Robust PID controller design for processes with stochastic parametric uncertainties

ContentslistsavailableatSciVerseScienceDirect

Journal

of

Process

Control

j our na l h o me p ag e :w ww . e l s e v i e r . c o m / l o c a t e / j p r o c o n t

Robust

PID

controller

design

for

processes

with

stochastic

parametric

uncertainties

Pham

Luu

Trung

Duong, Moonyong

Lee

∗

SchoolofChemicalEngineering,YeungnamUniversity,Gyeongsan,RepublicofKorea

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received13June2011

Receivedinrevisedform8June2012 Accepted14June2012

Available online 11 August 2012 Keywords:

Polynomialchaos Robustcontrollerdesign PIDcontroller Statisticalanalysis Stochasticuncertainty

a

b

s

t

r

a

c

t

Thestabilityandperformanceofasystemcanbeinferredfromtheevolutionofstatisticalcharacteristics

ofthesystem’sstates.Wiener’spolynomialchaoscanprovideanefficientframeworkforthe

statisti-calanalysisofdynamicalsystems,computationallyfarsuperiortoMonteCarlosimulations.Thiswork

proposesanewmethodofrobustPIDcontrollerdesignbasedonpolynomialchaosforprocesseswith

stochasticparametricuncertainties.Theproposedmethodcangreatlyreducecomputationtimeand

canalsoefficientlyhandlebothnominalandrobustperformanceagainststochasticuncertaintiesby

solvingasimpleoptimizationproblem.Simulationcomparisonwithothermethodsdemonstratedthe

effectivenessoftheproposeddesignmethod.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Mostengineeringapplicationsinvolvesolvingphysical prob-lemsbyconvertingthemintodeterministicmathematicalmodels withfixedphysicalparameters.Inrealityhowever,these param-etersshowsomerandomness,whichinfluencesthebehaviorof thesolution.Thisrandomnessisnotincorporatedinto determinis-ticmodels.Severalprobabilisticmethodshavebeendevelopedto includethisuncertaintyinmathematicalmodels.

Arepresentativetraditionalprobabilisticapproachfor uncer-taintyquantification is the Monte Carlo (MC) method[1,2]. Its brute-forceimplementationinvolvesfirstgeneratinganensemble ofrandomrealizationswitheachparameterrandomlydrawnfrom itsuncertaintydistribution.Deterministicsolversarethenapplied toeachmembertoobtainanensembleofresults.Theensembleof resultsisthenpost-processedtoestimaterelevantstatistical prop-erties,suchasthemeanandstandarddeviation.Theestimationof themeanconvergeswiththeinversesquarerootofthenumberof runs,makingMCmethodscomputationallyexpensive.

Polynomialchaos (PC)is a modern approach toquantifying uncertaintyinsystemmodels.A PCexpansion,originatingfrom Wienerchaos[3],isa spectralrepresentation ofarandom pro-cessby orthonormalpolynomials of a random variable.For PC expansions of infinite smooth functions(i.e.,analytic, infinitely differentiable),exponentialconvergenceisexpected.Ghanemand

∗ Correspondingauthor.Tel.:+82538102512;fax:+82538113262. E-mailaddress:mynlee@yu.ac.kr(M.Lee).

Spanos[4]showedthatPCisaneffectivecomputationaltoolfor engineering purposes. Karniadakis and Xiu[5] further general-ized PC for usewith non-standard distributions. Puvkov et al. [6]proposedthatiftheWiener–Askeypolynomialchaos expan-sion is chosen according to the probability distribution of the random input, then it can be used to construct simple algo-rithms for the statistical analysis of dynamic systems. Several PCexpansion-basedmethodshavebeenproposedincluding non-intrusivepolynomialchaosexpansions[7],thestochasticresponse surface method [8], and thedeterministic equivalentmodeling method(DEMM)[6,9,10],alsoknownastheprobabilistic collo-cation method (PCM). Controllers for processes withstochastic uncertaintiescanbedesignedbasedonthestatistical character-isticsoftheresponse.Eachdistributionhasanassociatedoptimal polynomial,whichgivesoptimalconvergence.GiventhatthePCM cancomputeresponsestatisticseffectively,bi-levelapproachesto controllerdesignunderuncertaintyhavebeenpursued[6].

