Ch5 transient and steady state response analyses(control)

Chapter 5 Transient and steady-state response analyses. From the book (Ogata Modern Control Engineering 5th).
5-1 introduction.
5-2 First-Order System.
5-3 second-order system.
5-6 Routh’s stability criterion.
5-8 Steady-state errors in unity-feedback control systems.

1. CHAPTER 5
TRANSIENT AND STEADY -STATE RESPONSE ANALYSES

2. 5-1 INTRODUCTION

3. 5-1 INTRODUCTION
•Thefirststepinanalyzingacontrolsystemwastoderiveamathematicalmodelofthesystem.
Oncesuchamodelisobtained,variousmethodsareavailablefortheanalysisofsystem
performance.
•Inpractice,theinputsignaltoacontrolsystemisnotknownaheadoftimebutisrandomin
nature,andtheinstantaneousinputcannotbeexpressedanalytically.Onlyinsomespecialcasesis
theinputsignalknowninadvanceandexpressibleanalyticallyorbycurves,suchasinthecaseof
theautomaticcontrolofcuttingtools.
•Inanalyzinganddesigningcontrolsystems,wemusthaveabasisofcomparisonofperformanceof
variouscontrolsystems.Thisbasismaybesetupbyspecifyingparticulartestinputsignalsandby
comparingtheresponsesofvarioussystemstotheseinputsignals.

4. 5-1 INTRODUCTION (CONT.)
•Many design criteria are based on the response to such test signals or on the response of
systems to changes in initial conditions (without any test signals).
•The use of test signals can be justified because of a correlation existing between the
response characteristics of a system to a typical test input signal and the capability of the
system to cope with actual input signals.

5. 5-1 INTRODUCTION (CONT.)
Typical Test Signals.
•The commonly used test input signals are step functions, ramp functions, acceleration functions, impulse
functions, sinusoidal functions, and white noise.
•With these test signals, mathematicaland experimentalanalyses of control systems can be carried out
easily, since the signals are very simple functions of time.
•Which of these typical input signals to use for analyzing system characteristics may be determined by the
form of the inputthat the system will be subjected to most frequently under normal operation.
•If the inputs to a control system are gradually changing functions of time, then a ramp function of time
may be a good test signal.

6. 5-1 INTRODUCTION (CONT.)
Typical Test Signals.(Count.)
•Similarly, if a system is subjected to sudden disturbances, a step function of time may be a
good test signal.
•for a system subjected to shock inputs, an impulse function may be best.
•Once a control system is designed on the basis of test signals, the performance of the
system in response to actual inputs is generally satisfactory. The use of such test signals
enables one to compare the performance of many systems on the same basis.

7. 5-1 INTRODUCTION (CONT.)
Transient Response and Steady-State Response.
•The time response of a control system consists of two parts: the transient response and the steady-state
response.
•By transient response, we mean that which goes from the initial state to the final state.
•By steady-state response, we mean the manner in which the system output behaves as t approaches infinity.
•Thus the system response c(t) may be written as
where the first term on the right-hand side of the equation is the transient response and the second term is the
steady-state response

8. 5-1 INTRODUCTION (CONT.)
Absolute Stability, Relative Stability, and Steady-State Error.
•Indesigningacontrolsystem,wemustbeabletopredictthedynamicbehaviorofthesystemfromaknowledgeofthe
components.
•Themostimportantcharacteristicofthedynamicbehaviorofacontrolsystemisabsolutestability—thatis,whetherthe
systemisstableorunstable.
•Acontrolsystemisinequilibriumif,intheabsenceofanydisturbanceorinput,theoutputstaysinthesamestate.
•Alineartime-invariantcontrolsystemisstableiftheoutputeventuallycomesbacktoitsequilibriumstatewhenthesystemis
subjectedtoaninitialcondition.
•Alineartime-invariantcontrolsystemiscriticallystableifoscillationsoftheoutputcontinueforever.
•Itisunstableiftheoutputdivergeswithoutboundfromitsequilibriumstatewhenthesystemissubjectedtoaninitial
condition.

9. 5-1 INTRODUCTION (CONT.)
Absolute Stability, Relative Stability, and Steady-State Error.(Count.)
•Actually,theoutputofaphysicalsystemmayincreasetoacertainextentbutmaybelimited
bymechanical“stops,”orthesystemmaybreakdownorbecomenonlinearaftertheoutput
exceedsacertainmagnitudesothatthelineardifferentialequationsnolongerapply.
•Importantsystembehavior(otherthanabsolutestability)towhichwemustgivecareful
considerationincludesrelativestabilityandsteady-stateerror.
•Sinceaphysicalcontrolsysteminvolvesenergystorage,theoutputofthesystem,when
subjectedtoaninput,cannotfollowtheinputimmediatelybutexhibitsatransientresponse
beforeasteadystatecanbereached.

