AA UNIVERSIAIS Srs: Mchancs Aoac onrol and Robocs Vol. 6 Scal Iss 7. 7-98 PHASE PLANE MEHOD APPLIED O OPIMAL ONROL IN NONLINEAR DYNAMIAL SYSEMS WIH ROGGER O OUPLED SINGULARIIES UD 5.5 Kaca R. Svanovć Hdrh acly od Mchancal Engnrng Unvrsy of Nš Mahacal Ins SANU Blgrad 8 - Nš l. Vojvod anosća / Srba lfa: 8 8 66 / Mobl 6 8 75 75 99 -al: aca@asfa.n.ac.rs * -alho: hdrh@n.rs Absrac. Shor Man da of h has lan hod ald o h oal conrol n nonlnar dynacal syss wh rggrs of a cold snglars and wh on dgr of frdo s rconsdrd. Par rsns a shor rvw of h ahor's rvos blshd rsls conanng srs of h scal cass of h oal conrol n nonlnar dynacal syss wh rggr of cold snglars oran for ngnrng alcaons. hs ar analyss h conrollably of oon of consrvav or nonconsrvav nonlnar dynacal syss n whch rggrs of cold snglars s or aar. I s shown ha h has lan hod s sfl for h analyss of nonlnar dynacs of consrvav and nonconsrvav syss wh on dgr of frdo of conrol srags and also shows h way can b sd for conrollng h rlav oon n rhonoc syss havng qvaln sclronoc consrvav or nonconsrvav sys. or h sys wh on gnralzd coordna dscrbd by nonlnar dffrnal qaon of nonlnar dynacs wh rggrs of cold snglars h fncons of sys onal nrgy and consrvav forc s sasfy so condons dfnd by a hor on h snc of a rggr of cold snglars and h sarar n h for of nbr gh. as of h dfnd dynacal consrvav sys oal conrol s: by sng conrollng forc acng o h sys ransfr nal sa of h nonlnar dynacs of h sys no h fnal rna sa of h nonlnar dynacs n h nal for ha oal conrol as. So rsarch rsls of fascnang nonlnar dynacs of a havy aral arcl along crcls wh cold roaons wh any dffrn rors of nonlnar dynacs and oal conrol of hs dynacs ar rsnd. A vsalzaons of nonlnar dynacal rocsss n sch rhonoc syss wr ad. Ky words: has lan hod oal conrol nonlnar dynacs consrvav sys nonconsrvav sys has lan orra rggrs of cold snglars hor cold roaon aral arcl hooclnc ons hooclnc orb bfrcaon sararc layrng sararc n h for of nbr gh. Rcvd brary 6 8
7 K.R. SEVANOVIĆ HEDRIH I. INRODUION h conrollably of oon of dynacal syss n whch s or aar rggrs of cold snglars s n h focs of or anon. I s shown ha h has lan hod s sfl for h analyss of nonlnar dynacs of syss wh on dgr of frdo conrol srags. h dffrnal qaon of h nonlnar dynacs of a sys wh ossbls of h rggr of h cold snglars snc s n h basc and gnral for of: g ] f whr s gnralzd coordna and n h scal cass n sa rlav coordna. or ha cas whn n h sys sd a rggr of h cold snglars han h fncons f and g ] s sasfy so condons dfnd by a hor on h snc of a rggr of h cold snglars and h sarar n h for of nbr gh. as of h dfnd dynacal sys oal conrol s: By sng conrollng forc acng o h sys ransfr nal nc sa of h nonlnar dynacs of h sys dfnd by α and β no h fnal or rna nc sa of h nonlnar dynacs dfnd by γ and χ whr s nal for ha oal conrol as. In an ngnrng sys 5] 7] 8] ] ] 6] 7] 8] wh ass dvaon rors o h as of roaon n h gravaon fld whr col cold roaon oon ss n h cas of so nc arar vals h hnonon of bfrcaon qlbr oson s Rf. ] ] ] 6] 9] ] or aaranc of wo nw rlav dynac sabl qlbr osons n rlaon o rcsson roaon oon ar nonlnar rors of sch sys. In sch syss wh h chang of so nc arars of h sys h rocss of losng sably of on sac qlbr oson s followd wh aarng wo clos sabl dynacal qlbr osons s Rf. ] ] ] ] 6] 9] ]. Also hs wo nw aard snglars wh h rvos sabl oson whch los s sably a a rggr of cold snglars. rggr of cold snglars wh corrsondng choc of nc arars of h sys dynacs whch dgnras no on hrfold rl snglar on corrsonds o sabl qlbr oson. old snglars snc s cold wh snc of wo ys of hooclnc orbs sarar rajcors: nw aard scond hooclnc orb n h for of nbr gh whch aar wh bfrcaon of qlbr oson nsd of rvos hooclnc orb whos sha s dford n h for of on ar of syrc "wavs". Hooclnc orb cold wh rggr of cold snglars s n h for of nbr "gh". Sch rggr of cold snglars s a characrsc nonlnar hnonon of dynacal syss whch s h sorc of snsv dndnc of h sys dynacs bhavor wh rsc o a sall chang of nal condons n h vcny of hs cold snglars. In sch sys ndr h acon of h r rodc caon bhavor of sys dynacs wh sochasc-l and chaoc-l rocsss aars. In on of classcal onograhs ] by A. A. Andronov A. A. W and S. E. Hajn whch has a gra nbr of dons so classcal als of nonlnar syss wh on dgr of frdo of oscllaory oon and hr has orras c gnral hory of nonlnar oscllaons ar rsnd and sch als can also b fond n boos by J. J. Sor ] as wll as by D.P. Rašovć ]. Escally n onograh by Gcnhr J. and Hols Ph. ] rsls of rsarch on nonlnar syss and
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 7 rors of varos nds of bfrcaons ar ond o. A srs of onograhs by Mroolysy Y. A ] dals wh hory hods and robls of asyoc hory of non-sady nonlnar oscllaons of nonlnar syss and rggr of cold snglars n h rsonan rang. onrol of dynacs n sch sys s vry oran for ngnrng alcaons. hs ar analyss h conrollably of oon of dynacal syss n whch s or aar rggrs of cold snglars. S srs of h sara ass arlr blshd s rsnd s Rfs. 7] ] ] 5]. I s shown ha h has lan hod s sfl for h analyss of nonlnar dynacs of syss wh on dgr of frdo conrol srags and also shos h way can b sd for conrollng h rlav oon n rhonoc syss havng qvaln sclronoc consrvav or nonconsrvav sys. In onograhs s 5] V. Vjčć gav a odfcaon of h analycal chancs of rhonoc syss wh h a o ncld h nflnc of nonsaonary consrans no h laws of oon. In hs aroach rhonoc coordna s chosn as a fncon drvd fro rhonoc consrans. On hs bass an ndd sys of Lagrangan qaons was forlad wh an addonal qaon corrsondng o rhonoc coordna. In h ar by Vjčć and Hdrh 6] h rhonoc consrans gnralzd forc n h nson of h Lagrang's sys of dffrnal qaons of oon of h frs and h scond nd s nrodcd. In h ar by K. S. Hdrh 9] h owr of h rhonoc consrans chang s nrodcd. A srs of ars -9] by ahor of hs ar rsns rsls of orgnal rsarch on nonlnar dynacs of chancal syss wh rors of rodc changs whch hav alcaon n ngnrng syss. In h ars 9 ] so rsls and nw nowldg abo rhonoc nonlnar syss whch hav qvaln holonoc sclronoc consrvav nonlnar sys ar rsnd. Par 9] consdrd a class of nonlnar syss wh cold roaon oons no sys wh wo dgrs of obly b wh on dgr of frdo of oon dfns on gnralzd coordna and on dgr of oly ha s dfnd by rhonoc coordna lnarly dndng on. or sch scal class of rhonoc nonlnar syss an qvaln holonoc sclronoc consrvav nonlnar sys and qvaln nc and onal nrgy ar dfnd. hor of snc of rlav qlbr osons n sch a class of rhonoc consrvav syss by sng qvaln holonoc consrvav sys s dfnd as wll as rovn 6]. hor of snc of hooclnc orbs n h for of nbr gh and rggr of cold snglars n h has orra of nonlnar dynacs of rlav oons n h class rhonoc sys s dfnd and rovn. L basc rors of lnar oscllaory sys wh on and or dgrs of oscllaory frdo ar h own frqncs of dscr aral arcl sys who hav lnar oscllaory oon also for nonlnar sys wh on dgr of oscllaory frdo w ay consdr has orra wh snglary srcr whch gvs nonlnar dynac's rors of sys n has lan. hrfor for nonlnar dynac syss and hr sbsyss 's oran o sdy a srcr of has orras hr sably as wll as hr ransforaons and ransforaons of has rajcors whch w oband changng any arars of syss. Ahor dfns a rggr of cold snglars hor and snc of hooclnc orbs and hr ransforaon shad by nbr gh as wll as hr alcaon on syss rlvan for chncal racc n hrs arcl 6]. Also sh consrcs has orras and arclarly consdrs hnona of hooclnc orbs ransforaons and hr dsngraon aaranc and dsaaranc hooclnc orbs shad by nbr gh l a rggr of cold snglars.