ThisworkproposesanefficientmethodforrobustPIDcontroller designthatusesthePCMforprocesseswithstochastic uncertain-ties.Theproposedmethodgreatlyreducescomputationaltimefor controller designand alsoefficiently handlesbothnominaland robustperformanceagainstuncertainties.

Thispaperproceedsasfollows:Section2brieflyintroducesthe basicconceptsandtheoryofthePCmethod.ThePCmethodisthen appliedtothestatisticalanalysisofintervalsystemsandrobustPID controllerdesign.InSection3,robustPIDcontrollersaredesigned forseveralrepresentativesystemswithstochasticuncertaintiesby judgingthestatisticalcharacteristicsoftheresponse.Simulation studycomparestheproposedPIDcontrollerwithothermethods. 0959-1524/$–seefrontmatter © 2012 Elsevier Ltd. All rights reserved.

2. Statisticalanalysisusingpolynomialchaostheory 2.1. Generalizedpolynomialchaos(gPC)theory

Considera control systemgovernedbydifferentialalgebraic equations[10]:



F(t,y,y,...,y(l)_,)=0 g(t0,y(t0),...,y(l)(t0),)=0

(1) where=(1,2,...,N)isarandomvectorofmutually indepen-dentrandom componentswithprobability density functionsof i(i):i→R+;ydenotesastatevariable.

Thus,thejointprobabilitydensityoftherandomvector,,is ␳=



Ni=1i,andthesupportofis≡



N

i=1i∈RN.Thesetof one-dimensionalorthonormalpolynomials,{i(i)dmi=0},canbedefined inthefinitedimensionspace,i,withrespecttotheweight,i(). Based ontheone-dimensional setof polynomials, anN-variate orthonormalsetcanbeconstructedwithPtotaldegreesinthespace  byusingthetensorproductoftheone-dimensionalpolynomials, thebasisfunctionofwhichsatisfies:



 ˚m()˚n()␳()d=



1 m=n 0 m/=n (2)

Consideringaresponsefunctionf(y(t,)),withstatistics(e.g., mean,variance)ofinterest;theN-variatePthorderapproximation oftheresponsefunctioncanbeconstructedas:

fP N(y(t,))= M



i=1  fi(t)˚i(); M+1=



_N+P N



= (N_N!P!+P)! (3)

wherePistheorderofpolynomialchaos,andf_mthecoefficientof thegPCexpansionthatsatisfies(2)as:



f_i=E[˚if(y)]=





f(y)˚i()()d (4)

whereE[]denotestheexpectationoperator.

2.2. Probabilisticcollocationforstatisticalanalysisofcontrol systems

In this work, a probabilistic collocation approach [10] is employedfor thestatistical analysisof controlsystems totake advantageofitscapabilitytodealwithcomplexresponsefunctions effectively.Itsalgorithmisbriefly:

• Chooseacollocationset,{(m) i ,w

(m) i }

qi

m=1,foreachrandom com-ponent, i, for every direction i=1,..., N, and construct a one-dimensionalintegrationrule:

Qq(i)i[g]= qi



j=1

g(i(j))w(j)_i (5)

whereQ []denotesthequadratureapproximationofthe univari-ateintegration,g()isaresponsefunctionofsystemstate,statistics ofwhichareneededtoestimate;w_i(j)and_i(j)arethejthweight andnodetakenfromthecollocationsetforrandomcomponenti. AGaussianquadrature(oraGaussiancollocationset)[11]is usu-allyusedastheone-dimensionalintegralruleinclassicalspectral methodssuchasthePCM(DEMM).

• ObtainanN-dimensionalintegrationrule(cubaturenodesand weights)bythetensorizationoftheone-dimensional integral

rule: Q[g]=(Qq(1)1 ⊗···⊗Q (N) qN )[g] = q1



j1=1 ··· qN



jN=1 g((j1) j ,..., (jN) j )(w (j1) 1 ···w1(jN))



 g()()d (6) where⊗denotesthetensorproduct,andQ_[]denotesthe mul-tivariatecubatureapproximation.

