P - საქართველოს ტექნიკური უნივერსიტეტი

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P - საქართველოს ტექნიკური უნივერსიტეტი

საქართველოს ტექნიკური უნივერსიტეტიინფორმატიკისა და მართვის სისტემების ფაკულტეტიჩვენ, ქვემორე ხელისმომწერნი ვადასტურებთ, რომ გავეცანითნანა მაღლაკელიძის მიერ შესრულებულ სადისერტაციო ნაშრომსდასახელებით: ”არაწრფივი დინამიკური ობიექტების მართვის სისტემებისსინთეზი სინერგეტიკული მეთოდებით” და ვაძლევთ რეკომენდაციასსაქართველოს ტექნიკური უნივერსიტეტის ინფორმატიკისა და მართვისსისტემების ფაკულტეტის სადისერტაციო საბჭოში მის განხილვას დოქტო-რის აკადემიური ხარისხის მოსაპოვებლად.ხელმძღვანელი:ვალიდა სესაძევლადიმერ კეკენაძერეცენზენტი:რეცენზენტი:ზურაბ გასიტაშვილიდავით გორგიძეii


საქართველოს ტექნიკური უნივერსიტეტი2008ავტორი:დასახელება:ნანა მღლაკელიძეარაწრფივი დინამიკური ობიექტების მართვისსისტემების სინთეზი სინერგეტიკულიმეთოდებითფაკულტეტი :ინფორმატიკისა და მართვის სისტემებისფაკულტეტიაკადემიური ხარისხი: დოქტორისხდომა ჩატარდა: 6 .12.08 სტუ-ს მე-6 კორპუსი, II სართული, ოთახი 210ინდივიდუალური პიროვნებების ან ინსტიტუტების მიერ ზემოთმო-ყვანილი დასახელების დისერტაციის გაცნობის მიზნით მოთხოვნისშემთხვევაში მისი არაკომერციული მიზნებით კოპირებისა და გავრცელებისუფლება მინიჭებული აქვს საქართველოს ტექნიკურ უნივერსიტეტს.ავტორის ხელმოწერაავტორი ინარჩუნებს დანარჩენ საგამომცემლო უფლებებს და არცმთლიანი ნაშრომის და არც მისი ცალკეული კომპონენტების გადაბეჭდვა ანსხვა რაიმე მეთოდით რეპროდუქცია დაუშვებელია ავტორის წერილობითინებართვის გარეშე.ავტორი ირწმუნება, რომ ნაშრომში გამოყენებული საავტორო უფლე-ბებით დაცული მასალებზე მიღებულია შესაბამისი ნებართვა (გარდა იმმცირე ზომის ციტატებისა, რომლებიც მოითხოვენ მხოლოდ სპეციფიურმიმართებას ლიტერატურის ციტირებაში, როგორც ეს მიღებულია სამეცნი-ერო ნაშრომების შესრულებისას) და ყველა მათგანზე იღებს პასუხისმგებ-ლობას.iii


დისერტაციას დიდი სიყვარულით ვუძღვნი ჩემი ოჯახის წევრებს.iv


AbstractIn the present period the new science-Synergetics is developed, which studies the generalquestions of the self-organisations and covers practically all branches of the science.It is the general science based on thermodynamics of irreversible processes anddynamics of nonlinear systems. The essence of this integrated science is that in opensystems the exchange of energy, substance and information with environment is carriedout. There are processes of self-organising. It means that from physical(biological,economic and social) chaos steady ordered structures with new propertieswill be formed.The synergetics-is a derivative from ancient Greek “synergos”, which means cooperation.In synengetics the main subject of studying is mutually coordinated processesbetween system elements that is a basis for a statement of uniform structure. The appliedtheory of management is close to synenmetics as studying behaviour of nonlinearsystems in a distance from an equilibrium condition at change of same operatingparameres on the ideology to the science. The basic concepts of synergetics are: bifurcation,subordination, order parameter, parameter of management and attractor, As aresult of changes of operating parameters the nonlinear system can lose a pile stabilityin linear approach, That is dynamic system in the distance from balance there wereordered structures, on this system should arrive continuous streams of energy, substancesand information.Self-organising is the general property of diverse systems,, which consist of elementsand various subsystems-atoms, molecules, cages, animals etc.Self-organising gives possibility to study properties of dynamic systems different inthe nature from united mathematical positions and united concepts.Synthesis of closed optimum dissipative operating systems and privately the problemof analytical designing of optimum regulators is solved by the use of synergetic methods.It is shown, that while solving this, considerable and independent problem isformation of corresponding qualities. It is described that the decision of the problemof optimum algorithms synthesis has resulted nonlinear objects to creation of themethod of analytical designing of aggregated regulations, where optimizing functionalhas a semicertain appearance.In this case a synthesis problem is lead to the decision of private linear equation thatgives possibility for finding managements of same nonlinear objects to construct digitalprocedures of synthesis.For classification of closed optimum dissipative systems are cinsiderer\d also a case ofscalar and vector control separately. Are considered a number of examples, whichgive the chance to estimate efficiecy of a method of synthesis optimum dissipativesystems of control in comparison with Classical metod of analytical designing of optimumregulators. In this work fundamental physical propertiesof closed dissipativesystems are studied, witch give the cgance to bring up thoroughly problem of the decisionof analytical designing of optium regulators. Obviously, this approach demandsthe subsequent development in relation to different classes of control systems.vii


Problem decisions of synergetic method, wich is based on carried in functional dependencein space of positions of attrectors on wich natural properties are arrangedwith technological requirements of control. By the synergetic approachis establishedconformity between invariant diverses and optimizing functions.The generalized method of analytical designing of the nonlinear aggregated regulatorswich is based on carried in phase space of sequence drawind variety is a guaranteeafasymtotive stability of the synthesized systems raising movement. The problemof stabilization of nonlinear systems is solved. Necessary time of repayment of transientsis provided etc.Features of the method of analytical designin of the aggregated regulations at the decisionof problems of synthesis of systems of scalar management for nonlinear objectsof the different nature are considered.viii


Sinaarsinaxazebis nusxa.............................................................................................................................................. XISesavali............................................................................................................................................................... 131. sinergetikuli koncefcia marTvis TeoriaSi.................................................. 181.1. erTiani midgoma marTvis meTodebisadmi ...................................................... 181.2 dinamikuri sistemebis divergencia....................................................................... 231.3 sinergetikis ZiriTadi cnebebi.................................................................................. 311.4 arawrfivi TviTorganizacia da disipatiuri strqturebi........... 601.5 sinergetika da marTvis procesebi........................................................................ 652. Caketili optimaluri disipatiuri marTvis sistemebissinTezi sinergetikuli Teoriis gamoyenebiT................................................. 702.1 Caketili optimaluri marTvis sistemebis disipaturoba............ 702.2 optimaluri disipatiuri sistemebis klasifikacia............................. 752.3 Sekruli optimaluri marTvis disipatiuri sistemebissinTezi............................................................................................................................................... 812.4 optimaluri disipatiuri marTvis sistemebis analizurikonstruireba............................................................................................................................. 883. arawrfivi marTvis sistemebis sinTezi sinergetikuliTeoriis gamoyenebiT.......................................................................................................................... 1013.1 marTvis sistemebis sinTezis amocanა................................................................. 1023.2. arawrfivi agregirebuli regulatorebis analizuri konstruirebisganzogadoebuli meTodi................................................................. 1223.3. agregirebuli regulatorebis analizuri konstruirebismeTodi mimdevrobiT CarTuli invariantuli mravalsaxeobismixedviT............................................................................................................................................. 1263.4. invariatuli mravalsaxeobebis paralelur – mimdevrobiTierToblioba da mravalkavSiriani sistemebis sinTezi................. 1333.5. arawrfivi dinamikuri obieqtebis skalaruli regulatorebisanalizuri konstruireba............................................................................... 139daskvna .................................................................................................................................................................. 157danarTi.................................................................................................................................................................. 158ix


gamoyenebuli literatura................................................................................................................ 160x


