Analysis and Design of Control System using Root Locus

Root locus analysis is a powerful tool in control systems engineering used to analyze the behavior of a system's closed-loop poles as a function of a parameter, typically a controller gain. It provides engineers with valuable insights into how changing system parameters affect stability and performance, helping them design robust and stable control systems. Let's explore the key concepts, techniques, and practical implications of root locus analysis. At its core, root locus analysis focuses on the movement of the closed-loop poles in the complex plane as a control parameter varies. These poles represent the characteristic equation's roots, which determine the system's stability and transient response. By examining the pole locations as the parameter changes, engineers can gain a deeper understanding of the system's behavior and make informed design decisions.

1. Analysis and Design
Via Root Locus

2. Introduction
•Rootlocus,agraphicalpresentationoftheclosed-looppolesasa
systemparameterisvaried,isapowerfulmethodofanalysisand
designforstabilityandtransientresponse(Evans.1948;1950).
•Therootlocuscanbeusedtodescribequalitativelytheperformanceof
asystemasvariousparametersarechanged.Forexample,theeffectof
varyinggainuponpercentovershoot,settlingtime,andpeaktimecan
bevividlydisplayed.Thequalitativedescriptioncanthenbeverified
withquantitativeanalysis.
•Besidestransientresponse,therootlocusalsogivesagraphic
representationofasystem'sstability.Wecanclearlyseerangesof
stability,rangesofinstability,andtheconditionsthatcauseasystemto
breakintooscillation.

3. Root Locus
Consider the following system along with equivalent transfer function;
s
s

4. Root Locus
Let,
and
Theopenlooptransferfunctionis,??????����,andit’spolescanbe
determinedsincetheyarisefromsimplecascadedfirstandsecondorder
subsystems.Butitisdifficulttodeterminethepolesoftheclosedloop
transferfunction,�
�??????s=
??????�??????
1+??????�??????�(??????)
,where??????≥0���??????��
Which can be written as,
�
�??????s=
????????????
�(�)??????
�(�)
??????
��??????
��+????????????
��??????
��
The root locus will be used to give us a vivid picture of the poles of
�
�??????sas K varies.
where N and D are factored polynomials and
signify numerator and denominator terms,
respectively.

5. Root Locus
Therootlocustechniquecanbeusedtoanalyzeanddesigntheeffectof
gainuponthesystem'stransientresponseandstability.Asthegain
varies,theclosedlooppolesmoveonacomplexplanei.e.itcanbereal
anddistinct,realandrepeated,complexorpurelyimaginary.Itisthis
representationofthepathsoftheclosed-looppolesasthegainis
variedthatwecallarootlocus.
Example 1: plot a root locus diagram for the following unity feedback
system.

6. Root Locus
Propertiesoftherootlocushelpustomakearapidsketchoftherootlocusfor
higher-ordersystemswithouthavingtofactorthedenominatoroftheclosed-
looptransferfunction.
A pole, s of the closed loop transfer function �
�??????s=
??????�??????
1+??????�??????�(??????)
exists when the characteristic polynomial in the denominator becomes zero, or
when ??????����become -1.
Alternatively, s, is a closed loop pole if,
??????����=1,���=
1
�??????�??????
????????????��������������????????????��
??????������??????����=2k+1180
0
,�=0,±1,±2,±3(??????�����������??????)
The values of sthat fulfill both the angle and magnitude conditions are the roots
of the characteristic equation, or the closed-loop poles.

7. Root Locus
Wehavejustfoundthatapoleoftheclosed-loopsystemcausestheangleof
KG(s)H(s),orsimplyG(s)H(s)sinceKisascalar,tobeanoddmultipleof
180
0
.Inotherwords,giventhepolesandzerosoftheopen-looptransfer
function,KG(s)H(s),apointinthes-planeisontherootlocusforaparticular
valueofgain,K,iftheanglesofthezerosminustheanglesofthepoles,all
drawntotheselectedpointonthes-plane,addupto(2k+1)180
0
where,
�=0,±1,±2,±3…
Furthermore,themagnitudeofKG(s)H(s)mustbeunity,implyingthatthe
valueofKisthereciprocalofthemagnitudeofG(s)H(s)whenthepolevalue
issubstitutedfors.
Where magnitude of G(s)H(s) is;
����=
??????����������
�����������

8. Root Locus
Sketching the root locus:
Itappearsfromourpreviousdiscussionthattherootlocuscan
beobtainedbysweepingthrougheverypointinthes-planeto
locatethosepointswhichtheangles,aspreviouslydescribed,
adduptoanoddmultipleof180
0
.Althoughthistaskistedious
withouttheaidofacomputer,theconceptcanbeusedto
developrulesthatcanbeusedtosketchtherootlocuswithout
theeffortrequiredtoplotthelocus.
Onceasketchisobtained,itispossibletoaccuratelyplotjust
thosepointsthatareofinteresttousforaparticularproblem.
Therulesyieldasketchthatgivesintuitiveinsightintothe
behaviorofacontrolsystem.

