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Estimation of region of attraction for polynomial nonlinear systems a numerical method

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This paper introduces a numerical method to estimate the region of attraction for polynomial nonlinear systems using sum of squares programming. This method computes a local Lyapunov function and an invariant set around a locally asymptotically stable equilibrium point. The invariant set is an estimation of the region of attraction for the equilibrium point. In order to enlarge the estimation, a subset of the invariant set defined by a shape factor is enlarged by solving a sum of squares optimization problem. In this paper, a new algorithm is proposed to select the shape factor based on the linearized dynamic model of the system. The shape factor is updated in each iteration using the computed local Lyapunov function from the previous iteration. The efficiency of the proposed method is shown by a few numerical examples.

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Estimation of region of attraction for polynomial nonlinear systems a numerical method

  1. 1. ResearchArticle Estimation ofregionofattractionforpolynomialnonlinearsystems: A numericalmethod Larissa Khodadadi a,n,1, BehzadSamadi b,2, HamidKhaloozadeh c,3 a Department ofElectricalEngineering,ScienceandResearchBranch,IslamicAzadUniversity,Tehran,Iran b Department ofElectricalEngineering,AmirkabirUniversityofTechnology,Tehran,Iran c Department ofSystemsandControl,IndustrialControlCenterofExcellence,K.N.ToosiUniversityofTechnology,Tehran,Iran a rticleinfo Article history: Received11July2012 Receivedinrevisedform 12June2013 Accepted8August2013 Availableonline3September2013 This paperrecommendedbyDr.Q.-GWang Keywords: Regionofattraction Sum ofsquaresprogramming Lyapunovstability Polynomial nonlinearsystems Van derPolequation a b s t r a c t Thispaperintroducesanumericalmethodtoestimatetheregionofattractionforpolynomialnonlinear systemsusingsumofsquaresprogramming.ThismethodcomputesalocalLyapunovfunctionandan invariantsetaroundalocallyasymptoticallystableequilibriumpoint.Theinvariantsetisanestimation of theregionofattractionfortheequilibriumpoint.Inordertoenlargetheestimation,asubsetofthe invariantsetdefinedbya shapefactor is enlargedbysolvingasumofsquaresoptimizationproblem.Inthis paper,anewalgorithmisproposedtoselectthe shapefactor basedonthelinearizeddynamicmodelof thesystem.TheshapefactorisupdatedineachiterationusingthecomputedlocalLyapunovfunctionfrom thepreviousiteration.Theefficiencyoftheproposedmethodisshownbyafewnumericalexamples. & 2013ISA.PublishedbyElsevierLtd.Allrightsreserved. 1. Introduction Regionofattraction(ROA)ofalocallyasymptoticallystable equilibriumpointisaninvariantsetsuchthatalltrajectories starting insidethissetconvergetotheequilibriumpoint.ROAis an importanttoolinthestabilityanalysisofsystems,becausethe size oftheROAshowsthathowmuchtheinitialpointscanbefar awayfromtheequilibriumpointandtrajectoriescanstillcon- verge.FindingtheexactROAisingeneralaverydifficult problem. An alternativeistoestimatetheROAbycomputingthelargest possible invariantsubsetoftheROA.Inmanyapplications, finding the stableequilibriumpointsforanonlinearsystemisnot sufficient toanalysisthebehaviorofsystem.Because,inpractice, a stableequilibriumpointwithaverysmallneighborhoodmay not besomuchdifferentcomparingtoanunstableequilibrium point. Besides,theautonomousnonlineardynamicalsystemcan haveseveralequilibriumpointsorlimitcyclessuchthatthe trajectoriesmightconvergetoeachofthesepointsorcyclesin case theyarestable.Therefore,estimatingthestabilityregionof a nonlinearsystemisatopicofsignificant importanceandhas been studiedextensivelyforexamplein [1–15]. Mostcomputa- tional methodsaimtocomputetheboundaryofaninvariantset inside theROA.ThesemethodscanbesplitintoLyapunovand non-Lyapunovmethods.LyapunovmethodscomputeaLyapunov