naxazebis nusxanax. 1.1 bifurkacia ,,unagira kvanZi”............................................................... 33nax. 1.2 bifurkaciuli diagrama.......................................................................... 34nax. 1.3 bifurkaciul diagramis simetriuli forma..................... 35nax. 1.4 andropov-hopfis bifurkacia.............................................................. 36nax 1.5 bifurkaciuli diagrama.......................................................................... 39nax. 1.6 struqturuli mdgradobis grafiki............................................ 42nax 1.7 zRvruli cikli - harmoniuli atraqtori.......................... 45nax. 1.8. zRvruli ciklis fazuri portreti......................................... 52nax. 1.9 potencialuri funqciis grafiki.................................................. 53nax 1.10 lorencis modelis fazuri traeqtoria............................... 54nax. 1.11. stacionaluri mdgomareobis grafiki...................................... 56nax. 1.12 sinTezirebuli sistemis gardamavali procesi............. 59nax. 2.1 aerodinamikuri damuxruWeba.............................................................. 95nax. 2.2 aerodinamikuri damuxruWebis gardamavali procesi 99nax. 3.1 gardamavali procesi Caketili sistemis moZraobistraeqtoriebi ( β = 1, a = 1, T1= 1)............................................... 141nax. 3.2 gardamavali procesi Caketili sistemis moZraobistraeqtoriebi ( b= 1, T = 1, A= 1 )............................................... 142nax. 3.3 gamosaxulia Caketili sistemis moZraobis traeqtoriaroca ( β = 1; T = 1, a= 2. )................................................... 144nax. 3.4 gardamavali procesis mrudi............................................................ 147nax. 3.5 fazuri portreti.......................................................................................... 147nax. 3.6 gardamavali procesis mrudi............................................................ 149nax. 3.7 fazuri portreti.......................................................................................... 150nax. 3.8 gardamavali procesis mrudi............................................................ 151nax. 3.9 fazuri portreti.......................................................................................... 151nax. 3.10 gardamavali procesis mrudi............................................................ 153nax 3.11 fazuri portreti.......................................................................................... 153nax. 3.12 Caketili marTvis sistemis gardamavali procesismrudi........................................................................................................................... 155nax. 3.13 Caketili marTvis sistemis gardamavali procesismrudi.........................................................................................................................156 xi


მადლიერებამადლობა მინდა გადავუხადო ყველა იმ ადამიანს, ვინც მეხმარებოდასადისერტაციო ნაშრომის შესრულობის პროცესში.უპირველეს ყოვლისა, მინდადიდი მადლიერების გრძნობითავღნიშნო აწ გარდაცვლილი, ბატონ ალმასხან გუგუშვილის უდიდესიმხარდაჭერა და წვლილი, ვისი თანადგომაც ჩემთვის ფასდაუდებელი იყო.უნდა აღვნიშნო ჩემი ხელმძღვანელების, ქალბატონ ვალიდა სესაძისდა ბატონ ვლადიმერ კეკენაძის, მხარდაჭერა და დახმარება, რისთვისაცმადლიერების გრძნობით მინდა გამოვხატო მათდამი უდიდესიპატივისცემა.დიდი მადლობა ჩემს მეგობარს და კოლეგას, ია მოსაშვილს,დახმარებისა და თანადგომისათვის.xii


SesavaliTanamedrove periodSi swrafad viTardeba axali disciplinaTSorisimimarTuleba bunebismetyvelebaSi-mecniereba TviTorganizaciisprocesebis Sesaxeb, romelic moicavs Cveni garemocvisada arsebobis yvela sferos. saerTaSoriso maTematikur literaturaSiam fundamentalur mimarTulebas sul ufro xSirad uwodeben“arawrfiv mecnierebas” (nonlinear science), xolo CvenSi -“arawrfivdinamikas”. sinergetikis arsi mdgomareobs sxvadasxva bunebisarawrfivi dinamikuri sistemebSi (fizikuri, qimiuri, biologiuri,ekologiuri, teqnikuri da a.S.) universaluri kanonzomierebebisgamovlenaSi. am zogadi TviTorganizaciis principebis gamovlenisaTvissaWiroa sistemuri midgoma da sxvadasxva mecnierebisgaerTianeba.sinergetikis ZiriTad cnebebs warmoadgens: bifurkacia, daqvemdebareba,wesrigis parametri, marTvis parametri da atraqtori.aRmoCnda, rom wonasworobis dakargvas wrfiv miaxloebaSi,wesrigis parametrebis warmoSobasa da daqvemdebarebis principisrealizacias Soris arsebobs mniSvnelovani Sinagani urTierTkav-Siri. mmarTveli parametrebis cvlilebis Sedegad arawrfivmasistemam SeiZleba dakargos wonasworoba wrfiv miaxloebaSi. sinergetikulsistemebSi SeiZleba warmoiSves, rogorc mowesrigebuli,aseve qaosuri rxevebi. imisaTvis rom dinamikur sistemaSiwonasworobidan moSorebiT arsebobdnen mowesrigebuli struqturebiam sistemebze mudmivad unda moedinebodes energiis, nivTierebebisada informaciis uwyveti nakadi. swored TviTorganizaciawarmoadgens zemoT CamoTvlili mravalferovani sistemebissaerTo Tvisebas, romlebic Sedgebian elementebisagan da sxvadasxvabunebis qvesistemebisagan-atomebis, molekulebis, ujredebis,cxovelebis da a.S TviTorganizacia rac ar unda pirveli SexedviTmogveCvenos ucnaurad saSualebas gvaZlevs SeviswavloTTavisi bunebiT sxvadasxvanairi dinamikuri sistemebis Tvisebebi,erTiani maTematikuri poziciebidan da erTiani cnebebiT.13


aqtualoba: sadisertacio naSromi eZRvneba axali integrirebulimecnierebis-sinergetikis safuZvlebis gamoyenebas, romelicmowodebulia Seiswavlos koleqtiuri TviTorganizaciisprocesebi da praqtikulad moicvas Tanamedrove mecnierebis yveladargi.praqtikuli Rirebuleba. sadisertacio naSromSi ganxiluliarawrfivi dinamikuri sistemebis sinTezis sakiTxebs gaaCniaTpraqtikuli Rirebuleba, radganac, masSi naCvenebia sinergetikulimidgomis warmatebuli gamoyenebis SesaZebloba ekonomikis,ekologiis, bioteqnologiis da sxv. sistemebis marTvis amocanebSi.disertaciaSi motanilia sinergetikuli midgomis gamoyenebismravali magaliTi sxvadasxva arawrfivi dinamikuri obieqtebissinTezisaTvis.siaxle. sinergetika, romelic Seiswavlis arawrfivi sistemebisyofaqcevas mmarTveli parametris cvlilebisas, TavisiideologiiT yvelaze ufro axloa marTvis gamoyenebiT Teorias-Tan. amdenad, Zalze perspeqtiulia Tanamedrove marTvis TeoriisSeswavlis mizniT sinergetikuli sistemebis Tvisebebis gadatanaarawrfivi marTvis teqnikuri sistemebis konstruirebisaTvis. naSromSinaCvenebia, rom arawrfivi sistemebis sinTezis amocanisgadawyveta sinergetikuli principebis gamoyenebiT daiyvaneba ricxviTdamokidebulebebze.kvlevis meTodebi: marTvis klasikuri meTodebi, arawrfivisistemebis marTvis meTodebi, optimaluri marTvis meTodebi.sadisertacio naSromi Sedgeba 3 Tavisagan, daskvnisagan, danarTisada gamoyenebuli literaturis siisagan.sadisertacio naSromis pirvel TavSi gadmocemulia sinergetikulikoncefciis roli marTvis TeoriaSi. naCvenebia, romsinergetikul sistemebSi TviTorganizebis da disipatiuri struqturebis(atraqtorebis) warmoqmnis procesebSi adgili aqvs Tavisuflebisxarisxis Semcirebas makrocvladebis, xarisxis parametrebis,gamoyofis xarjze, romlebic gansazRvraven sistemis dinamikisTaviseburebebs. aRwerilia, rom TviTorganizebis procesis14


