The Sine Wave Tuning method: Robust PID controller design in the frequency domain

The

Sine

Wave

Tuning

method:

Robust

PID

controller

design

in

the

frequency

domain

ˇS.

Bucz

,

A.

Kozáková

FacultyofElectricalEngineeringandInformationTechnology,SlovakUniversityofTechnologyinBratislava,Slovakia

Availableonline2December2015

Abstract

ThepaperpresentsanovelrobustPIDcontrollerdesignmethodfornominalperformancespecifiedintermsofmaximumovershoot andsettlingtime.ThePIDcontrollerdesignprovidesguaranteedgainmarginGM.Theparameterofthetuningrulesisasuitably chosenpointoftheplantfrequencyresponseobtainedbyasine-wavesignalwithexcitationfrequencyωn.Then,thedesigned controllermovesthispointintothephasecrossoverwiththerequiredgainmarginGM.Thecouple(ωn;GM)isspecifiedwithrespect toclosed-loopperformancerequirementsintermsofηmax (maximumovershoot)and ts (settling time)accordingtodeveloped parabolicdependences.Thenewapproachhasbeenverifiedonavastbatchofbenchmarkexamples;subsequently,thedeveloped algorithmhasbeenextendedtorobustPIDcontrollerdesignforplantswithunstablezeroandunstructureduncertainties. ©2016ElectronicsResearchInstitute(ERI).ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Gainmargin;PIDtuning;Guaranteedperformance;Unstablezero

1. Introduction

Control system designers frequently have to cope with plants exhibiting inverse response dynamics; such plantsusually modeled as systems withunstable zeroare encountered inpower engineering, servosystems, auto-motive and hydraulic plants. It is a well known difficulty to control the class of non-minimum phase systems G(s)=(1−αs)/(1+Ts)nwithunstablezeroz=+1/α,evenforsmallvaluesofα;moreover,controlcomplexityincreases withincreasingα(Yu,2006).

Theproposednewmethodisapplicableforcontroloflinearsingle-input-single-outputnon-minimumphasesystems evenwithunknownmathematicalmodelwithunstructureduncertainties.AsurveyonPIDcontrollertuningcanbe foundinÅströmandHägglund(1995),Visioli(2006)andYu(2006).Thecontrol objectiveistoproviderequired nominalmaximumovershootηmax and settlingtimets ofthe controlledprocessvariabley(t). Thekeyideabehind guaranteeingspecifiedvaluesηmax andts consistsinextendingvalidityof therelations ηmax=f(GM)andts=f(ωn) derivedfor2ndordersystems(Reinisch,1974)forarbitraryplantorders;two-parameterquadraticdependenceswere

∗_{Correspondingauthor.}

E-mailaddresses:[email protected]( ˇS.Bucz),[email protected](A.Kozáková).

PeerreviewunderresponsibilityofElectronicsResearchInstitute(ERI).

http://dx.doi.org/10.1016/j.jesit.2015.11.001

2314-7172/©2016ElectronicsResearchInstitute(ERI).ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCC

Fig.1.LoopforthedesignedSWTmethod.

obtainedforboththemaximumovershootηmax=f(GM,ωn)andsettlingtimets=f(GM,ωn).Theresultingplotscalled B-parabolasenablethedesignerchoosingsuchacouple(GM,ωn)thatguaranteesfulfillmentofspecifiedperformance requirements thusallowing consistentandsystematicshaping of the closed-loopstep response withregardtothe controlledplant(BuczandKozáková,2012).ThedesignmethodhasbeencalledSineWaveTuning(SWT)method, thenamebeingderivedfromthegeneratorofsinusoidsignalusedtomeasureplantparametersnecessaryforthePID controllerdesign.

2. PIDcontrollerdesignobjectivesforprocesseswithunstablezero

Duetosignificantchangesofthegainmarginoftheplantbroughtaboutbythenon-minimumphasebehavior,itis beneficialtousegainmarginGMasaperformancemeasurewhendesigningthePIDcontroller(BuczandKozáková, 2012).ConsideramultipurposeloopshowninFig.1(theswitchinpositionSW=1).LetG(s)betransferfunctionof anuncertainnon-minimumphaseplant,andGR(s)thePIDcontroller.

