Nonlinear Systems and Control Lecture Exponential Stability Region of Attraction p - PDF document

NonlinearSystemsandControlLecture#11ExponentialStability&RegionofAttraction –p.1/18 ExponentialStability: Theoriginof_x=f(x)isexponentiallystableifandonlyifthelinearizationoff(x)attheoriginisHurwitz Theorem: Letf(x)bealocallyLipschitzfunctiondenedoveradomainDRn;02D.LetV(x)beacontinuouslydifferentiablefunctionsuchthatk1kxkaV(x)k2kxka_V(x)k3kxkaforallx2D,wherek1,k2,k3,andaarepositiveconstants.Then,theoriginisanexponentiallystableequilibriumpointof_x=f(x).Iftheassumptionsholdglobally,theoriginwillbegloballyexponentiallystable–p.2/18 Proof: Choosec�0smallenoughthatfk1kxkacgDV(x)c)k1kxkac\nc=fV(x)cgfk1kxkacgD\nciscompactandpositivelyinvariant;8x(0)2\nc_Vk3kxkak3 k2VdV Vk3 k2dtV(x(t))V(x(0))e(k3=k2)t–p.3/18 kx(t)kV(x(t)) k11=a"V(x(0))e(k3=k2)t k1#1=a"k2kx(0)kae(k3=k2)t k1#1=a=k2 k11=ae\rtkx(0)k;\r=k3=(k2a)–p.4/18 Example _x1=x2_x2=h(x1)x2c1y2yh(y)c2y2;8y;c1�0;c2�0V(x)=1 2xT"1112#x+2Zx10h(y)dyc1x212Zx10h(y)dyc2x21_V=[x1+x2+2h(x1)]x2+[x1+2x2][h(x1)x2]=x1h(x1)x22c1x21x22Theoriginisgloballyexponentiallystable–p.5/18 RegionofAttractionLemma: Ifx=0isanasymptoticallystableequilibriumpointfor_x=f(x),thenitsregionofattractionRAisanopen,connected,invariantset.Moreover,theboundaryofRAisformedbytrajectories–p.6/18 Example _x1=x2_x2=x1+(x211)x2 -4 -2 0 2 4 -4 -2 0 2 4 x1x2 –p.7/18 Example _x1=x2_x2=x1+1 3x31x2 -4 -2 0 2 4 -4 -2 0 2 4 x1 x2 –p.8/18 EstimatesoftheRegionofAttraction :Findasubsetoftheregionofattraction Warning: LetDbeadomainwith02Dsuchthatforallx2D,V(x)ispositivedeniteand_V(x)isnegativedeniteIsDasubsetoftheregionofattraction? NO Why? –p.9/18 Example: Reconsider_x1=x2_x2=x1+1 3x31x2V(x)=1 2xT"1112#x+2Rx10(y1 3y3)dy=3 2x211 6x41+x1x2+x22_V(x)=x21(11 3x21)x22D=fp 3x1p 3g IsDasubsetoftheregionofattraction? –p.10/18 ThesimplestestimateistheboundedcomponentoffV(x)cg,wherec=minx2@DV(x)ForV(x)=xTPx,whereP=PT&#x-320;&#x.324;0,theminimumofV(x)over@DisgivenbyForD=fkxkrg;minkxk=rxTPx=min(P)r2ForD=fjbTxjrg;minjbTxj=rxTPx=r2 bTP1bForD=fjbTixjri;i=1;:::;pg;Takec=min1ipr2i bTiP1biminx2@DxTPx–p.11/18 Example(Revisited) _x1=x2_x2=x1+(x211)x2V(x)=1:5x21x1x2+x22_V(x)=(x21+x22)(x31x22x21x22)_V(x)0for0kxk22 p 5def=r2Takec=min(P)r2=0:6912 p 5=0:618fV(x)cgisanestimateoftheregionofattraction–p.12/18 x1=cos;x2=sin_V=2+4cos2sin(2sincos)2+4jcos2sinj
j2sincosj2+40:38492:23612+0:86140;for21 0:861Takec=min(P)r2=0:691 0:861=0:803Alternatively,choosecasthelargestconstantsuchthatfxTPxcgisasubsetoff_V(x)0g–p.13/18 -2 -1 0 1 2 -2 -1 0 1 2 x1x2(a) -2 0 2 -3 -2 -1 0 1 2 3 x1x2(b) (a)Contoursof_V(x)=0(dashed)V(x)=0:8(dash-dot),V(x)=2:25(solid)(b)comparisonoftheregionofattractionwithitsestimate–p.14/18 IfDisadomainwhereV(x)ispositivedeniteand_V(x)isnegativedenite(or_V(x)isnegativesemideniteandnosolutioncanstayidenticallyintheset_V(x)=0otherthanx=0),thenaccordingtoLaSalle'stheoremanycompactpositivelyinvariantsubsetofDisasubsetoftheregionofattraction Example _x1=x2_x2=4(x1+x2)h(x1+x2)h(0)=0;uh(u)0;8juj1–p.15/18 V(x)=xTPx=xT"2111#x=2x21+2x1x2+x22_V(x)=(4x1+2x2)_x1+2(x1+x2)_x2=2x216(x1+x2)22(x1+x2)h(x1+x2)2x216(x1+x2)2;8jx1+x2j1=xT"8666#x_V(x)isnegativedeniteinfjx1+x2j1gbT=[11];c=minjx1+x2j=1xTPx=1 bTP1b=1–p.16/18 =x1+x2d dt2=2x2822h()2x282;8jj1On=1,d dt22x280;8x24On=1,d dt22x280;8x24c1=V(x)jx1=3;x2=4=10;c2=V(x)jx1=3;x2=4=10=fV(x)10andjx1+x2j1gisasubsetoftheregionofattraction–p.17/18 -5 0 5 -5 0 5 (-3,4)(3,-4)x2x1V(x) = 10V(x) = 1 –p.18/18