Sedegs warmoadgens atraqtorebis warmoqmna, romlebic miizidebiansistemis traeqtoriisaken. miTiTebul atraqtorebs aqvT ufromcire ganzomileba sistemis sawyis ganzomilebasTan SedarebiT,rac ganapirobebs sistemis mier sawyisi mdgomareobis “daviwyebas”,Tu saidan iwyeba sistemis moZraoba atraqtorisaken. amisSedegad miiReba arawrfivi diferencialuri sistemebis invariantuliamonaxsnebi. TiToeul atraqtors fazur sivrceSi gaaCniaTavisi mizidvis are da SesaZlebelia gamoyofili iqnes am areebisgamyofi sazRvari. amitomac am sazRvarTan mcire cvlilebamsawyis pirobebSi SeiZleba gamoiwvios arawrfivi sistemis gansxvavebuliyofaqceva am sazRvarTan. am movlenas uwodeben TviTorganizebisprocess disipatiur sistemebSi. am TavSi ganmartebuliasinergetikis ZiriTadi cnebebi, kerZod qaosi da bifurkacia.sadisertacio naSromis meore TavSi ganxilulia Caketilioptimaluri disipatiuri marTvis sistemebis sinTezi, kerZod optimaluriregulatorebis analizuri konstruirebis problemisgadawyveta sinergetikuli meTodebis gamoyenebiT. naCvenebia, romam amocanis gadawyvetisas damoukidebel problemas warmoadgensSesabamisi xarisxis kriteriumis formireba. aRwerilia, rom mar-Tvis arawrfivi obieqtebis optimaluri algoriTmebis sinTezisproblemis gadawyvetam migviyvana optimaluri regulatorebis analizurikonstruirebis meTodis SeqmnasTan, sadac optimizirebulfunqcionals gaaCnia naxevradgansazRvruli saxe. am SemTxvevaSisinTezis amocana daiyvaneba kerZowarmobulebiani wrfivigantolebis amoxsnamde, rac saSualebas iZleva zogierTi klasisarawrfivi obieqtebis marTvis mosaZebnad avagoT sinTezis ricxviTiprocedurebi.Caketili optimaluri disipatiuri marTvis sistemebis klasifikaciismizniT ganxilulia, agreTve, skalaruli da veqtorulimarTvis SemTxveva cal-calke. motanilia rigi magaliTebi,romlebic gvaZleven saSualebas SevafasoT marTvis optimaluridisipatiuri sistemebis sinTezis meTodis efeqturoba optimaluriregulatorebis analizuri konstruirebis klasikur meTod-15


Tan SedarebiT. am TavSi Seswavlilia Caketili disipatiuri sistemebisfundamentalur fizikuri Tvisebebi, romlebic saSualebasiZlevian Zireulad wamovwioT optimaluri regulatorebisanalizuri konstruirebis gadawyveta. rasakvirvelia es midgomamoiTxovs Semdgom ganviTarebas marTvis sistemebis sxvadasxvaklasebTan mimarTebaSi.sadisertacio naSromis mesame TavSi ganxilulia arawrfiviregulatorebis analizuri konstruirebis amocanis gadawyvetasinergetikuli meTodiT, romelic emyareba mdgomareobaTa sivrceSisistemis koordinatebs Soris funqcionaluri damokidebulebis-atraqtorebisSemoRebas, romlebzedac obieqtis bunebriviTvisebebi saukeTesod eTanadeba marTvis teqnologiur moTxovnebs.sinergetikuli midgomis mixedviT damyarebulia Sesabamisobainvariantul mravalsaxeobebsa da saoptimizirebel funqcionalebsSoris. ganxilulia axali sinergetikuli midgomis SesaZleblobani,romlebic dafuZnebulia Semotanili invariantulimravalsaxeobebis-atraqtorebis koncepciaze, raTa gamovlenil iqnesaxali perspeqtiuli mimarTulebebi farTo klasis arawrfivi,mravalganzomilebiani da mravalkavSiriani obieqtebis marTvissistemebis konstruirebis problemebis gadawvetisaTvis. amproblemis arsi mdgomareobs uaryofiTi da dadebiTi arawrfiviukukavSirebis generaciis analizuri meTodebis damuSavebaSi.ganxilulia arawrfivi agregirebuli regulatorebis analizurikonstruirebis ganzogadoebuli meTodi, romelic dafuZnebuliafazur sivrceSi mimzidavi mravalsaxeobebis Tanmimdevrobis Semotanaze,warmoadgens sinTezirebuli sistemebis SeSfoTebuli moZraobisasimptoturi mdgradobis garantias e.i. gadavwyvetiliaarawrfivi sistemebis stabilizaciis amocana. uzrunvelvyofiliagardamavali procesebis milevadobis saWiro dro da a.S.SemoTavazebuli sinTezis meTodi Tavisi arsiT efuZnebasawyisi n rigis amocanis ganzomilebaTa klebadi r raodenobisqveamocanis(n− r)rigis qveamocanaze mimdevrobiT dekompozicias.amasTan, sawyis fazur sivrceSi saerTo moZraobis dayofa16


1. sinergetikuli koncefcia marTvis TeoriaSi1.1 erTiani midgoma marTvis meTodebisadmiCvens garemomcvel samyaroze ATanamedrove xolistikuri(erTiani) Sexedulebebi xasiaTeba sistemuri midgomiT, romelSicklasikur bunebismetyvelebaSi tradiciul reduqcionistul midgomasTanSedarebiT upirvelesia sinTezis procesebi. rasakvirvelia,xolistikuri midgoma ar aris xistad antagonisturi reduqcionisturisadmi,am sityvis pirdapiri gagebiT, igi miiswrafisSeunarCunos misi dadebiTi mxareebi, mianiWos maT didi originalobada sistemuroba. AmniSvnelovania aRvniSnoT, rom klasikurbunebismetyvelebaSi, romelSic mTavaria reduqcionisturimidgoma, xolistikuri azris elementi bunebriv movlenaze yovel-Tvis iyo Setanili TviT mecnierebis struqturaSi, misi fundamentaluriprincipebis formis saxiT (magaliTad fizikis variaciuliprincipebi, monand leibnicis Teoria da sxv. naTelia ganekuTvnebianxolistikur midgomas).Tanamedrove xolistikuri xedva samyaros bunebriv mecnierulsuraTze, romelic moicavs fizikur, biologiur da sxvaprocesebs, efuZvneba bunebriv movlenebis sayovelTao kavSirisfundamentalur principsa da ganviTarebis princips. amasTan er-Tad gamoiyofa bunebrivi sistemebis fizikuri (biologiuri) bir-Tvi, rogorc erToblioba, materiis dabali reduqcionistuliformebisa Tavisi moZraobis kanonebiT. umaRlesi xolistikuriwarmodgenebi efuZneba udables formebs, akuTvnebs ra pirvelxarisxobrivrols struqturul maxasiaTebleblebsa da bunebrivmovlenebs Soris kavSirebs da maT Tvisebebs. naTelia rom es orimidgoma-reduqcionistuli da xolistikuri SefardebiTia da ur-TierTSeqcevadia, radganac es midgomebi Tavisi SinaarsiTa da azriTmiiswrafvian erTi da igive miznisaken-gamoavlinos integraciulida sinTezirebadi mdgomareobebi mecnierebaSi da amiTmiRweul iqnes misi erTianoba da mTlianoba. mniSvnelovania Seve-18