function (LF)asalocalstabilitycertificate andsublevelsetsof this LFprovideinvariantsubsetsoftheROA.Withtherecent advancesinpolynomialoptimizationbasedonsumofsquares (SOS) relaxations,itispossibletosearchforpolynomialLFsfor systemswithpolynomialand/orrationaldynamics.Inthelitera- ture, thefollowingformsofLyapunovcandidatefunctionshave been employedtoestimatetheROAfornonlinearsystems: Rational LFs,PolyhedralLFs,Piecewiseaffine LFs,PolynomialLFs such asSingleLFsandCompositepolynomialLFs.RationalLFsthat approachinfinity ontheboundaryoftheROAareconstructed iterativelyin [1] motivatedbyZubov'swork.References [1,2] presentedmethodsbasedontheconceptofamaximalLF,for estimating theROAofanautonomousnonlinearsystem.In [3] a nonlinear quadraticsystemwithalocallyasymptoticallystable equilibriumpointinoriginwasconsidered.Amethodisthen proposedtodeterminewhetheragivenpolytopebelongstothe ROAoftheequilibriumusingpolyhedralLFs.Theauthorsof [4] proposedamethodtoconstructpiecewiseaffine LFs.They suggestedafan-liketriangulationaroundtheequilibrium.They showedthatifatwodimensionalsystemhasanexponentially stable equilibrium,therewillbealocaltriangulationschemesuch that apiecewiseaffine LFexistforthesystem.Themethod Contents listsavailableat ScienceDirect journalhomepage: www.elsevier.com/locate/isatrans ISATransactions 0019-0578/$-seefrontmatter & 2013ISA.PublishedbyElsevierLtd.Allrightsreserved. http://dx.doi.org/10.1016/j.isatra.2013.08.005 n Corresponding author.Tel.: þ98 9144210986. E-mail addresses: lkhodadadi@iaut.ac.ir (L.Khodadadi), bsamadi@mechatronics3d.com (B.Samadi), h_khaloozadeh@kntu.ac.ir (H.Khaloozadeh). 1 Postal address:Hesarak,Ashrafi Esfahani Blv.,PoonakSq.,Tehran,Iran. 2 Postal address:HafezAve,Tehran,Iran. 3 Postal address:Resaalat,SeyyedKhandanBridge,ShariatiStreet,Tehran,Iran. ISA Transactions53(2014)25–32
  2. 2. proposed in [5] consists ofestimatingtheROAviatheunionofa continuous familyofpolynomialLyapunovestimatesratherthan via oneLyapunovestimate.Thismethodisformulatedasaconvex Linear MatrixInequality(LMI)optimizationbyconsideringstabi- lity conditionsforallcandidateLFsatthesametime [6,16]. In [7], stability analysisandcontrollersynthesisofpolynomialsystems based onpolynomialLFswereinvestigated.TosearchforLFsas wellasthecontrollersandtoprovelocalstability,theproblemwas formulated asiterativeSOSoptimizationproblems.Inthiswork, the sizeoftheROAofthesystemwasalsoestimated.Theproposed method in [7] uses avariable-sizedregiondefined bya shape factor to enlargetheestimationoftheROA.Thegoalisto find thelargest sublevel setofalocalLFthatincludesthelargestpossibleshape factor region.Following [7], theauthorsof [8] proposedusing bilinear SOSprogrammingforenlargingaprovableROAofpoly- nomial systemsbypolynomialLFs.Similarto [7], apolynomialwas employedasa shape factor to enlargetheROAestimation.Forthe same objective,thelevelsetsofapolynomialLFofhigherdegree wereemployedbecausetheirlevelsetsarericherthanthatof quadraticLFs.However,thenumberofoptimizationdecision variablesgrowsextremelyfastasthedegreeofLFandthestate dimension increases.Inordertokeepthenumberofdecision variableslow,usingpointwisemaximumorminimumofafamily of polynomialfunctionswasproposed.In [9], amethodologyis proposed togenerateLFcandidatessatisfyingnecessaryconditions for bilinearconstraintsutilizinginformationfromsimulations. Qualified candidateswereusedtocomputeinvariantsubsets of theROAandtoinitializevariousbilinearsearchstrategies for furtheroptimization.InadditiontoLyapunov-basedmethods, there arenon-Lyapunovmethodssuchas [10] that focusontopo- logical propertiesoftheROA.Forasurveyofresults,aswellasan extensivesetofexamples,thereaderisreferredto [11]. In thelastyears,duetotheimportanceofestimatingtheROA in several fields suchasclinical [17,18], economy [19], traffics [20], biological systems [21], chemicalprocesses [22] etc., theROA estimation hasreceivedconsiderableattention. This