the system shown below the closed-loop transfer function is
For the system shown below the closed-loop transfer function is
Steady State Error
Steady State Error constants
Inagivensystem,theoutputmaybetheposition,velocity,pressure,
temperature,orthelike.Thephysicalformoftheoutput,however,is
immaterialtothepresentanalysis.Therefore,inwhatfollows,weshall
calltheoutput“position,”therateofchangeoftheoutput“velocity,”
andsoon.
Theerrorconstants??????
�,??????
�,and??????
??????describetheabilityofaunity-
feedbacksystemtoreduceoreliminatesteady-stateerror.Therefore,
theyareindicativeofthesteady-stateperformance.Itisgenerally
desirabletoincreasetheerrorconstantsbyaddingintegratorstothefeed
forwardpath,whilemaintainingthetransientresponsewithinan
acceptablerange.
Steady State Error constants
Inagivensystem,theoutputmaybetheposition,velocity,pressure,
temperature,orthelike.Thephysicalformoftheoutput,however,is
immaterialtothepresentanalysis.Therefore,inwhatfollows,weshall
calltheoutput“position,”therateofchangeoftheoutput“velocity,”
andsoon.
Theerrorconstants??????
�,??????
�,and??????
??????describetheabilityofaunity-
feedbacksystemtoreduceoreliminatesteady-stateerror.Therefore,
theyareindicativeofthesteady-stateperformance.Itisgenerally
desirabletoincreasetheerrorconstantsbyaddingintegratorstothefeed
forwardpath,whilemaintainingthetransientresponsewithinan
acceptablerange.
Steady State Error
Steady State Error Constants
Steady State Error Constants
Steady State Error
This shows type 0 system is incapable of following a ramp inputin the steady state.
The type 1 system with unity feedback can follow the ramp input with a finite error.
Steady State Error Constants
This shows type 0 system is incapable of following a ramp inputin the steady state.
The type 1 system with unity feedback can follow the ramp input with a finite error.
Steady State Error Constants
Steady State Error
Steady State Error Constants
Steady State Error Constants
Steady-State Error, System type and Error
constants
constants
Example
Figure shows a mechanical vibratory system. When 9Nof force (step
input) is applied to the system, the mass oscillates, as shown in Figure
below.
a) Find transfer function
b) Determine m, b, and k of the system from this response curve. The
displacement x is measured from the equilibrium position.
Figure shows a mechanical vibratory system. When 9Nof force (step
input) is applied to the system, the mass oscillates, as shown in Figure
below.
a) Find transfer function
b) Determine m, b, and k of the system from this response curve. The
displacement x is measured from the equilibrium position.
Stability
InChapter1wehaveseentherequirementsinthedesignofacontrol
system:transientresponse,stability,andsteady-stateerrors.Thusfarwe
havecoveredtransientresponseandsteadystateerror.Wearenow
readytodiscussthenextrequirement,stability.Stabilityisthemost
importantsystemspecification.Ifamootsystemisunstable,transient
responseandsteady-stateerrorsarepoints.
Physically,anunstablesystemwhosenaturalresponsegrowswithout
boundcancausedamagetothesystem,toadjacentproperty,orto
humanlife.Manysystemsaredesignedwithlimitstopstopreventtotal
runaway.Fromtheperspectiveofthetimeresponseplotofaphysical
system,instabilityisdisplayedbytransientsthatgrowwithoutbound
andconsequently,atotalresponsethatdoesapproachasteady-state
valueorotherforcedresponseisstable.
InChapter1wehaveseentherequirementsinthedesignofacontrol
system:transientresponse,stability,andsteady-stateerrors.Thusfarwe
havecoveredtransientresponseandsteadystateerror.Wearenow
readytodiscussthenextrequirement,stability.Stabilityisthemost
importantsystemspecification.Ifamootsystemisunstable,transient
responseandsteady-stateerrorsarepoints.
Physically,anunstablesystemwhosenaturalresponsegrowswithout
boundcancausedamagetothesystem,toadjacentproperty,orto
humanlife.Manysystemsaredesignedwithlimitstopstopreventtotal
runaway.Fromtheperspectiveofthetimeresponseplotofaphysical
system,instabilityisdisplayedbytransientsthatgrowwithoutbound
andconsequently,atotalresponsethatdoesapproachasteady-state
valueorotherforcedresponseisstable.
Stability
There are many definitions for stability, depending upon kind of system or the
point of view.In this course we limit ourselves to linear, time-invariant
systems.
