small-gain theorem for iss systems

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1、Math.ControlSignalsSystems(1994)7:95-12091994Springer-VerlagLondonLimitedMathematicsofControl,Signals,andSystemsSmall-GainTheoremforISSSystemsandApplications*Z.-P.Jiang,tA.R.Teel,$andL.PralytAbstract.Weintroduceaconceptofinput-to-outputpracticalstability(IOpS)wh

2、ichisanaturalgeneralizationofinput-to-statestabilityproposedbySontag.Itallowsustoestablishtwoimportantresults.ThefirstonestatesthatthegeneralinterconnectionoftwoIOpSsystemsisagainanIOpSsystemifanappropriatecompositionofthegainfunctionsissmallerthantheidentityfunc

3、tion.Thesecondoneshowsanexampleofgainfunctionassignmentbyfeedback.Asanillustrationoftheinterestoftheseresults,weaddresstheproblemofglobalasymptoticstabilizationviapartial-statefeedbackforlinearsystemswithnonlinear,stabledynamicperturbationsandforsystemswhichhavea

4、particulardisturbedrecurrentstructure.Keywords.Input-to-statestability,Nonlinearsystems,Partial-statefeedback,Globalstability.I.IntroductionStudyinguncertaindynamicalsystemsisnotonlypracticalbutameansofad-dressingthecontrolproblemforalargeclassofnonlinearsystemsb

5、asedonasimplifiedmodel(see[BCL],[CL],[JP],andthereferencestherein).Inthispaperweintroducesomenewdesigntoolswhich,whencombinedtogether,allowustoaddresstheproblemofstabilizingsystemswithintricatestructure.Inparticularweprovethatthefollowinguncertaindynamicalsystemc

6、anberobustlystabilizedbymeansofpartial-statefeedback:"si=xi+1+fi(x1.....xi,Zi),1

7、satisfythefollowingassumption:*Datereceived:April14,1993.Daterevised:March23,1994.ThefinalversionofthisworkwasfinishedwhenZ.-P.JiangheldavisitingpositionatI.N.R.I.A.,SophiaAntipolis.tCentreAutomatiqueetSyst6mes,EcoledesMinesdeParis,35rueStHonor6,77305Fontaineblea

8、uc6dex,France.zpjl01@deakin.anu.edu.anandpraly@cas.ensmp.fr.:~DepartmentofElectricalEngineering,UniversityofMinnesota,4-174EE/CSBuilding,200UnionSt.SE,Minneapo