paperismotivatedbytheworkin [7] thatusesa shapefactor toenlargetheestimationoftheROA.Itwillbeshownthatthechoice of aproper shape factor is veryimportant.However,nosystematic methodhasbeenproposedtoselectorupdatethe shape factor. Therefore,inthispaper,wepresent ageneralalgorithmforusinga propershapefactortoenlargetheROAestimationfornonlinear systemswithpolynomialvector fields. Itwillbeshownthatthe proposedmethodisabletocomputeanestimationoftheROAofa benchmarkproblemthat,tothebestofourknowledge,islargerthan theresultsobtainedbyexistingmethods. This paperisorganizedasfollows: Section 2 contains mathe- matical preliminaries.ProblemstatementandaLyapunov-based method toestimatetheROAfornonlinearsystemsisexplainedin Section 3. Then,weproposethemainresult,analgorithmfor selecting ashapefactortoimprovetheestimationoftheROAin Section 4. Somenumericalexamplesandsimulationresultshave been shownin Section 5 to showtheefficiency oftheproposed algorithm.Thepapercloseswithaconclusionandoutlookin Section 6. 2. Mathematicalpreliminaries Let thenotationbeasfollows: ℝ, Zþ : therealnumbersetandthepositiveintegerset. ℝn: an n-dimensional vectorspaceoverthe field ofthereal numbers. ℜn: thesetofallpolynomialsin n variables. Consider theautonomousnonlineardynamicalsystem _x¼ f ðxÞ ð1Þ where xAℝn is thestatevectorand xð0Þ ¼ x0 is theinitialstateat t ¼ 0 and f Aℜn is avectorpolynomialfunctionof x with f ð0Þ ¼ 0. The originisassumedtobelocallyasymptoticallystable. Definition 1. When theoriginisasymptoticallystable,theROAof the originisdefined as Ω :¼ fx0j lim t-1 φðt; x0Þ ¼ 0g ð2Þ where φðt; x0Þ is asolutionofEq. (1) that startsatinitialstate x0. Definition 2. A monomial ma in n variablesisafunctiondefined as ma :¼ xa1 1 xa2 2 …xan n ð3Þ for aiAZþ. Thedegreeof ma is definedasdegma :¼Σni ¼ 1ai. Definition 3. A polynomial f in n variablesisa finite linear combination ofmonomials, f :¼Σa cama ¼Σa caxa ð4Þ with caAℝ. Thedegreeof f is defined asdegf :¼ max a degma (ca is non-zero). Definition 4. Define Σn to bethesetofSOSpolynomialsin n variables. Σn :¼ fpAℜnjp ¼ Σ k i ¼ 1 f2i ; f iAℜn; i ¼ 1;…; k:g ð5Þ Obviouslyif pAΣn, then pðxÞZ0 8xAℝn. Apolynomial, pAΣn if (0rQ Aℝrr such that pðxÞ ¼ zT ðxÞQzðxÞ ð6Þ with zðxÞ a vectorofsuitablemonomials [7]. Definition 5. Given fpigmi ¼ 0Aℜn, generalizedS-procedurestates: if thereexist fsigmi ¼ 1AΣn such that p0Σmi ¼ 1sipiAΣn, then [23] mi ¼ 1fxAℝnjpiZ0gDfxAℝnjp0Z0g ð7Þ 3. Estimatingtheregionofattraction In general,exactcomputationoftheROAisadifficult task [9]. Hence, oneshouldlookforanumericalmethodto find thebest possible estimationoftheROA.SincetheLyapunovtechnique is apowerfulmethodininvestigatingthestabilityofnonlinear systems [24,25], inthissection,aLyapunov-basedmethodis described toestimatetheROAbyaLFsublevelset.Thenumerical algorithminthissectionisbasedonalemmafrom [26], thatwill be describedinthefollowing. Ifforanopenconnectedset S in ℝn containing 0,thereexists a function V : ℝn-ℝ suchthat Vð0Þ ¼ 0 andthefollowingconditions hold V 0 ð Þ40; _V ðxÞo0; 8xa0 in S ð8Þ theneveryinvariantsetcontainedin S is alsocontainedintheROAof equilibriumpoint,but S itself neednotbecontainedinROAof0.For findingsuchinvariantsets,aneasywayistouseso-calledlevelsetsof the(local)LF V. Let c be apositivevalue,andconsidertheset MV ðcÞ ¼ fxAℝnjVðxÞrcg ð9Þ Now,theconnectedlevelset MV ðcÞ containing 0,isasubsetof the ROAof0 [26]. L. Khodadadietal./ISATransactions53(2014)2526 –32