• ApproximatethegPCcoefficientsin(4)usingthenumerical inte-grationrulein(6). ˜  fj=Q[f(y,)˚()␳]= Q



m=1 f((m))˚j((m))w(m) for j=1,...,M (7) where f˜ represents the numerical approximation of f and f()˚j()playtheroleofg()inEq.(6).

• Construct an N-variate Pth order gPC approximation of the responsefunctionintheform: ˜fP

N=

M j=1˜

 fj˚j().

Once all the gPC coefficients have been evaluated, a post-processingprocedureisthencarriedouttoobtainthestatisticsof theresponsefunctionf(y(t,)).

Themeanvalueisthefirstexpansioncoefficient:

E[˜fP N]=f =



 ˜fP N()d=





⎡

⎣



M j=1 ˜  f_j()˚_j()

⎤

⎦

()d₌˜f₁ (8) Thevarianceoftheresponsefunctionf(y(t,))canbeevaluated as: Df = 2f =E[(f −f)2]=





⎛

⎝



M j=1 ˜  f_j()˚j()−˜  f₁

⎞

⎠

×

⎛

⎝



M j=1 ˜  fj()˚j()−˜  f1

⎞

⎠

()d= M



j=2 ˜  f2_j (9)

Eqs.(8)and(9)employthepropertythatthepolynomialset beginswith˚1()=1.Theweightfunctionofthepolynomialis theprobabilitydensityfunction.Ifanf(y)=yresponsefunctionis chosen,themeanandvarianceofthesystem’sstatesare approxi-matelygivenby(8)and(9),respectively.ThesurrogategPCseries ˜fP

N=

M j=1˜



f_j˚j()canbesampledforobtainingtheprobability den-sityfunctionfortheresponsefunction.

Theset{i}di=1i istheorthonormalpolynomialofiwithweight functioni(i).Theweightfunctionistheprobabilitydensity func-tionofrandomvariablei.Thislinksthedistributionofrandom variableiandthetypeoftheorthonormalpolynomialinitsgPC basis.

2.3. Polynomialchaosforarbitrarydistributionanditsassociate quadrature

Let()betheprobabilitydensityfunctionofascalarrandom variable,,whichhasfinitemomentsoforderupto2m,m∈N.Let Pdenotethespaceofarealpolynomial,withPm⊂Pdenotingthe spaceofthepolynomialwithdegreeuptom.Theinnerproductof twopolynomials,pandq,relativetod=()disdefinedby: (p,q)d=





(

)

R

s

-+ ( ) Y s G(s) C(s) + +

(

s

)

D

Fig.1.Closed-loopcontrolsystem.

where isthesupportdomainoftherandomvariable.

Orthonormalpolynomialsrelativetotheprobabilitymeasure, d,canbegivenbythreetermrecurrence[12]:



ˇ_k+1_k+1()=(−˛k)k()−k−1()



ˇk, k=0,1,2..., ₋₁()=0, 0()= 1



ˇ0 (11)

withrecurrencecoefficientsgivenby: ˛k(d)= (k,k)d (k,k)d k=0,1,2..., ˇ_k(d)₌ (k,k)d (k−1,k−1)d ˇ0=



 d() k₌1,2..., (12)

Theindexrangeisinfinite(k≤∞)orfinite(k≤m)depending onwhethertheinnerproductispositivedefiniteonPorPm.Hence, thefirstmrecursioncoefficientpairs,˛k,ˇk,areuniquely deter-minedbythefirst2mmomentsof .Classicalmethodssuchas Chebyshev’s,forobtainingtherecurrencecoefficientsarebasedon moments.However,ingeneralobtainingthesecoefficientsfrom momentsbecomesseverelyill-conditionedandisthusnotuseful even for well-behavedmeasures forwhich there are no classi-calorthogonalpolynomials. Alternatively,approximatemethods basedonthediscretizationofanarbitrarymeasure(e.g.the dis-cretizationprocedure)canbeemployedtoobtaintherecursion coefficients. Discretizationis basedonapproximating theinner productin(10)withadiscretemeasure:

(p,q)_d=



 p()q()()d=(p,q)_dS= S



=1 w()_p((v)_)q((v)₎₍₁₃₎

ThenumberofdiscreteSpointmeasurescanbeincreaseduntil therecurrencecoefficientsareachievedatthedesiredaccuracy. Thegeneral-purposeFejer’squadrature[13]canbeusedforthe discretizationin(13).Gautschi[12]showedthattheperformance ofdiscretizationcanbeincreasedthroughmulti-component dis-cretization.Afterconstructinganorthonormalsetwithrespectto thearbitrarymeasure,thecollocationpointanditscorresponding weightintheGaussianquadraturearegivenby[11]:

w(i)=− an+1 ann+1((i))n((i))

i=1,...,n (14)

wherea_n+1isthecoefficientofn_in

n()and(i)denotesthezeros ofn().

2.4. RobustPIDcontrollerdesign

ConsidertheclosedloopcontrolsysteminFig.1,whereC(s)is aPIDcontroller:

C(s)=Kp+Ki

s +Kds (15)

andG(s)isaprocesswithstochasticparametricuncertainties: G(s)= bmsm+bm−1sm−1+···+b0

ansn+an−1sn−1+···+a0

e−Ls (16)

whereai,bj,Larerandomvariableswithgivendistributions. ThePIDcontroller’sparametersareobtainedbyoptimizingthe costfunction:

min

Kp,Ki,KdJ=Kp,Ki,Kdmin



T 0

|Me(t)|dt (17)

subjecttoaconstraintonthevarianceoftheoutput: max

0≤t≤TDy(t)≤Dmax (18)

whereMandDdenotethemeanandthevarianceofthesignal, respectively.

Sincevarianceisameasureofthevariabilityofarandom pro-cess,alargevarianceofsystemoutputimpliesalargedeviation fromthenominalresponse underparametricuncertainties,and thuscanleadtoeitherverysluggishoroscillatoryresponses.The upperbound,Dmax,ofthevarianceofthesystemoutputcanbe con-sideredthemaximumallowablesensitivityofsystemresponseto theparametricuncertainties.Hence,adjustingDmaxallows trading-off betweenthe robustness ofthe response touncertainty and theresponsespeed.Themeanerrorin(17)canbeinterpretedas theweightedaverageerrorformulti-modelsystems,wheremore importantmodelsaremoreheavilyweighted(withhighervalue probabilitydensities).

Notethatalthoughtheglobaloptimalsolutionisnot guaran-teedduetoitsnonlinearproperty,theconstrainedoptimization problemby(17)and(18)canbereadilysolvedwithconventional non-linearprogrammingmethodswithoutsignificantdifficulties in convergence. The resulting PID parameters have the prop-ertyof afinite variancesystemoutputandMy(t)=1. Fromthe finite of output variance and mean, the Chebychev inequality [14] becomes Pr{|y(t)−My(t)|≥}≤(Dy(t))/2, forany >0, whichindicatesthattheprobabilityofthedeviationfromthemean ofthesystemresponsesisbounded.Thus,althoughtheproposed approachforoptimalPIDdesigndoesnotexplicitlyspecifythe sta-bilizingcondition,theconstraintonthevariancetakesintoaccount therequirementforstabilizingthesystemimplicitlyin stochas-ticsense.Forexample,supposethatDy(t)=0.01; =1atsteady state,thenPr{|y(t)−1|≥1}≤(Dy(t))/1=0.01,whichimpliesthe probabilityofthesystemresponsebeingoutsidethebounddefined by|y(t)−1|=1is just1%and thustheresponse isboundedin stochasticsense.NotethattheboundderivedbytheChebychev inequalityisexpectedtobesomewhatconservative.More accu-rateboundonthesystemoutputcanbefoundbyconstructingthe probabilitydensityfunction(pdf)oftheoutputusingeithertheMC methodortheproposedPCMmethod.

Theoverallprocedureandalgorithmsfortheproposeddesign methodaresummarizedinFig.2.

Note that Dmax is selected by trialand error,and the opti-mizationproblemmaybecomeinfeasibledependingonthechoice ofDmax.Moreover,itisassumedthatthePIDparametersarein boundedintervalsasintheotherexistingdesignmethods[14,15] usingthescenarioapproach (MonteCarlo) forstochastic uncer-tainties.Acrude approximationforthoseboundedintervalscan beobtainedfromtherecentstabilizationalgorithmforarbitrary system[16–18].