10. 5-1 INTRODUCTION (CONT.)
Absolute Stability, Relative Stability, and Steady-State
Error.(Count.)
•Thetransientresponseofapracticalcontrolsystemoftenexhibitsdampedoscillations
beforereachingasteadystate.Iftheoutputofasystematsteadystatedoesnotexactly
agreewiththeinput,thesystemissaidtohavesteadystateerror.
•Thiserrorisindicativeoftheaccuracyofthesystem.Inanalyzingacontrolsystem,we
mustexaminetransient-responsebehaviorandsteady-statebehavior.

11. 5-1 INTRODUCTION (CONT.)
transient response of a practical control system.

12. 5-1 INTRODUCTION (CONT.)

13. 5-2 FIRST-ORDER SYSTEM

14. 5-2 FIRST-ORDER SYSTEM
•Consider the first-order system shown below. Physically, this system may represent an
RC circuit, thermal system, or the like.
•A simplified block diagram is shown below.

15. 5-2 FIRST-ORDER SYSTEM (CONT.)
•The input-output relationship is given by
•Inthefollowing,weshallanalyzethesystemresponsestosuchinputsastheunit-step,
unit-ramp,andunit-impulsefunctions.Theinitialconditionsareassumedtobezero.
•Notethatallsystemshavingthesametransferfunctionwillexhibitthesameoutputin
responsetothesameinput.
•Foranygivenphysicalsystem,themathematicalresponsecanbegivenaphysical
interpretation.

16. 5-2 FIRST-ORDER SYSTEM
Unit-Step Response of First-Order Systems.
•Since the Laplace transform of the unit-stepfunction is 1/s, substituting R(s)=1/s into
Equation:
•Expanding C(s) into partial fractions gives
•Taking the inverse Laplace transform of Equation, we obtain

17. 5-2 FIRST-ORDER SYSTEM
Unit-Step Response of First-Order Systems. (Cont.)
•Equation(inverseLaplace)statesthatinitiallytheoutputc(t)iszeroandfinallyitbecomes
unity.Oneimportantcharacteristicofsuchanexponentialresponsecurvec(t)isthatatt=T
thevalueofc(t)is0.632,ortheresponsec(t)hasreached63.2%ofitstotalchange.
•Thismaybeeasilyseenbysubstitutingt=Tinc(t).Thatis,
•Note that the smaller the time constant T, the faster the system response. Another important
characteristic of the exponential response curve is that the slope of the tangent line at t=0 is
1/T, since

18. 5-2 FIRST-ORDER SYSTEM
Unit-Step Response of First-Order Systems. (Cont.)
•Theoutputwouldreachthefinalvalueatt=Tifitmaintaineditsinitialspeedofresponse.
•weseethattheslopeoftheresponsecurvec(t)decreasesmonotonicallyfrom1/Tatt=0tozeroat
t=∞.
•Inonetimeconstant,theexponentialresponsecurvehasgonefrom0to63.2%ofthefinalvalue.Intwo
timeconstants,theresponsereaches86.5%ofthefinalvalue.Att=3T,4T,and5T,theresponsereaches
95%,98.2%,and99.3%,respectively,ofthefinalvalue.Thus,fort4T,theresponseremainswithin2%of
thefinalvalue.
•Inpractice,however,areasonableestimateoftheresponsetimeisthelengthoftimetheresponsecurve
needstoreachandstaywithinthe2%lineofthefinalvalue,orfourtimeconstants.

19. 5-2 FIRST-ORDER SYSTEM
Unit-Ramp Response of First-Order Systems.
•Since the Laplace transform of the unit-ramp function is 1/s2 , we obtain the output of
the system
•Expanding C(s) into partial fractions gives
•Taking the inverse Laplace transform

20. 5-2 FIRST-ORDER SYSTEM
Unit-Ramp Response of First-Order Systems.
•The error signal e(t) is then

21. 5-2 FIRST-ORDER SYSTEM
Unit-Ramp Response of First-Order Systems.
•As t approaches infinity, e
–t/T
approaches zero, and thus the error signal e(t) approaches
T or
•The error in following the unit-ramp input is equal to Tfor sufficiently large t. The smaller
the time constant T, the smaller the steady-state error in following the ramp input.