Thecorrespondingclosed-loopcharacteristicequationc(s)=1+L(s)=1+G(s)GR(s)=0expressestheclosed-loop stabilitycaneasilybebrokendownintothemagnitudeandphaseconditions



G(jωp∗)GR(jωp∗) =1/GM, argG(ω∗p)+argGR(ωp∗)=−π, (1) whereGMisrequiredgainmargin,L(jω)istheopen-looptransferfunction,andωp*istheopen-loopphasecrossover frequency.Denoteϕ=argG(ωp*),Θ=argGR(ωp*),andconsidertheidealPIDcontrollerintheform

GR(s)=K  1+ 1 Tis+Tds  , (2)

whereKistheproportionalgain,andTi,Tdareintegralandderivativetimeconstants,respectively.Aftercomparing thetwoformsofthePIDcontrollerfrequencytransferfunctions

GR(jω∗_p)=K+jK  Tdω∗_p− 1 Tiω∗p  , (3) GR(jω∗p)=GR(jω∗p)  cosΘ+j sinΘ. (4)

PIDcoefficientscanbeobtainedfromthecomplexequationatω=ωp*

K+jK  Tdω∗_p− 1 Tiω∗p  = cosΘ GMG(jω∗p) +j sinΘ GMG(jω∗p) (5)

usingthesubstitution|GR(jωp*)|=1/[GM|G(jωp*)|]resultingfrom(1a).Thecomplexequation(5)isthensolvedasa setoftworealequations

K= cosΘ GMG(jωp∗) , K  Tdω∗p− 1 Tiω∗p  = sinΘ GMG(jωp∗) , (6)

LP Θ Θ 0 L(jω) -1 1 G(jω) G ωn ϕϕ

Fig.2.GraphicalrepresentationoftheSWT-typePIDcontrollertuning.

where(6a)isageneralruleforcalculatingthecontrollergainK;substituting(6a)into(6b),aquadraticequationinTd isobtained T_d2 ω∗_p 2 −Tdωp∗tgΘ− 1 β =0,β= Ti Td . (7)

ExpressionforcalculatingTdisthepositivesolutionof Td= tgΘ 2ω∗_p + 1 ω∗_p  tg2Θ 4 + 1 β. (8)

Hence,(6a),(7b)and(8)aretheresultingPIDtuningrules,wheretheangleΘisobtainedfromthephasecondition

(1b) Θ=−180◦−argG ω∗_p =−180◦_{−ϕ (9)}

3. Plantidentificationbyasinusoidalexcitationinput

ConsideragainFig.1;ifSW=2,asinusoidalexcitationsignalu(t)=Unsin(ωnt)withmagnitudeUnandfrequency ωnisinjectedintotheplantG(s).Theplantoutputy(t)=Ynsin(ωnt+ϕ)isalsosinusoidalwithmagnitudeYn,where ϕisthephaselagbetweeny(t)andu(t).AfterreadingthevaluesYnandϕfromtherecordedvaluesofu(t)andy(t),a particularpointoftheplantfrequencyresponsecorrespondingtotheexcitationfrequencyωn

Θ=−180◦−argG

ω∗_p

=−180◦−ϕ (10)

canbeplottedinthecomplexplane.

Excitationfrequencyωnistakenfromtheinterval

ωn ∈ 0.5ωc,1.25ωc , (11)

wheretheplantcriticalfrequencyωc canbeobtainedbythewell-knownrelayexperiment(ÅströmandHägglund, 1995),i.e.forSW=3.

UsingthePIDcontrollerwiththecoefficients{K;Ti=βTd;Td},theidentifiedpointG(jωn)withcoordinates(10)

canbemovedintothephasecrossoverLP L(jωp*_{)onthenegativerealhalf-axis,wheretherequiredgainmarginGM} isguaranteed(Fig.2),ifthefollowingidentitybetweentheexcitationandphasecrossoverfrequenciesωnandωp*, respectively,isfulfilled

ω_p∗=ωn (12)

Considering(11),thefollowingrelationsresult 

G(jω∗p) = |G(jωn)| , argG(ω∗p)=argG(ωn)=ϕ, (13)

Θ=−180◦−argG(ωn) (14)

Fig.3.(a)–(c)Closed-loopstepresponsesofG2(s)withT2=0.75,α2=1.3forvariousGMandωn;(d)timeresponsesofG2(s)forα/T=1and

α/T=0.1duringtherelaytest.