cadoT gamovavlinoT ZiriTadi reduqcionistuli Taviseburebebida xolestikuri tendenciebi marTvis TeoriaSi.cneba ,,marTva” ZvelTaganve atarebda pasiur, damkvirveblurxasiaTs e.i. is ar iTvaliswinebda Sesaswavli sistemis dinamikazemizanmimarTul zemoqmedebebs. niutonisebri da gansakuTrebiTmecnierebaze Tanamedrove Sexedulebebi didad aris damokidebulimmarTvelobiT midgomaze, amasTan erTad, sistemis mdgomareobisdamoukidebeli koordinatebi sawyisi mniSvnelobidan mar-Tvis zemoqmedebiT SeiZleba gadaiqces nawilobriv an mTlianaddamokidebulad sistemis saWiro traeqtoriaze moZraobis miznisuzrunvelsayofad. SemdgomSi Cven ZiriTadaT ganvixilavT mar-Tvad dinamikur sistemebs da maT Tvisebebs.gadavideT marTvis TeoriaSi reduqcionistul da xolistikuritendenciebis gamovlenaze. Tanamedrove marTvis Teorieb-Si procesebis gavrcelebuli aRwera Gmdgomareobis sivrcis gantolebebissaxiT, ufro axloa reduqcionistur midgomasTan. Eesaixsneba imiT, rom gamoyenebuli mdgomareobis koordinatebi faqtiuradmniSvnelovnebiT utoldeba erTmaneTs da maT Soris armyardeba raime kavSiri an ierarqiuli daqvemdebareba, aq koordinatebiar arian dajgufebuli funqcionalur blokebSi an qvesistemebSi,rac mogvcemda saSualebas gagvexorcielebina dekompoziciamisi fizikuri struqturis safuZvelze. magaliTad,amAazriT sistemis garegani aRwera ,,Sesasvleli_gamosasvleli”kavSiris saxiT ufro axlos aris xolistikur midgomasTan, radganis ar Seicavs informacias lokalur procesebze da dafuZnebuliamxolod im asaxvasTan, romelic akavSirebs sistemis SesasvlelsgamosasvlelTan. Tumca sistemis Sinagani aRwera SeicavsarsebiTad ufro met informacias misi moqmedebis saSualebebze,radganac aseTi aRwera qmnis mis garegan aRweras. cnobilia,rom sistemis modelis agebisas daismeba egreTwodebulirealizaciis amocana, romlis Tanaxmadac aucilebelia gairkvesSinagani da garegani aRweris srulyofileba [1].19


marTvis TeoriaSi, magaliTad kriterialuri midgomis gamoyeneba,axlos aris xolistikur SexedulebasTan, radgan uSualoddakavSirebulia mecnierebis variaciul principebTan. arCeulioptimizirebuli funqcionali (an xarisxis kriteriumi) undaasaxavdes sistemis globalur Tvisebebs, romlebic uwesebs Sez-Rudvebs mis nebismier lokalur moZraobebs. es moZraobebi ucilobladunda akmayofilebdnen romelime funqcionalis eqstremums.aqedan gamomdinare marTvis sistemebis Tvisebebis Sefasebagardamavali procesebis konkretuli parametrebiT, ganekuTvnebareduqcionistul midgomas, radganac mdebareobs ierarqiulikibis yvelaze dabal safexurze: funqcionali (xarisxiskriteriumi)-mdgomareobis gantoleba, romlebic aniWebs funqcionalseqstremums moZraobis traeqtoriebze-gardamavali procesebi,romlebic warmoadgenen am gantolebebis amonaxsnebs sistemiskerZo sasazRvro pirobebisaTvis.gavagrZeloT marTvis TeoriaSi xolistikuri tendenciebisSeswavla da SevecadoT gamovavlinoT maTi arsi zogad fizikurikanonzomierebebze dayrdnobiT. erT-erTi yvelaze gavrcelebulixolistikuri cnebaa ,,kavSiri”, radganac is warmoadgens erT-erTsabazo xarisxobriv maCvenebels. cneba ,,sistema” gulisxmobskavSirs zogierTi elementebis erTobliobisa, romlebic qmniansistemis struqturas da amitomac struqturuli kavSirebis rRvevisasqreba TviT sistemac. marTvis sistemebSi, romlebic aRiwerebadiferencialuri gantolebebiT, kavSirurTierToba asaxavsdinamikuri urTierTobebis xasiaTs komponentebs Soris, romlebicierarqiulad ganekuTvneba Sesabamis sistemas [2].rogorc cnobilia, marTvis sistemebis ganmasxvavebel Taviseburebaswarmoadgens maTi dinamikuri aRweriloba, e.i. fizikuri(qimiuri, biologiuri) obieqtis warmodgena moZraobaSi. am sistemebSimimdinare procesebi asaxaven zogierTi lokaluri im qvesistemebis(elementebis), urTierTzemoqmedebis reaqcias, romlebisganacSedgeba marTvis sistema.20


amrigad, Cveni garemomcveli realuri samyaros aRweraSimniSvnelovani roli ukavia marTvis sistemaSi komponentebs SorisurTierTqmedebas. amasTan, nebismieri bunebrivi movlena Secnobadiamxolod sxva movlenebTan urTierTkavSiriT. es dakav-Sirebuli movlenebi SeiZleba aRweril iqnes xolistikuri xedviTrogorc erToblivi warmodgena bunebriv procesze. yvela bunebrivisistema, maT Soris cocxali organizmebic, calkeuli populaciidanbiosferomde-ekologiur kompleqsebamde organizebulniarian garkveul funqcionalur erTianobaSi, romlebicawarmoeben urTierTgacvlas erTmaneTTan nivTierebebiT, energiiTda informaciiT. cxadia, rom amaTgan informacia warmoadgensmarTvis wyaros rogorc calkeuli komponentebis aseve mTlianisistemis marTvisa da mdgomareobisa.amJamad warmoiSva aucilebloba gamovleniliyo marTvisiseTi meqanizmebi, romlebic moqmedeben bunebriv sistemebSi dasafuZvlad udevs maT funqcionirebas da ganviTarebas. TvalnaTlivSeiZleba warmovidginoT, rom mocemuli meqanizmebi undabazirebdnen, bunebrivi sistemebis, nivTierebebis energiisa da informaciismarTvadi urTierTqmedebis koncefcias.SevCerdeT iseT mniSvnelovan xolistikur cnebaze, rogoricaa,simetria. Bsimetria bunebriv sistemebSi warmoadgens fizikuriurTierTqmedebis Tanamedrove Teoriis erT-erT fundamentalurcnebas. is arsebobs yvelgan sadac ki arsebobs kavSirebiromelime obieqtis an sistemis nawilebs Soris [1,3,7].simetriis Zalovan velebsa da nawilakebs Soris urTierTkavSirisTeoriis safuZvelze Tanamedrove fizikosebi mividnenaraordinalur daskvnamde,rom Cven vcxovrobT mravalganzomilebian(ufro zustad TerTmetganzomilebian) samyaroSi. am TeoriisTanaxmad, samganzomilebian samyaros, romelic Cven garSemo gvakravsemateba sivrcul-droebiTi ganzomilebebi, romlebic warmoadgenenZalebs an zemoqmedebebs. simetriaSi gamovlindeba struqturebisada sistemebis Tvisebebis erToblioba. amave dros, simetriaes aris garkveuli saxis akrZalva bunebrivi procesebis21