7 K.R. SEVANOVIĆ HEDRIH II. HEOREM ON HE EXISENE O A RIGGER O HE OUPLED SINGULARIIES AND HE SEPARARIX IN HE ORM O NUMBER EIGH IN HE ONSERVAIVE SYSEM By sng nonlnar dynac analyss of syss wh dscrbd nonlnar hnonon of h rggr of cold snglars and corrsondng fals of has orras and onal nrgs s Rfs. ] ] 6] as wll as h corrsondng rnal nvsgaons of sch nonlnar dynacs n chancal ngnrng syss wh cold roaon oons 5] ] was asy o dfn and o rov h hor of h snc of a rggr of cold snglars n nonlnar dynacal syss wh rodcal srcr rvos blshd n h Rfrnc ]. hor *: In h sys whos dynacs can b dscrbd wh h s of non-lnar dffrnal qaon n h for: g ] f and whos onal nrgy s n h for: E g ] f d G ] n whch h fncons f and g ar: f d and G g d and sasfy h followng condons: f f g n g f n f g g f g r ] for... f s s s s... r r r... < g ] for... and boh fncons f and g hav on a or n n h nrval bwn wo zro roos: a* for arars vals... osd of h nrvals h rggr of snglars n h local ara dos no s. b* for arars vals... nsd of h nrvals h srs of rggrs of cold snglars n h local doans s. h roof of dfnd hor and dals ar blshd n h Rfrncs ] and 6]. hn h rggr of cold snglars ss n h has orra n h nrvals dfnd by: s s s... Ingral nrgy of h sys dscrbd by s: G ] G ] cons 5
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 75 Eqaon of hooclnc orb n h for nbr "gh" rogh hooclnc on s: G ] G ] h hc cons 6 for g r ] for... n whch h fncons and G ar n h for and sasfy h condons. Eqaon of hooclnc orb ohr for n h connos closd for rogh hoolnc on s s: G ] G ] h cons 7 s sc III. LINEARIZED APPROXIMAION Now l s consdr scal cass of h dynacs dscrbd by dfnd nonlnar dffrnal qaon n h vcny of h qlbr osons. or bgnnng w consdr sall oscllaons arond sabl and also nsabl qlbr osons corrsond o h sabl and nsabl saddl y snglars cold n ons s wh rggr of hr cold snglars. By s h cass consdrd n h rvos hor of h rggr of h snc of h cold snglars n h s of h rggr of h cold snglars w can sar wh hs nonlnar dffrnal qaon n h cas whn on sabl snglar on cnr y ss n h has lan and corrsond o sabl qlbr oson. a* for arars vals... osd of h nrvals... h rggr of snglars n h local ara dos no s. Only a snglar on cnr y ss. hn for sall vbraons n h vcny of hs snglar on s asy o oban rgh and accabl lnarzd dffrnal qaon n h for: ω 8 whr own gn crclar frqncy of h sall oscllaons s rssd n h followng for: g ] ω f ] g ] f > and ang no accon rors of h fncons consdrd n h rvos hor w can wr sls rsson: ω { g ] f } >.hs s condon for no snc of h rggr of cold snglars. b* for arars vals... nsd of h nrvals... h srs of rggrs of cold snglars n h local doans s. hs rggr of cold snglars conans on nsabl snglar on saddl y and wo sabl ons cnr ys. hn for sall rrbaons n h vcny of hs nsabl snglar saddl on s asy o oban an naccabl lnarzd dffrnal qaon for long rod of h oon. In h vcny of h nsabl saddl on corrsondng o nsabl qlbr oson lnarzd qaon as h followng for: 9
76 K.R. SEVANOVIĆ HEDRIH whr { g ] f } > In h vcny of h wo corrsondng sabl snglar ons of cnr y corrsondng o sabl qlbr osons lnarzd qaon s n h for 8 whr g ] ω ] ] r f g f r > and ang no accon rors of h fncons consdrd n h rvos hor * for h cas of h rggr of cold snglars snc w can wr sls rsson: g r ] ωr f r ] > for h own gn crclar frqncs of h sall oscllaons of h wo sabl qlbr osons. IV. OPIMAL ONROL O NONLINEAR DYNAMIS IN HE ONSERVAIVE SYSEM or oal conrol of h nonlnar dynacs of hs consdrd sys dfnd by nonlnar dffrnal qaon w can nrodc wo has coordna n h sac sa and and by h dffrnal qaon scond ordr w can nrodc wo nonlnar dffrnal qaons frs ordr n h followng for: g ] f wh nal condons α and β. as of h dfnd dynacal sys oal conrol s: By sng conrollng forc acng o h sys ransfr nal sa of h nonlnar dynacs of h sys dfnd by α and β no h fnal rna sa of h nonlnar dynacs dfnd by γ and χ whr s nal for ha oal conrol as. han w can wr wo nw nonlnar dffrnal qaons of h frs ordr for oal conrol as n h followng for: g ] f 5 wh nal condons sa n h for α and β. 6 and wh fnal rna condons n h for γ and χ 7 whr s ncssary for hs oon.
