Nonlinear Systems and Control Lecture Converse Lyapunov Functions Time Varying Systems - PDF document

NonlinearSystemsandControlLecture#12ConverseLyapunovFunctions&TimeVaryingSystems –p.1/18 ConverseLyapunovTheorem–ExponentialStability Letx=0beanexponentiallystableequilibriumpointforthesystem_x=f(x),wherefiscontinuouslydifferentiableonD=fkxkrg.Letk,,andr0bepositiveconstantswithr0r=ksuchthatkx(t)kkkx(0)ket;8x(0)2D0;8t0whereD0=fkxkr0g.Then,thereisacontinuouslydifferentiablefunctionV(x)thatsatisestheinequalities–p.2/18 c1kxk2V(x)c2kxk2@V @xf(x)c3kxk2\r\r\r\r@V @x\r\r\r\rc4kxkforallx2D0,withpositiveconstantsc1,c2,c3,andc4Moreover,iffiscontinuouslydifferentiableforallx,globallyLipschitz,andtheoriginisgloballyexponentiallystable,thenV(x)isdenedandsatisestheaforementionedinequalitiesforallx2Rn–p.3/18 Ideaoftheproof: Let (t;x)bethesolutionof_y=f(y);y(0)=xTakeV(x)=Z0 T(t;x) (t;x)dt;�0–p.4/18 Example: Considerthesystem_x=f(x)wherefiscontinuouslydifferentiableintheneighborhoodoftheoriginandf(0)=0.ShowthattheoriginisexponentiallystableonlyifA=[@f=@x](0)isHurwitzf(x)=Ax+G(x)x;G(x)!0asx!0GivenanyL�0,thereisr1�0suchthatkG(x)kL;8kxkr1Becausetheoriginof_x=f(x)isexponentiallystable,letV(x)bethefunctionprovidedbytheconverseLyapunovtheoremoverthedomainfkxkr0g.UseV(x)asaLyapunovfunctioncandidatefor_x=Ax–p.5/18 @V @xAx=@V @xf(x)@V @xG(x)xc3kxk2+c4Lkxk2=(c3c4L)kxk2TakeLc3=c4;\rdef=(c3c4L)&#x-320;&#x.323;0)@V @xAx\rkxk2;8kxkminfr0;r1gTheoriginof_x=Axisexponentiallystable–p.6/18 ConverseLyapunovTheorem–AsymptoticStability Letx=0beanasymptoticallystableequilibriumpointfor_x=f(x),wherefislocallyLipschitzonadomainDRnthatcontainstheorigin.LetRADbetheregionofattractionofx=0.Then,thereisasmooth,positivedenitefunctionV(x)andacontinuous,positivedenitefunctionW(x),bothdenedforallx2RA,suchthatV(x)!1asx!@RA@V @xf(x)W(x);8x2RAandforanyc�0,fV(x)cgisacompactsubsetofRAWhenRA=Rn,V(x)isradiallyunbounded–p.7/18 Time-varyingSystems _x=f(t;x)f(t;x)ispiecewisecontinuousintandlocallyLipschitzinxforallt0andallx2D.Theoriginisanequilibriumpointatt=0iff(t;0)=0;8t0Whilethesolutionoftheautonomoussystem_x=f(x);x(t0)=x0dependsonlyon(tt0),thesolutionof_x=f(t;x);x(t0)=x0maydependonbothtandt0–p.8/18 ComparisonFunctions Ascalarcontinuousfunction(r),denedforr2[0;a)issaidtobelongtoclassKifitisstrictlyincreasingand(0)=0.ItissaidtobelongtoclassK1ifitdenedforallr0and(r)!1asr!1 Ascalarcontinuousfunction(r;s),denedforr2[0;a)ands2[0;1)issaidtobelongtoclassKLif,foreachxeds,themapping(r;s)belongstoclassKwithrespecttorand,foreachxedr,themapping(r;s)isdecreasingwithrespecttosand(r;s)!0ass!1–p.9/18 Example (r)=tan1(r)isstrictlyincreasingsince0(r)=1=(1+r2)�0.ItbelongstoclassK,butnottoclassK1sincelimr!1(r)==21 (r)=rc,foranypositiverealnumberc,isstrictlyincreasingsince0(r)=crc1�0.Moreover,limr!1(r)=1;thus,itbelongstoclassK1 (r)=minfr;r2giscontinuous,strictlyincreasing,andlimr!1(r)=1.Hence,itbelongstoclassK1–p.10/18 (r;s)=r=(ksr+1),foranypositiverealnumberk,isstrictlyincreasinginrsince@ @r=1 (ksr+1)2�0andstrictlydecreasinginssince@ @s=kr2 (ksr+1)20Moreover,(r;s)!0ass!1.Therefore,itbelongstoclassKL (r;s)=rces,foranypositiverealnumberc,belongstoclassKL–p.11/18 Denition: Theequilibriumpointx=0of_x=f(t;x)is uniformlystableifthereexistaclassKfunctionandapositiveconstantc,independentoft0,suchthatkx(t)k(kx(t0)k);8tt00;8kx(t0)kc uniformlyasymptoticallystableifthereexistaclassKLfunctionandapositiveconstantc,independentoft0,suchthatkx(t)k(kx(t0)k;tt0);8tt00;8kx(t0)kc globallyuniformlyasymptoticallystableiftheforegoinginequalityissatisedforanyinitialstatex(t0)–p.12/18 exponentiallystableifthereexistpositiveconstantsc,k,andsuchthatkx(t)kkkx(t0)ke(tt0);8kx(t0)kc globallyexponentiallystableiftheforegoinginequalityissatisedforanyinitialstatex(t0)–p.13/18 Theorem: Lettheoriginx=0beanequilibriumpointfor_x=f(t;x)andDRnbeadomaincontainingx=0.Supposef(t;x)ispiecewisecontinuousintandlocallyLipschitzinxforallt0andx2D.LetV(t;x)beacontinuouslydifferentiablefunctionsuchthatW1(x)V(t;x)W2(x)(1)@V @t+@V @xf(t;x)0(2)forallt0andx2D,whereW1(x)andW2(x)arecontinuouspositivedenitefunctionsonD.Then,theoriginisuniformlystable–p.14/18 Theorem: Supposetheassumptionsoftheprevioustheoremaresatisedwith@V @t+@V @xf(t;x)W3(x)forallt0andx2D,whereW3(x)isacontinuouspositivedenitefunctiononD.Then,theoriginisuniformlyasymptoticallystable.Moreover,ifrandcarechosensuchthatBr=fkxkrgDandcminkxk=rW1(x),theneverytrajectorystartinginfx2BrjW2(x)cgsatiseskx(t)k(kx(t0)k;tt0);8tt00forsomeclassKLfunction.Finally,ifD=RnandW1(x)isradiallyunbounded,thentheoriginisgloballyuniformlyasymptoticallystable–p.15/18 Theorem: Supposetheassumptionsoftheprevioustheoremaresatisedwithk1kxkaV(t;x)k2kxka@V @t+@V @xf(t;x)k3kxkaforallt0andx2D,wherek1,k2,k3,andaarepositiveconstants.Then,theoriginisexponentiallystable.Iftheassumptionsholdglobally,theoriginwillbegloballyexponentiallystable.–p.16/18 Example: _x=[1+g(t)]x3;g(t)0;8t0V(x)=1 2x2_V(t;x)=[1+g(t)]x4x4;8x2R;8t0Theoriginisgloballyuniformlyasymptoticallystable Example: _x1=x1g(t)x2_x2=x1x20g(t)kand_g(t)g(t);8t0–p.17/18 V(t;x)=x21+[1+g(t)]x22x21+x22V(t;x)x21+(1+k)x22;8x2R2_V(t;x)=2x21+2x1x2[2+2g(t)_g(t)]x222+2g(t)_g(t)2+2g(t)g(t)2_V(t;x)2x21+2x1x22x22=xT"2112#xTheoriginisgloballyexponentiallystable–p.18/18