  3. 3. The followingcorollaryisadirectresultofthisfact. Corollary 1. If thereexistsacontinuouslydifferentiablefunction V : ℝn-ℝ such that [8] V ispositivedefinite ð10Þ Ω :¼ fxAℝnjVðxÞr1g is bounded ð11Þ xAℝn VðxÞr1 f0gD xAℝn ∂V ∂x f o0 ð12Þ Then forall xð0ÞAΩ, thesolutionofEq.(1)existsand lim t-1 x t ð Þ¼0. As such, Ω is asubsetoftheROAforEq.(1).Thecontinuously differentiable function V is calledalocalLF. In ordertoenlargethe Ω (by choiceof V), theauthorof [7] defines avariablesizedregion Pβ :¼ fxAℝnjpðxÞrβg ð13Þ and maximizes β while imposingtheconstraint PβDΩ. Here β is a positivevalueand pðxÞ is apositivedefinite polynomial,chosen to reflect therelativeimportanceofstates,anditiscalledthe shape factor. Withtheapplicationof Corollary 1, theproblemcan be posedasthefollowingoptimizationproblem [7]: Subject to : VðxÞ40 Forall xAℝn max V Aℜn β =f0g and Vð0Þ ¼ 0 ð14Þ Theset xAℝn V x ð Þr1 is bounded ð15Þ xAℝn pðxÞrβ D xAℝn VðxÞr1 ð16Þ xAℝn VðxÞr1 =f0gD xAℝn ∂V ∂x f o0 ð17Þ Using the S-procedureandSOSprogramming,thefollowing sufficient conditionscanbeobtained max β V Aℜn; Vð0Þ ¼ 0; s1;s2 AΣn Subject to : Vl1AΣn ð18Þ βp s1þ V1 AΣn ð19Þ ð1VÞs2þ ∂V ∂x f þl2 AΣn ð20Þ where s1 and s2 areSOSpolynomialsand liðxÞ is apositivedefinite polynomial oftheform [7] liðxÞ ¼ Σ n j ¼ 1 εijx2j ð21Þ for i ¼ 1; 2 and εij are positivenumbers.UsingSOSpolynomials for solvingtheproblemdecreasesgeneralityoftheproblem. But insteadpossibilityofconvertingtheproblemintoSDPand solvingitefficiently withnumericalsolversisprovided [27]. Inthis paper,wehaveusedYalmip [28] to convertSOSproblemstoSDP problems.Then,theSeDuMisolver [29] has beenusedtosolve the resultingSDPproblems.Sincethereareproductsofdecision variablesinEqs. (19,20), thisisnotalinearSOSprogramming. Here weencounterSOSprogramsthatarebilinearinthedeci- sion polynomials.Wloszekhasproposedaniterationalgorithm, holding onesetofdecisionpolynomials fixedandoptimizingover the otherset(whichisanSDP),thenswitchingoverthesetswhich areoptimizedandholdingtheothers fixed [7]. Weillustratethismethodwithanexampletakenfrom [30]. Example 3.1. Consider thefollowingVanderPolsystem _x1 ¼x2 _x2 ¼ x1þ5ðx21 1Þx2 ð22Þ It hasanunstablelimitcycleandastableequilibriumpointat the origin.Theproblemof finding theROAoftheVanderPol systemshasbeenstudiedextensively,forexample,in [1,7,8,9]. The ROAforthissystemistheregionenclosedbyitslimitcycle. In thefollowing,numericalestimationsoftheROAusing different shapefactorsarepresented. Here wewilltrytwodifferentpolynomialsfor p x ð Þ and estimate the ROA.Firstwechoose pðxÞ ¼ xTx ð23Þ For asixthdegreeLF,theresultisshownin Fig. 1. Thesecond shape factorisapolynomial pðxÞ that alignsbetterwiththeshape of theROA.Thus,thefollowing pðxÞ is chosen pðxÞ ¼ 0:2828x21 0:1290x1x2þ0:0359x22 ð24Þ The resultisshownin Fig. 2. Itisseenthatbyusingtheshape factor inEq. (24), theestimationisquitelargerthanthe first one. As aresult,thecomputedestimationcanpotentiallybeenhanced by choiceofapropershapefactor.Tothebestofourknow- ledge, noalgorithmhasbeenproposedforchoosingaproper Fig. 1. ROAestimationof Example 3.1. BoundaryoftrueROA(gray),levelsetof pðxÞ in theformofEq. (23) (dotted)andestimatedROA(solid). Fig. 2. ROAestimationof Example 3.1. BoundaryoftrueROA(gray),levelsetof pðxÞ in theformofEq. (24) (dotted)andestimatedROA(solid). L. Khodadadietal./ISATransactions53(2014)25–32 27
  4. 4. shape factor.Toaddressthisissue,inthenextsection,wewill propose analgorithm. 4. Mainresult In theprevioussection,itwasshownthatchoosingtheshape factor greatlyaffectstheshapeandthesizeoftheROAestimation. Forsecondordersystems,itmightbepossibletosearchforbetter choices of pðxÞ by lookingatthephaseportraitofthesystem. However,thisishardlypossibleforsystemsinhigherdimensions where weareunabletovisualizetheROA [7]. In [7,31], a completelyuninspiredpolynomialas Eq. (23) waspickedto enlarge theROAestimationfordifferentsystemswithROAsof different shapes.Althoughwhenwedonothaveanyinformation about theROA,anuninspiredchoiceof p x ð Þ seems tobeagood choice, abetteroptionistochoosetheshapefactorbasedonthe dynamics ofeachsystem,inordertocomputeabetterestimation of theROA.Inthefollowing,wepresentanalgorithmforchoosing the shapefactor.Assumingthattheoriginisalocallyasymptoti- cally stableequilibriumpointforthesysteminEq.