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 77 Ponrjagn's a rncl s Rfs. 5] and ] s sd. or nzaon of h o h rvos sys dynacs dfnd by nonlnar dffrnal qaons 5 w add o hs qaons h followng fnconal I d 8 as a crron of h oaly nzaon or "crron of qaly" of h oon conrol wh addon n h for of h conrollng forc consrans n h for:. 9 h conc of conrollably of oon or sys dynacs ls h ossbly ha h chancal sys oon or dynacs s ralzd accordng o a gvn rogra ndr h caon of scal gnralzd forcs. Moon or dynacs conrollng forc s hr consdrd as gnralzd forc of conrollably corrsondng o gnralzd coordna of h sys. h hras "sys oon or sys dynacs conrol" ls h rocss of ralzng a gvn or rograng oon or dynacs. Progras can b of a gra vary. hs sdy nclds h rogra of h ahways and h rogra of vlocs. or h oon on h drvd anfolds s also ncssary o now and a no consdraon h rlaons of hr gnraon. orlaon of h hor * s Rf. 5] gvs h followng lanaon: "h chancal sys oon or dynacs s conrollabl accordng o h rogra gvn n advanc f hr ar sch conrollng forcs of sch agnd dndng on h rogra arars whch ar by hr absol val grar han ohr rscv acv forcs f h conrollng forc drc h oon oos o h oon drcon ndr h nflnc of h ohr forcs". onc of oal oon of sys dynacs ls hr oon or dynacs of h chancal syss whos arclar arbs hav r vals wh rsc o so dynacs arars. Now l's drn h conrollng forc ha can conrol h dynacs of h sys of accordanc wh dfnd conrol ass 5-9. h robl of oal conrollabl dynacs s o fnd dynac arars ha s hos forcs ha ransla a conrollabl chancal sys dynacs dscrbd by sys dffrnal qaon fro nal sa α β] o fnal rna sa of h nonlnar dynacs dfnd by γ χ] whr s nal so ha fnconal 8 achv s r val. I s ossbl crron of h oal conrol of dynacs o wr n h followng way: I Η ] d or n h fors: I H ] { g ] f } d.a hs rvos fnconal.a s for h cas wh nscfc nrval of. As h on of h bondary of nrval s n h ngral of fnconal hn s ncssary o a no accon nosochroos varaons of h fnconal.
78 K.R. SEVANOVIĆ HEDRIH On hs bass w can o wr: ] ] Η Η Η Δ Η Δ d I Oal dynacs s dfnd by solvng h followng sys of dffrnal qaons wh corrsondng addonal condons of h oal conrol of oon: Η ; Η ; Η ; ; Η. By sng hory rsnd n Rf. 5] as wll as rvos oband condons of h oal conrol of oons w can wr h followng Halon fnconal n h for: } ] { ] f g H Scond ar of h canoncal dffrnal qaons s: ] ] ] f g f g H H or ] ] ] f g f g * and H } ] { ] f g H ] } { ] f g H 5 rs ngral of h sys of oal conrol of h sys dynacs ] f g 6 w can oban by followng for: ] ] G G 7
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 79 ang no accon rsson for h sys onal nrgy and corrsondng nal condon: s ossbl o wr h rvos ngral 7 of nrgy n h followng for E E E E 8 hn w hav a s of wo has rajcors wh dffrn nal ons frs hrogh h nal condon on α and β and scond hrogh fnal rna condon on γ and χ corrsondng o h fnal and rna dynac sa of h sys and wh forc wh alrnav drcons: E E E E 9 or E E E E ] ] G G 9.a G ] G ].a Prvos dfnd rajcors by qaons 9.a-a s b wh las on coon on cross scon rrsnav on N n h has lan of dynacal sa as a cross scon of h rvos rajcors frs hrogh h nal condon on α and β and scond hrogh fnal rna condon on γ and χ corrsond o h fnal rna dynac sa of h sys and wh forc wh alrnav drcons: If hs cross scon s ral conrol of h oon s ossbl and sys s conrollabl f hs cross scon dos no s conrol of hs oon s no ossbl. As h rajcors ar n h cas f h consrvav chancal sys ar crvs of h consan nrgy hn for oal conrol of oon wh rsc o h rod nzaon hn oal conrol s wh consan nrgy. W s fnd consan nrgy crv - rajcory assng hrogh h nal has sa of sys dynacs and canal has sa of h sys dynacs. W hav cas of conrol by sng a forc wh consan val by wh chang of h drcon bng-bang solon. In hs sys h conrol forc occrs lnarly and n accordanc wh a rncl aans s r and/or lowd bonds n h gnral cas. Solon of h as s by solvng basc sys of dffrnal qaons and corrsondng arclar solons whch conan h sa of h dfnd dynacs n h nal on and n h fnal rna on of. A on w s chang h conrol forc drcon. In h consdrd cas h dynacs of h sys s dscrbd by nonlnar dffrnal qaons b s ossbl o solv h robl n analycal for. Now for obanng has coordna of h cross scon bwn has rajcors w s wo has rajcors hrogh hs on and for dffrn drcon of h conrol forc by whch w can fnd on of n whch s ncssary o chang conrollng forc drcon. or ha rason w can wr ha boh has rajcors conan hs has dynacal sa on:
8 K.R. SEVANOVIĆ HEDRIH ] ] G G G ] G ] Phas coordnas of h dynac sa n whch s oal o chang drcon of h oal conrol forc ar n h for: { G ] G ]} G ] G ] Oal rod for ransfr nonlnar sys fro on sa dynacs along on has rajcory o h ohr along ohr has rajcory sccssv assng hrogh coon sa on h boh has rajcors whch corrsond o cass wh dffrn conrol oon forc drcon w can oban as s of h s O and as s of h oon of h sys has rrsnav on along on frs and scond rajcory. d G ] G ] f d G ] G ] 5 Oal rod o for ransfr nonlnar sys fro on sa dynacs N along on has rajcory o h ohr N along ohr has rajcory sccssv assng hrogh coon sa s o d G ] G ] d G ] G ] 6 cos cos cos cos ] c :.8 Labda : cos cos cos cos ] c :.8 cos.58 cos cos.58 cos ] ca :.8.58 Labda : cos.58 cos cos.58 cos ] ca :.8.58
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 8 E P ϕ f f f ϕ c c ca ca ϕ 5 5 5 5 a* b* g.. a* Ponal nrgy crvs for wo cass of h basc sys loadd by conrol forc wh oos drcons. b* Phas rajcors for h basc nonlnar dscr sys dynacs loadd by conrol forcs wh oos drcons. Grahcal rsnaon of h cross scons bwn wo rajcors for h cass of h conrol forc oos drcons. In hs al hr s on cross scon bwn corrsondng rajcors and bwn nal dynacal sa and rna fnal dynac sa. or solon of h robl s ncssary o fnd cross scon O al has dynac sa bwn rvos has rajcors n whch h conrol forc changs drcon. Inal branch of h has rajcory conans h rrsnav on and fnal branch of h has rajcory conans h on. By sng has rajcors of h nonlnar dynacs w can fnd h on n whch w s chang conrol forc drcon and fnal nal for oal conrol oon. Grahcal rsnaon s rsnd n gr and. In gr. a* onal nrgy crvs for wo cass of h basc sys loadd by conrol forc wh oos drcons ar rsnd. In gr. b* h has rajcors for h basc nonlnar dscr sys dynacs loadd by conrol forcs wh oos drcons ar rsnd. Grahcal rsnaon of h cross scons bwn wo rajcors for h cass of h conrol forc oos drcons s vsbl. In hs al hr s on cross scon bwn corrsondng rajcors and bwn nal dynacal sa and rna fnal dynac sa. In gr. a* and b* wo als of h has rajcors for h basc nonlnar dscr sys dynacs loadd by conrol forcs wh oos drcons ar rsnd. Grahcal rsnaon of h cross scons bwn wo rajcors for h cass of h conrol forc oos drcons s vsbl. In boh als hr ar wo cross scon bwn corrsondng rajcors and bwn nal dynacal sa and rna fnal dynac sa. cos.58 cos cos.58 cos ] a :.8.58 Labda/ cos.58 cos cos.58 cos ] a :.8.58 cos.8 cos cos.8 cos ] : a.8.8 cos.8 cos cos.8 cos ] a :.8.8 Labda : cos.58 cos cos.58 cos ] aa :.8.58 Labda/ cos.58 cos cos.58 cos ] aa :.8.58 cos.58 cos cos.58 cos ] aa :.8.58 cos.58 cos cos.58 cos ] aa :.8.58 Labda :
