the separation principle

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1、Experiment7TheSeparationPrinciple7.1IntroductionWhenaplantismodeledbyastateequation=+(7.1)onepopularcontrolstrategyistoclosethefeedbackloopbyapplyingtheinput=−+(7.2)whereisamatrixofgainsandistheexternalinputtothefeedbacksystem.Thisstructureiscalledstatefeedback.Comb

2、ining(7.1)and(7.2),weseethattheclosed-loopsystemisgovernedby=(−)+(7.3)Thedynamicsoftheclosed-loopsystemcanbeadjustedbychoosing.Forexample,canbechosensothat+hassomespecifiedeigenvalues;infact,thisproblemisessentiallythesameastheproblemofplacingobservereigenvalues,enco

3、unteredinExperiment6.Toseethis,notethat(−)=−(7.4)Sinceformingthetransposeofamatrixdoesnotchangeitseigenvalues,thetwoproblemscoincide,with,,andplayingtherolesof,,and,respectively.Themaindrawbacktothestatefeedbackapproachisthat,inmostapplications,someofthestates

4、aresimplynotmeasureddirectly.Forexample,thespring-masssystemofExperiment6hasfourstates,onlytwoofwhicharemeasuredandavailableasoutputs.7.2TheSeparationPrincipleInthoseapplicationswherenotallstatesareaccessible,anobviousapproachtocompensatordesignistobuildanobserverfortheplant,aswe

5、didinExperiment6,andfeedbacktheestimatedstatesasiftheywereexact.Sincestateestimatesoftensufferfromconsiderableestimationerror,onewouldbejustifiedinmaintainingadegreeofskepticismthatsuchasimpleapproachwouldactuallywork.Asitturnsout,however,thesefearsareputtorestbyoneofthecentraltheo

6、remsofmoderncontroltheory,theseparationprinciple.Supposethattheplantismodeledbyastateandoutputequation=+(7.5)=(7.6)5556EXPERIMENT7.THESEPARATIONPRINCIPLEAsinExperiment6,wemayconstructastateestimatorb=(−)b++(7.7)Toclosetheloop,weapply(estimated)statefeedback=

7、−b(7.8)(Forthisexperimentwillnotneedtheexternalinputpresentin(7.2).)Combining(7.7)and(7.8),thecompensatorisgivenbyb=(−−)b+(7.9)withoutputequation(7.8).Thecompensatorhasthe(1×2)transferfunctionmatrix()=−(−++)−1(7.10)Fromequations(7.5)-(7.8),weobtainthecl

8、osed-loopsystem∙¸∙¸∙¸−=(7.11)b