3. Examples

Construct optimal polynomial with respect

to stochastic uncertainties

Obtained nodes and weights for numerical

integration from Gaussian quadrature using

(14)

Estimated the coefficient for gPC expansion

of system output using mutli-dimensional

numerical integration by using (6),(7)

Obtain the mean and variance of system

output using (8) and (9)

Robust optimization with

objective and constraint

in (16 ) and (17)

Output optimal controller

parameters

Process with known uncertainties, order of

polynomial chaos, controller parameters

Fig.2.Flowchartoftherobustoptimalcontrollerdesignusingpolynomialchaos.

3.1. Case1

Todemonstratetheproposedmethodbasedforstatistical anal-ysis,thefollowingfirstorderplusdead-timesystemoftheheated tankin[19]isconsidered.Ithastheclosedloopconfiguration out-linedinFig.1.

G(s)= 1 Ts+1e

−Ls ₍₁₉₎

where T,L are random variables with triangular distributions Tr(0.5,1,1.5)andTr(0,0.25,0.5),respectively.Thetriangular distri-butionisgivenbythefollowingcumulativedistributionfunction [20]: Tr(a,c,b)=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

(x−a)2 (b−a)(c−a) if a≤x≤c 1− (b−x) 2 (b−a)(b−c) if c≤x≤b (20) withaPIcontrollerC(s)=2.5+1.67/s.

ThestepinputR(t)=1isintroducedtothesystematt=0.The PCMapproachiscomparedwithanMCmethodforpredicting out-putmeanandvarianceforastepreferenceinput(Fig.3).Thetwo orthonormalpolynomialsets withrespecttoTr(0.5, 1,1.5) and Tr(0,0.25,0.5)areobtainedusing(11)withrecurrencecoefficients givenin Table1.The recurrencecoefficientsfor theorthogonal

0

2

4

6

8

10

0

0.5

1

1.2

0

2

4

6

8

10

0

0.02

0.04

0.05

Halt

on sampli

ng

gPC

Halt

on sampli

ng

gPC

M_y(t) D_y(t)

Fig.3.Statisticalcharacteristicsincase1.

polynomialsetswithrespecttothesetriangulardistributionsare computedwiththemroutinemcdis.m[12],whichrealizesthe algo-rithmdescribedin(11)–(13)inSection2.Thenodesandweightsof cubaturearecalculatedusingtheDEMMlibraryasin(14)[6].The coefficientsofgPCexpansionareestimatedfrom(7)byutilizing thenodesandweights.Ananalytical representationofthe out-putistheneffectivelyobtainedintermsof=(1,2)=(T,L)as ˜yP

N(t)=

M j=1˜



yj(t)˚j().Theprobabilitydensityfunctionofoutput canbeobtainedbyasimplesamplingofthisanalytical representa-tion.NotethatintheMCmethodoneneedstosolvethedynamic equation(1)thousandtimesatsamplingpointsforobtainingthe requiredprobabilitydensity,whileinthePCMmethodonlyata smallnumberofcollocationnodesforobtainingtheanalytical rep-resentation.In thePCMmethod,thecollocationnodesplaythe sameroleasthesamplepointsintheMCmethod.Themeanand varianceareefficientlyestimatedby(8)and(9)fromthe coeffi-cientsofthegPCexpansion.Forasystemwithasmoothresponse, exponentialconvergenceisexpectedforthePCMmethod[10].It isfoundfromextensivesimulationstudythatingeneralonlyfour termsaresufficientineachrandomvariableforpresentingthe ran-domoutput.ThisfastconvergencepropertyofthePCMcanalsobe foundin[10,21].However,inthisstudy,forguaranteeingthe accu-racyrequiredforoptimization,15termswereusedforthetime delaywhileonly5termsforpresentingtherandomtimeconstant, whichledto75collocationnodes.Theprobabilitydensityfunctions ofoutputat0.4s,wherethepeakofthevarianceoccurs,obtained bythePCMandMCmethodsareshowninFig.4.Thecomputation timesrequiredbybothmethodstoobtainthestatistical character-isticsandsimulationparametersarelistedinTable2.Theproposed methodbasedonthePCMismoreefficientthantheMCmethod, whilethegeneratedstatisticalcharacteristicsfrombothmethods

Table1

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 MC PCM y(0.4)

Fig.4.Probabilitydensityfunctionofsystemoutputat0.4sforcase1.