8 K.R. SEVANOVIĆ HEDRIH a*.5 a a aa aa.5.5.5 8 6 6 8.5.5 ϕ O ϕ b* a a aa aa 6 8 5 ϕ O ϕ g.. Phas rajcors for h basc nonlnar dscr sys dynacs loadd by conrol forcs wh oos drcons. Grahcal rsnaon of h cross scons bwn wo rajcors for h cass of h conrol forc oos drcons. In boh als hr ar wo cross scon bwn corrsondng rajcors and bwn nal dynacal sa and rna fnal dynac sa. If n h sys dynacs a rggr of cold snglars ss and aar and dsaar wh arar chang hn s ncssary o nvsga oson of h dynacal sa corrsond o has rrsnav on O. If hs sa s nar or n h vcny of h nsabl saddl ons or n h vcny of h hooclnc orbs n h for of h nbr gh hn s ncssary o nvsga ohr ossbls for h chang drcon of h conrol oon forc far of hs crcal ara. I s ncssary bcas dynacal sa n h vcny of h hooclnc nsabl ons s vry snsv o h sall chang of h nal condons and sys can los sably and wh hs oal conrol of h sys s no ossbl. hn hr ar ohr ossbls o s wo ss n h oal conrol of oon sng rvos hod. By chosng a ranson dynacal sa bwn nal and fnal dynacal sa s ncssary o dsan osd fro crcal dynacal sa clos o nsabl saddl on on h hooclnc orb n h for of nbr gh. hs nvsgaon shows s ha s ncssary o nvsga for frs basc consrvav nonlnar sys and rors and afr ha ossbls for oal oon conrol accordng o h rsn condons of h oal conrol of oon. V. HEOREM ON HE EXISENE APPEARANE O A RIGGER O HE OUPLED SINGULARIIES AND HE SEPARARIX IN HE ORM O "OPEN SPIRAL NUMBER EIGH" IN HE SYSEM WIH URBULEN DAMPING hor. In h sys wh rbln dang whos dynacs can b dscrbd wh h s of non-lnar dffrnal qaon n h for: g ] f 7
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 8 and whos onal nrgy s n h for n whch h fncons f and g ar n h for and sasfy h condons and boh fncons f and g hav on a or n n h nrval bwn wo zro roos: a* for arars vals... osd of h nrvals... h rggr of cold snglars n h local ara dos no s. b* for arars vals... nsd of h nrvals... h srs of rggrs of cold snglars n h local doans s as wll as corrsondng hooclnc orb - h sarar n h for of "on sral nbr gh". h roof of dfnd hor and dals ar blshd n h Rfrncs ] and 6]. or h sys wh rbln dang ] whos dynacs can b dscrbd wh h s of non-lnar dffrnal qaon n h for 7 w can solv has rajcors by ngrang rvos qaon by nrodcng followng noaon: v : dv v g ] f 8 d and afr ngraon w oban: v G ] 8 G ] d 9 whr ngral consan s n h followng for: G ] 8 G ] d dndng on nal condon of oons. Eqaon of hooclnc orb n h for "on sral nbr gh" rogh hooclnc on s: v G ] 8 whr consan of ngraon s: G ] d G ] {G } for g r ] for... Eqaon of hooclnc orb ohr for n h connos closd for rogh hoolnc on s s: whr s: v G ] 8 G ] d s s G s ] 8 G ] s d
8 K.R. SEVANOVIĆ HEDRIH VI. OPIMAL ONROL O NONLINEAR DYNAMIS IN HE NO ONSERVAIVE SYSEM or oal conrol of h nonlnar dynacs of hs consdrd no consrvav sys dfnd by nonlnar dffrnal qaon 7 w can nrodc wo has coordna n h sac dynac sa and and by h dffrnal qaon scond ordr w can nrodc wo nonlnar dffrnal qaons frs ordr n h followng for for dals s Rf. 5]: g ] f 5 wh nal condons α and β. as of h dfnd dynacal sys oal conrol s: By sng conrollng forc acng o h sys ransfr nal sa of h nonlnar dynacs of h sys dfnd by α and β no h fnal sa of h nonlnar dynacs dfnd by γ and χ whr s nal for ha oal conrol as. han w can wr wo nw nonlnar dffrnal qaons frs ordr for oal conrol as n h followng for: g ] f 6 wh nal condons sa n h for 6 and wh fnal condons n h for 7 whr s ncssary for hs oon. Ponrjagn's a rncl s Rf. 5] s sd. or nzaon of h o h rvos sys dynacs dfnd by nonlnar dffrnal qaons 7 w add o hs qaons h followng fnconal n h for 8as a crron of h oaly nzaon or "crron of qaly" of h oconrol wh addon n h for of h conrollng forc consrans n h for 9:. I s ossbl o wr h crron of h oal conrol of dynacs n h rvos sd forlaon : I H ] { g ] f } d 7 hs rvos fnconal 7 s for h cas wh nscfc nrval of. As h on of h bondary of nrval s n h ngral of fnconal hn s ncssary o a no accon nosochroos varaons of h fnconal. On hs basc w can o wr: ΔI ] Η Η ] Δ Η Η d In hs cas oal conrol dynacs s dfnd by solvng h corrsondng sys of dffrnal qaons wh corrsondng addonal condons of h oal conrol of oon as n rvos consdrd cas dfnd by sys. 8
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 85 By sng hory fro Rf. 5] and 5] as wll as rvos oband condons of h oal conrol of oons w can wr h followng Halon fnconal n h for: { } ] ] f g H 9 Scond ar of h canoncal dffrnal qaons fro sys of qaons s: ] ] ] f g f g H H 5 and H { } ] ] f g H 5 } ] { ] f g H or h sys wh rbln dang loadd by conrol forc whos dynacs can b dscrbd wh h s sys of non-lnar dffrnal qaons n h for 6 qaons of h has rajcors by ngrang rvos sys of dffrnal qaons by nrodcng followng noaon: v. hn w oban: ] f g v d dv 5 and afr ngraon w oban h followng qaons of h has rajcors: ] 8 ] d G G v 5 whr consan of ngraon s: ] 8 ] d G G 5 dndng on nal condon of oons. Prvos dfnd rajcors s b wh las on coon on cross scon rrsnav on N n h has lan of dynacal sa as a cross scon of h rvos rajcors. rs hrogh h nal condon on α and β and scond hrogh fnal condon on γ and χ corrsond o h fnal dynac sa of h sys and for h forc wh alrnav drcons. If hs cross scon s ral dfnd oal conrol of h oon s ossbl and sys s
86 K.R. SEVANOVIĆ HEDRIH conrollabl f hs cross scon no ss conrol of hs oon n h dfnd condons s no ossbl. As h has rajcors n h cas of h corrsondng consrvav chancal sys ar consan nrgy crvs of h sys hn for oal conrol of oon wh rsc o h rod nzaon hn oal conrol s wh consan nrgy. W s o fnd consan nrgy crv rajcory assng hrogh h nal has dynac sa of h sys dynacs and fnal rna has sa of h sys dynacs. W hav cas of conrol by sng a conrol forc wh consan val b by wh chang of h drcon bng-bang solon. In hs sys loadd by h conrol forc occrs lnarly and n accordanc wh a rncl aans s r and/or lowd bonds n h gnral cas. Solon of h as s by solvng basc sys of dffrnal qaons and corrsondng arclar solons whch conan h dynac sa of h dfnd dynacs n h nal on and n h fnal rna on of. A on w s chang h conrol forc drcon. In h consdrd cas h dynacs of h sys s dscrbd by nonlnar dffrnal qaons b s ossbl o solv h robl n analycal for. Now for obanng has coordna of h cross scon bwn has rajcors w s wo has rajcors hrogh hs on and for dffrn drcon of h conrol forc by whch w can fnd on of n whch s ncssary o chang conrollng forc drcon. or ha rason w can wr ha boh has rajcors conan hs has dynacal sa on and w can consdr wo cass. I* rs cas ondons ar: v ] G ] 8 ] G d v ] G ] 8 ] G d 55 whr ngral consans ar n h fors: d G ] 8 G ] ] 8 G G ] d 56 Phas coordnas of h dynac sa n whch s oal o chang drcon of h oal conrol forc ar n h for: ln G ] 8 G ] 8 G G ] ] d d 57