(1),the proposed algorithmtocomputeanestimationoftheROAis described inthefollowing: Algorithm1 Initialization: Compute theJacobianmatrixof f evaluatedat x ¼ 0: A ¼ ∂f ðxÞ ∂x x ¼ 0 ð25Þ Then solvetheequation ATPþPA¼I ð26Þ for apositivedefinite matrix P. Define VðxÞ¼xTPx ð27Þ pðxÞ¼xTPx ð28Þ γ Step: Hold V fixedandsolvethefollowingSOSprogramming for s2ðxÞ Subject to : γV s2þ ∂V ∂x f þl2 max s2 A Σn; γ A ℝ γn AΣn ð29Þ where l2 is defined inEq. (21). Thereareproductsofdecision variables, γ and s2, inthisstepofthealgorithm.Since γ is a positivescalarvariableand s2 is apolynomial,thereforewe can usethebisectionmethoddescribedin [33] for solvingthe bilinear SOSproblem. β Step: Hold V and p fixedandsolvethefollowingSOS programmingfor s1ðxÞ Subject to : ½ðβpÞs1þðVγÞ max s1 A Σn; β Aℝ βn AΣn ð30Þ Similar to γ Step,thebilinearprobleminthisstepisalsosolved by bisectionmethod. V Step: Hold s1, s2, βn, γn and p fixedandcompute V such that ∂V ∂x f þl2þs2 γV AΣn ð31Þ Vγ þs1 βp AΣn ð32Þ Vl1AΣn; Vð0Þ¼0 ð33Þ where l1 is defined intheformofEq. (21). Scale V: Replace V with V=γn. Inthisstepanew V is obtained which satisfies theconstraintsoftheproblem. Update p: Replacethequadraticpartofthenew V as new p. Repeat: Nowwithnew V and new p repeatthealgorithmto reach tothemaximumnumberofiterations.Intheotherword, the differenceoftwoconsecutiveestimationparametersisless than aspecified tolerance. Remark1. There aretwomajordifferencesbetweentheproposed algorithmandthe V-s iterationin [9] that potentiallyleadto a largerROAestimation.The first differenceisthatinthe initialization stepoftheproposedalgorithm,weset p as theLF for thelinearizedsystem(Eq. (28)), whichisobtainedfromEq. (26) insteadofanuninspiredshapefactoroftheformEq. (23). Updatingtheshapefactorineachiterationisthesecondmajor difference betweenthetwoalgorithms.Weupdate p by the quadratictermoftheLF,whichsatisfies thelocalstability constraints.Itisshownbyvariousnumericalexamplesthat updating p in thisway,greatlyimprovestheROAestimation.The resultingROAestimationhasbeenshowntobelargerthanthe resultsofthepreviousworkssuchas [7,9,31] thatusea fixed shape factortoestimatetheROA. Remark2. V has tobepositivedefinite and Vð0Þ¼0, hence, it doesnotcontainconstantandlinearterms.Alsothedegree of V should beeven.Thelackofaconstanttermin VðxÞ imposes some constraintsontheselectionofmonomialsintheSOSmulti- pliers. Forsatisfyingconstraintin Eq. (19), wedefine thedegreeof s1 such thatdeg pþdeg s1Zdeg V. Asweknow, f ð0Þ¼0 causes ð∂V=∂xÞf not tocontainanyconstantorlinearterms.SoforSOS conditions thatrequire ð∂V=∂xÞf o0 ontheset fxjVðxÞr1g, theSOS multiplier associatedwiththeterm ð1VÞ should nothavea constant term.Therefore,inordertosatisfyEq. (20), wetakethe degree of s2 such thatdegV þdegs2 be largerthanthemaximum degree of s2, l2 and ð∂V=∂xÞf [7,32]. Remark3. For theinitializationstepofthealgorithm,any Pq that is asolutionof ATPqþPqA¼Q ð34Þ with Q40 couldhavebeenchosen.Inthefollowing,weexplain why Q ¼ I is chosen.ThedynamicequationofthesysteminEq. (1) can bewrittenas: _x¼ Axþf 1ðxÞ ð35Þ where Ax is thelinearizedpartofthedynamics.Let f 1ðxÞ ¼ f ðxÞAx ð36Þ lim jjxjj-0 jjf 1ðxÞjj jjxjj ¼ 0 ð37Þ Define VðxÞ ¼ xTPqx ð38Þ where Pq is thesolutionofEq. (34). In [26], itisshownthat: _V ðxÞrjjxjj½2λmaxðPqÞjjf 1ðxÞjjλminðQÞjjxjj ð39Þ If r40 ischosensuchthat jjf 1ðxÞjj jjxjj o λminðQÞ 2λmaxðPqÞ ; 8xABr ð40Þ where Br is theballofradius r, then _V ðxÞo0 whenever xABr and xa0. Asitissaidin [26], everyboundedlevelsetof VðxÞ contained in Br is alsocontainedintheROA. L. Khodadadietal./ISATransactions53(2014)2528 –32