Substituting(13a)into(6a)and(12)into(8),thePIDcontrollercoefficientsguaranteeingtherequiredgainmargin GMareobtainedusingthesine-wavetypetuningrulesexpressedinthefollowingform

K= cosΘ GM|G(jωn)| , Td = tgΘ 2ωn + 1 ωn  tg2_Θ 4 + 1 β, (15) β=4, Ti=βTd, Θ=−180◦−ϕ. (16)

4. Closed-loopperformanceunderthedesignedPIDcontroller

Thissectionanswersthefollowingquestion:howtotransformthemaximumovershootηmaxandsettlingtimetsas requiredbythedesignerintothecoupleoffrequency-domainparameters(ωn,GM)neededforidentificationandPID controllertuning?ConsidertypicalgainmarginsGMgivenbytheset

 GMj

= {3dB,5dB,7dB,9dB,11dB,13dB,15dB,17dB} , (17)

j=1,...,8;letussplit(11)into5equalsectionsofthesize ωn=0.15ωcandgeneratethesetofexcitationfrequencies

{ωnk}={0.5ωc,0.65ωc,0.8ωc,0.95ωc,1.1ωc,1.25ωc}. (18)

k=1,...,6;itselementsdividedbytheplantcriticalfrequencyωcdetermineexcitationlevelsσk=ωnk/ωcgivenby theset

{σk}={0.5,0.65,0.8,0.95,1.1,1.25}, (19)

k=1,...,6.Fig.3showstheclosed-loopstepresponseshapingfordifferentGMandωnusingthePIDcontrollerdesign fortheplant(20b)withparametersT2=0.75,α2=1.3,andrequiredgainmarginsGM=5dB,9dB,11dBand13dBat

Fig.4.B-parabolas:(a)ηmax=f(GM);(b)τs=ωcts=f(GM)foridentificationlevelsωnk/ωc,k=1,2,3,4,5,6validfornon-minimumphasesystems

withtheratioα/T>0.3.

Considerthefollowingbenchmarkplants(ÅströmandHägglund,1995) G2(s)= −α2 s+1 (T2s+1)n2 , G3(s)= −α3 s+1 (s+1)(T3s+1)(T32s+1)(T33s+1) . (20)

TheproposedmethodhasbeenappliedforeachelementoftheCartesianproductωnk×GMjofthesets(18)and(17) forj=1,...,8andk=1,...,6.Significantdifferencesbetweendynamicsofindividualcontrolloopsunderdesigned PIDcontrollerscanbeobservedforthebenchmarksystems(20).Thesettlingtimetscanbeexpressedbytherelation

ts= γπ

ωn, (21)

whereγisthecurvefactorofthestepresponse.Toexaminesettlingtimesofclosed-loopsforvariousplantdynamics, itisadvantageoustodefinetherelativesettlingtimeτs=tsωc.Substitutingωn=σωcweobtainrelationfortherelative settlingtime

tsωc=π

σγ⇒τs=

π

σγ, (22)

wheretsisrelatedtotheplantcriticalfrequencyωc.Duetointroducingωc,thel.h.s.of(22a)isconstantforthegiven plantandindependentofωn.Thedependence(22b)obtainedempiricallyfordifferentexcitationfrequenciesωnkis depictedinFig.4bandFig.5b,respectively;itisevidentthatwithincreasedphasemarginGMateveryexcitationlevel σtherelativesettlingtimeτsfirstdecreasesandafterachievingitsminimumτsmin,itincreasesagain.

ConsiderthebenchmarkplantsG1(s)andG2(s)withfollowingparameters:G1.1(s):(T1,n1,α1)=(0.75,8,0.2);G1.2(s):

(1,3,0.1);G1.3(s):(0.5,5,1);G2(s):T2=0.5,α2=1.3.Couplesofexaminedplants[G2(s),G1.3(s)]and[G1.2(s),G1.1(s)]

differprincipally by the ratio α/T, whichfor the 1st couple is [α2/T2=2.6, α1.3/T1.3=2] andfor the 2nd couple

[α1.2/T1.2=0.1,α1.1/T1.1=0.27].Hence,theratiooftheparameterαandthe(dominant)timeconstantToftheplant

issignificantfortheclosed-loopperformanceassessmentunderthePIDcontrollerdesignedforaplantwithunstable zero.

Basedonthepreviousanalysisofdesignresultsofaseriesofbenchmarkexamples,unknownplantswithunstable zerocanbeclassifiedaccordingtotheratioα/Tinfollowingtwogroups:

1.plantswiththeratioα/T<0.3; 2.plantswiththeratioα/T>0.3.