variantebis SesaZlo raodenobaze. naCvenebi akrZalvebi realizdebabunebriv movlenebis Sesabamisi Senaxvis kanonebis meSveobiT.magaliTad, fizikaSi simetriis idea safuZvlad udevs elementarulnawilakebis klasifikacias. qimiaSi is gamovlindeba periodulikanonis saxiT, biologiaSi - memkvidreobis SenarCunebiskanonSi, maTematikaSi ki - jgufebis TeoriaSi da a.Szogadad, simetria warmoadgens nawilakebis garkveuli mowesrigebulobas,romlebic qmnian mTlians. Tavis mxriv mowesrigebulobaiZleva SesaZleblobas SekumSos informacia bunebriviobieqtebis struqturis Sesaxeb masSi garkveuli blokebisgamoyofis gziT da am blokebis agebis wesis codniT. Tumca bunebrivsistemebSi simetriis Tvisebas sistemaSi yovelTvis erTvisasimetria e.i rRveva. simetriis gamovlindeba bunebrivi procesebisTvisebebis erTgvari ganzogadeba, xolo asimetriaSi ki gansxvavebada mravalferovneba. Cveni garemomcveli samyaros yvelamovlena moicavs simetriisa da asimetriis dieleqtrikulerTobas, romelic asaxavs Senaxvisa da cvlilebis Tvisebebs dawarmoadgens wesrigisa da uwonasworobis mizezebs, kanonzomierebisada SemTxveviTobis erTianobas. mniSvnelovania aRvniSnoT,rom bunebriv sistemebSi simetriisa da asimetriis fundamentaluriTviseba aris ara mxolod zogadmecnieruli koncefcia,romelsac gaaCnia garkveuli filosofiuri Sinaarsi. isgvaZlevs saSualebas gamovavlinoT raRac konstruqciuli sawyisiromelic udevs safuZvlad gansaxilvel bunebriv process.aseTi mniSvnelovania simetriis da struqturis cneba. simetriisda struqturis cneba gansakuTrebiT mniSvnelovania arawrfividisipatiuri sistemebis gamokvlevisas romelTaTvisac damaxasiaTebeliaintensiuri dinamikuri urTierTqmedebebi. aseT sistemebSiSeiZleba warmoiSvas e.w TviTmoZraobis procesebi (TviTorganizacia),romlebic SeuZlebelia arsebobdes wrfiv sistemebSi.aseTi sistemebisaTvis, romlebic aRiwerebian arawrfivi diferencialurigantolebebiT, rogorc cnobilia ar arsebobs maTiamoxsnis zogadi meTodebi. swored aq simetriis Tviseba gvaZ-22


levs saSualebas gamovyoT zogierTi kerZo e.w invariantuliamonaxsnebi, romlebic rogorc aRmoCnda xSirad Seicavs mniSvnelovanda mdidar informacias bunebrivi sistemebis TvisebebisSesaxeb. TviTorganizebis movlenebis gamokvlevebi, gvaZlevssaSualebas miuTiToT bunebrivi sistemebis agebulebis principebisgagebis axal gzebze. viTvaliswinebT ra rom am principebisgadatana adamianis konkretul saqmianobaze gvaZlevs sa-Sualebas gamovavlinoT axali midgomebi (qimiuri, biologiuri)obieqtebis, bunebrivi sistemebis TviTorganizciis movlenis ufrodawvrilebiT Seswavlaze. amisaTvis mizanSewonilia klasikurTanamedrove mecnierebaSi SeviswavloT TviTorganizaciis ideisganviTarebis gzebi da tendenciebi.1.2 dinamikuri sistemebis divergenciaTanamedrove mecnierebam miaRwia TvalsaCino warmatebebs bunebismetyvelebisada teqnikis mraval dargSi. magram bunebrivimovlenebisa da teqnikuri procesebis umetesoba SeiZleba aRiwerosmaTematikur enaze. mniSvnelovania is faqti, rom maTematikisefeqturi gamoyeneba uSualod dakavSirebulia maTematikuri modelebisagebasTan, romlebic ama Tu im xarisxiT adeqvaturadaRweren Sesabamis movlenebsa da procesebs. bunebismetyvelebisganviTarebis istoria gviCvenebs, rom maTematikuri aRwera gvaZlevssaSualebas movaxdinoT racionaluri axsna uamravi sxvadasxvaformis moZraobebisa, romlebic gansazRvraven garemomcvelsamyaros.moZraobis aRmwer gantolebebis agebas safuZvlad udevs mizezobriobisprincipi, romlis Tanaxmadac yvela sxeulisa ansistemis mdgomareoba Tanmimdevrulad viTardeba droSi, radganacsxeulis esa Tu is mdgomareoba drois mocemul momentSiwarmoadgens wina mdgomareobebis Sedegs: 300 welze meti xniT23


adre niutonma da misgan damoukideblad leibricma SemogvTavazesmoZraobis aRweris meTodi, romelic imis Semdeg yvelaze ufrogavrcelebulia [1]. am meTodis arsi mdgomareobs bunebaSi mo-Zraobebis kanonebis aRwera diferencialuri gantolebebis saxiT:sadacdxf ( x , x ,..., x ), i 1,2,..., n,ii 1 2 ndt = = (1.1)– Sesabamisi sistemis mdgomareobis koordinatebia.x i(1.1) diferencialuri gantolebebis saSualebiT aRwerilidinamikuri sistemis fundamentalur Tvisebas warmoadgens determinireba,regulireba da Seqcevadobas. miTiTebul debulebebisgamoyenebiT SeiZleba nebismieri bunebis dinamikuri sistemismoZraobis traeqtoriis gamoTvla. amisaTvis saWiroa mxolod sawyisix i0 -mdgomareobis dakveTa, romelic aRwers sistemis momentalurmdgomareobas. sxva sityvebiT, viciT ra moZraobis zogadikanonebi, romlebic Cawerilia determinirebuli diferencialurigantolebebis saxiT (1.1) SegviZlia mocemuli sawyisi pirobebidanyovelTvis zustad gamoviyvanoT mdgomareobaTa isusasrulo simravle, romelsac gadis sistema droTa ganmavlobaSi.Ee.i. Tu viciT ra moZraobis kanonebi (1.1), sistemis mdgomareobisabsoluturi da zusti aRwerisaTvis rogorc momavalSi,aseve warsulSi sakmarisia mxolod erTaderTi sawyisi mdgomareobiscodna. yvelaferi mocemulia da yvelaferi SesaZlebeliaaseTiaabstratuli dinamikis fuZemdebluri daskvna, romelic s-afuZvlad udevs klasikur mecnierebas [ 6,7].klasikur meqanikaSi didi xania dadgenilia, rom diferencialurgantolebebis invariantul–jgufur Tvisebebs, romlebicaRweren meqanikuri sistemis moZraobas, da fizikur Senaxvis kanonebsSoris arsebobs Rrma da metad aratravialuri kavSiri. esniSnavs, rom ama Tu im sistemis fizikuri Tvisebebis aRwera misiinvariantebis saxiT saSualebas gvaZlevs warmatebulad gadavwyvitoTsistemebis sinTezis problemebi.invariantobis principTan uSualod aris dakavSirebuliwesrigis ideebis ganviTareba rogorc maTematikaSi ise sxva ne-24


ismier mecnierebaSi. am principidan gamomdinareobs zogadi fizikuriSenanaxvis kanoni. zogadad, invariantobis Tvisebis arsebobamigviTiTebs bunebriv sistemebSi wesrigis warmoSobaze.(1.1) sistemis integraluri mrudebis erToblioba aRwers x 1 ,x 2 , ... x n fazur sivrceSi moZraobas rac gvaZlevs mis fazur portrets.Tu (1.1) sistema organzomilebiania (n=2), maSin misi yofaqcevaaRiwereba x 1 , x 2 fazur sibrtyeSi. aqedan gamomdinareobsrom nazrdebis namravli Δx 1 , Δx 2 ganisazRvros rogorc fazurisibrtyis farTobis elementebi. cxadia, rom cneba `farTobi~ SeiZlebaganzogadebul iqnes mravalganzomilebiani sistemebisaTvisac(n≥3) moculobis cnebaze fazur sivrceSi. (1.1) sistemisdinamikuri Tvisebebi principialurad damokidebulia imaze, Turogori saxiT icvlebian farTobebi (n=2) an moculobebi (n≥3)maT fazur sivrceSi. zogad SemTxvevaSi n-uri rigis sistemismoculobis cvlilebis fardobiTi siCqare fazur sivrceSirogorc cnobilia ganisazRvreba lis warmoebulis mixedviT:n1 dV ∂xi= ∑ & ,(1.2)V dt ∂xi=1sadac x i da x i (t) Sesabamisad sistemis i-uri kordinataa da misiwarmoebulia droSi.(1.2) gamosaxulebis marjvena nawils uwodeben (1.1) sistemasfazuri siCqaris veqtoris divergencias.divxini 1& =(1.3)∑i=1∂f ( x ,..., x n)∂xromelic warmoadgens kerZo warmoebulebis (1.1) sistemis marjvenanawilebis Sesabamisi warmoebulebis srul jams. (1.3) divergenciauSualod dakavSirebulia (1.3) sistemis Sesabamisi bunebisfizikur da dinamikur TvisebebTan. amasTan, saxeldobr divergencia(ufro zustad misi niSani) bunebrivi da teqnikuri sistemebisusasrulo mravalsaxeobebs yofs or klasad, romlebsac gaaCniaTprincipialurad gansxvavebuli Tvisebebi. pirvel klass ganekuTvnebiansistemebi, romlebisTvinaci25