Definitions of stability for linear, time-invariant systems; using natural
response
1.A system is stableif the natural response approaches zero as time approaches infinity.
2.A system is unstableif the natural response approaches infinity as time approaches
infinity
3.A system is marginally stable if the natural response neither decays nor grows but
remains constant or oscillates.
Thus, the definition of stability implies that only the forced response remains
as the natural response approaches zero.
Bounded-input, Bounded Output (BIBO)) definition of stability; using the
total response (BIBO):
1.A system is stable if every bounded input yields a bounded output.
2.A system is unstable if bounded input yields an unbounded output.
There are many definitions for stability, depending upon kind of system or the
point of view.In this course we limit ourselves to linear, time-invariant
systems.
Definitions of stability for linear, time-invariant systems; using natural
response
1.A system is stableif the natural response approaches zero as time approaches infinity.
2.A system is unstableif the natural response approaches infinity as time approaches
infinity
3.A system is marginally stable if the natural response neither decays nor grows but
remains constant or oscillates.
Thus, the definition of stability implies that only the forced response remains
as the natural response approaches zero.
Bounded-input, Bounded Output (BIBO)) definition of stability; using the
total response (BIBO):
1.A system is stable if every bounded input yields a bounded output.
2.A system is unstable if bounded input yields an unbounded output.
Stability
Stable system Unstable system
Stable system Unstable system
Stability
Iftheclosed-loopsystempolesareinthelefthalfofthes-planeand
hencehaveanegativerealpart,thesystemisstable.Andunstable
systemshaveclosed-looptransferfunctionswithatleastonepoleinthe
righthalf-planeand/orpolesofmultiplicitygreaterthanoneonthe
imaginaryaxis.
Marginally stable systems have closed-loop transfer functions with only
imaginary axis poles of multiplicity 1and poles in the left halfplane.
Unfortunatelyitisnotalwaysasimplemattertodetermineifa
feedbackcontrolsystemisstableduetodifficultyofsolvingforthe
rootsofhighordertransferfunction.Thereis,however,another
methodstotestforstabilitywithouthavingtosolvefortherootsofthe
denominator.
Iftheclosed-loopsystempolesareinthelefthalfofthes-planeand
hencehaveanegativerealpart,thesystemisstable.Andunstable
systemshaveclosed-looptransferfunctionswithatleastonepoleinthe
righthalf-planeand/orpolesofmultiplicitygreaterthanoneonthe
imaginaryaxis.
Marginally stable systems have closed-loop transfer functions with only
imaginary axis poles of multiplicity 1and poles in the left halfplane.
Unfortunatelyitisnotalwaysasimplemattertodetermineifa
feedbackcontrolsystemisstableduetodifficultyofsolvingforthe
rootsofhighordertransferfunction.Thereis,however,another
methodstotestforstabilitywithouthavingtosolvefortherootsofthe
denominator.
Routh-HurwithCriterion
Using this method, we can tell how many closed-loop system poles are
in the left half-plane, in the right half-plane, and on the �??????-axis, but we
cannot find their coordinates.
The method requires two steps:
(I)Generate a data table called a Routh table and
(II)interpret the Routh table to tell how many closed-loop system poles
are in the left half-plane, the right half-plane, and on the on �??????-axis.
Using this method, we can tell how many closed-loop system poles are
in the left half-plane, in the right half-plane, and on the �??????-axis, but we
cannot find their coordinates.
The method requires two steps:
(I)Generate a data table called a Routh table and
(II)interpret the Routh table to tell how many closed-loop system poles
are in the left half-plane, the right half-plane, and on the on �??????-axis.
Routh-HurwithCriterion
Generating a Basic Routh Table
Consider the equivalent closed-loop transfer function;
Since we are interested in the system poles, we focus our
attention on the denominator.
WefirstcreatetheRouthtableshown.Beginbylabelingthe
rowswithpowersofsfromthehighestpowerofthe
denominatoroftheclosed-looptransferfunctionto�
�
.
Nextstartwiththecoefficientofthehighestpowerofsin
thedenominatorandlist,horizontallyinthefirstrow,every
othercoefficient.Inthesecondrowlisthorizontally,starting
withthenexthighestpowerofs,everycoefficientthatwas
skippedinthefirstrow.
Initial layout for routhable
Generating a Basic Routh Table
Consider the equivalent closed-loop transfer function;
Since we are interested in the system poles, we focus our
attention on the denominator.
