非线性系统第四讲输入输出稳定性.pdf

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1、NonlinearSystemsTheoryHuJiangpingSchoolofAutomationEngineeringUniversityofElectronicScienceandTechnologyofChinaE-mail:hjp_lzu@163.comUs()Ys()G(s)orft(,x)ut()yt()Topic4Input-Outputstability4.1.L-stability4.2.L-stabilityofstatemodel4.3.L2gain4.4.ThesmallgaintheoremT

2、opic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-OutputstabilityRemark:ifthesystemiscausal,wecandropTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-Outputstability4.1.L-stabil

3、ity4.2.L-stabilityofstatemodel4.3.L2gain4.4.ThesmallgaintheoremTopic4Input-Outputstability4.2.L-stabilityofstatemodelTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-OutputstabilityTopic4Input-

4、OutputstabilityTopic4Input-Outputstability4.1.L-stability4.2.L-stabilityofstatemodel4.3.L2gain4.4.ThesmallgaintheoremTopic4Input-Outputstability4.3L2gainut()yt()Measure:energyofsignalTututdt()()FinitegainL2stabilityTopic4Input-Outputstability4.3L2gainAccordingtoP

5、arseval’sRelationFouriertransformoftheoutputsignal:Topic4Input-Outputstability4.3L2gainTopic4Input-OutputstabilityExample:checkthefinitegainL2stabilityofthefollowingsystemxfx()Gxux(),(0)x0yhx()nmxfR,uR,(0)0,()h00.Topic4Input-OutputstabilityRemark:inordert

6、oobtainV(x),weneedsolvethefollowingHamilton-JacobiinequalityTopic4Input-OutputstabilityTopic4Input-Outputstabilityx20fx()3,Gax12kxu1Topic4Input-Outputstability11()kkRemark:Theorem5.1cannotbeappliedsincetheoriginofthesystemisnotexponentiallysta

7、bleTopic4Input-OutputstabilityExample:IfthereexistsaW(x)suchthatWecantakeTopic4Input-OutputstabilityTopic4Input-OutputstabilityLemma5.1SupposetheassumptionsofTheorem5.5aresatisfiedonandomainthatcontainstheorigin,f(x)iscontinuouslyDRdifferentiable,andx=0isanasympt

8、oticallystableequilibriumpointofThen,thereisxfx().kl>0suchthatforeachx0withthesystem

9、

10、x01

11、

12、kissmall-signalfinite-gainL2stable.Topic4Input-Outputstabi