输入-状态稳定性-input-to-state-stability-非线性控制-英文材料

Chapter7:Input-to-StateStability1MotivationConsideragainthesystem˙x=f(x,u)(1)Assumingthat˙x=f(x,0)hasauniformlyasymptoticallystableequilibriumpointattheorigin,westudywhathappenswhenu
=0.Example1:Considerthefollowingfirst-ordernonlinearsystem:˙x=−x+(x+x3)u.Ifu=0,weobtain˙x=−x,(asymptoticallystableequilibriumpoint).Whenu(t)=1,however,weobtain˙x=x3,whichresultsinanunboundedtrajectoryforanyinitialcondition.
⇒Asymptoticstabilityofx=0doesnotsaymuchabouttheforcedsystem.2DefinitionsThenotionofinput-to-statestability(ISS)atteptstocapturethenotionof“boundedinput—boundedstate”.Definition1:Thesystem(1)issaidtobelocallyinput-to-state-stable(ISS)ifthereexistaKLfunctionβ,aclassKfunctionγandconstantsk1,k2∈R+suchthatx(t)≤β(x0,t)+γ(uT(·)L∞),∀t≥0,0≤T≤t(2)forallx0∈Dandu∈Dusatisfying:x0k1andsupt0uT(t)=uTL∞k2,0≤T≤t.Itissaidtobeinput-to-statestable,orgloballyISSifD=Rn,Du=Rmand(2)issatisfiedforanyinitialstateandanyboundedinputu.1Implications(Assumethat˙x=f(x,u)isISS)•Unforcedsystems:consider˙x=f(x,0).Theresponseof(1)withinitialstatex0satisfiesx(t)≤β(x0,t)∀t≥0,x0k1,⇒x=0isuniformlyasymptoticallystable.•Interpretation:ifu∞δ,trajectoriesremainboundedbytheballofradiusβ(x0,t)+γ(δ),i.e.,x(t)≤β(x0,t)+γ(δ).Astincreases,β(x0,t)→0andtrajectoriesapproachtheballofradiusγ(δ),i.e.,limt→∞x(t)=LL≤γ(δ)γ(·)iscalledtheultimateboundofthesystem(1).•AlternativeDefinition:AvariationofDefinition1istoreplaceequation(2)withthefollowingequation:x(t)≤max{β(x0,t),γ(uT(·)L∞)},∀t≥0,0≤T≤t.(3)Definition2:AcontinuouslydifferentiablefunctionV:D→RissaidtobeanISSLocalLyapunovfunctiononDforthesystem(1)ifthereexistclassKfunctionsα1,α2,α3,andXsuchthat:α1(x)≤V(x(t))≤α2(x)∀x∈D,t0(4)∂V(x)∂xf(x,u)≤−α3(x)u∈Du:x≥X(u).(5)VissaidtobeanISSGlobalLyapunovfunctionifD=Rn,Du=Rm,andα1,α2,α3∈K∞.Remarks:thismeansthatVisanISSLyapunovfunctionif(a)ItispositivedefiniteinD.(b)Itisnegativedefinitealongthetrajectoriesof(1)wheneverthetrajecto-riesareoutsideoftheballdefinedbyx∗=X(u).23Input-to-StateStability(ISS)TheoremsTheorem1:(LocalISSTheorem)Considerthesystem(1)andletV:D→RbeanISSLyapunovfunctionforthissystem.Then(1)isinput-to-state-stableaccordingtoDefinition1withγ=α−11◦α2◦X(6)k1=α−12(α1(r))(7)k2=X−1(min{k1,X(ru)}).(8)i.exr,uru(9)Theorem2:(GlobalISSTheorem)IfthepreceedingconditionsaresatisfiedwithD=RnandDu=Rm,andifα1,α2,α3∈K∞,thenthesystem(1)isgloballyinput-to-statestable.3Example2:Considerthefollowingsystem:˙x=−ax3+ua0.WeproposetheISSLyapunovfunctioncandidateV(x)=12x2.ThisVispositivedefiniteandsatisfies(4)withα1(x)=α2(x)=12x2.Wehave˙V=−x(ax3−u).(10)Weneedtofindα3(·)andX(·)∈Ksuchthat˙V(x)≤−α3(x),wheneverx≥X(u).Letθ:0θ1˙V=−ax4+aθx4−aθx4+xu=−a(1−θ)x4−x(aθx3−u)≤−a(1−θ)x4=−α3(x)providedthatx(aθx3−u)0.Thiswillbethecase,providedthataθ|x|3|u|or,equivalently|x||u|aθ1/3χ(u)=|u|aθ1/3Itfollowsthatthesystemisgloballyinput-to-state-stablewithγ(u)=|u|aθ1/3.
4Example3:Nowconsiderthefollowingsystem,whichisaslightlymodifiedversionoftheoneinExample2:˙x=−ax3+x2ua0UsingthesameISSLyapunovfunctioncandidateusedinExample2,wehavethat˙V=−ax4+x3u=−ax4+aθx4−aθx4+x3u0θ1=−a(1−θ)x4−x3(aθx−u)≤−a(1−θ)x4,providedx3(aθx−u)0or,|x||u|aθ.Thus,thesystemisgloballyinput-to-statestablewithγ(u)=|u|aθ.
54Input-to-StateStabilityRevisitedTheorem3:AcontinuousfunctionV:D→RisanISSLyapunovfunc-tiononDforthesystem(1)ifandonlyifthereexistclassKfunctionsα1,α2,α3,andσsuchthatthefollowingtwoconditionsaresatisfied:α1(x)≤V(x(t))≤α2(x)∀x∈D,t0(11)∂V(x)∂xf(x,u)≤−α3(x)+σ(u)∀x∈D,u∈Du(12)VisanISSGlobalLyapunovfunctionifD=Rn,Du=Rm,andα1,α2,α3,andσ∈K∞.Remarks:Noticethat,givenru0,thereexistpointsx∈Rnsuchthatα3(x)=σ(ru).Thisimpliesthat∃d∈R+suchthatα3(d)=σ(ru),ord=α−1(σ(ru)).DenotingBd={x∈Rn:x≤d},wehavethatforanyxdandanyu:uL∞ru:∂V∂xf(x,u)≤−α(x)+σ(u)≤−α(d)+σ(uL∞).Thus,thetrajectoryx(t)resultingfromaninputu(t):uL∞ruwilleventuallyentertheregionΩd=maxx≤dV(x).Onceinsidethisregion,itistrappedinsideΩd,becauseoftheconditionon˙V.6u
Σ2z
Σ1
xFigure1:CascadeconnectionofISSsystems.5Cascade-ConnectedSystemsThroughoutthissectionweconsiderthecompositesystemshowninFigure1,whereΣ1andΣ2aregivenbyΣ1:˙x=f(x,z)(13)Σ2:˙z=g(z,u)(14)whereΣ2isthesystemwithinputuandstatez.ThestateofΣ2servesasinputtothesystemΣ1.Theorem4:ConsiderthecascadeinterconnectionofthesystemsΣ1andΣ2.Ifbothsystemsareinput-to-state-stable,thenthecompositesystemΣΣ:u→xzisinput-to-state-stable.7