  5. 5. Now Eq. (40) makes itclearthatthelargertheratio λminðQÞ=λmaxðPqÞ is, thelargerthepossiblechoiceof r becomes. Hence the “best” choice of Q is onethatmaximizestheratio μðQÞ ¼ λminðQÞ λmaxðPqÞ ð41Þ where Pq of coursesatisfies Eq. (34). Itisshownin [26] that μðUÞ is notaffectedbyscaling Q and theidentitymatrixisthebest choice for Q among positivedefinite matriceswith λminðQÞ ¼ 1. Therefore P, thesolutionofEq. (26) allowsforthelargestball Br in which V_ ðxÞ is negative. Choosing Q ¼ I providesonlythelargestballsothischoicecan be conservative. Remark4. In ordertoguaranteetheexistenceofaquadratic LF allowingtoinitializethealgorithm,weconsider A in theEq. (25) tobeHurwitz.Thereforethealgorithmwillalwaysremainfeasible if itisstartedfromafeasiblepoint.ThefeasiblepointisaLFsuch thatthereexist s1 and s2 thatmaketheproblemfeasible [7]. Remark5. In theproposedalgorithm,thequadraticpartofthe computed V is employedtoupdate p. Tojustifythischoice,the Van derPolexamplein section 3 is explainedwithmoredetails in thefollowing.IntheVanderPolsystemin Eq. (22), the boundary oftheROAcanbeplottedusingsimulationdata.The best ellipsoidthatapproximatestheboundaryoftheROAofthis systemis pðxÞ ¼ 1 withthe p in Eq. (24). Usingthisellipsoidasthe shape factorintheROAestimationproblemdescribedbyEqs. (18)–(20), alocalLFofdegreesixwasobtained: VðxÞ ¼ 0:1008x61 þ0:08008x51 x2þ0:01761x41 x22 0:01027x31 x32 þ0:00254x21 x42 0:0007x1x52 þ0:00013x62 0:51883x41 0:1535x31 x2þ0:06856x21 x22 0:00214x1x32 0:00028x42 þ1:0696x21 0:38864x1x2þ0:05370x22 ð42Þ The ellipsoidandtheestimatedROA VðxÞ ¼ 1 wereshownin Fig. 2. When x is inthecloseneighborhoodoftheorigin, VðxÞ can be approximatedbyitsquadraticpart: V2ðxÞ ¼ 1:0696x21 0:38864x1x2þ0:05370x22 ð43Þ V2ðxÞ can berewrittenas V2ðxÞ ¼ 3ð0:3565x21 0:1295x1x2þ0:0179x22 Þ ð44Þ This showsthat V2ðxÞ ¼ 3 isanellipsoidsimilartotheellipsoid that wasusedastheshapefactor: pðxÞ ¼ 1 inEq. (24). This similarity ledustotheideathatthequadraticpartofthelocal LF canbeusedastheshapefactorforthenextiterationofthe algorithm. Althoughtheinspirationcamefromapproximatingthe exactROAwithanellipsoid,thealgorithmdoesnotneedtoknow the exactROAandthereforesimulationdataisnotused.Itwillbe shown inthenextsectionthatthisideagreatlyimprovesthe performance oftheROAestimationalgorithm. The proposedalgorithmcanpotentiallyimprovetheROAesti- mation ofsystemswithanunboundedstabilityregionbyincreas- ing thenumberofiterations.Inpreviousworksuch as [8], itcanhappenthatafterafewiterations,evenwhenthe stability regionisunbounded,theestimationdoesnotimprove because theshapefactoris fixedandenlargementoftheROA estimation islimitedwiththat fixedshapefactor.Whereasinthe proposed approachtheshapefactorcanchangeitsdirectionto coveralargerareainsidetheROAineachiteration.Atthe initialization stepoftheproposedalgorithm,thereisstillno computed localLFavailable.Therefore,theLFcorrespondingto the linearizeddynamicsofthesystemisusedastheLF.ThisLFis quadraticsotheinitialshapefactorischosenequaltothisLF.Since this choiceisbasedonthedynamicsofthesystem,itseemstobe more efficient thananuninspiredchoiceof pðxÞ as Eq. (23) for this example.Thenextsectionshowsefficiency oftheproposed algorithm. 5. Examplesandsimulationresults In thissection,numericalexamplesarepresentedtoillustratethe proposedmethodforenlargingtheROAestimation.Theseexam- ples aresecondordersystemswithboundedandunbounded ROAs. First, wepresentthesystemswhichtheirexactstability boundaries areknownandcaneasilybeshownonphaseportraits. Van derpolequationinreversetimeisagoodexampleofthistype of systems.WeshallbenchmarkinnerboundsontheROA obtained withourmethodsagainsttheexactstabilityboundaries. Then, weconsidersecondordersystemswithunboundedROAs. At theendofeachexample,wecompareourmethodwith previouswork. 