1 dVdivx& = ≡ 0,(1.4)V dte.i. maTi fazuri moculoba rCeba ucvleli. es Tviseba gamomdinareobisluikilis cnobili Teoremis mixedviT da gamomdinareobs(1.2)-dan, radganac (1.4) pirobebidan sidide v=const. aseT sistemebsewodeba konservatiuli [13].klasikuri da kvanturi fizikis yvela kanoni aRwers sworedaseTi konservatuli sistemebis yofaqcevas. dinamikuri sistemebisTviseba SeinarCunos fazuri moculoba warmoadgens maTumTavres Tvisebas. am Tvisebis Sedegs warmoadgens energiis Senaxvada dinamikis gantolebebis Seqcevadoba droSi da agreTve,maT fazur sivrceSi mizidvis an izolirebuli fazuri traeqtoriebisar arseboba. sxvanairad rom vTqvaT am sistemebSi ganmsazRvrelimniSvneloba Rebulobs sawyisi pirobebi mniSvnelobas.konservatuli sistemebisa ganzomikleba (n) aris luwi. ase,rom klasikuri dinamikis sistemebis divergencia tolia nulisda Sesabamisad, inarCuneben fazur moculobas droSi.aRwerili sistemebis tipebis garda, bunebaSi ufro gavrcelebuliamniSvnelovani klasi dinamikuri sistemebisa, romelTacewodebaT disipatiuri. zogadad, bunebaSi yvela sistema disipatiuria.xolo am SemTxvevaSi rodesac disipatia Zalian mcireada Sesabamisad drois garkveul monakveTze SesamCnevad ver avlensTavs, maSin Sesabamisi sistemebi faqtiurad iqcevian iseverogorc konservatulebi.disipatiuri sistemebi SeiZleba dayofili iqnen pasiurisistemebis klasad, romlebic ar Seicaven energiis wyaros daaqtiuri sistemebis klasad, romlebsac gaaCniaT energiis mudmivian cvladi wyaroebi [15]. disipatiuri sistemebis principul ganmasxvavebelniSans warmoadgens droTa ganmavlobaSi fazurimoculobis Semcireba e.w. divergenciisdivx& () t < 0(1.5)mniSvneloba am sistemebSi-uaryofiTia. disipatiuri sistemebisTvisebebi upirispirdeba konservatiuli sistemebis dinamikur26


Tvisebebs. kerZod. fazuri moculobebi ikumSebian, energia arSeinaxeba xolo maTi moZraobis gantolebebi Seuqcevadia droSi.amasTan dakavSirebiT, gamosaxuleba (1..2) SeiZleba CavweroT SemdegisaxiT.1 dVV dt= − aa , > 0(1.6)(1.6) gantolebis integrirebiT, miviRebTVt ( ) = Vexp( − at)0e.i. roca t→∞ moculoba V(t)→0-ken da gamomdinareobs ZalianmniSvnelovani daskvna:• disipatiur sistemebSi yvela traeqtoria (roca t→0) miiswrafisromeliRac kompaqtur simravlemde-atraqtorisakenfazur sivrceSi.• atraqtors (mimzidvels) gaaCnia nulovani moculoba. Tumcamis mizidulobis ares fazur sivrceSi gaaCnia sasrulo moculoba.maTematikaSi mizidulobis ared miCneulia sawyisimonacemebis iseTi are x in ∈R, rom misgan gamomavali traeqtoriebiaucileblad miiswafian atraqtorisken.• atraqtorTan mizidulobis gamo xdeba disipaturi sistemismaxsierebis dakargva misi sawyisi monacenebis Sesaxeb.• atraqtoris zomis ganzomilebis maCvenebeli r yovelTvisnaklebia gamomavali disipatiuri sistemis fazuri sivrcisn.-ganzomilebis maCvenebelze r


[13]. sxvanairad rom vTqvaT, sistemis sawyisi mdgomareobebi (1.1)romelic imyofeba atraqtors gareT, raRac drois Semdeg aucilebladmoxvdebian atraqtorze. atraqtorTan aseTi mizidulobisSedegad warmoiqmnebian aRwerili disipatiuri sistemebismniSvnelovani maxasiaTeblebi. principialur ganmasxvavebel TaviseburebasromlebisTvisac warmoadgens fazuri moculobis drozedamokidebuleba. divergencia mTlianobaSi axasiaTebs sistemisfazuri moculobis (SekumSvis) siCqares.disipatiuri sistemebis zemoT aRniSnul Tvisebebidan gamomdinareobs,rom maTi moZraoba SeiZleba daiyos or arsebiTad gansxvavebulklasad: gardamavali procesebis klasi sistemis konkretulisawyisi pirobebidan zRvrul simravlemde-atraqtoramdeda atraqtorebis gaswvriv stacionaruli moZraobebis klasi [12].saerTo SemTxvevaSi (1.1) sistemis (1.3) divergencia garkveulSezRudul droSi SeiZleba iyos dadebiTi, xolo fazuri moculobaSeiZleba gaizardos. fizikuri TvalTaxedvidan aseTi xangrZlivistacionaluri reJimis arseboba SeuZlebelia, radganacamisaTvis saWiroa usasrulo energia. Tumca bevr realur arawrfivsistemebSi drois sasrulo intervalebSi fazuri moculobaSeiZleba gaizardos, rac niSnavs axal damyarebul mdgomareobaSigadasvlis process.naTelia, rom sistemis (1.1) divergencia dakavSirebuliaiakobianis fundamentalur matricasTan [9]∂ f∂ x∂ f∂ x1 11 21 2. . .∂ f∂ x1n... ... ... ...(1.7)∂ f ∂ f. . .∂ f∂ x ∂ x ∂ xn n nase rom, sistemebi romlebisTvinac fazuri sivrcis nebismierwertilSi sruldeba piroba∂ fi= 0∂xin(1.8)28


uwodeben sistemas pirdapiri kavSiris gareSe. sxva klasi sistemebisa,romelTac gaaCniaT `diagonaluri~ kerZo warmoebulebi∂ fi= const,∂xiewodebaT sistemebi pirdapiri arawrfivi kavSirebiT.(1.1)-sistemas (1.4), (1.6), (1.8), (1.9) Tvisebebi SeiZleba mivani-WoT, Tu CavrTavT mis marjvena nawilSi U 2 (x 1 ...., x n ) marTvis funqcias.maSin sistema (1.1) SeiZleba warmodgenil iqnes saxiT.x i (t)=f i (f 1 ,..., x n , U 1 ,...,U r ) i=1,2,..., n, r=1,2..., m≤n (1.10)gantoleba (1.10) aRwers marTvadi dinamikuri sistemebisklass.Tu SevirCevT Sesabamis marTvas U 1 (x 1 ...., x n ) SeiZleba miviRoTsistema sasurveli dinamikuri TvisebebiT mis fazur sivrceSi.disipatiuri sistemebis fazur sivrceSi SeiZleba warmoiSvasatraqtorebi da repelerebi. SegaxsenebT [15], rom atraqtoradiwodeba kompaqturi simravle, romliskenac miiswrafis yvelatraeqtoria misi mizidvis areSi, da mas gaaCnia sasrulo fazurimoculoba. es are warmoadgens sistemis iseTi sawyisi pirobebiserTobliobas, rom misgan gamomavali fazuri traeqtoriebi aucilebladikribebian atraqtorTan. arsebobs atraqtorebis oriZiriTadi saxe:• umartivesi atraqtorebi romlebic arseboben sistemisstacionaluri wertilebis saxiT; zRvruli ciklissaxiT, romelsac gaaCnia sakuTari amplituda da periodi:da, bolos toris saxiT, romelic aRwers kvaziperiodulprocess ramodenime damoukidebeli six-SiriT.• `ucnauri~ e.w. qaoturi atraqtorebi, romelTac rigSigaaCniaT ganmasxvavebeli Tvisebebi, maRal mgrZnobialuridamokidebulebiT sistemis sawyisi pirobebisadmi,fraqtaluri (arasrulricxviania) ganzomileba daTavisi yofaqcevis winaswarganusazRvreloba. saocariarom `ucnauri~ atraqtorebi aRiwerebian dabali xa-(1.9)29


