5 Modeling and Control of Vibration in Mechanical Systems

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1、[6]ConsiderasystemLEMMA5.1nativestakingadvantageofastate-spacerepresentationoforequivalently[2]halfplane)isdefinedasThe5.2.15.2controlschemesinthelaterchapters.backcontrolsystemisdiscussed,whichisnecessarytohelpparametrization,willbeinvestigated.Moreover,performancelimitationofa

2、feed-certainty,andtheparametrizationofallstablecontrollers,i.e.,theso-calledYoulaapproach.Robustcontrolofsystemswithmultiplicativeuncertaintyandadditiveun-Hmeasuresofperformancespecificationsofacontrolsystem.Whatfollowarethenormsandtheircalculationarefirstintroducedsincetheyaretw

3、ocommonlyusedThischapterintendstobrieflyreviewtheoptimalandrobustcontrol.5.1IntroductionIntroductiontoOptimalandRobustControl5Although2Hand2HnormofamatrixtransferfunctionH2HG2and∞G2normcontrolproblemsandcontrollerdesignviaalinearmatrixinequality:=2Hcanbecomputedfromitsdefini

4、tion,therearesomesimplealter-G∞2σ>sup=normsG0{(s2π1)2π1withastate-spacerepresentation−∞+−∞∞+∞TraceTrace[GG[G∗(s∗(σ)(jω+analyticin)jωG()GjωG)](σunderstandthevibration(sdω.+Re).jω(s)(A,B,C,D)]>dω0},(openright-H2and).If(5.2)(5.1)H89∞tainedfromthefollowingLyapunovequations:whe

5、rethesystemissubjecttoawhitenoisehavingzeromeanandunitvariance.ThatisHwhereByParseval’stheorem,outputdisturbanceinput.Letwherelowingstate-spacerepresentation:beobtainedfromthefollowingLyapunovequations:whereA90The2LettheinputWealsoconsideradiscrete-timelineartime-invariantsyste

6、misstableandnormofYzwTHg(z.ThenthexX22k∈R2andnormforthediscrete-timesystem)istheimpulseresponseofand2TX=zwn2wTxYDarethereachabTracecanalsobeinterpretedastheRMSvalueoftheoutputtothesystembeawide-bandstationarystochasticprocess.ThezwHisthestate,2arethecontrollabilityandobse

7、rvabilityGramiansthatcanG=022T=normisdefinedasT2=2,thenwehave(DzwzwTzwModelingandControlofVibrationinMechanicalSystemsxTraceAAYT22π1denotethetransferfunctionfromtheinput(kAAXTDT=2zz+1)=YTX2A+zwcanequivalentlybedefinedasTrace(22ilityandobservabilityGramiansthatcanbeob-∈

8、Rk2+(BATB)=+X−−T2TracenYYTYY=T[zCxAx22AX22zw−A2B2+B