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 87 ] 8 ] d G G v 58 Oal rod for ransfr of nonlnar sys fro on sa of dynacs N along on has rajcory o h ohr N along ohr has rajcory sccssv assng hrogh coon sa on boh has rajcors whch corrsond o cass wh dffrn conrol oon forc drcon w can oban as s of s O and as s of h oon of h sys has rrsnav on along h frs and h scond rajcory. Oal rod o for ransfr of nonlnar sys fro on nal sa dynacs N along on has rajcory o h ohr fnal dynac sa N along ohr has rajcory sccssv assng hrogh coon sa s oband n h for: o G d G d G d G d ] ] 8 ] ] 8 59 or solon of h robl s ncssary o fnd cross scon O al has dynac sa bwn rvos has rajcors n whch h conrol forc changs drcon. Inal branch of h has rajcory conans h rrsnav on and fnal branch of h has rajcory conans h on. By sng has rajcors of h nonlnar dynacs w can fnd h on n whch w s chang conrol forc drcon and fnal nal for oal conrol oon: II* Scond cas ondons ar: ] 8 ] ] d G G v ] 8 ] ] d G G v 6 whr ngral consans ar n h fors: ] 8 ] d G G ] 8 ] d G G 6
88 K.R. SEVANOVIĆ HEDRIH In hs scond cas boh has coordnas of h dynac sa n whch s oal o chang drcon of h oal conrol forc ar no ossbl o oban n lc for: 6 ] ] G Sh d 6 G ] d v G ] 8 6 If rvos sys of qaons 6-6 hn n hs cas oal conrol of oon ndr dfnd condons and by oband alrnav conrol forc s ossbl. In h cas ha rvos sys of qaons 6-5 havn' ral cross scon hn oal conrol s no ossbl wh alrnaon of h conrollng forc wh consan nnsy hrogh on chang of h conrol forc drcon. hn w can rconsdr as of h oal conrol by a fw coonn ass of h oal conrol. Oal rod for ransfr nonlnar sys fro on sa dynacs N along on has rajcory o h ohr N along ohr has rajcory sccssv assng hrogh coon sa on h boh has rajcors whch corrsond o cass wh dffrn conrol oon forc drcon w can oban as s of h s O and as s of h oon of h sys has rrsnav on along on frs and scond rajcory. O 8 G ] d d G ] 6 8 d G ] d G ] 65 Oal rod o for ransfr nonlnar sys fro on sa dynacs N along on has rajcory o h ohr N along ohr has rajcory sccssv assng hrogh coon sa s o. Oal conrol of h nonlnar dynacs of a consdrd consrvav and nonconsrvav nonlnar syss wh dynacs dscrbd by ordnary dffrnal qaons s sdd by sng varaonal hod cobnd wh h has lan hod. Phas rajcors orras for ransforaon ndr h nflnc of h conrol forc and coarson wh basc nonlnar sys dynacs ar ond o.
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 89 VII. NONLINEAR DYNAMIS O A HEAVY MAERIAL PARILE ALONG IRLE WHIH ROAES AND OPIMAL ONROL VII.. Moon of h havy aral arcl along crcls As al w wll consdr a dscr sys s g.. of a havy aral arcl wh ass along a crcl whch roas arond h swly osond angl α o h horzonal drcon as ornd by n vcor n r and n h cas ha h ccnrcy s or no qal o zro or. Rlav oson of h aral arcl along h crcl wh rads l s drnd by angl ϕ as a gnralzd coordna. h rhonoc coordna θ Ω s h angl of crcl's roaon arond h as ornd by h n vcor n r. horzon A O ϕ l lsnϕ h r θ G g α B h n r α θ Ω θ Ω 8 8 a* b* g.. Moon of h havy aral arcl along a crcl whch roas abo a fd as. Sl odl of h nonlnar dynacs. Phas orra of basc odl nonlnar dynacs fro Rf. 7]. Usng h noaon n g.. h ra of onal nrgy chang E rssd by gnralzd coordna ϕ and rhonoc coordna θ and rsson of nc nrgy E of h sys rsnd n h blshd Rfrnc 7] and nrodcng h followng noaon g sn α g sn α θ Ω λ ε and Ω rz 66 lω l l h dffrnal qaon of h rlav oon of h havy aral arcl w can wr n h followng for: f f f f f5 f6 f7 f8 f9 f f f f f f5 f6 ϕ Ω λ cosϕsn ϕ Ω εcosϕ Ω λcgα cosϕcosω 67 ro ndd Lagrang's sys qaon for rhonoc coordna θ s Rfs. 5 6 9 ] w can oban h rsson of h col M θ corrsondng o h rhonoc coordna n h followng for: d g M θ l ε sn ϕ Ω] sn θcosα ε sn ϕ 68 d l.... ϕ ϕ
9 K.R. SEVANOVIĆ HEDRIH VII.. Lnarzd Aroaon Now l s consdr scal cass of h havy aral arcl dynacs n h cas ha h ccnrcy s qal o zro. or bgnnng w consdr sall oscllaons arond sabl rlav oson corrsond o h sabl rlav qlbr osons of h hoognos dffrnal qaon 67 for h cass: * λ > ϕ and * λ < ϕ arccos λ and whn cg α s sall. * or h cas ha λ > w sdy sall oscllaons arond sabl rlav qlbr oson ϕ by sng corrsondng lnarzaon of h dffrnal qaon n h for of: ϕ Ω λ ϕ Ω λcgα cosω 69 whn w a ha aroaon ar: sn ϕ ϕ... and cos ϕ... w can concld ha hs sall oscllaons arond ϕ for λ > ar sl forcd wo frqncy oscllaons wh followng frqncs: "own" frqncy ω Ω λ and forcd frqncy Ω wh alds dndng of nal condons and of h forcd caon ald Ω λ cg α as fncon of h as angl α and ha cg α s sall. W can concld ha n hs cas h rsonanc rg s ossbl whn Ω Ω rz and for λ hn λ > and ha anglar vlocy of h crcl s crcal. B hs conclson s oos wh h asson ha oscllaons ar wh sall ald. In ha cas lnarzaon s no corrc. * or h cas whn λ < w sdy sall oscllaons arond h sabl rlav qlbr oson ϕ s arccos λ by alyng corrsondng lnarzaon of h dffrnal qaon w a chang of h gnralzd coordna ϕ by followng ϕ s ϕ. Afr lnarzaon w oban h followng lnarzd qaon: cg λ α ϕ Ω λ cosω ϕ Ω λcgα cosω 7 λ hs lnarzd qaon s of Mah-Hll y. ro hs lnarzd dffrnal qaon w can concld ha hs sall oscllaons arond ϕ s arccos λ for λ < ar no sl forcd wo frqncy oscllaons b ar of y of h Mah ararc and forcd oscllaons. In hs cas w can on o ossbls of aaranc of ararc rsonanc. In h frs lnarzd aroaon arond h rlav qlbr oson ϕ s arccos λ for λ < sall oscllaons ar dscrbd by Mah's oscllaor wh crclar frqncy and arars: ω Ω λ λ λ and γ λ λ cgα h λ τ Ω and rnal forcd caon ald h λ cg α and frqncy qal o h anglar vlocy of h roaon of h crcl. or dal s Rfrnc ] Mah's dffrnal qaon or loq Annals d l'ecol Noral 88. On h bass of hs rfrncs w can wr ϕ A μ B μ whr A and B ar ngral consans; μ s characrsc onn and ar rodc fncons wh rod π dndng of arars: λ and γ. Man and rncal robl s nvsgaon of h sably of solons s Rfs. ] ] and ]. By Inc and Sr ] h doans of sably and nsably grahcally ar rsnd n h for of Inc- Sr's sably car for dffrn vals of h arars λ and γ.