Table2

Simulationparametersandtimeprofilesforobtainingthestatisticalcharacteristic forthePCMandtheMonteCarlo(MC)methodincase1.

Simulation parameters

Computation time QuasiMonteCarlo

(HaltonSamling)

PCM QuasiMonteCarlo

(HaltonSamling)

PCM

8100samples 75collocation points

515.7s 5.9s

arealmostidentical.ThisdemonstratesthatthePCMgivessimilar

accuracywithfarlesscomputationaleffortthantheMCmethod.

3.2. Cases2and3

3.2.1. Case2

ThealgorithminSection2isappliedtothedesignofrobust

controllersfortheFOPDTsystemwithstochasticuncertaintiesin theaboveexample.

OptimalPIDparametersforrobustdesignareobtainedby min-imizingtheobjectivefunction:

min

Kp,Ki,KdJ=Kp,Ki,Kdmin



T 0 |Me(t)|dt (21) subjectto: max 0≤t≤TDY(t)≤0.02 (22)

TheresultingoptimalPIcontrolleris:

C1(s)=1.445+1.465/s (23)

Themeanandvarianceofthesystemwiththeoptimalcontroller C1areshowninFig.5.

3.2.2. Case3

The proposed method is also applied to design a con-trollerfor a FOPDT systemwithT and Luniformly distributed: T∈U[0.5,1.5],L∈U[0,0.5]andthesameobjectiveandconstraint as in case 2.The result ofthe optimizationproblemis C2(s)= 0.989+1.104/s+0.007s.Themeanand varianceof thesystem withoptimalC2anduniformrandomuncertaintiesarealsoshown inFig.5.Thestochasticuncertaintiesin cases2and3have the samesupportdomain,butthevarianceoftheuniformdistribution islargerthanthatofthetriangledistribution.Asexpected,the opti-malcontrollerforuniformstochasticuncertaintiesismorecautious thanthatforthetriangulardistributionincase2.Forcomparison, aboundedregionobtainedfrom2000possibleresponsesofthe FOPDTsystemwithcontrollerC1andeachofthetwodistributions areshowninFig.6.Fig.7showsaboundedregionofoutputfor2000 possibleresponsesofstochasticFOPDTsystemswithcontrollerC2. Fig.6showsthattheoptimalcontrollerC1designedforthe trian-gulardistributionsuffersfromperformancedegradationbecause triangulardistributionsassumethatparameterswithvaluesnear thepeakofthedistributionaremoreheavilyweightedin(21).The C2controller,designedforuniformdistributions,canbehavewell fortriangulardistributions,asexpected.Byminimizingthe objec-tivefunction(21),theC1controller treatsthenominalvalueas themostprobablevalue,leadingtobetternominalperformance, demonstratedbythemeanofthesystemoutputinFig.5.TheC2 controllertendstotreatallthevaluesinsidethesupportdomain

0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 0 0.01 0.02

C1 with triangular distribution C2 with uniform distributions

C1 with triangular distribution C2 with uniform distribution M

y(t)

D

y(t)

t

0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 (a) (a) (b) (b) t y(t)

Fig.6.Boundedregionobtainedfrom2000possibleresponsesoftheuncertain FOPDTsystemswiththeC1controller.(a)TandLareoftriangulardistributions; (b)TandLareofuniformdistributions.

thesamebyminimizationoftheobjectivefunction(21),leadingto morerobustperformance.