isxis n≥3 arawrfivi determinirebuli diferencialurigantolebebiT da warmoadgenen qaosis wyarosdinamikur sistemebSi [15].sistemis fazur sivrceSi SeiZleba iyos ramodenime atraqtori.roca maTi mizidvis areebi iSlebian zogierTi aramdgradisimravleebiT-repelerebiT, romlebisganac ganizidebian sistemismezobeli fazuri traeqtoriebi. repelerebis umartives tipiurmagaliTs warmoadgens aramdgradi gansakuTrebuli wertili, aramdgradizRvruli cikli da aramdgradi tori fazur sivrceSi.saWiroa gansakuTrebiT aRvbiSnoT, rom, traeqtoriebi romlebicganTavsebulia atraqtorebze da repelerebze, ikaveben nulovanfazur moculobas. aqedan gamomdinareobs rom atraqtorebis darepelerebis ganzomilebani yovelTvis naklebia sawyisi sistemisfazuri sivrcis ganzomilebebTan SedarebiT. es Tviseba ZalianmniSvnelovania mravalganzomilebiani arawrfivi dinamikuri obieqtebismarTvis problemebis gadawyvetisaTvis, radganac warmoiqmnebaerTmaneTSi Cawyobili sasurveli atraqtorebis sinTezisSesaZlebloba. aseT SemTxvevaSi SeiZleba ugulvebelyofili iqnesgardamavali movlenebi da yuradReba ZiriTadad unda mivaqcioTmoZraobis (asimptotur) reJimebs, rasac mivyavarT arawrfiviobieqtebis sinTezis problemis arsebiT gamartivebamde.Yufro dawvrilebiTatraqtorebis Tvisebebi ganxilul iqna Semdeg TavSi.amrigad, fizikur, qimiur da biologiur sistemebSi nebismieriprocesebi SeiZleba davyoT or arsebiT gansxvavebul klasad.pirvel klass ganekuTvneba procesebi izolirebul (TermodinamikuriazriT) sistemebSi. maT mivyevarT wonasworul mdgomareobamde,romelsac Seesabameba fizikuri qaosi. es procesebi mar-Tvadia garegani zemoqmedebis gziT. meore klass unda mivakuTvnoTprocesebi Ria sistemebSi, romelTa mimdinareobisas SeiZlebawarmoiSvas mowesrigebuli struqturebi – atraqtorebi, racdamaxasiaTebelia TviTorganizaaciis procesebisaTvis. amasTanerTad, mizanSewonilia SemoviRoT wesrigis xarisxis raodenobriviSefaseba-sistemis sxvadasxva mdgomareobebis TviTorga-30


nizacia, sistebis efeqturi struqturuli TviTorganizaciis gzebisasarCevad. mowesrigebuli struqturebi Ria sistemebSi warmoiSvebianmmarTveli parametrebis cvlilebis Sedegad, redesaces parametrebi ramodenimea, maSin cxadia gamovlindeba TviTorganizaciissxvadasxva gza.1.3 sinergetikis ZiriTadi cnebebirTuli sistemebis farTo klasis gamokvlevisas aRmoCeniliqna SesaniSnavi movlena-Tavisuflebis xarisxis mniSvnelovaniSemcireba, romelic efeqturad aRwers sistemis yofaqcevas moZraobisfinalur etapze. amasTan aRmoCnda, rom SesaZlebelia gamovyoTTavisuflebis ramodenime xarisxi _ wesrigis parametrebi,romlebsac bifurkaciis movlenis Sedegad moergebian sxva danar-Ceni cvladebi, romelTa raodenobac SeiZleba iyos sakmaod didi.zustad wesrigis parametrebiT, ganisazRvreba mravalganzomilebianwrfiv sistemebSi mimdinare procesebis dakvirvebis qveSmyofi dinamika. am faqtis aRmoCena SeiZleba mivakuTvnoT Tanamedrovearawrfivi mecnierebis fundamentalur aRmoCenas, rac gvaZlevssaSualebas gavakeToT didi nabiji disipatiur (disipatogafantva)sistemebSi moulodneli movlenebis SeswavlaSi.wesrigis parametrebi warmoadgens mravalkomponentur makrocvlads,romelic aRwers sistemis fazur gadasvlebs garkveulmowesrigebul moZraobaSi. is axasiaTebs komponentebs Soriskorelacias da rac ufro Zlieria korelacia, miT metia wesrigisxarisxi makrosistemaSi. mocemuli wesrigi sistemaSi warmoiqmnebaSesabamisi Sinagani da garegani Zalebis moqmedebiT misienergiis cvladebiT.Tumca wesrigis parametri asaxavs ara wesrigis warmoqmnissazoms sistemaSi raime fazuri gadasvlis Sedegad, aramed aRwersmikrosistemis safiniSo mowesrigebuli mdgomareobis umniS-31


vnelovanes Tvisebebs. Amas mivyavarT sistemis ganzomilebis mniSvnelovanSemcirebamde e.i. xdeba Tavisuflebis xarisxis maCvenebeliricxvis reduqcia. wesrigis parametrebi mWidrod aris dakavSirebulisistemis Sinagan struqturasTan, romelic warmoiqmnebamisi evoluciis Sedegad. arsebobs mWidro dinamikuri kavSirisistemis wesrigsa da simetrias Soris [11,12].gadavideT zemoT naxsenebi sinergetikis sabazo cnebebis ma-Tematikur aRweraze, arawrfivi sistemebis tipiuri, evoluciurigantolebebis tipiur magaliTebze, romelTac gaaCniaT spontanuriTviTorganizaciis unari. mokled ganvixiloT arawrfivisistemebis bifurkaciis Teoriis ZiriTadi cnebebi.arawrfivi dinamikuri sistemebis bifurkaciis Tvisebasswrafad ganviTarebadi Tanamedrove arawrfivi mecnierebisaTvisgaaCnia fundamentaluri mniSvneloba, romelic sul ufro da ufrometi xarisxiT aRwevs dargebSi - Ria sistemebis fizikidan,biologiidan da maTematikidan fsiqologiamde, ekonomikamde dasociologiamde. miTumetes es cneba mniSvnelovania rogorc gamoyenebiTimecnierebisa da teqnikis sxvadasxva sferoebisaTvis, asevemarTvis TeoriisaTvis.ganvixiloT diferencialuri veqtoruli gantolebaX&( t )n= F ( X , μ ), X ∈ R , μ ∈ R , μ ≤μμn, (1.11)romelic aRwers sxvadasxva obieqtebis qcevas da marTvis sistemebs.igi SeiZleba interpretirebuli iqnes rogorc romeliRacaavtonomiuri nakadi fazur sistemebSinR, romelic damokidebuliagarkveuli bifurkaciuli parametrebzeμk( k = 1,2,..., k < n). fizikuradam parametrebs SeiZleba warmoadgendnen magaliTad temperatura,moculoba, koncentracia da a.S. (1.11) nakadis stacionalurireJimi ganisazRvreba algebruli gantolebebis sistemebisamonaxsnebiTF ( x,μ)= 0,(1.12)μ32