9. Sketching the root locus:
Rules for sketching the root locus
Rule-1: The number of branches of the root locus equals the number of
closed-loop poles.
Rule-2: The root locus is symmetrical about the real axis.
Rule-3: On the real axis, for K > 0 the root locus exists to the left of an
odd number of real-axis, finite open-loop poles and/or finite open-loop
zeros.
Rule-4: The root locus begins at the finite and infinite polesof open
looptransfer function, G(s)H(s) and endsat the finite and infinite zeros
of open loop transfer function, G(s)H(s).

10. Sketching the root locus:
Rules for sketching the root locus
Rule-5:Thepointwherethelocusleavestherealaxis,iscalledthebreakaway
point,andthepointwherethelocusreturnstotherealaxis,iscalledthebreak-in
point.Therootlocusbreaksawayfromtherealaxisatapointwherethegainis
maximumbetweenopenlooppoles,andbreaksintotherealaxisatapointwhere
thegainisminimumbetweentwozeros.
Thispointisfoundbyrearrangingtheclosedlooptransferfunctioncharacteristics
equationforgain,Kandsolveforthepointbydifferentiatingitwithrespecttosand
equatingittozero.
Also,breakawayandbreakinpointssatisfytherelationship;

�
�+??????
�
=
�
�+�
�
Where??????
??????and�
??????arenegativeofopenloopzerosandpolesrespectively
Atthebreakawayorbreak-inpoint,thebranchesoftherootlocusformanangleof
180
0
�withtherealaxis,wherenisthenumberofclosed-looppolesarrivingator
departingfromthesinglebreakawayorbreak-inpointontherealaxis.

11. Sketching the root locus:
Rules for sketching the root locus
Rule-5: Cont…
Ifarootlocusliesbetweentwoadjacentopen-looppolesonthereal
axis,thenthereexistsatleastonebreakawaypointbetweenthetwo
poles.Similarly,iftherootlocusliesbetweentwoadjacentzeros(one
zeromaybelocatedatinfinity)ontherealaxis,thentherealwaysexists
atleastonebreak-inpointbetweenthetwozeros.Iftherootlocuslies
betweenanopen-looppoleandazero(finiteorinfinite)ontherealaxis,
thentheremayexistnobreakawayorbreak-inpointsortheremayexist
bothbreakawayandbreak-inpoints.

12. Sketching the root locus:
Rules for sketching the root locus
Example 2: plot a root locus diagram and assign critical points for the
following unity feedback system.

13. Sketching the root locus:
Rules for sketching the root locus
Rule-6:Therootlocusapproachesstraightlinesasasymptotesasthe
locusapproachesinfinity.Furthermore;theequationoftheasymptotes
isgivenbythereal-axisintercept,andangleinasfollows.
??????
??????=
��������??????������− ��������??????�??????����
#�����������−#������??????����
??????
??????=
(��+�)�??????�
#�����������−#������??????����
where,�=0,±1,±2,±3…
Here,k=0correspondstotheasymptoteswiththesmallestanglewith
therealaxis.Althoughkassumesaninfinitenumberofvalues,askis
increasedtheanglerepeatsitself,andthenumberofdistinctasymptotes
is#�����������−#������??????����−1.

14. Sketching the root locus:
Rules for sketching the root locus
Rule-7:The�??????-axiscrossingisapointontherootlocusthatseparatesthe
stableoperationofthesystemfromtheunstableoperation.Therootlocus
crossesthe�??????-axisatthepoint;
where??????����������=2k+1180
0
(??????�����������??????).
Routh-Hurwitzcanbeusedtofindthe�??????-axiscrossing.
Thevaluesof??????thusfoundgivethefrequenciesatwhichrootlocicrossthe
imaginaryaxis.The??????valuecorrespondingtoeachcrossingfrequencygivesthe
gainatthecrossingpoint.

15. Sketching the root locus:
Rulesforsketchingtherootlocus
Rule-8:Therootlocusdepartsfromcomplex,openlooppolesand
arrivesatcomplex,open-loopzerosatanglesthatcanbecalculatedas
follows.
Angleofdeparture:??????
�=±180
�
2�+1+ ??????���??????����− ����??????�����
Angleofarrival:??????
??????=±180
�
2�+1− ??????���??????����+ ����??????�����

16. Sketching the root locus:
Example 3: plot a root locus diagram and assign critical points for the
following unity feedback system.