WefirstcreatetheRouthtableshown.Beginbylabelingthe
rowswithpowersofsfromthehighestpowerofthe
denominatoroftheclosed-looptransferfunctionto�
�
.
Nextstartwiththecoefficientofthehighestpowerofsin
thedenominatorandlist,horizontallyinthefirstrow,every
othercoefficient.Inthesecondrowlisthorizontally,starting
withthenexthighestpowerofs,everycoefficientthatwas
skippedinthefirstrow.
Initial layout for routhable
Routh-HurwithCriterion
Generating a Basic Routh Table
Theremainingentriesarefilledinasfollows.Each
entryisanegativedeterminantofentriesintheprevious
tworowsdividedbytheentryinthefirstcolumn
directlyabovethecalculatedrow.Theleft-handcolumn
ofthedeterminantisalwaysthefirstcolumnofthe
previoustworows,andtheright-handcolumnisthe
elementsofthecolumnaboveandtotheright.Thetable
iscompletewhenalloftherowsarecompleteddownto
�
�
.
Routh-Hurwitzcriteriondeclaresthatthenumberof
rootsofthepolynomialthatareintherighthalf-plane
isequaltothenumberofsignchangesinthefirst
column.
Iftheclosed-looptransferfunctionhasallpolesinthe
lefthalfofthes-plane,thesystemisstable.Thus,a
systemisstableiftherearenosignchangesinthefirst
columnoftheRouthtable.
Completed routhable
Generating a Basic Routh Table
Theremainingentriesarefilledinasfollows.Each
entryisanegativedeterminantofentriesintheprevious
tworowsdividedbytheentryinthefirstcolumn
directlyabovethecalculatedrow.Theleft-handcolumn
ofthedeterminantisalwaysthefirstcolumnofthe
previoustworows,andtheright-handcolumnisthe
elementsofthecolumnaboveandtotheright.Thetable
iscompletewhenalloftherowsarecompleteddownto
�
�
.
Routh-Hurwitzcriteriondeclaresthatthenumberof
rootsofthepolynomialthatareintherighthalf-plane
isequaltothenumberofsignchangesinthefirst
column.
Iftheclosed-looptransferfunctionhasallpolesinthe
lefthalfofthes-plane,thesystemisstable.Thus,a
systemisstableiftherearenosignchangesinthefirst
columnoftheRouthtable.
Completed routhable
Routh-HurwithCriterion
Example
Decide whether the following system is stable or not using Routh
stability criterion
Dorm work: Decide whether the following system is stable or not using
Routh stability criterion if of the closed loop transfer function is
??????
��(s)=
1
�
4
+8�
3
+18�
2
+16�+5
Example
Decide whether the following system is stable or not using Routh
stability criterion
Dorm work: Decide whether the following system is stable or not using
Routh stability criterion if of the closed loop transfer function is
??????
��(s)=
1
�
4
+8�
3
+18�
2
+16�+5
Routh-HurwithCriterion: Special cases
First case: Zero only in the first column
Ifthefirstelementofarowiszero,divisionbyzerowouldberequired
toformthenextrowwhichwillresultininfinity.Toavoidthis
phenomenon,anepsilon,�isassignedtoreplacethezerointhefirst
column.Thevalue�isthenallowedtoapproachzerofromeitherthe
positiveorthenegativeside,afterwhichthesignsoftheentriesinthe
firstcolumncanbedetermined.
Example: Determine the stability of closed loop transfer function;
??????
��s=
10
�
5
+2�
4
+3�
3
+6�
2
+5�+3
First case: Zero only in the first column
Ifthefirstelementofarowiszero,divisionbyzerowouldberequired
toformthenextrowwhichwillresultininfinity.Toavoidthis
phenomenon,anepsilon,�isassignedtoreplacethezerointhefirst
column.Thevalue�isthenallowedtoapproachzerofromeitherthe
positiveorthenegativeside,afterwhichthesignsoftheentriesinthe
firstcolumncanbedetermined.
Example: Determine the stability of closed loop transfer function;
??????
��s=
10
�
5
+2�
4
+3�
3
+6�
2
+5�+3
Routh-HurwithCriterion: Special cases
Second case: Entire Row is Zero
SometimeswhilemakingaRouthtable,wefindthatanentirerow
consistsofzerosbecausethereisanevenpolynomialthatisafactor
oftheoriginalpolynomialwithrootsthataresymmetricalaboutthe
origin.Thiscasemustbehandleddifferentlyfromthecaseofazeroin
onlythefirstcolumnofarow.