22. 5-2 FIRST-ORDER SYSTEM
Unit-Impulse Response of First-Order Systems.
•For the unit-impulse input, R(s)=1 and the output of the system
•The inverse Laplace transform of Equation gives

23. 5-2 FIRST-ORDER SYSTEM
Unit-Impulse Response of First-Order Systems.
•For the unit-impulse input, R(s)=1 and the output of the system
•The response curve given by Equation (inverse Laplace) is shown below.

24. 5-2 FIRST-ORDER SYSTEM
An Important Property of Linear Time-Invariant Systems.
•In the analysis above, it has been shown that for the unit-ramp input the output c(t) is
•For the unit-step input, which is the derivative of unit-ramp input, the output c(t) is
•Finally, for the unit-impulse input, which is the derivative of unit-step input, the output c(t)
is

25. 5-2 FIRST-ORDER SYSTEM
An Important Property of Linear Time-Invariant Systems.
•Comparingthesystemresponsestothesethreeinputsclearlyindicatesthatthe
responsetothederivativeofaninputsignalcanbeobtainedbydifferentiatingthe
responseofthesystemtotheoriginalsignal.Itcanalsobeseenthattheresponsetothe
integraloftheoriginalsignalcanbeobtainedbyintegratingtheresponseofthesystemto
theoriginalsignalandbydeterminingtheintegrationconstantfromthezero-output
initialcondition.Thisisapropertyoflineartime-invariantsystems.Lineartime-varying
systemsandnonlinearsystemsdonotpossessthisproperty.

26. 5-3 SECOND-ORDER
SYSTEM

27. 5-2 FIRST-ORDER SYSTEM
•weshallobtaintheresponseofatypicalsecond-ordercontrolsystemtoastepinput,
rampinput,andimpulseinput.Hereweconsideraservosystemasanexampleof
second-ordersystem.
ServoSystem.
•TheservosystemshowninFigure(a)consistsofaproportionalcontrollerandload
elements(inertiaandviscous-frictionelements).Supposethatwewishtocontrolthe
outputpositioncinaccordancewiththeinputpositionr.

28. 5-2 FIRST-ORDER SYSTEM
ServoSystem.(Cont.)
•Theequationfortheloadelementsis
•whereTisthetorqueproducedbytheproportionalcontrollerwhosegainisK.Bytaking
Laplacetransformsofbothsidesofthislastequation,assumingthezeroinitialconditions,
weobtain
•SothetransferfunctionbetweenC(s)andT(s)is

29. 5-2 FIRST-ORDER SYSTEM
ServoSystem.(Cont.)
•Byusingthistransferfunction,Figure(a)canberedrawnasinFigure(b),whichcanbe
modifiedtothatshowninFigure(c).

30. 5-2 FIRST-ORDER SYSTEM
ServoSystem.(Cont.)
•Theclosed-looptransferfunctionisthenobtainedas
•Suchasystemwheretheclosed-looptransferfunctionpossessestwopolesiscalleda
second-ordersystem.(Somesecond-ordersystemsmayinvolveoneortwozeros.)

31. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.
•Theclosed-looptransferfunctionofthesystemshowninFigure(c)is
•whichcanberewrittenas

32. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Theclosed-looppolesarecomplexconjugatesif??????
2
-4JK≥0.Inthetransient-response
analysis,itisconvenienttowrite
•wheresiscalledtheattenuation;??????
??????,theundampednaturalfrequency;and,the
dampingratioofthesystem.ThedampingratiozistheratiooftheactualdampingBto
thecriticaldamping or

33. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thisformiscalledthestandardformofthesecond-ordersystem.

34. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thedynamicbehaviorofthesecond-ordersystemcanthenbedescribedintermsoftwo
parameters??????and??????
??????.If0<??????<1,theclosed-looppolesarecomplexconjugatesandlieinthe
left-halfsplane.
•Thesystemisthencalledunderdamped,andthetransientresponseisoscillatory.If??????=0,the
transientresponsedoesnotdieout.If??????=1,thesystemiscalledcriticallydamped.
Overdampedsystemscorrespondtoz>1.Weshallnowsolvefortheresponseofthesystem
showninFigureinpreviouspagetoaunit-stepinput.Weshallconsiderthreedifferentcases:
theunderdamped(0<??????<1),criticallydamped(??????=1),andoverdamped??????>1)cases.

35. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•(1)Underdampedcase(0<??????<1):Inthiscase,C(s)/R(s)canbewritten
•where??????
??????=??????
??????1−??????
2
Thefrequency??????
??????iscalledthedampednaturalfrequency.Fora
unit-stepinput,C(s)canbewritten.

36. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•TheinverseLaplacetransformofEquationcanbeobtainedeasilyifC(s)iswritteninthe
followingform:
•Laplacetransform,

37. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•HencetheinverseLaplacetransformofEquationisobtainedas
•FromEquation,itcanbeseenthatthefrequencyoftransientoscillationisthedampednatural
frequency??????
??????andthusvarieswiththedampingratio??????.

38. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Theerrorsignalforthissystemisthedifferencebetweentheinputandoutputandis
•Thiserrorsignalexhibitsadampedsinusoidaloscillation.Atsteadystate,oratt=∞,no
errorexistsbetweentheinputandoutput.

39. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Ifthedampingratio??????isequaltozero,theresponsebecomesundampedandoscillations
continueindefinitely.Theresponsec(t)forthezerodampingcasemaybeobtainedby
substituting??????=0in,yielding

40. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thus,fromEquation,weseethat??????
??????representstheundampednaturalfrequencyofthe
system.Thatis,??????
??????isthatfrequencyatwhichthesystemoutputwouldoscillateifthedamping
weredecreasedtozero.Ifthelinearsystemhasanyamountofdamping,theundamped
naturalfrequencycannotbeobservedexperimentally.
•Thefrequency
•thatmaybeobservedisthedampednaturalfrequency??????
??????,whichisequaltoThisfrequencyis
alwayslowerthantheundampednaturalfrequency.Anincreasein??????wouldreducethe
dampednaturalfrequency??????
??????.If??????isincreasedbeyondunity,theresponsebecomes
overdampedandwillnotoscillate.

41. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•(2)Criticallydampedcase(??????=1):IfthetwopolesofC(s)/R(s)areequal,thesystemis
saidtobeacriticallydampedone.
•Foraunit-stepinput,R(s)=1/sandC(s)canbewritten
•TheinverseLaplacetransformofEquationmaybefoundas

42. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thisresultcanalsobeobtainedbyletting??????approachunityinEquationandbyusingthe
followinglimit:
•(3)Overdampedcase(??????>1):Inthiscase,thetwopolesofC(s)/R(s)arenegativereal
andunequal.Foraunit-stepinput,R(s)=1/sandC(s)canbewritten

43. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•TheinverseLaplacetransform:

44. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•where and Thus,theresponsec(t)includes
twodecayingexponentialterms.
•When??????isappreciablygreaterthanunity,oneofthetwodecayingexponentialsdecreases
muchfasterthantheother,sothefaster-decayingexponentialterm(whichcorresponds
toasmallertimeconstant)maybeneglected.

45. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thatis,if–s2islocatedverymuchclosertothejwaxisthan–s1,thenforan
approximatesolutionwemayneglect–s1.Thisispermissiblebecausetheeffectof–s1on
theresponseismuchsmallerthanthatof–s2,sincetheterminvolvings1inEquation
decaysmuchfasterthantheterminvolvings2.
•Oncethefaster-decayingexponentialtermhasdisappeared,theresponseissimilarto
thatofafirst-ordersystem,andC(s)/R(s)maybeapproximatedby

46. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thisapproximateformisadirectconsequenceofthefactthattheinitialvaluesandfinal
valuesofboththeoriginalC(s)/R(s)andtheapproximateoneagreewitheachother.
•WiththeapproximatetransferfunctionC(s)/R(s),theunit-stepresponsecanbeobtainedas:
•Thetimeresponsec(t)isthen

47. 5-2 FIRST-ORDER SYSTEM
StepResponseofSecond-OrderSystem.(Cont.)
•Thisgivesanapproximateunit-stepresponsewhenoneofthepolesofC(s)/R(s)canbe
neglected.

48. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.
•Frequently,theperformancecharacteristicsofacontrolsystemarespecifiedintermsofthe
transientresponsetoaunit-stepinput,sinceitiseasytogenerateandissufficientlydrastic.(If
theresponsetoastepinputisknown,itismathematicallypossibletocomputetheresponse
toanyinput.)
•Thetransientresponseofasystemtoaunit-stepinputdependsontheinitialconditions.For
convenienceincomparingtransientresponsesofvarioussystems,itisacommonpracticeto
usethestandardinitialconditionthatthesystemisatrestinitiallywiththeoutputandall
timederivativesthereofzero.Thentheresponsecharacteristicsofmanysystemscanbeeasily
compared.

49. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.(Cont.)
•Thetransientresponseofapracticalcontrolsystemoftenexhibitsdampedoscillations
beforereachingsteadystate.Inspecifyingthetransient-responsecharacteristicsofa
controlsystemtoaunit-stepinput,itiscommontospecifythefollowing:
•1.Delaytime,td
•2.Risetime,tr 3.Peaktime,tp
•4.Maximumovershoot,Mp 5.Settlingtime,ts

50. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.(Cont.)
•ThesespecificationsaredefinedinwhatfollowsandareshowngraphicallyinFigure.

51. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.(Cont.)
•1.Delaytime,td:Thedelaytimeisthetimerequiredfortheresponsetoreachhalfthe
finalvaluetheveryfirsttime.
•2.Risetime,tr:Therisetimeisthetimerequiredfortheresponsetorisefrom10%
to90%,5%to95%,or0%to100%ofitsfinalvalue.Forunderdampedsecondorder
systems,the0%to100%risetimeisnormallyused.Foroverdampedsystems,the10%to
90%risetimeiscommonlyused.

52. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.(Cont.)
•3.Peaktime,tp:Thepeaktimeisthetimerequiredfortheresponsetoreachthefirstpeakofthe
overshoot.
•4.Maximum(percent)overshoot,Mp:Themaximumovershootisthemaximumpeakvalueofthe
responsecurvemeasuredfromunity.Ifthefinalsteady-statevalueoftheresponsediffersfromunity,
thenitiscommontousethemaximumpercentovershoot.Itisdefinedby
Theamountofthemaximum(percent)overshootdirectlyindicatestherelativestabilityofthesystem.

53. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.(Cont.)
•5.Settlingtime,ts:Thesettlingtimeisthetimerequiredfortheresponsecurvetoreach
andstaywithinarangeaboutthefinalvalueofsizespecifiedbyabsolutepercentageofthe
finalvalue(usually2%or5%).Thesettlingtimeisrelatedtothelargesttimeconstantofthe
controlsystem.Whichpercentageerrorcriteriontousemaybedeterminedfromthe
objectivesofthesystemdesigninquestion.
•Thetime-domainspecificationsjustgivenarequiteimportant,sincemostcontrolsystemsare
time-domainsystems;thatis,theymustexhibitacceptabletimeresponses.(Thismeansthat,
thecontrolsystemmustbemodifieduntilthetransientresponseissatisfactory.)

54. 5-2 FIRST-ORDER SYSTEM
DefinitionsofTransient-ResponseSpecifications.(Cont.)
•Notethatnotallthesespecificationsnecessarilyapplytoanygivencase.
Forexample,foranoverdampedsystem,thetermspeaktimeand
maximumovershootdonotapply.

55. 5-2 FIRST-ORDER SYSTEM
AFewCommentsonTransient-ResponseSpecifications.
•Exceptforcertainapplicationswhereoscillationscannotbetolerated,itisdesirablethatthe
transientresponsebesufficientlyfastandbesufficientlydamped.Thus,foradesirable
transientresponseofasecond-ordersystem,thedampingratiomustbebetween0.4and0.8.
•valuesof??????(thatis,??????<0.4)yieldexcessiveovershootinthetransientresponse,andasystem
withalargevalueof??????(thatis,??????>0.8)respondssluggishly.
•Weshallseelaterthatthemaximumovershootandtherisetimeconflictwitheachother.In
otherwords,boththemaximumovershootandtherisetimecannotbemadesmaller
simultaneously.Ifoneofthemismadesmaller,theothernecessarilybecomeslarger.

56. 5-6 ROUTH’S
STABILITY CRITERION

57. 5-6 ROUTH’S STABILITY CRITERION
•The most important problem in linear control systems concerns stability.
•That is, under what conditions will a system become unstable? If it is unstable, how
should we stabilize the system?
•Most linear closed-loop systems have closed-loop transfer functions of the form
where the a’sand b’sare constants and m <= n

58. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
•A simple criterion, known as Routh’s stability criterion, enables us to determine the
number of closed-loop poles that lie in the right-half s plane without having to factor the
denominator polynomial. (The polynomial may include parameters that MATLAB cannot
handle.).

59. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Routh’s Stability Criterion.
•Routh’sstabilitycriteriontellsuswhetherornotthereareunstablerootsinapolynomial
equationwithoutactuallysolvingforthem.
•Thisstabilitycriterionappliestopolynomialswithonlyafinitenumberofterms.
•Whenthecriterionisappliedtoacontrolsystem,informationaboutabsolutestabilitycanbe
obtaineddirectlyfromthecoefficientsofthecharacteristicequation.
•TheprocedureinRouth’sstabilitycriterionisasfollows:
1.Thepolynomialinsinthefollowingform:
wherethecoefficientsarerealquantities.WeassumethatanZ;thatis,anyzeroroothasbeen
removed.

60. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Routh’s Stability Criterion.
2.Ifanyofthecoefficientsarezeroornegativeinthepresenceofatleastonepositivecoefficient,aroot
orrootsexistthatareimaginaryorthathavepositiverealparts.Therefore,insuchacase,thesystemisnot
stable.
•Ifweareinterestedinonlytheabsolutestability,thereisnoneedtofollowtheprocedurefurther.
•Notethatallthecoefficientsmustbepositive.Apolynomialinshavingrealcoefficientscanalwaysbe
factoredintolinearandquadraticfactors,suchas(s+a)and(+bs+cB),wherea,b,andcarereal.
•Thelinearfactorsyieldrealrootsandthequadraticfactorsyieldcomplex-conjugaterootsofthe
polynomial.
??????
2

61. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Routh’s Stability Criterion.
•Thefactor(??????
2
+bs+c)yieldsrootshavingnegativerealpartsonlyifbandcarebothpositive.
•Forallrootstohavenegativerealparts,theconstantsa,b,c,andsoon,inallfactorsmustbe
positive.
•Theproductofanynumberoflinearandquadraticfactorscontainingonlypositive
coefficientsalwaysyieldsapolynomialwithpositivecoefficients.
•Itisimportanttonotethattheconditionthatallthecoefficientsbepositiveisnotsufficient
toassurestability.

62. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Routh’s Stability Criterion.
3.Ifallcoefficientsarepositive,arrangethecoefficientsofthepolynomialinrowsand
columnsaccordingtothefollowingpattern:

63. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Routh’s Stability Criterion.
•Theprocessofformingrowscontinuesuntilwerunoutofelements.(Thetotalnumber
ofrowsisn+1.)Thecoefficients??????
1,??????
2,??????
3andsoon,areevaluatedasfollows:

64. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Routh’s Stability Criterion.
•Theevaluationoftheb’siscontinueduntiltheremainingonesareallzero.Thesame
patternofcross-multiplyingthecoefficientsofthetwopreviousrowsisfollowedin
evaluatingthec’s,d’s,e’s,andsoon.Thatis,

65. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
EXAMPLE 5–11

66. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
EXAMPLE 5–12
•Consider the following polynomial:
•Let us follow the procedure just presented and construct the array of coefficients. (The first two rows can be obtained directlyfrom the
given polynomial. The remaining terms are obtained from these. If any coefficients are missing, they may be replaced by zerosinthe
array.).
•In this example, the number of changes in sign of the coefficients in the first column is 2. This means that there are two rootswith
positive real parts. Note that the result is unchanged when the coefficients of any row are multiplied or divided by a positive number in
order to simplify the computation.

67. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Special Cases.
1. If a first-column term in any row is zero, but the remaining terms are not zero or there is
no remaining term, then the zero term is replaced by a very small positive number € and
the rest of the array is evaluated. For example, consider the following equation:
The array of coefficients is

68. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Special Cases (Cont).
•If the sign of the coefficient above the zero (€) is the same as that below it, it indicates
that there are a pair of imaginary roots.
•If, however, the sign of the coefficient above the zero (€) is opposite that below it, it
indicates that there is one sign change. For example, for the equation.
•the array of coefficients is

69. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Special Cases (Cont.).
There are two sign changes of the coefficients in the first column. So there are two roots in
the right-half s plane. This agrees with the correct result indicated by the factored form of
the polynomial equation.

70. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Special Cases (Cont.).
2.Ifallthecoefficientsinanyderivedrowarezero,itindicatesthattherearerootsofequal
magnitudelyingradiallyoppositeinthesplane—thatis,tworealrootswithequalmagnitudes
andoppositesignsand/ortwoconjugateimaginaryroots.Insuchacase,theevaluationofthe
restofthearraycanbecontinuedbyforminganauxiliarypolynomialwiththecoefficientsofthe
lastrowandbyusingthecoefficientsofthederivativeofthispolynomialinthenextrow.Such
rootswithequalmagnitudesandlyingradiallyoppositeinthesplanecanbefoundbysolving
theauxiliarypolynomial,whichisalwayseven.Fora2n-degreeauxiliarypolynomial,therearen
pairsofequalandoppositeroots.Forexample,considerthefollowingequation:

71. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Special Cases (Cont.).
•The array of coefficients is
•Thetermsinthes3rowareallzero.(Notethatsuchacaseoccursonlyinanoddnumbered
row.)Theauxiliarypolynomialisthenformedfromthecoefficientsofthes4row.The
auxiliarypolynomialP(s)is

72. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Special Cases (Cont.).
The terms in the ??????
3
row are replaced by the coefficients of the last equation—that is, 8 and
96. The array of coefficients then becomes

73. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Application of Routh’s Stability Criterion to Control-
System Analysis.
In the following, we shall consider the problem of determining the stability range of a
parameter value. Let us determine the range of K for stability.

74. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Application of Routh’s Stability Criterion to Control-
System Analysis.
The closed-loop transfer function is
The characteristic equation is

75. 5-6 ROUTH’S STABILITY CRITERION (CONT.)
Application of Routh’s Stability Criterion to Control-System Analysis.
The array of coefficients becomes
For stability, K must be positive, and all coefficients in the first column must be positive. Therefore,
When the system becomes oscillatory and, mathematically, the oscillation is sustained at constant amplitude
Note that the ranges of design parameters that lead to stability may be determined by use of Routh’s stability
criterion.

76. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS
•Errorsinacontrolsystemcanbeattributedtomanyfactors.
•Changesinthereferenceinputwillcauseunavoidableerrorsduringtransientperiodsandmayalsocause
steadystateerrors.
•Imperfectionsinthesystemcomponents,suchasstaticfriction,backlash,andamplifierdrift,aswellas
agingordeterioration,willcauseerrorsatsteadystate.
•Inthissection,however,weshallnotdiscusserrorsduetoimperfectionsinthesystemcomponents.
Rather,weshallinvestigateatypeofsteady-stateerrorthatiscausedbytheincapabilityofasystemto
followparticulartypesofinputs.
•(Theonlywaywemaybeabletoeliminatethiserroristomodifythesystemstructure.)Whethera
givensystemwillexhibitsteady-stateerrorforagiventypeofinputdependsonthetypeofopen-loop
transferfunctionofthesystem.

77. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
ClassificationofControlSystems.
•Controlsystemsmaybeclassifiedaccordingtotheirabilitytofollowstepinputs,ramp
inputs,parabolicinputs,andsoon.Thisisareasonableclassificationscheme,because
actualinputsmayfrequentlybeconsideredcombinationsofsuchinputs.
•The magnitudes of the steady-state errors due to these individual inputs are indicative of
the goodness of the system.
•Consider the unity-feedback control system with the following open-looptransfer
function G(s):

78. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
ClassificationofControlSystems.(Count.)
•Itinvolvestheterm??????
??????
inthedenominator,representingapoleofmultiplicityNattheorigin.
•Thepresentclassificationschemeisbasedonthenumberofintegrationsindicatedbythe
open-looptransferfunction.
•Asystemiscalledtype0,type1,type2,…,ifN=0,N=1,N=2,...,respectively.Notethatthis
classificationisdifferentfromthatoftheorderofasystem.
•Asthetypenumberisincreased,accuracyisimproved;however,increasingthetypenumber
aggravatesthestabilityproblem.
•Acompromisebetweensteady-stateaccuracyandrelativestabilityisalwaysnecessary.

79. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
Steady-StateErrors.
•Thetransferfunctionbetweentheerrorsignale(t)andtheinputsignalr(t)is

80. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
Steady-StateErrors.(Count.)
•wheretheerrore(t)isthedifferencebetweentheinputsignalandtheoutputsignal.The
final-valuetheoremprovidesaconvenientwaytofindthesteady-stateperformanceofa
stablesystem.SinceE(s)is
•thesteady-stateerroris

81. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
Steady-StateErrors.(Count.)
•Thestaticerrorconstantsdefinedinthefollowingarefiguresofmeritofcontrolsystems.The
highertheconstants,thesmallerthesteady-stateerror.
•Inagivensystem,theoutputmaybetheposition,velocity,pressure,temperature,orthelike.
•Thephysicalformoftheoutput,however,isimmaterialtothepresentanalysis.
•Therefore,inwhatfollows,weshallcalltheoutput“position,”therateofchangeofthe
output“velocity,”andsoon.Thismeansthatinatemperaturecontrolsystem“position”
representstheoutputtemperature,"velocity”representstherateofchangeoftheoutput
temperature,andsoon.

82. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
1.StaticPositionErrorConstant??????
??????
•Thesteady-stateerrorofthesystemforaunit-stepinputis
•ThestaticpositionerrorconstantKpisdefinedby

83. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
1.StaticPositionErrorConstant??????
??????.(Cont.)
•Thus,thesteady-stateerrorintermsofthestaticpositionerrorconstantKpisgivenby
•Foratype0system,

84. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
1.StaticPositionErrorConstant??????
??????.(Cont.)
•Foratype1orhighersystem,
•Hence,foratype0system,thestaticpositionerrorconstantKpisfinite,whileforatype
1orhighersystem,Kpisinfinite.
•Foraunit-stepinput,thesteady-stateerroressmaybesummarizedasfollows:

85. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
1.StaticPositionErrorConstant??????
??????.(Cont.)
•Fromtheforegoinganalysis,itisseenthattheresponseofafeedbackcontrolsystemtoa
stepinputinvolvesasteady-stateerrorifthereisnointegrationinthefeedforwardpath.
•Ifzerosteady-stateerrorforastepinputisdesired,thetypeofthesystemmustbeone
orhigher.

86. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticVelocityPositionErrorConstant??????
??????.(Cont.)
•Thesteady-stateerrorofthesystemwithaunit-rampinputisgivenby
•Thestaticvelocityerrorconstant??????
??????isdefinedby

87. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticVelocityPositionErrorConstant??????
??????.(Cont.)
•Thus,thesteady-stateerrorintermsofthestaticvelocityerrorconstant??????
??????isgivenby
•Thetermvelocityerrorisusedheretoexpressthesteady-stateerrorforarampinput.
Thedimensionofthevelocityerroristhesameasthesystemerror.Thatis,velocity
errorisnotanerrorinvelocity,butitisanerrorinpositionduetoarampinput.Fora
type0system,

88. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticVelocityPositionErrorConstant??????
??????.(Cont.)
•Foratype1system,

89. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticVelocityPositionErrorConstant??????
??????.(Cont.)
•Foratype2orhighersystem,
•Thesteady-stateerroressfortheunit-rampinputcanbesummarizedasfollows:

90. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticVelocityPositionErrorConstant??????
??????.(Cont.)
•Theforegoinganalysisindicatesthatatype0systemisincapableoffollowingaramp
inputinthesteadystate.
•Thetype1systemwithunityfeedbackcanfollowtherampinputwithafiniteerror.
•Insteady-stateoperation,theoutputvelocityisexactlythesameastheinputvelocity,but
thereisapositionalerror.
•Thiserrorisproportionaltothevelocityoftheinputandisinverselyproportionaltothe
gainK.

91. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticAccelerationErrorConstantKa.(Cont.)
•Thesteady-stateerrorofthesystemwithaunit-parabolicinput(accelerationinput),
whichisdefinedby
•isgivenby

92. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticAccelerationErrorConstantKa.(Cont.)
•ThestaticaccelerationerrorconstantKaisdefinedbytheequation
•Thesteady-stateerroristhen
•Notethattheaccelerationerror,thesteady-stateerrorduetoaparabolicinput,isan
errorinposition.

93. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticAccelerationErrorConstantKa.(Cont.)
•ThevaluesofKaareobtainedasfollows:Foratype0system,
•Foratype1system,

94. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticAccelerationErrorConstantKa.(Cont.)
•Foratype2system,
•Foratype3orhighersystem,
•Thus,thesteady-stateerrorfortheunitparabolicinputis

95. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticAccelerationErrorConstantKa.(Cont.)

96. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
2.StaticAccelerationErrorConstantKa.(Cont.)
•Notethatbothtype0andtype1systemsareincapableoffollowingaparabolicinputin
thesteadystate.Thetype2systemwithunityfeedbackcanfollowaparabolicinputwith
afiniteerrorsignal.

97. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
Summary.
Rememberthatthetermspositionerror,velocityerror,andaccelerationerrormeansteady-state
deviationsintheoutputposition.

98. 5-8 STEADY-STATE ERRORS IN UNITY-FEEDBACK
CONTROL SYSTEMS. (CONT.)
Summary.(Cont.)
•Afinitevelocityerrorimpliesthataftertransientshavediedout,theinputandoutputmoveatthesame
velocitybuthaveafinitepositiondifference.
•TheerrorconstantsKp,Kv,andKadescribetheabilityofaunity-feedbacksystemtoreduceor
eliminatesteady-stateerror.Therefore,theyareindicativeofthesteady-stateperformance.
•.Itisgenerallydesirabletoincreasetheerrorconstants,whilemaintainingthetransientresponsewithin
anacceptablerange.Itisnotedthattoimprovethesteadystateperformancewecanincreasethetype
ofthesystembyaddinganintegratororintegratorstothefeedforwardpath.This,however,introduces
anadditionalstabilityproblem.Thedesignofasatisfactorysystemwithmorethantwointegratorsin
seriesinthefeedforwardpathisgenerallynoteasy.

99. Eng. Elaf Ahmed Saeed
Email: [email protected]
LinkedIn: https://www.linkedin.com/in/elaf-a-saeed
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GitHub: https://github.com/ElafAhmedSaeed