5.1.SystemswithboundedROAs Example 5.1. In section 3 we discussedtheVanderPolequation in reversetimeandestimateditsROA.HerewecomputetheROA for thatsystemwiththeproposedalgorithm. _x1 ¼x2 _x2 ¼ x1þ5ðx21 1Þx2 ð45Þ In initializingstepofalgorithm,matrices A and P arecomputedby using Eqs. (25) and (26) respectively: A ¼ 0 1 1 5 ð46Þ P ¼ 2:7 0:5 0:5 0:2 ð47Þ we initializethefunctionsasfollows: VðxÞ ¼ 2:7x21 x1x2þ0:2x22 ð48Þ pðxÞ ¼ 2:7x21 x1x2þ0:2x22 ð49Þ Then, wehold V fixedandsolvetheEq. (29). Forsolvingthe bilinear problem,weapplybisectionmethod,describedin [33]. To this aim,wetake γA½0; 10 and define thebisectiontolerance equalto103. Wealsotake l2 ¼ 106 ðx21 þx22 Þ. Inthisstep,we obtain γ ¼ 1:1115 and s2ðxÞ ¼ 0:5768x41 þ0:28121x31 x2þ0:34161x21 x22 0:18304x1x32 þ0:13742x42 þ0:53518x21 0:55863x1x2þ0:42883x22 ð50Þ In thenextstep,wehold V and p fixedandsolvetheEq. (30). In thisstep,wealsoapplythebisectionmethodbyconsidering βA½0; 10 and thebisectiontoleranceequalto103. So,weobtain: s1ðxÞ ¼ 0:29828x41 0:06448x31 x2þ0:22045x21 x22 0:07636x1x32 þ0:01596x42 þ0:06618x21 0:08892x1x20:07466x22 þ0:92221 ð51Þ and β ¼ 1:1115. Wetake l1 ¼ 106 ðx21 þx22 Þ. In V step, wehold s1 and s2 and β and γ fixed.Forachieving richer levelsetsofLFs,wesearchforasixthdegreepolynomialLFs. By solvingtheEq. (31)-(33), wehave: VðxÞ ¼ 0:20506x61 0:00441x51 x2þ0:03987x41 x22 0:01754x31 x32 þ0:01075x21 x42 0:00434x1x52 þ0:00083x62 0:20809x41 0:1323x31 x2þ0:07401x21 x22 þ0:01021x1x32 0:00863x42 L. Khodadadietal./ISATransactions53(2014)25–32 29
  6. 6. þ1:5876x21 0:67042x1x2þ0:17587x22 ð52Þ WereplacethequadratictermsofLFasshapefactor: pðxÞ ¼ 1:5876x21 0:67042x1x2þ0:17587x22 ð53Þ These stepsarethenrepeatedtoachieveanacceptableestima- tion. Hereweexperimentallysetthenumberofiterationsto30. In thisexample,byusinginformationofLyapunovfunctionfrom the 29thiteration,wechoosetheshapefactoras: pðxÞ ¼ 1:1046x21 0:37796x1x2þ0:042117x22 ð54Þ Then wehave: VðxÞ¼0:09720x61 þ0:04312x51 x20:00141x41x22 0:00890x31 x32 þ0:00232x21 x42 0:00025x1x52 0:52973x41 0:10206x31 x2 þ0:10739x21 x22 0:01492x1x32 þ0:00056x42 þ1:1112x21 0:37955x1x2þ0:04222x22 ð55Þ Finally thelevelsetofthe V in Eq. (55) is theROAestimationof this system.Theresultsareshownin Fig. 3. Comparingwith Fig. 1, the ROAestimationisenhancedsaliently. Example 5.2. The nextexampleistakenfrom [5]: _x1 ¼ x2 _x2 ¼2x13x2þx21 x2 ð56Þ Chesi estimatedtheROAforthissystembyusingaunionof quadraticLFs(Fig. 4). UsingasixthdegreeLFtoestimatetheROA, the resultingROAisshownin Fig. 5. Thisisagainasignificant improvementcomparingto Fig. 4. 5.2. SystemswithunboundedROAs Example 5.3. Consider thefollowingsystem [34]: _x1 ¼4x31 þ6x21 2x1 _x2 ¼2x2 ð57Þ Phase portraitofthissystemshowsthattheROAofthe equilibriumintheoriginisnotbounded. Authorsof [34] estimated theROAofthissystemusingaquartic LF.WealsoestimatetheROAofthissystemusingaquarticLF. The resultafter30iterationsisshownin Fig. 6. Itisseenthatthe estimation isbetterthanthatof [34]. Inthiscase,theROA estimation canbeimprovedbyincreasingthenumberofiterations without increasingthedegreeofthecandidateLF.Theresultfor 150iterationsisshownin Fig. 7. Fig. 3. ROAestimationof Example 5.1. BoundaryoftrueROA(gray),levelsetof pðxÞ in theformofEq. (54) (dotted)andestimatedROA(solid). Fig. 4. ROAestimationof Example 5.2. BoundaryoftrueROA(gray),ROA estimation providedbyChesiin [5] (solid). Fig. 5. ROAestimationof Example 5.2. BoundaryoftrueROA(gray),levelsetof pðxÞ (dotted)andestimatedROAprovidedbytheproposedalgorithm(solid). Fig. 6. ROAestimationof Example 5.3. ROAestimationtakenfrom [23] (gray),Level set of pðxÞ (dotted)andestimatedROAprovidedbyproposedalgorithm(solid). L. Khodadadietal./ISATransactions53(2014)2530 –32
  7. 7. Example 5.4. Consider thefollowingsecond-orderquadraticsys- tem [3]: _x1 ¼50x116x2þ13:8x1x2 _x2 ¼ 13x19x2þ5:5x1x2 ð58Þ The originisalocallyasymptoticallystableequilibriumforthis system.TheROAestimationcomputedin [3] and theresultfrom the proposedalgorithmareshownin Fig. 8. Remark6. Although theproposedalgorithmimprovestheesti- mated ROAforthebothofsystemswithboundedandunbounded ROAs,theestimationisbounded.Alsoforgloballystablesystems we cannotconcludethattheROAisthewholestatespace. Remark7. The mainadvantageofstudyingmultivariatepolyno- mial systemsisthat,polynomialsystemscanbeusedtoapprox- imate manynonlinearsystems.Moreover,severalmethodsexist for approximatingthenon-polynomialsystemswithpolynomial ones. Ontheotherhand,therearevariousavailabletoolsfor stability analysisanddesign.Forexample,somepowerfuland promising relaxationtechniquessuchasSOSprogrammingare suitable forpolynomialsystemswithpolynomialLFs.Also,some softwaressuchasSOSTOOLS [35] and Gloptipoly [36] make the analysisofpolynomialsystemsstraightforward.Allofthese provideourmotivationforinvestigatingsystemswithpolynomial descriptions. 