Accordingtothisclassification,empiricaldependencesηmax=f(GM),τs=f(GM)fornon-minimumphasesystems withanunstable zerowere constructedfor different open-loopgainmargins GM andexcitation levelsσ, andare depictedinFig.4a(forα/T>0.3),andFig.5a(forα/T<0.3).Thenetworkofdependencesshowsthatincreasinggain marginGMbringsaboutdecreasingofηmax.

Fig.5.B-parabolas:(a)ηmax=f(GM);(b)τs=ωcts=f(GM)foridentificationlevelsωnk/ωc,k=1,2,3,4,5,6validfornon-minimumphasesystems

withtheratioα/T<0.3.

AstheempiricaldependencesinFigs.4and5wereapproximatedbyquadraticregressioncurvestheyarecalled

B-parabolas (Bucz and Kozáková, 2012). B-parabolas are a useful design tool to carry out the transformation

:(ηmax,ts)→(ωn,GM) that enablestochoose appropriate valuesof gain marginGM andexcitationfrequencyωn, respectively,toguaranteetheperformancespecifiedbythedesignerintermsofmaximumovershootηmaxandsettling timets (BuczandKozáková,2012).Notethat pairsofB-parabolasatthesamelevel(Fig.4a, Fig.4b)or(Fig.5a,

Fig.5b)aretobeused.

When areal plantwith an unstablezero is tobe controlled, the ratio α/T cannot be specified exactly dueto unavailabilityof the plant model.To decide towhichcategory agivenplantbelongs (α/T>0.3or α/T<0.3)it is sufficienttoanalyzetheriseportionoftheoutputvariableduringtherelaytestforfindingωc.Ify(t)hasanS-form withatinyundershoot,theplantisincludedinthecategoryα/T<0.3andB-parabolasfromFig.5aretobeused.Ifa considerableundershootofy(t)occurshavinga“squarerootsign”form(Fig.3dinthereddashedellipse),theplant belongstothecategoryα/T>0.3anditsperformancewillbeassessedusingB-parabolasinFig.4.

5. RobustPIDcontrollerdesignusingSWTM

Themainideaoftheuncertainplantidentificationconsistsinrepeatingthesine-wavetypeexcitationforindividual uncertaintychangesusingtheexcitationsignalfrequencyωnyieldingasetofidentifiedpointsGioftheuncertainplant frequencyresponses

Gi(jωn)= |Gi(jωn)| ejargGi(ωn)=ai+jbi, i=1,2,...,N. (23)

Plantparameterchangesarereflectedinmagnitudeandphasechanges|Gi(jωn)|andargGi(ωn),wherei=1,...,N; N=2pisthenumberofidentificationexperimentsandpisthenumberofvaryingtechnologicalquantitiesoftheplant. ThenominalplantmodelG0(jωn)atωnisobtainedasmeanvaluesofrealandimaginarypartsofGi(jωn),respectively

G0(jωn)=a0+jb0= 1 N N  i=1 ai+j 1 N N  i=1 bi, i=1,2,...,N. (24)

The pointsGi representingunstructureduncertaintiesof theplant canbe enclosedinthecircle MG centeredin G0(jωn),where|G0(jωn)|=(a02+b02)0.5,ϕ0(ωn)=argG0(ωn)=arctg(b0/a0)withtheradiusRG≡RG(ωn)obtainedas amaximumdistancebetweentheithidentifiedpointGi(jωn)andthenominalpointG0(jωn)

RG=max i  (ai−a0)2+(bi−b0)2  , i=1,2,...N. (25)

Fig.6.DispersioncirclesMGandML.

ThedispersioncircleMGcenteredinthenominalpointG0withtheradiusRGencirclesallidentifiedpointsGiof

theuncertainplant(Fig.6).

TheproposedcontrollawgeneratedbytherobustcontrollerGRrob(s)designedforthenominalpointG0(jωn)actually carriesoutthetransformation℘:{RG→RL:RL=|GRrob|RG}ofthesetofidentifiedpointsGi(jωn)encircledbyMG withtheradiusRG intothesetof pointsLi(jωn)delimited byML,andalsocalculatestheradiusRL≡RL(ωn)ofthe dispersioncircleMLcorrespondingtothepointsLi(jωn)oftheNyquistplotsoastoguaranteefulfillmentoftherobust stabilitycondition.

