8.6 Linearization of Nonlinear Systems nonlinear ...

1 Willstartwitha simplescalarfirst-ordernonlineardynamics ystem Assumethatunderusualworkingcircumstances thissystemoperatesalongthetrajectory whileit isdrivenbythesysteminput . We call and , respectively, , 83 Onthenominaltrajectorythefollowingdiffer entialequationissatisfied Assumethatthemotionofthenonlinearsystemi sintheneighborhoodofthenominalsystemtraj ectory,thatis whererepresentsa isnaturaltoassumethatthesystemmotionincl oseproximitytothenominaltrajectorywillbe sustainedbya systeminputwhichisobtainedbyaddinga smallquantitytothenominalsysteminput Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, 84 Forthesystemmotionincloseproximitytothen ominaltrajectory,wehave Sinceandaresmallquantities,theright-hand sidecanbeexpandedintoa Taylorseriesaboutthenominalsystemtraject oryandinput,whichproduces Cancelinghigher-orderterms(whichcontainv erysmallquantities )

2 , thelineardifferentialequationisobtained Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, 85 Thepartialderivativesin thelinearizationprocedure are evaluatedat thelinearizedsystemcanberepresentedas Ingeneral, , remainstofindtheinitialconditionfortheli nearizedsystem,whichcanbeobtainedfrom Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, 86 Similarly,wecanlinearizethesecond-ordern onlineardynamicsystem byassumingthat andexpanding intoa Taylorseriesaboutnominalpoints , whichleadsto Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, 87wherethecorrespondingcoefficientsareev aluatedatthenominalpointsas Theinitialconditionsforthesecond-orderli nearizedsystemareobtainedfrom :Themathematicalmodelofa stick-balancingproblemiswhereis thehorizontalforceofa fingerandrepresentsthestick , 88 Thissecond-orderdynamicsystemislinearize datthenominalpoints , producing !

3 " ThelinearizedequationisgivenbyNotethatsi nce . Itisimportanttopointoutthatthesamelinear izedmodelcouldhavebeenobtainedbysetting, whichis , 89We canextendthepresentedlinearizationproced uretoan-ordernonlineardynamicsystemwitho neinputandoneoutputina , , #where,, andare,respectively,the-dimensionalsyste mstatespacevector,the-dimensionalinputve ctor, (operating)systemtrajectory$isknownandth atthenominalsysteminputthatkeepsthesyste monthenominaltrajectoryisgivenby$.Thesli descontainthecopyrightedmaterialfromLine arDynamicSystemsandSignals, 90 Usingthesamelogicasforthescalarcase, ,startingwith%%and%%%weexpandtheright-ha ndsideintotheTaylorseriesasfollows%%%%%& "')(+* , -.(*/, -& '0(1* , -.(* , -Theslidescontainthecopyrightedmaterialf romLinearDynamicSystemsandSignals, 91 Higher-ordertermscontainatleastquadratic quantitiesofand.)))

4 Sinceandaresmalltheirsquaresareevensmall er, ,anapproximationisobtained2"35416 7 8946 7 82 354:6;7 8946 7 8 ThepartialderivativesrepresenttheJacobia nmatricesgivenby2"354 6!7 8946;7 8<>=?<@BADC@FEC@BADC@FEBG@HAIC@FE4@BAG@FEC@HAG@FE4@HAKJ@FEML@HA4@FEC@HA4@FEBG@BA4@FE42 346 7 894+6 7 8 TheslidescontainthecopyrightedmaterialfromLinearDynamicSystemsandSignals, 92N O5P Q R STP Q R SU>V?WXFY[ZX]\ZXHYIZX^\`_XHYIZX^\baXHY_X^\` ZXHY_X^\aXHYKcXd\feXHYPX^\ZXHYPX^\_XHYPX ^\aNOPQ!R STPQ!R SNotethattheJacobianmatriceshavetobeeval uatedatthenominalpoints,thatis,atUandU. Withthisnotation,thelinearizedsystemhast heformggUgTheoutputofa nonlinearsystemsatisfiesa nonlinearalgebraicequation,thatisTheslid escontainthecopyrightedmaterialfromLinea rDynamicSystemsandSignals, 93 Thisequationcanalsobelinearizedbyexpandi ngitsright-handsideintoaTaylorseriesabou tnominalpointshandh.

5 Thisleadstohhhi"j5k l!m nokl!m ni"j5k+l;m nokl!m ,thelinearizedpartoftheoutputequationisg ivenbywheretheJacobianmatricesandsatisfy Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, 94prqtsu"v)w:x;y z{w x y z|F}^~| ~| }d~|F B | }d~| w|F} | ~| } | w|F}B |F : | } |F F~| }: |F | }: | wuvwx;y z{wx;y zprq u"vwx y z{wx y z| }~|^ ~|F}~|d |F}~|d | }F |^ ~|F} |d | } |^ | } |^ ~|F} |d |F} |^ u vwx y z{wx y zTheslidescontainthecopyrightedmaterialf romLinearDynamicSystemsandSignals, :Leta nonlinearsystemberepresentedby K B b 5 Assumethatthevaluesforthesystemnominaltr ajectoriesandinputareknownandgivenby f and . Thelinearizedstatespaceequationofthisnon linearsystemisobtainedas f b [ f [ f Havingobtainedthesolutionofthislinearize dsystemunderthegivensysteminput, thecorrespondingapproximationofthenonlin earsystemtrajectoriesisTheslidescontaint hecopyrightedmaterialfromLinearDynamicSy stemsandSignals, 96 :Considerthemathematicalmodelofa single-linkroboticma-nipulatorwitha flexiblejointgivenby where areangularpositions,aremomentsofinertia, andare,respectively,linkmassandlength, themanipulator sstatespacenonlinearmodelisgivenbyThesli descontainthecopyrightedmaterialfromLine arDynamicSystemsandSignals, 97 Takethenominalpointsas f b ` ` , thenthematricesandare.]]

6 + ` + F Assumingthattheoutputvariableisequaltoth elink sangularposition,thatis , thematricesandaregivenbyTheslidescontain thecopyrightedmaterialfromLinearDynamicS ystemsandSignals, 98