EE Winter Lecture Analysis of systems with sector nonlinearities Sector nonlinearities - PDF document

EE363Winter2008-09Lecture16AnalysisofsystemswithsectornonlinearitiesSectornonlinearitiesLur'esystemAnalysisviaquadraticLyapunovfunctionsExtensiontomultiplenonlinearities16{1 Sectornonlinearitiesafunction:R!Rissaidtobeinsector[l;u]ifforallq2R,p=(q)liesbetweenlqanduq qpp=(q)p=lqp=uqcanbeexpressedasquadraticinequality(puq)(plq)0forallq;p=(q)Analysisofsystemswithsectornonlinearities16{2 examples:sector[1;1]meansj(q)jjqjsector[0;1]means(q)andqalwayshavesamesign(graphinrst&thirdquadrants)someequivalentstatements:isinsector[l;u]iforallq,(q)u+l 2qul 2jqjisinsector[l;u]iforeachqthereis(q)2[l;u]with(q)=(q)qAnalysisofsystemswithsectornonlinearities16{3 Nonlinearfeedbackrepresentationlineardynamicalsystemwithnonlinearfeedback pq(t;
)_x=Ax+Bpq=Cxclosed-loopsystem:_x=Ax+B(t;Cx)acommonrepresentationthatseparateslinearandnonlineartime-varyingpartsoftenp,qarescalarsignalsAnalysisofsystemswithsectornonlinearities16{4 Lur'esystema(singlenonlinearity)Lur'esystemhastheform_x=Ax+Bp;q=Cx;p=(t;q)where(t;
):R!Risinsector[l;u]foreachthereA,B,C,l,anduaregiven;isotherwisenotspeciedacommonmethodfordescribingtime-varyingnonlinearityand/oruncertaintygoalistoprovestability,orderiveabound,usingonlythesectorinformationaboutifwesucceed,theresultisstrong,sinceitappliestoalargefamilyofnonlineartime-varyingsystemsAnalysisofsystemswithsectornonlinearities16{5 StabilityanalysisviaquadraticLyapunovfunctionslet'strytoestablishglobalasymptoticstabilityofLur'esystem,usingquadraticLyapunovfunctionV(z)=zTPzwe'llrequireP�0and_V(z)V(z),where�0isgivensecondconditionis:_V(z)+V(z)=2zTP(Az+B(t;Cz))+zTPz0forallzandallsector[l;u]functions(t;
)sameas:2zTP(Az+Bp)+zTPz0forallz,andallpsatisfying(puq)(plq)0,whereq=CzAnalysisofsystemswithsectornonlinearities16{6 wecanexpressthislastconditionasaquadraticinequalityin(z;p):zpTCTCCTC1zp0where=lu,=(l+u)=2so_V+V0isequivalentto:zpTATP+PA+PPBBTP0zp0wheneverzpTCTCCTC1zp0Analysisofsystemswithsectornonlinearities16{7 by(lossless)S-procedurethisisequivalentto:thereisa0withATP+PA+PPBBTP0CTCCTC1orATP+PA+PCTCPB+CTBTP+C0anLMIinPand(2;2blockautomaticallygives0)byhomogeneity,wecanreplaceconditionP�0withPIournalLMIisATP+PA+PCTCPB+CTBTP+C0;PIwithvariablesPandAnalysisofsystemswithsectornonlinearities16{8 hence,canecientlydetermineifthereexistsaquadraticLyapunovfunctionthatprovesstabilityofLur'esystemthisLMIcanalsobesolvedviaanARE-likeequation,orbyagraphicalmethodthathasbeenknownsincethe1960sthismethodismoresophisticatedandpowerfulthanthe1895approach:{replacenonlinearitywith(t;q)=q{chooseQ�0(e.g.,Q=I)andsolveLyapunovequation(A+BC)TP+P(A+BC)+Q=0forP{hopePworksfornonlinearsystemAnalysisofsystemswithsectornonlinearities16{9 Multiplenonlinearitiesweconsidersystem_x=Ax+Bp;q=Cx;pi=i(t;qi);i=1;:::;mwherei(t;