TherobustPIDcontrollerisdesignedusingtheSWTmethoddescribedinSections2and3;theinputdataforthe nominalmodelG0(jωn)areitscoordinates: {|G0(jωn)|; ϕ0=argG0(ωn)}. Substitutingtheminto(15)and(16)the followingexpressionsforcalculatingrobustPIDcontrollerparametersareobtained

Krob= cosΘ0 GM|G(jωn)| , Tdrob= tgΘ0 2ωn + 1 ωn  tg2_Θ 0 4 + 1 β, (26) Tirob=βTdrob, β=4, Θ0=−180◦−ϕ0. (27)

ItcanbeseenthatthegainmarginGMappearingin(26a)isatthesametimearobustPIDcontrollertuningparameter requiredforguaranteeingrobuststability.

Theorem1. SufficientconditionofrobuststabilityunderaPIDcontroller

Consideranuncertaincontinuous-timestabledynamicsystemdescribedbyunstructureduncertainty.The closed-loopsystemT(s)underthecontrollerGR(s)isrobustlystableifthenominalclosed-loopsystem(G0(s)underaPID

controllerGR(s))isstable,and GM> 1+χL  RG(ωn)/|G0(jωn)|  1−χS(GS−1)/GS  , (28)

whereGMistherequiredgainmargin,ωnistheexcitationfrequency,χListhesafetyfactor,RG(ωn)istheradiusof thedispersioncircleoftheNyquistplotsoftheplantatωn,andG0(jωn)isapointontheNyquistplotofthenominal plantatωn.

Proof

TheproofcaneasilybeperformedaccordingtoFig.6.Ifthenominalopen-loopL0(s)=G0(s)GR(s)isstable,then accordingtotheNyquiststabilitycriteriontheclosed-loopwiththeuncertainplantwillbestableifthedistancebetween L0andthepoint(−1,j0),i.e.|1+L0(jωn)|isgreaterthantheradiusRL(ωn)ofthecircleMLcenteredinL0,i.e.

|RL(jωn)| < |1+L0(jωn)| , (29)

whereωnisthesine-wavegeneratorfrequency.Thedistance|1+L0(jωn)|isacomplementarydistance|0,L0|=|L0|to

theunitvalue.Thus

|L0(jωn)| + |1+L0(jωn)| =1, |1+L0(jωn)| =1− |L0(jωn)| . (30) FromtheprinciplesoftheproposedPIDcontrollertuningmethodresultsthattherobustcontrollershiftsthenominal pointoftheplantfrequencyresponseG0(ωn)toapointL0onthenegativerealhalf-axisofthecomplexplane.Thus,

themagnitude|L0(jωn)|=|G0(jωn)||GR(jωn)|=1/GMyieldingtheratio|GR(jωn)|=1/[GM|G0(jωn)|]betweentheradii RGandRL=|GR|RGofthecirclesMGandML,respectively.

Real Axis Im ag inar y Ax is -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

R

G

(jω)

G

(jω)

G

(jω

)

L

(jω

)

R

M

L

(j

ω)

L

(jω

)

M

L

(jω)

1/10

1/10

Fig.7.NyquistplotsofG30(jω),G3N(jω),L30(jω),L3N(jω):forrequiredperformanceηmax0=5%andτs0=12.

TheradiusRLofthedispersioncircleMLiscalculatedas

RL=RG 1

GM|G0(jωn)|

. (31)

Substituting(30b)and(31)intothegeneralrobuststabilitycondition(29)andconsideringthesafetyfactorχL,the followinginequalityholds

GM−1

GM

> χLRG

GM|G0(jωn)|

, (32)

whichaftersomemanipulationsisidenticaltotheprovencondition(28).LetχL=1.2.Accordingtotherobuststability conditionthechosenvalueGMissubstitutedinto(26a)andafterwardstherobustPIDcontrollerparametersareobtained from(26)and(27).Asetupoftheproposedmethodisextensivelyillustratedonthefollowingexample.

6. VerificationoftheproposedrobustPIDcontrollerdesignmethod

ConsiderthefollowinguncertainplantG3(s)withanunstablezero

G3(s)= K3(−α3s+1) (T3s+1)3 , (33) G30(s)= K30(−α3s+1) (T30s+1)3 = 0.8(−7.5s+1) (27.5s+1)3 (34)

withparametersK3,T3andα3varyingwithin±15%aroundthenominalvalues;G30(s)isthenominalmodel.Forthe

aboveplant,arobustPIDcontrolleristobedesignedtoguaranteeamaximumovershootηmax0=5%andamaximum relative settling timeτs0=12for the nominal model(34), andstability of the family ofplantsG3(s)(33) (robust

stability).