3.3. Case4

Theproposedmethodisappliedtotheintervalsystemstudied in[22]:

G(s)= 1

a2s2+a1s+a0

where 1.8≤a2≤4.2, 0.8≤a1≤1.2,

1.4≤a0≤2.6 (24)

TheoptimalPIDparametersforrobustdesignareobtainedby minimizingtheobjectivefunction:

min

Kp,Ki,KdJ=Kp,Ki,Kdmin



T 0 |Me(t)|dt (25) subjectto: max 0≤t≤TDY(t)≤0.015 (26)

Fig.8compares2000possibleresponsesoftheuncertain feed-back system using a constrained optimal PID controller (Kp= 54.0; Ki=38.0; Kd=120.0)obtainedbysolving(25)and(26) andan unconstrained optimalPIDcontroller (Kp=125.6; Ki= 256.3; Kd=383.2)obtainedbyconsidering(25)only.Thespace ofstabilizationsetforPIDparametersisunlimitedinthisexample case.TofindtheoptimumparametersofPIDcontrol,thesearch

0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (b) (b) (a) (a) t y(t)

Fig.7.Boundedregionobtainedfrom2000possibleresponsesoftheuncertain FOPDTsystemwiththeC2controller.(a)TandLareofuniformdistributions;(b)T andLareoftriangulardistributions.

0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5

t

y(t)

(a)

(b)

Fig.8.2000possibleresponsesoftheuncertainsystemincase4using:(a)The constrainedoptimalPIDcontrolleremploying(25)and(26);(b)theunconstrained optimalPIDcontrolleremploying(25)only.

spacewaslimitedtothebox[0,500]3_{forsakeofsimplicityasin} otherworksbasedonprobabilisticapproach[14,15].Themeanand varianceofthesystemoutputsinbothcasesareshowninFig.9. Morerobustandinsensitiveresponsescanbeobtainedfromthe constrainedoptimalPIDcontrollerbytakingoutputvarianceinto accountduringcontrollerdesign;however,thisisattheexpense ofresponsespeed.

Thesameprocessmodel,thoughwithanactuatorsaturation of|u(t)|≤100,isconsideredtoevaluatetheperformanceofthe proposedPIDcontrollerwithactuatorsaturation.Forthis,aPID controllerwithaback-calculationstructure[23]isemployed.The sameprocedurecanbeappliedforfindingtheoptimalPID parame-tersandback-calculationcoefficient.TheresultingPIDparameters are Kp=252.3; Ki=66.1; Kd=71.8, with a back-calculation coefficientof1/Tt=0.3.Fig.10depicts2000possibleresponses fromtheproposedPIDcontrollerwithbackcalculation.Themeans andvariancesofsystemoutputfromthesecontrollersaredepicted inFig.11,whichshowsthattheconstraintonthevarianceofsystem outputisnotactivated;hence,theresultisequivalentto minimiza-tionoftheobjectivefunction(25)only.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.01 0.02 0.025

Constrained optimal PID Unconstrained optimal PID

Constrained optimal PID Unconstrained optimal PID M_y(t)

D_y(t)

t

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 y(t) t

Fig.10.2000possibleresponsesoftheuncertainsystemincase4withacontroller outputsaturationof|u(t)|≤100.

3.4. Case5

Thefollowinghighorderintervalsystem[17]isconsidered: G(s)= 5.2(s+2)

s(s3_+a2s2_+a1s+a0) where 3.5≤a2≤4.8,

12≤a1≤15, 9.5≤a0≤11.5 (27)

AconstrainedoptimalPIDcontrollercanbedesignedby mini-mizingtheobjectivefunction:

min

Kp,Ki,KdJ=Kp,Ki,Kdmin



T 0 |Me(t)|dt (28) subjectto max 0≤t≤TDY(t)≤0.009 (29)

Toevaluateits performance, theproposedoptimalPID con-troller (Kp=2.006; Ki=0.002; Kd=1.846) is compared with otherexistingPIDcontrollers:one(Kp=1.72;Ki=0.05;Kd=0.01) designed based onKharitonovtheorem [17] and another(Kp= 1.6909;Ki=0.5791;Kd=0.7827)fromthemixedH∞/H2method [24].