17. Control System Design Using Root Locus
Settingthegainataparticularvalueyieldsthetransientresponsedictated
bythepolesatthatpointontherootlocus.Butwearelimitedtothose
responsesthatexistalongtherootlocus.
Weaugmentorcompensate,thesystemwithadditionalpolesandzeros
intheforwardpath,sothatthecompensatedsystemhasarootlocusthat
goesthroughthedesiredpolelocationforsomevalueofgain
Additionofcompensatingpolesandzerosneednotinterferewiththe
poweroutputrequirementsofthesystemorpresentadditionalloador
designproblems.Compensatingpolesandzeroscanbegeneratedwitha
passiveoranactivenetwork.

18. Control System Compensation Using Root Locus
Transientresponseisimprovedwiththeadditionofdifferentiation,and
steady-stateerrorisimprovedwiththeadditionofintegrationinthe
forwardpath.
Systemsthatfeedtheerrorforwardtotheplantarecalledproportional
controlsystems.Systemsthatfeedtheintegraloftheerrortotheplantare
calledintegralcontrolsystems.Finally,systemsthatfeedthederivativeof
theerrortotheplantarecalledderivativecontrolsystems.
Proportional Controller

19. Compensation Techniques
•Cascade Compensation
•Parallel Compensation

20. Ideal integral compensator and Lag Compensator
Steadystateerrorcanbeimprovedbyplacinganopen-looppoleatthe
origin,becausethisincreasesthesystemtypebyone.
Thefirsttechniqueisidealintegralcompensation(implementedwithactive
networks,suchasamplifiers),whichusesapureintegratortoplaceanopen-
loop,forward-pathpoleattheorigin,thusincreasingthesystemtypeand
reducingtheerrortozero.Wealsoaddazeroclosetothepoletokeepthe
rootlocusgothroughthedesiredpointandkeepthegainthesameasbefore.
Thesecondtechniquedoesnotusepureintegration.Thiscompensation
techniqueplacesthepoleneartheorigin(canbeimplementedwithaless
expensivepassivenetworkthatdoesnotrequireadditionalpowersources),
andalthoughitdoesnotdrivethesteady-stateerrortozero,itdoesyielda
measurablereductioninsteady-stateerror.
Implemented by PI controller

21. PI Controller or Ideal Integral Compensator
Thesignal(u)justpastthecontrollerisnowequaltotheproportional
gain(??????
�)timesthemagnitudeoftheerrorplus(??????
??????)timestheintegralof
theerror.
where ??????
�= Proportional gain, ??????
??????= integral gainand ??????
??????is integral time constant.

22. Ideal derivative compensator and Lead Compensator
Theresultofaddingdifferentiationistheadditionofazerotothe
forward-pathtransferfunctionwhichimprovesthetransientresponse.
•Thefirsttechniqueisanidealderivativecompensator,inwhichapure
differentiatorisaddedtotheforwardpathofthefeedbackcontrol
system(requireactivenetworkforitsrealization).
The ideal derivative compensator is implemented with PD controller.
Differentiation of high frequencies can lead to large unwanted signals of
saturation of amplifiers and other components.
•Thesecondtechniqueapproximatedifferentiationwithpassive
networkbyaddingzeroandmoredistantpoletotheforwardpath
transferfunction.

23. PD Controller or Ideal derivative compensator
Judiciouschoiceofthepositionofthecompensatorzerocan
quickentheresponseovertheuncompensatedsystem.
where ??????
�= Proportional gain, ??????
�= Derivative gainand ??????
�is derivative time constant.

24. Ideal integral-derivative compensator and
Lag-lead compensator
Ideal integral-derivative compensator
-Implemented by PID controller
Lead-lag compensator

25. PID controller and Lead-lag compensator
Theerrorsignale(t)willbesenttothePIDcontroller,andthe
controllercomputesboththederivativeandtheintegralofthis
errorsignal.Thesignal(u)justpastthecontrollerisnowequalto
theproportionalgain(Kp)timesthemagnitudeoftheerrorplus
theintegralgain(Ki)timestheintegraloftheerrorplusthe
derivativegain(Kd)timesthederivativeoftheerror.The
proportionalpartactsonthepresentvalueoftheerror,theintegral
representsanaverageofpasterrorsandthederivativecanbe
interpretedasapredictionoffutureerrorsbasedonlinear
extrapolation
??????
�+
??????
??????
�
+??????
��=
??????
��
2
+??????
��+??????
??????
�

26. Compensator design procedure

27. PID controller and Lead-lag compensator
Example-1,[Norman Nise]
Giventhesystemoffigurebelow,
designaPDcontrollersothatthe
systemcanoperatewithapeaktime
thatistwothirdsofthatof
uncompensatedsystem(0.3sec)at
20%overshoot.Therootlocusof
uncompensatedsystemisshownto
theright.

28. PID controller and Lead-lag compensator
Example-2 [Norman Nise]
Upgrade the controller in example-1 to a PID controller that reduce the
steady state error to zero for a step input.