1. The measured critical frequencyof the nominal model is ωc=0.0488s−1. From requirements on the nominal closed-loopperformanceresultsts=τs0/ωc=12/0.0488=245.9s.

2. To achieve the expected nominal performance (ηmax0,τs0)=(5%,12), the gain margin and excitation fre-quency are chosen (GM,ωn)=(18dB,0.65ωc) using the “pink” B-parabolas in Fig. 5 as according to (34)

0 200 400 600 800 1000 1200 -0.5 0 G_M=18 dB, ωω_n=0.65ωω_c Co nt ro lled va Time (s) 0 200 400 600 800 1000 1200 -0.5 0 0.5 1 1.5 ηηmaxN_obt.=13.5%, tsN_obt.=301 s G+MN=13.1 dB, ωωn=0.65ωωc

Closed-loop time responses for the worst plant model G3N(s)

C ont ro lled va ri ab le y (t) Time (s)

Fig.8.Closed-loopstepresponseswiththeuncertainplantG3(s)andrequiredvaluesηmax0=5%andτs0=12.

α30/T30=7.5/27.5=0.27<0.3. Uncertaintiesof the plantare included in threeparameters: K3, T3 andα3, the

numberofidentificationexperimentsisN=23=8.

3. Using the sine-wave method, eight points of Nyquist plots of the uncertain plant were identified at ωn=0,65ωc=0,65.0,0488=0,0317s−1:G31(jωn)...G38(jωn)(depictedbyblue“x”inFig.7).Thenominalpoint G30(jωn),whichpositionwascalculatedfromthecoordinatesofidentifiedpointsG3i(jωn),i=1,...,8,islocatedon theNyquistplotofthenominalmodelG30(jωn)(bluecurve)thusprovingcorrectnessoftheidentification.Radius ofthedispersioncircleMGdrawnfromthenominalpointG30(jωn)isRG=0.164.

4. As GM=18dB and the r.h.s. of (27) G0RS=3.52dB, the robust stability condition (26) GM>G0RS is sat-isfied. The designed robust PID controller moves the nominal point G30(jωn) on the negative half-axis into L30(jωn)=G30(jωn)GRrob(jωn)=0.12e−j180

◦

,through whichpasses the Nyquist plot of the nominalopen-loop L30(jωn) (Fig. 7 in green), where the gain margin GM=18dB is guaranteed. The nominal closed-loop step response (Fig. 8a, green curve) proves achieving the required nominal performance ηmax0obtained=4.55%, τs0obtained=ωcts0obtained=0.0488.243=11.86.

5. ThedispersioncircleML (ingreen)radiusRL=0.0573encompassesallpointsL3i(jωn)=G3i(jωn)GRrob(jωn)for i=1,...,8.ThePIDcontrollerhasmovedtheworstpointG3N(jωn)oftheplant(bluesymbol“+”inFig.7)into L3N(jωn)=0.16e−j197

◦

,accordingtoittheestimatedworstgainmarginisGMN=14.9dB.

6. ThesmallestgainmarginwiththeworstpointG3N(jωn)oftheplant(bluesymbol“+”inFig.7)isspecifiedbythe intersectionoftheredNyquistplotwiththenegativerealaxis,wheretheopen-loopgainmarginisG+MN=13.1dB; hereηmaxN=25% and the relativesettling timeτsN=16are expected (accordingto “pink”curves inFig.5 at ωn=0.65ωc).AchievedperformanceηmaxNobtained=13.5%,tsNobtained=301s(redstepresponseinFig.8b)prove thisfact.

Acknowledgment

This research work has been supported by the Slovak Research and Development Agency under grant APVV-0772-12.

Åström,K.J.,Hägglund,T.,1995.PIDControllers:Theory,DesignandTuning,2nd_{edition.InstrumentSocietyofAmerica,ISBN1556175167.}

Bucz, ˇS.,Kozáková,A.,2012.PIDcontrollerdesignforspecifiedperformance.In:IntroductiontoPIDControllers:Theory,TuningandApplication

toFrontierAreas.DepartmentofChemicalEngineering,CLRI,Adyar,India,ISBN978-953-307-927-1.

Reinisch,K.,1974.KybernetischeGrundlegenundBeschreibungKontinuierlicherSystems.VEBVerlagTechnik,Berlin.

Visioli,A.,2006.PracticalPIDControl,AdvancesinIndustrialControl.SpringerLondonLimited,ISBN1846285852.