):R!Rissector[li;ui]foreachtweseekV(z)=zTPz,withP�0,sothat_V+V0lastconditionequivalentto:zpTATP+PA+PPBBTP0zp0whenever(piuiqi)(piliqi)0;i=1;:::;mAnalysisofsystemswithsectornonlinearities16{10 wecanexpressthislastconditionaszpTcicTiicieTiieicTieieTizp0;i=1;:::;mwherecTiistheithrowofC,eiistheithunitvector,i=liui,andi=(li+ui)=2nowweuse(lossy)S-proceduretogetasucientcondition:thereexists1;:::;m0suchthatATP+PA+PPmi=1iicicTiPB+Pmi=1iicieTiBTP+Pmi=1iieicTiPmi=1ieieTi0Analysisofsystemswithsectornonlinearities16{11 wecanwritethisas:ATP+PA+PCTDFCPB+CTDGBTP+DGCD0whereD=diag(1;:::;m);F=diag(1;:::;m);G=diag(1;:::;m)thisisanLMIinvariablesPandD2;2blockautomaticallygivesusi0byhomogeneity,wecanaddPItoensureP�0solvingtheseLMIsallowsusto(sometimes)ndquadraticLyapunovfunctionsforLur'esystemwithmultiplenonlinearities(whichwasimpossibleuntilrecently)Analysisofsystemswithsectornonlinearities16{12 Example x1x2x31=s1=s1=s1(
)2(
)3(
)2weconsidersystem_x2=1(t;x1);_x3=2(t;x2);_x1=3(t;2(x1+x2+x3))where1(t;
);2(t;
);3(t;
)aresector[1;1+]givesthepercentagenonlinearityfor=0,wehave(stable)linearsystem_x=2422210001035xAnalysisofsystemswithsectornonlinearities16{13 let'sputsysteminLur'eform:_x=Ax+Bp;q=Cx;pi=i(qi)whereA=0;B=2400110001035;C=2410001022235thesectorlimitsareli=1,ui=1+dene=liui=12,andnotethat(li+ui)=2=1Analysisofsystemswithsectornonlinearities16{14 wetakex(0)=(1;0;0),andseektoboundJ=Z10kx(t)k2dt(for=0wecancalculateJexactlybysolvingaLyapunovequation)we'llusequadraticLyapunovfunctionV(z)=zTPz,withP0LyapunovconditionsforboundingJ:if_V(z)zTzwheneverthesectorconditionsaresatised,thenJx(0)TPx(0)=P11useS-procedureasabovetogetsucientcondition:ATP+PA+ICTDCPB+CTDBTP+DCD0whichisanLMIinvariablesPandD=diag(1;2;3)notethatLMIgivesi0automaticallyAnalysisofsystemswithsectornonlinearities16{15 togetbestboundonJforgiven,wesolveSDPminimizeP11subjecttoATP+PA+ICTDCPB+CTDBTP+DCD0P0withvariablesPandD(whichisdiagonal)optimalvaluegivesbestboundonJthatcanbeobtainedfromaquadraticLyapunovfunction,usingS-procedureAnalysisofsystemswithsectornonlinearities16{16 UpperboundonJ 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 100 101 P11(upperboundonJ)boundistightfor=0;for0:15,LMIisinfeasibleAnalysisofsystemswithsectornonlinearities16{17 Approximateworst-casesimulationheuristicmethodfornding`bad'i's,i.e.,onesthatleadtolargeJndVfromworst-caseanalysisasaboveattimet,choosepi'stomaximize_V(x(t))subjecttosectorconstraintsjpiqijjqijusing_V(x(t))=2xTP(Ax+Bp),wegetp=q+diag(sign(BTPx))jqjsimulate_x=Ax+Bp;p=q+diag(sign(BTPx))jqjstartingfromx(0)=(1;0;0)Analysisofsystemswithsectornonlinearities16{18 Approximateworst-casesimulationAWCsimulationwith=0:05:Jawc=1:49;Jub=1:65forcomparison,linearcase(=0):Jlin=1:00 0 5 10 15 20 25 30 35 40 45 50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 tx2(t)Analysisofsystemswithsectornonlinearities16{19 Upperandlowerboundsonworst-caseJ 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 100 101 boundsonJlowercurvegivesJobtainedfromapproximateworst-casesimulationAnalysisofsystemswithsectornonlinearities16{20