0

2

4

6

8

10

0

0.5

1

0

2

4

6

8

10

0

0.5

1

x 10

-3

t

t

D

_y

(t)

M

_y

(t)

Fig.11.Meanandvarianceofoutputincase4withacontrolleroutputsaturation of|u(t)|≤100. 0 2 4 6 8 10 0 0.5 1 1.5 0 2 4 6 8 10 0 0.5 1 1.5 0 2 4 6 8 10 0 0.5 1 1.5 t 0 2 4 6 8 10 0 0.5 1 1.5 y(t) (a) (b) (c) (d)

Fig.12.1000possibleresponsesoftheuncertainsystemincase5using:the pro-posedcontroller(a)without,and(d)withaset-pointfilter;andthecontrollersof (b)Huangetal.,and(c)Goncalvesetal.

Fig.12 compares1000possible responsesby each PID con-troller.Themeansandthevariancesofthesystemoutputfrom thesemethodsareshowninFig.13.TheproposedoptimalPID con-trollergivesthemostrobustandfastestclosed-loopresponses.A set-pointfiltercanbeusedifoscillationoftheresponseneedstobe mitigated.AnoptimalPIDcontrollerwithasecondorderset-point filter(˛s+1)/(ˇ2s2+ˇ1s+1)canbedesignedbythesame proce-dure.Theoptimalparametersare(˛=0.0809;ˇ1=1.6184;ˇ2= 0.5897;Kp=3.4520;Ki=2.1724;Kd=1.1748).TheproposedPID controllerwiththeset-pointfiltergivessmootherresponsesandis lessinfluencedbyparametervariations.

3.5. Case6

Inthisexample,aregulatorycase,whichisparticularly impor-tantinprocesscontrol,isconsidered.Theplantistakenfromcase 3(thetimedelayandtimeconstantareofuniformdistributions) forarobustperformancedesignasdiscussedincase3.Thesystem

0

2

4

6

8

10

0

0.5

1

1.5

0

2

4

6

8

10

0

0.005

0.01

0.015

0.032

Proposed Goncalves et.al. Huang et.al

Proposed with set-point filter

Proposed Goncalves et.al. Huang et.al

Proposed with set-point filter

t

M

y

(t)

D

y

(t)

0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 9 10 0 0.005 0.01 D y(t) M y(t) t

Fig.14.Meanandvarianceofoutputincase6.

0 1 2 3 4 5 6 7 8 9 10 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 y(t) t

Fig.15.Boundedregionobtainedfrom2000possibleresponsesoftheuncertain FOPDTincase6.

isnowsubjectedtoinputdisturbanceD(t)=1att=0,whileR(t)=0. Theoptimizationproblemis:

min

Kp,Ki,KdJ=Kp,Ki,Kdmin



T 0 |Me(t)|dt (29a) subjectto: max 0≤t≤TDY(t)≤0.01 (30)

The resulting optimal parameters are: Kp=1.957;Ki= 2.012;Kd=0.32. The mean and variance with this controller isshowninFig.14.Fig.15showsaboundedregionfor2000 pos-sibleresponsesofuncertainsystemwiththeproposedcontroller. Itcanbeseenfromtheresponseoftheuncertainsystemsthatthe proposedcontrolstillcanachieveasatisfactoryresponseforthe regulatorycase.

4. Conclusions

ArobustPIDcontrollerwasdesignedanditsstatisticalanalysis wasstudiedforprocesseswithstochasticparametric uncertain-ties. The use of the polynomial chaos methodgreatly reduced computationtimeoverthetraditionalQuasiMonteCarlomethod, without a loss of accuracy. The stability of uncertain systems

canbeinferredfromthestatisticalevolutionofthesystem out-put,andhencerobustPIDcontrollerscanbeefficientlydesigned withrespecttostochasticuncertainties.Depending onwhether robustnessornominalperformanceisofprimaryconcern,different distributioncanbeassumedforthestochasticuncertainties.With differentassumptionsofthedistributionofuncertainties,optimal PIDparameterswerecomputedusingasimplenonlinear optimiza-tion.Severalsimulationexamplesshowedthattheproposeddesign methodgivesrobustclosed-loopresponsesforvariousstochastic systems.

Acknowledgment

ThisresearchwassupportedbytheHumanResource Develop-mentProgramofKoreaInstituteofEnergyTechnologyEvaluation andPlanning(KETEP)grant(No.20104010100580)fundedbythe KoreanMinistryofKnowledgeEconomy.

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