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 9 VII.. Oal conrol of nonlnar dynacs or conrol of oon of consdrd nonlnar sys w consdr on arar conrol by on gnralzd forc corrsondng o h gnralzd sys coordna. or oal conrol of h nonlnar dynacs of hs sys dfnd by nonlnar dffrnal qaon 7 w can nrodc wo has coordnas n h sa sac and and by h dffrnal qaon scond ordr w can nrodc wo nonlnar dffrnal qaons of frs ordr n h followng for: ϕ; v; τ λ cos sn εcos cgα cos cosτ ϕ wh nal condons ar: ϕ ϕ and v. Ω as of h dfnd dynacal sys oal conrol s: By sng conrollng forc acng on h sys h ransfr fro h nal sa of h nonlnar dynacs of h sys dfnd by and no h fnal rna sa of h nonlnar dynacs dfnd by and whr s nal for ha oal conrol as shold b rford hn w can wr wo nw nonlnar dffrnal qaons of frs ordr for h oal conrol as n h for and nal condons and fnal condons. As n h rvos cass for nzaon of h Ponrjagn's a rncl s sd. onc of oal oon of sys dynacs ls hr oon or dynacs of h chancal syss whos arclar arbs hav r vals wh rsc o so of h arars dynacs. Now l's drn h conrollng forc ha can conrol h dynacs of h sys of accordanc wh dfnd conrol as. By sng hory w can wr h followng Halon fncon n h for: Η λ cos sn εcos cgα cos cos τ ] 7 oal dynacs ar dscrbd by sys of qaons. W hav cas of conrol by sng a forc wh consan val by wh chang of h drcon bng-bang solon. In hs sys h conrol forc occrs lnarly and n accordanc wh a rncl aans s r and/or lowd bonds n h gnral cas. Solon of h as s by solvng basc sys of dffrnal qaons 7 and corrsondng arclar solons whch conan h sa of h dfnd dynacs n h nal on and n h fnal on of. A on w s chang h conrol forc drcon. In h consdrd cas h dynacs of h sys s dscrbd by nonlnar dffrnal qaons and s no ossbl o solv h robl n analycal for. B s convnn o s has lan hods for qalav analyss by sng h basc sclronoc holonoc consrvav sys corrsondng o rhonoc and forcd cd sys s Rfs. 9] ]. By follow rrsnav on n h has lan along has rajcors and ransforaon of h srcr of has orra by sng conrol forc s ossbl o larn abo nonlnar dynacs n h oal conrol dynacs. als of h consan nrgy crvs of h sys ar also a ool o larn h rors of conrollably of h nonlnar sys. 7
9 K.R. SEVANOVIĆ HEDRIH a* c* f l g h a a a a a5 a6 a7 a8 a9 a a a a a 8 6 8 6 6 8 8 6 6 8 E P ϕ ϕ ϕ b* d* a a a a a5 a6 a7 a8 a9 a a a a a 8.76 S 7.7 5 5.9 S.7 g.. Ponal nrgy crvs a* for dffrn sys arars has rajcors b* c* for h corrsondng basc nonlnar dscr sys dynacs s g.. a* of h havy aral arcl wh ass along crcl whch roas arond swly osond wh angl α 9 o h horzon as ornd by h n vcor n r and for cas ha h ccnrcy s qal o zro. Dffrn vals of h sys arars and for conrol basc sys wh conrol forc rgh sz. ransforaon of h srcr of has orra by conrol forc rgh sz. h sys arars λ and Ω ar vard n h grs. Phas rajcors d* for h corrsondng basc nonlnar dscr sys dynac whch roas arond swly osond wh angl α o h horzon. ϕ ϕ ϕ ϕ or h conrol of h rlav oon of h havy aral arcl along crcl whch roas abo a fd as by h rg of h corrsondng nbr of roaons n on and n oos drcon w can s has lan hods and vsalzaon of h nrcal Mahad rn. Vsalzaon of h has orra of h basc holonoc sclronoc consrvav sys gvs s nforaon abo snc of a rggr of cold snglars as wll as an hooclnc orb n h for of nbr gh s gr. b*. Also w can s n g. ha h srcr of h has orra changs ndr h nflnc and caon of h conrol forc. In gr onal nrgy crvs a* for dffrn sys arars ar rsnd as wll as has rajcors b* c* for h corrsondng basc nonlnar dscr sys dynacs s g.. a* of h havy aral arcl wh ass along crcl whch roas arond swly osond wh angl α 9 o h horzon as ornd by h n vcor n r and for cas ha h ccnrcy s qal o zro. Dffrn vals of h sys arars and for conrol basc sys wh conrol forc rgh sz. ransforaon of h srcr of has orra by conrol forc rgh sz. h sys arars λ and Ω ar vard n h grs. Phas rajcors d* for h corrsondng basc nonlnar dscr sys dynac whch roas arond swly osond wh angl α o h horzon
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 9 Eqaons of h has rajcors assng rogh nal on sa and fnal on sa for h consdrd cas and for horzonal roaon as of h sys ar n h for: λcos cos cos cos εsn sn λcos cos cos cos εsn sn 7 or solon of h robl s ncssary o fnd cross scon al has sa bwn rvos has rajcors n whch h conrol forc changs drcon. Inal branch of h has rajcory conans h rrsnav on and fnal branch of h has rajcory conans h on. By sng has rajcors of h nonlnar dynacs w can fnd h on n whch w s chang conrol forc drcon and fnal nal for oal conrol oon: λcos λcos cos cos cos cos cos d d cos εsn εsn sn sn 7 VIII. ON RHEONOMI SYSEMS WIH EQUIVALEN HOLONOMI ONSERVAIVE SYSEMS APPLIED O HE NONLINEAR DYNAMIS O HE WA'S REGULAOR Eqvaln holonoc sclronoc consrvav sys o h scal class of h rhonoc sys s consdrd n h Rfrncs 9] and ]. A nrcal rn has orras and hooclnc orbs of a Wa's rglaor nonlnar dynacs ar rsnd n hs sa Rfrncs. hs sys s ossbl o s for h oal conrol als of h rvos gnral cass of h oal conrol. or al of a rhonoc sys wh qvaln holonoc sclronoc consrvav sys w shall consdr h odl of a chans of a Wa's rglaor s g. 5. a* or wo havy aral arcls ovng along corrsondng syrcally conncd crcls ha roa arond vrcal as wh consan anglar vlocy. Ω n h gravaonal fld. Knc and onal nrgy of slfd chanss of Wa's rglaor whch roa arond vrcal as wh consan anglar vlocy Ω n h gravaonal fld ar: l ϕ θ a sn ϕ l ss ss E and gl cosϕ E 75 whr ϕ s gnralzd coordna q θ Ω rhonoc coordna asss of h balls - h aral arcls l and a consrcv arars. or holonoc sclronoc consrvav sys qvaln o a rhonoc sys whch s a chans of