6. Conclusion In thispaper,weproposedamethodforestimatingtheROA of nonlinearsystemswithpolynomialvector fields. Theproposed method containsanalgorithmforupdatingtheshapefactor. The resultsofthismethodwasdemonstratedforaVanderPol systeminreversetimeandalsoothersystemswithboundedand unbounded ROAs.Thesenumericalexamplesshowedthepractical importance oftheproposedmethod. Appendix A In thissection,weexplainbisectionmethodforsolving the existingnon-convexoptimizationproblem.Theinterval [γlowerγupper] isguaranteedtocontain γn. Given γlower rγn, γupperZγn and bisectiontolerance40 While ðγupperγlowerÞZbisection tolerance 1. γtry ¼ ðγlowerþγupperÞ=2 2. SolvetheconvexfeasibilityprobleminEq. (1) 3. Iftheproblemisfeasible, γlower ¼ γtry else γupper ¼ γtry 4. End. References [1] Vannelli A,VidyasagarM.MaximalLyapunovFunctionsandDomainsof AttractionforAutonomousNonlinearSystems.Automatica1985;21:69–80. [2] RozgonyiS,HangosKM,SzederkényiG.Improvedestimationmethodofregion of stabilityfornonlinearautonomoussystems.Technicalreport;2006. [3] Amato F,CalabreseF,CosentinoC,MerolaA.Stabilityanalysisofnonlinear quadraticsystemsviapolyhedralLyapunovfunctions.Automatica2011;47: 614–7. [4] Giesl P,HafsteinS.Existenceofpiecewiseaffine Lyapunovfunctionsintwo dimensions. JournalofMathematicalAnalysisandApplications2010;371: 233–48. [5] Chesi G.Estimatingthedomainofattractionviaunionofcontinuousfamilies of Lyapunovestimates.SystemsandControlLetters2007;56:326–33. [6] Chesi G.Domainofattraction:analysisandcontrolviaSOSprogramming. London: Springer;2011. [7] Jarvis-WloszekZW.LyapunovBasedAnalysisandControllerSynthesisfor PolynomialSystemsusingSum-of-SquaresOptimization.Berkeley:University of California;2003(Dissertation(Ph.D)). [8] TanW.,PackardA.Stabilityregionanalysisusingsumofsquaresprogram- ming. In:Proceedingsofthe2006Americancontrolconference;2006. p. 2297–302. [9] TopcuU,PackardA,SeilerP.Localstabilityanalysisusingsimulationsand sum-of-squaresprogramming.Automatica2008;44:2669–75. [10] Chiang HD,ThorpJS.Stabilityregionsofnonlineardynamicalsystems: a constructivemethodology.IEEETransactionsonAutomaticControl1989;34: 1229–41. [11] Genesio R,TartagliaM,VicinoA.Ontheestimationofasymptoticstability regions: stateoftheartandnewproposals.IEEETransactionsonAutomatic Control 1985;30:747–55. [12] Valmorbida G,TarbouriechS,GarciaG.Regionofattractionestimatesfor polynomial systems.In:Proceedingsofthe48hIEEEconferenceondecision and control.Shanghai,China;2009.p.5947–52. [13] Henrion D,PeaucelleD,ArzelierD, Šebek M.Ellipsoidalapproximationofthe stability domainofapolynomial.IEEETransactionsonAutomaticControl 2003;48:2255–9. [14] Ou L,ZhangW,GuD.Nominalandrobuststabilityregionsofoptimization- based PIDcontrollers.ISATransactions2006;45:361–71. [15] Zhai J,DuH.Semi-globaloutputfeedbackstabilizationforaclassofnon- linear systemsusinghomogeneousdominationapproach.ISATransactions 2013;52:231–41. [16] Chesi G,GarulliA,TesiA,VicinoA.LMI-basedcomputationofoptimal quadraticLyapunovfunctionsforoddpolynomialsystems.International Journal ofRobustandNonlinear2005;15:35–49. [17] Merola A,CosentinoC,AmatoF.Aninsightoftumordormancyequilibrium via theanalysisofitsdomainofattraction.BiomedicalSignalProcessing 2008;3:212–9. Fig. 8. ROAestimationof Example 5.4. Levelsetof pðxÞ (dotted),ROAestimation providedin [3] (gray)andestimatedROAbyusingtheproposedalgorithm(solid). Fig. 7. ROAestimationof Example 5.3. Levelsetof pðxÞ (dotted)andROAestimation by increasingtheiterationto150(solid). L. Khodadadietal./ISATransactions53(2014)25–32 31
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