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JOURNALOFTHEFRANKLININSTITUTE3382001781–809ENHANCEDROBUSTSTABILITYANALYSISOFLARGEHYDRAULICCONTROLSYSTEMSVIAABIFURCATIONBASEDPROCEDUREGREGORYGKREMERDEPARTMENTOFMECHANICALENGINEERING,OHIOUNIVERSITY,254STOCKERCENTER,ATHENS,OH457012979,USARECEIVED14AUGUST2001ABSTRACTBECAUSEOFTHEIRSIZEANDCOMPLEXITY,THEINITIALDESIGNOFMANYHYDRAULICSYSTEMSISBASEDPRIMARILYONSTEADYSTATEMODELSNONLINEARSYSTEMDYNAMICCHARACTERISTICSARENORMALLYCHECKEDBYSIMULATIONAND/ORPROTOTYPETESTINGOFTHEfiNALDESIGNCONfiGURATION,BUTEVENATTHISSTAGEONLYTHENOMINALSYSTEMDESIGNANDALIMITEDNUMBEROFOTHERPOSSIBLESYSTEMSCANBEANALYZEDDUETOTHEEXCESSIVECOSTOFEACHSYSTEMANALYSISEXHAUSTIVEPARAMETRICSTUDIESTHATVERIFYTHEPERFORMANCEANDSTABILITYOFALLPOSSIBLESYSTEMSAREGENERALLYNOTPRACTICALTHEDEfiCIENCYASSOCIATEDWITHTHISANALYSISLIMITATIONISTHATHYDRAULICCONTROLSYSTEMSTHATAREPREDICTEDTOBESTABLESOMETIMESEXHIBITNONLINEARPRESSUREOSCILLATIONSOFUNACCEPTABLYLARGEMAGNITUDETHISPAPERDOCUMENTSTHEDEVELOPMENTANDDEMONSTRATIONOFABIFURCATIONBASEDANALYSISPROCEDURETHATFOCUSESONPOTENTIALMODESOFOSCILLATIONRATHERTHANONANALYZINGALLPOSSIBLESYSTEMSTOYIELDA‘‘PRACTICALLYRIGOROUS’’ROBUSTSTABILITYANALYSISOFLARGENONLINEARSYSTEMSADDITIONALCONTRIBUTIONSOFTHISRESEARCHINCLUDE1PROPOSEDSOLUTIONSTOTHEMAINISSUESTHATCOMPLICATETHEROBUSTSTABILITYANALYSISOFLARGENONLINEARSYSTEMS,2DEMONSTRATIONOFTHEUSEOFTHERESULTSFROMABIFURCATIONANALYSISTOINFORMANDENABLEANEffiCIENTNONLINEARANALYSIS,AND3ADETAILEDDESCRIPTIONOFTHEPOSSIBLENONLINEARRESPONSESFORALARGEAUTOMATICTRANSMISSIONHYDRAULICSYSTEMWITHA9DIMENSIONALSTATESPACEANDA24DIMENSIONALPARAMETERSPACER2001THEFRANKLININSTITUTEPUBLISHEDBYELSEVIERSCIENCELTDALLRIGHTSRESERVEDKEYWORDSNONLINEARSYSTEMS；NONLINEARDYNAMICS；HYDRAULICCONTROLSYSTEMS；ROBUSTNESS；STABILITY；HOPFBIFURCATION；LIMITCYCLE；PERIODDOUBLINGBIFURCATION；PARTICIPATIONFACTORTEL17405931561；FAX17405930476EMAILADDRESSKREMEROHIOUEDUGGKREMER00160032/01/2000R2001THEFRANKLININSTITUTEPUBLISHEDBYELSEVIERSCIENCELTDALLRIGHTSRESERVEDPIIS001600320100031X1INTRODUCTIONHYDRAULICCONTROLSYSTEMSPRESENTINREALWORLDAPPLICATIONSSUCHASAUTOMATICTRANSMISSIONSANDPOSITIONCONTROLLERSARECOMPOSEDOFALARGENUMBEROFPRESSURECHAMBERSVOLUMESOFflUIDATASINGLEPRESSURESEPARATEDFROMOTHERCHAMBERSBYVALVEPORTSORfiXEDORIfiCESANDNUMEROUSMECHANICALCOMPONENTSIEVALVES,PUMPSANDACCUMULATORSTHEMATHEMATICALMODELSFORTHESEHYDRAULICSYSTEMSTHEREFOREHAVEALARGESTATESPACECOMPOSEDOFTHEVALVEPOSITION,VALVEVELOCITYANDCHAMBERPRESSURESTATEVARIABLESANDALARGEPARAMETERSPACECOMPOSEDOFTHESYSTEMDESIGNPARAMETERSALTHOUGHTHESEHYDRAULICSYSTEMSHAVEBEENSTUDIEDEXTENSIVELYOVERTHEYEARS,THECURRENTSTATEOFTHEARTANALYSISPROCEDURESNUMERICALSIMULATIONSANDRULESOFTHUMBBASEDONLINEARIZEDANALYSISSTILLHAVESOMESIGNIfiCANTLIMITATIONSONESUCHLIMITATIONISTHATWITHNUMERICALSIMULATIONSITISPRACTICALLYINFEASIBLETODOACOMPLETEPARAMETRICANALYSISOFTHELARGEMATHEMATICALMODELSTHATARENECESSARYTOADEQUATELYDESCRIBEREALHYDRAULICSYSTEMS1GENERALLYSPEAKING,APARAMETERSPACEWITHDIMENSIONOFFOURORGREATERISCLASSIfiEDASLARGEBECAUSEITSSIZEPROHIBITSACOMPREHENSIVEROBUSTNESSANALYSISFFORCONTEXTNOTETHATTHEDIMENSIONOFTHEPARAMETERSPACEFORAREALHYDRAULICSYSTEMISOFTENINTHEHUNDREDSANDNEARLYALWAYSEXCEEDS20THESIZEOFTHESTATESPACEISNOTASIGNIfiCANTLIMITINGFACTORFORROBUSTSTABILITYANALYSISBUTITISACOMPLICATINGFACTORBECAUSESYSTEMMODELSWITHALARGESTATESPACEAREMORELIKELYTOEXHIBITANUMBEROFDIffERENTRESPONSEMODESFORPARAMETERCOMBINATIONSWITHINTHEPOSSIBLEPARAMETERSPACE,INCLUDINGCOMPLEXNONLINEARBEHAVIORSSUCHASLIMITCYCLES,2FREQUENCYTORI,NOMENCLATUREDIRIAUNITVECTORTHATDEfiNESADIRECTIONINPARAMETERSPACEDSCALARDISTANCEPARAMETERJJACOBIANMATRIXLIEIGENVALUEOFTHEJACOBIANMATRIX,I1；Y；NVIEIGENVECTOROFTHEJACOBIANMATRIX,ASSOCIATEDWITHLINASNONASYMPTOTICALLYSTABLERESPONSEPMPARAMETERSPACESUBSETOFRMRESTRICTEDBASEDONPARAMETERLIMITSRMREAL,MDIMENSIONALSPACEXISTATEVARIABLEI1；Y；N,XSTATEVECTORMIBIFURCATIONPARAMETERI1；Y；M,MPARAMETERVECTORSUBSCRIPTSEEQUILIBRIUMVALUENOMVALUEFORTHENOMINALSYSTEMOVALUEATTHEBIFURCATIONPOINTCVALUEATTHECLOSESTBIFURCATIONPOINTGGKREMER/JOURNALOFTHEFRANKLININSTITUTE3382001781–809782ANDCHAOSTHEPOSSIBLEPARAMETERSPACEREFERSTOTHEMDIMENSIONALSPACEPMCONSISTINGOFALLPOSSIBLECOMBINATIONSOFVALUESFORTHESYSTEMPARAMETERS,WHERETHEPOSSIBLERANGEOFEACHPARAMETERISDETERMINEDBYITSTOLERANCERANGEORITSUNCERTAINTYRANGETHEDEfiCIENCYASSOCIATEDWITHTHEINABILITYOFCURRENTANALYSISMETHODSTOHANDLELARGESYSTEMSISTHATHYDRAULICCONTROLSYSTEMSTHATAREPREDICTEDTOBESTABLESOMETIMESEXHIBITNONLINEARPRESSUREOSCILLATIONSOFSUffiCIENTLYLARGEAMPLITUDETOSERIOUSLYDEGRADESYSTEMPERFORMANCE2ANIMPROVEDMETHODFORQUANTIFYINGTHESTABILITYROBUSTNESSOFLARGEHYDRAULICSYSTEMSUSINGABIFURCATIONBASEDPROCEDUREISTHETOPICOFCURRENTRESEARCHBIFURCATIONBASEDPROCEDURESHAVEBEENUSEDBYANUMBEROFRESEARCHERSTOANALYZETHESTABILITYOFVARIOUSTYPESOFSYSTEMS3–5,ANDKREMERANDTHOMPSON1HAVESHOWNTHATANANALYSISPROCEDUREBASEDONTHECOMPUTATIONOFTHECLOSESTHOPFBIFURCATIONCANEffiCIENTLYQUANTIFYTHESTABILITYROBUSTNESSOFAHYDRAULICPOSITIONCONTROLSYSTEMWITHALARGE7DIMENSIONALPARAMETERSPACESEVERALIMPORTANTISSUESFORTHEROBUSTSTABILITYANALYSISOFLARGESYSTEMSWEREADDRESSEDINTHATWORK,INCLUDINGNORMA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