9 K.R. SEVANOVIĆ HEDRIH a Wa's rglaor nc and onal nrgy n h sns of h dfnons n h Rfrncs 9] and ] w can wr: ss E l ϕ and ss a gl cosϕ l Ω snϕ E 76 l By nrodcng h followng noaons θ g a Ω λ ε w can wr h l Ω l followng dffrnal qaons of h rlav balls oon: ϕ Ω λ cosϕsn ϕ εcosϕ] 77 Ingral of nrgy of qvaln holonoc sclronoc consrvav sys o h rhonoc sys whch s a chans of Wa's rglaor s: ϕ ϕ Ω {λcosϕ cosϕ ε λsn ϕ ε λsnϕ ]} 78 whr ϕ and ϕ ar nal condons of h rlav balls oon. rs ngral of h consdrd Wa's rglaor nonlnar sys dynacs of oal conrol acvad by conrol forc dscrbd by dffrnal qaon ϕ Ω λ cosϕsn ϕ εcosϕ] 79 y of 6 w can oban by followng for: ϕ ϕ Ω λcosϕ cosϕ ε λsn ϕ ε λsn ϕ ]} ϕ ϕ 8 { K θ Ω M a θ Ω N ϕ l/ L ll EP EP EP EP.5 8 6 6 8.5.5 E ϕ EP5 A a γ B EP6 EP7 EP8.5.5 O P.5 a* b* g. 5. Nrcal slaons and grahcal rsnaons. Ponal nrgy grahs of h qvaln holonoc sclronoc consrvav sys o h rhonoc sys a* h chanss of h Wa's rglaor for dffrn sys nc arars: b* λ ; ε ; ; ; 5; 8; ; ; ; Las qaon 8 s qaon of h has rajcory of h consdrd Wa's rglaor nonlnar sys dynacs loadd by conrol forc wh alrnav drcons and n rocdr for oal conrol as s dfnd n IV by qaons 9 6 for h scal cas of h fncon df ss a ] E gl cos ϕ l Ω sn ϕ G. l
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 95 IX. ONLUDING REMARKS W can concld ha s vry sabl for nvsgaons of nonlnar rors of h oon of h scal class of rhonoc syss wh rhonoc coordna lnarly dndn of h n h for q Ω o s corrsondng qvaln holonoc sclronoc consrvav sys and corrsondng has orras of hs sys for oal conrol. By sng al of chanss of Wa's rglaor blshd n Rfrncs 9] and ] w show ha s ossbl o aly has orras for oal conrol n h nonlnar syss wh nonlnar oon rors and dffrn fors of hooclnc orbs as wll as h bfrcaons of h rlav rs osons n h consdrd class of h rhonoc syss and h ransforaon of h hooclnc orbs. W nvsga snc and nonsnc of hooclnc orbs n h sha of h nbr gh for dffrn vals of h sys nc arars: ccnrcy ε and vlocy of h sor roaon Ωλ and oal conrol n h syss wh rggr of cold snglars. or sch scal class of rhonoc nonlnar syss h qvaln holonoc consrvav nonlnar sys and qvaln nc and onal nrgs ar on h for 76. In gr 6. has rajcors for a corrsondng basc dscr sys nonlnar dynacs s g. 5. a* of a havy aral arcl wh ass along crcl whch roas arond swly osond as wh angl α o h horzon as ornd by n vcor n r and for cas ha ccnrcy qal zro ar rsnd. Phas orras ar rsnd for dffrn vals of h sys arars and for conrol sys wh conrol forc. ransforaon of h srcr of has orra by conrol forc s vsbl. a a a a a5 a6 a7 a a a a a5 a6 a7 a8 a9 8 6 a8 a9 8 6 8 6 a a a a a a a a a a 5 g. 6. Phas rajcors for a corrsondng basc dscr sys nonlnar dynacs s g. 5. a* of a havy aral arcl wh ass along crcl whch roas arond swly osond as wh angl α o h horzon as ornd by n vcor n r and for cas ha ccnrcy qal zro. Dffrn vals of h sys arars and for conrol sys wh conrol forc. ransforaon of h srcr of has orra by conrol forc. In gr 7. nflncs of h conrol forc and conrol forc drcon o onal nrgy crvs and o h has orras ar rsnd. Ponal nrgy crvs and has rajcors for a corrsondng basc dscr sys nonlnar dynacs s g. 5. a* of a havy aral arcl wh ass along crcl whch roas arond swly osond as wh angl α o h horzon as ornd by n vcor n r and for cas ha ccnrcy qal zro ar ond o o b vsbl. Phas orras ar rsnd
96 K.R. SEVANOVIĆ HEDRIH for dffrn vals of h sys arars and for conrol sys wh conrol forc. ransforaon of h srcr of has orra by conrol forc n on and oos drcon s ond o o b vsbl. a a a a a5 a6 a7 a8 a9 a a 5 5 a a a a a5 a6 a7 a8 a9 a a 5 5 a a a a a a f f 5 5 5 5 5 5 5 5 f f f f 5 5 5 5 5 5 5 5 a a.5 a a.5 b b b b c c c d.5 c d.5 d d 5 5 5 5 5 5 5 5.5.5 5 6 5 6 7 8.5 7 8.5 g. 7. Inflnc of h conrol forc and conrol forc drcon o Ponal nrgy crvs and o h has orras. Ponal nrgy crvs and has rajcors for a corrsondng basc dscr sys nonlnar dynacs s g. 5. a* of a havy aral arcl wh ass along crcl whch roas arond swly osond as wh angl α o h horzon as ornd by n vcor n r and for cas ha ccnrcy qal zro. Dffrn vals of h sys arars and for conrol sys wh conrol forc. ransforaon of h srcr of has orra by conrol forc n on and oos drcon.