Example:Decidewhetherthefollowingsystemisstableornotusing
Routhstabilitycriterionifcharacteristicsequationoftheclosedloop
transferfunctionis
�
5
+�
4
+4�
3
+24�
2
+3�+63
Second case: Entire Row is Zero
SometimeswhilemakingaRouthtable,wefindthatanentirerow
consistsofzerosbecausethereisanevenpolynomialthatisafactor
oftheoriginalpolynomialwithrootsthataresymmetricalaboutthe
origin.Thiscasemustbehandleddifferentlyfromthecaseofazeroin
onlythefirstcolumnofarow.
Example:Decidewhetherthefollowingsystemisstableornotusing
Routhstabilitycriterionifcharacteristicsequationoftheclosedloop
transferfunctionis
�
5
+�
4
+4�
3
+24�
2
+3�+63
Routh-HurwithCriterion: Special cases
An entire row of zeros will appear in the Routh
table when a purely even or purely odd
polynomial is a factor of the original polynomial.
Even polynomials only have roots that are
symmetrical about the origin. This symmetry can
occur under three conditions of root position:
(I)The roots are symmetrical and real,(A)
(II)the roots are symmetrical and imaginary, (B)
(III)the roots are quadrantal(C).
Each of these three case or combination of these
cases will generate an even polynomial.
An entire row of zeros will appear in the Routh
table when a purely even or purely odd
polynomial is a factor of the original polynomial.
Even polynomials only have roots that are
symmetrical about the origin. This symmetry can
occur under three conditions of root position:
(I)The roots are symmetrical and real,(A)
(II)the roots are symmetrical and imaginary, (B)
(III)the roots are quadrantal(C).
Each of these three case or combination of these
cases will generate an even polynomial.
Routh-HurwithCriterion: Special cases
AnothercharacteristicoftheRouthtableisthattherowprevioustotherow
ofzeroscontainstheevenpolynomialthatisafactoroftheoriginal
polynomial.Everyentryinthetablefromtheevenpolynomial'srowtothe
endofthechartappliesonlytotheevenpolynomial.Therefore,thenumber
ofsignchangesfromtheevenpolynomialtotheendofthetableequalsthe
numberofright-half-planerootsoftheevenpolynomial.Becauseofthe
symmetryofrootsabouttheorigin,thepolynomialmusthavethesame
numberofleft-half-planerootsasitdoesright-half-plane.Havingforthe
rootsintherightandlefthalf-planes,weknowtheremainingrootsmustbe
onthe�??????-axis.
EveryrowintheRouthtablefromthebeginningofthecharttotherow
containingtheevenpolynomialappliesonlytotheotherfactoroftheoriginal
polynomial.Forthisfactorthenumberofsignchanges,fromthebeginningof
thetabledowntotheevenpolynomial,equalsthenumberofright-half-plain
roots.Theremainingrootsareleft-half-planeroots.Therecanbeno�??????roots
containedintheotherpolynomial.
AnothercharacteristicoftheRouthtableisthattherowprevioustotherow
ofzeroscontainstheevenpolynomialthatisafactoroftheoriginal
polynomial.Everyentryinthetablefromtheevenpolynomial'srowtothe
endofthechartappliesonlytotheevenpolynomial.Therefore,thenumber
ofsignchangesfromtheevenpolynomialtotheendofthetableequalsthe
numberofright-half-planerootsoftheevenpolynomial.Becauseofthe
symmetryofrootsabouttheorigin,thepolynomialmusthavethesame
numberofleft-half-planerootsasitdoesright-half-plane.Havingforthe
rootsintherightandlefthalf-planes,weknowtheremainingrootsmustbe
onthe�??????-axis.
EveryrowintheRouthtablefromthebeginningofthecharttotherow
containingtheevenpolynomialappliesonlytotheotherfactoroftheoriginal
polynomial.Forthisfactorthenumberofsignchanges,fromthebeginningof
thetabledowntotheevenpolynomial,equalsthenumberofright-half-plain
roots.Theremainingrootsareleft-half-planeroots.Therecanbeno�??????roots
containedintheotherpolynomial.
Routh-HurwithCriterion: Special cases
Dorm work:Determine the number of right-half-plane poles in the
closed-loop transfer function;
??????
��s=
10
�
5
+7�
4
+6�
3
+42�
2
+8�+56
Dorm work:Determine the number of right-half-plane poles in the
closed-loop transfer function;
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5
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4
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3
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2
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