Phas Plan Mhod Ald o Oal onrol n Nonlnar Dynacal Syss... 97 Acnowldgns. Pars of hs rsarch wr sord by h Mnsry of Scncs and Envronnal Procon of Rblc of Srba hrogh Mahacal Ins SANU Blgrad Gran ON "horcal and Ald Mchancs of Rgd and Sold Body. Mchancs of Marals" and acly of Mchancal Engnrng Unvrsy of Nš. REERENES. Andronov A.A. V A.A. Hayn S.E. 98 orya olbany Naa Mosva.. 568. Grard I. and Danl J. Elnary Sably and BIfrcaon hory Srngr Vrlag 98.. Gcnhr J. and Hols Ph. 98 Nonlnar Oscllaons Dynacal Syss and Bfrcaons of lds Srngr-Vrlag. 6.. Hdrh Svanovć K. rggr of old Snglars nvd lnary lcr Dynacal Syss- hory and Alcaons Edd By J. Awrjcwcz and all Lodz. 5-78. 5. Hdrh Svanovć K. Nonlnar Dynacs of a Gyroroor and Snsv Dndnc on nal ondons of a Hav Gyroroor orcd Vbraon/Roaon Moon S-Plnary Invd Lcr Procdnngs: O Edd by.l. hrnoso and A.I. radov IEEE SS IUAM SPIS S. Prsbrg Ins. for Probls of Mch. Eng. of RAS. Vol. of. 59-66. 6. Hdrh Svanovć K. A rggr of old Snglars MEANIA Vol.9 No... 95-. DOI:./B:ME.99.89.5f 7. Hdrh Svanovć K. 5 Nonlnar Dynacs of a Havy Maral Parcl Along rcl whch Roas and Oal onrol haoc Dynacs and onrol of Syss and Procsss n Mchancs Eds: G. Rga and. Vsron. 7-5. IUAM Boo n Srs Sold Mchancs and Is Alcaons Edrd by G.M.L. Gladwll Srngr. 5 XXVI 5. ISBN: --67-6. 8. Hdrh Svanovć K. 5 Hooclnc Orbs Layrng n h old Roor Nonlnar Dynacs and haoc loc Modls SM7 Mlbody Dynacs M. Graldn and. Pfffr. L D - SM6 Mchancs of h s nry s IAM Warsaw - D ROM INLUDED dd by Wold Gows and oasz A. Kowalws IUAM Srngr 5 ISBN --56- Hardcovr.. D. ISBN- 978---56-5 HB ISBN- --559--boo ISBN- -978---559-9. Hdrh Svanovć K. On Rhonoc Syss wh Eqavaln Holonoc onsrvav Sys In. onf. INM-IV Edd by W. hn and all. Shangha. Nonlnar dynacs.. 6-5.. Hdrh Svanovć K. On Rhonoc Syss wh Eqvaln Holonoc onsrvav Syss Ald o h Nonlnar Dynacs of h Wa's Rglaor Procdngs Vol h lvnh world congrss n Mchans and achn scncs IoMM hna Machn rss anjn hna Arl -. 75-79. ISBN 7--7-7/H-8.. Hdrh Svanovć K. h Vcor Mhod of h Havy Roor Knc Parar Analyss and Nonlnar Dynacs Monograh Unvrsy of Nš. 5. YU ISBN 86-78-6-.. Hdrh Svanovć K. A rggr of old Snglars MEANIA Vol.9 No... 95-. Inrnaonal Jornal of h Ialan Assocaon of horcal and Ald Мchancs Mccanca Pblshr: Srngr ScncBsnss Mda B.V. orrly Klwr Acadc Pblshrs B.V. ISSN: 5-655 Par 57-968 Onln DOI:./B:ME.99.89.5f Iss: Vol 9 Nbr Da: Jn Pags: 95 - h://www.srngrln.co/a/ho/conrbon.as?wasddfddaad7bd7887cafbbcr frrrarnbacoss6;jornal57;lnngblcaonrsls:958. Hdrh Svanovć K. 7 h onrol n Nonlnar Dynacal Syss wh rggrs of a old Snglars Invd Parcaon h h Inrnaonal Worsho on Dynacs onrol o b hld on May 8 Jn 7 n Zvngorod Moscow Rssoa Absracs 7.. Ins for Probls n Mchancs of h h Rssan Acady of Scncs.and Slov Mahacal Ins of h Rssan Acady of Scncs.. Hdrh Svanovć K. h oal conrol n nonlnar chancal syss wh rggr of h cold snglars n h boo: Advancs n Mchancs : Dynacs and onrol : Procdngs of h h Inrnaonal Worsho on Dynacs and onrol / d. by.l. hrnoso G.V. Kosn V.V. Sarn] : A.Y. Ishlnsy Ins for Probls n Mchancs RAS. Moscow : Naa 8.. 7-8 ISBN 978-5--6667-. 5. Kaca Svanovć Hdrh Oal onrol of Dssav Nonlnar Dynacal Syss wh rggrs of old Snglars Jornal of Physcs: onfrnc Srs 96 8 do:.88/7-6596/96// IOP Pblshng h://www.o.org/ej/an/-lscrrn/ 6. Kaca Svanovć Hdrh and Ljljana Vljovć Nonlnar dynacs of h havy gyro-roor wh wo sw roang as Jornal of Physcs: onfrnc Srs 96 8 do:.88/7-6596/96// IOP Pblshng h://www.o.org/ej/an/-lscrrn/
98 K.R. SEVANOVIĆ HEDRIH 7. Hdrh Svanovć K. or oal of sdy: vcor and nsor hods n classcal chancs. Procdngs Nonlnar Dynacs Ddcad o h 5 h Annvrsary of A.M. Lyanov Polychnc Kharov 8.. 98-7. 8. Hdrh Svanovć K. Dynacs of cold syss Nonlnar Analyss: Hybrd Syss Vol Iss Jn 8 Pags - 9. Hdrh Svanovć K. 99 Snaga Pron Rononh Vza h Powr Of h Rhonoc onsrans hang Zborn Radova Jgoslovnsog Dr{va Za Mhan Sozj Iz Oš Mhan Nov Sad 99 P. 77-8.. Mah E. ors d Physq Mahéaq Pars 87.;. Mroolysy Y.A 96 Probl asochsoy or nsashonarnh olbany Naa Mosva... Rašovć D. orja osclacja hory of Oscllaons Načna njga Bograd 965.. 5.. Sor J. J. 95 Nonlnar Vbraons Inrscnc Pblshrs Nw Yor.7.. Sr M.J. O.: Laésch Mahsch nd vrwandl nonn n Phys nd chn Brln9. 5. Vjčć V. A. Prrncls of Mhancs Mahacal Ins SANU Bograd 999.. 6. Vjčć V. Hdrh Svanovć K. 99 h Rhonoc onsrans orc aca Unvrsas Srs Mchancs Aoac onrol and Robocs Vol. N 99 Nš.. -. MEODA AZNE RAVNI U PRIMENI NA OPIMALNO UPRAVLJANJE U NELINEARNIM DINAMIČKIM SISEMIMA SA RIGEROM SPREGNUIH SINGULARIEA Kaca R. Svanovć Hdrh Glavna dja od fazn ravn rn na oalno ravljanj nlnarn dnač ssa sa rgro srgnh snglara sa jdn sno slobod ranja j onovo razorna. Člana sadrž objdnjn rgld aorovh ranj blovanh rzlaa srj scjalnh slčajva oalnog ravljanja nlnarn dnač ssa sa rgro srgnh snglara oj s značajn za nžnjrs rn. U rad s analzra ravljvos ranj onzrvavnh ao nonzrvavnh nlnarnh dnačh ssa sa rgr snglara odgovarajćoj faznoj ravn. Poazano j da j oda fazn ravn orbljva rgodna za analz ravljačh sragja nlnarn dna onzrvavnh nonzrvavnh ssa sa jdn sno slobod ranja sa rgro srgnh snglara a aođ ao za ravljanj rlavn ranj ronon ssa oj aj vvalnn sclronon conzrvavn l nonzrvavn szs. Za ss oj j osa sa jdno gnralsano oordnao nlnarno dfrncjalno jdnačno sa rgro srgnh snglara fncja oncjaln nrgj onzrvavn sl oraj zadovoljava odrđn slov dfnsan oro o osojanj rgra srgnh snglara sarars obl broja osa. Zadaa dfnsanog oalnog ravljanja onzrvavn sso j sldć: orsć dsvo sl ravljanja na nlnarn ss s rvs z osaranog sanja nlnarn dna dfnsano sanj nlnarn dna roz ro nalnog vrna ao rrj oalnog ravljanja. N sražvač rzla "čdsn" nlnarn dna š arjaln ač dž rgova sa srgn roacjaa sa vš razlčh svojsava nlnarn dna ao oalno ravljanj dnaaa s aođ razan. Vzlzacja nlnarnh dnačh rocsa av ronon ssa s rdsavljn. Kljčn rč: Moda fazn ravn oalno ravljanjnlnarna dnaa onzrvavn ss nonzrvavn sys fazn orr ravn rggr srgnh snglara ora o rgr srgnh snglara srgn roacj ša arjalna ača hoolnča ača hoolnča orba bfracja raslojavanj orb sararsa obl broja osa