omlebic wamoadgenen nakadis uZrav stacionalur wertilebs. na-Telia rom nakads SeiZleba qondes sxva amonaxsnebic _ periodulitorebi da a.S. naCvenebi amonaxsnebi parametrebis sivrceSi(1 kμ μ ,..,. μ ) SeiZleba warmovidginoT grafikis saxiT, romelic asaxavszemoT moyvanili gantolebas (1.12). igi iwodeba bifurkaciuliparametr μ cvladebis mimarT amonaxsnis Stod. maSin wertilsparametrebis sivrceSi, romlisganac iSlebian Stoebi, ewodebasistemis bifurkaciis wertili. Pparametrebis sivrcis umciresiganzomileba, romelSic SesaZlebelia bifurkacia, iwodebaSesabamisi tipis koganzomilebis bifurkaciad. bifurkaciis TeoriaSikoganzomilebisMcnebis SemotaniT saSualeba gveZleva movaxdinoTabstraqcireba sivrcis parametrebis konkretuli ganzomilebidanda gamoviyenoT mxolod Sesabamisi sivrcis geometriuliTvisebebi.ganvixiloT bifurkaciis tipiuri saxeebi:bifurkacia ,,unagira kvanZi”. aseT SemTxvevaSi sistemisnormalur gantolebas aqvs Semdegi saxe.x ( t ) = μ − x2, (1.13)nax. 1.1 bifurkacia ,,unagira kvanZi”.33


misi stacionaluri amonaxsnix = ± μ gansazRvrulia roca μ >0sda pirvelad warmoiqmneba pirobisas μ = 0 . roca μ < 0 amonaxsnebiar arsebobs. nax.1.1-ze naCvenebia (1.13) gantolebis Sesabamisiμ bifurkaciuli diagrama. rogorc nax.1.1-dan Cans bifurkaciis( x = 0, μ = 0) wertilidan gamodis stacionaluri mdgomareobisori Sto, romelTaganac erTi mdgradia (mTliani wiri), xolomeore aramdgradi_(wyvetili wiri) [13].transkritikuli bifurkacia. aseT tipis bifurkaciisaTvissistemis normalur gantolebas eqneba Semdegi saxe:.x ( t ) = μ x − x2(1.14)(1.14) gantolebas gaaCnia ori stacionaluri amonaxsni x s= 0 dax = μ . pirveli amonaxsni mdgradia roca < 0sμ da aramdgradia,roca μ > 0 meore amonaxsnisaTvis-piriqiT. orive amonaxsni mdgradobascvlian bifurkaciis wertilSi. nax.-ze 1..2 naCvenebia Sesabamisibifurkaciuli diagrama. normaluri gantoleba.x( t)= μx− x3(1.15)nax. 1.2 bifurkaciuli diagrama.34


nax. 1.3 bifurkaciul diagramis simetriuli formaam saxis bifurkaciisaTvis arsebobs sami stacionaluri amonaxsnixs= 0daxs= ±μ . Sesabamis bifurkaciul diagramas gaaCnia simetriuliforma da warmodgenilia nax. 1.3-ze [13]andronov-hopfis bifurkacia. SedarebiT am rTul SemTxvevaSinormalur gantolebas aqvs Semdegi saxe:z2z& ( t)= ( μ + iy)z − z z(1.16)sadac kompleqsuri cvladia, xolo y - romeliRac konstantaa,romelic ar TamaSobs bifurkaciuli parametris rols,i − aris warmosaxviTi erTiani. gantoleba (1.16) warmoadgens ,,Canglis’’(1.15) saxis kompleqsur bifurkaciis analogs. imisaTvisrom moZebnil iqnes (1.16) gantolebis stacionaluri amonaxsnebi,mizanSewonilia gadavideT namdvil marTkuTxa an polarul cvladebze,maSin, Tu davuSvebT, romz = x 1+ ix 2 , (1.17)miviRebT namdvili koordinatebian evoluciuri gantolebis normalurformas:x& [ 2 ] 2121( t)= μ − ( x + x ) x1− yx2x ( t)= yx + [ μ−(x 1 2 ] 2 + x2 x2& . (1.18)2 1)35


Eevoluciuri gantolebebi (1.8) dakavSirebuli arian erTmaneTTan(1.17) kompleqsuri cvladis saSualebiT da gaaCniaT Semdegiori saxis stacionaluri amonaxsnix = x 0 Tu z = 0(1.19)1 2=22122z = x + x = μ(1.20)nax. 1.4 andropov-hopfis bifurkaciapirveli amonaxsni (1.19) warmoadgens aramdgrads da emTxvevabifurkaciis wertilebs, xolo meore amonaxsni (1.20) gansazRvravswrewiris rkals radiusiTμ . nax. 1.4-ze gamosaxulia Sesabamisibifurkaciuli diagrama, romelzedac isrebi uCveneben sistemisveqtoruli velis Zalovani wirebis mimarTulebas.bifurkaciuli gantolebebis (1.13)–(1.16) normaluri formebsuwodeben superkritikuls. es niSnavs rom Sesabamisi gantolebebisarawrfivi wevrebi x da x axdenen uwonasworobis aram-23dgradobis Tvisebis sawinaaRmdego zemoqmedebaze, romlebic aRiZvrebianufro dabali rigis wevrebis moqmedebis Sedegad. Tumcaarawrfivi wevrebis win niSnis Secvlisas isini ukve ganaxorcielebenmadestabilizirebel zemoqmedebas sistemaze. aseT Sem-TxvevebSi warmoiqmnebian subkritikuli an ukubifurkaciebi.36


zemoT aRwerili bifurkaciebi yvelaze ufro gavrcelebuliasxvadasxva bunebis mqone arawrfiv mravalganzomilebian dinamikursistemebSi. rasakvirvelia es bifurkaciebi ar moicaven ase-Ti tipis arawrfivi movlenebis mravalsaxovnebebs.bifurkaciuli diagramebi asaxaven arawrfivi dinamikuri sistemebisstatikur mdgomareobebs bifurkaciuli (mmarTveli) parametrebiscvlilebisas. sainteresoa sistemebis bifurkaciul mdgomareobazegadasvlis procesebis Seswavla drosa da sivrceSi.amasTan dakavSirebiT ganvixiloT dinamikuri procesebis iseTiarawrfivi sistemebis magaliTebi, romelTa evoluciuri gantolebebia“Cangali” da “andropov-hopfis bifurkacia’’[15].,,Canglis” tipis bifurkacia. maTematikuri TvalsazrisiTDdaqvemdebarebisprincipi sinergetikaSi eyrdnoba adiabaturimiaxlovebis meTods. an sxva sityvebiT, sawyisi sistemebis dayofisideaze nel da swraf qvesistemebad. amasTan erTad, xorcieldebacvladebis adiabaturi gamoricxvis procedura. daqvemdebarebisprincipis garda, sinergetikisaTvis Zalze mniSvnelovania,agreTve, wesrigis parametris cneba. am cnebebis arsi ganvixiloTmeore rigis arawrfivi sistemebis konkretul magaliTze, romelicaRiwereba Semdegi saxis gantolebiT [10]:& ( t)= λ Χ − ΧΥΧ1Υ&( t)= −λΥ + Χ22(1.21)sadac λ ≥ 10 ; λ2>0. aseTi diferencialuri gantolebebiT aRiwerebarigi procesebi fizikaSi, qimiaSi, ekologiaSi da a.S. vivaraudoTrom koeficienti λ 1Zalian mcirea da λ 2>> λ 1. Tu cvladebi Χda Υ mcireebia, maSin SeiZleba kvadratuli forma ΧΥ ugulebelvyoT,maSin Χ cvladi Seicvleba Zalze nela. Mmeore gantolebidanCans, rom nazrdi Υ ganisazRvreba wevriT da radganaccvladi Χ icvleba Zalian nela, unda velodoT, romΥ -icSeiZleba sakmaod nela Seicvalos. radganac λ 2>0 da bevrad metiaλ 1-ze, amitom warmoebuliΥ & (t)SeiZleba ugulebelvyoTλ 2Υ37


sididesTan SedarebiT. Camoyalibebuli analizi arsebiTad aRniSnavs,rom sistema SeiZleba warmodgenil iqnes, rogorc neli sistemaaRwerili pirveli gantolebiT da swrafi sistema aRwerilimeore gantolebiT. neli da swrafi qvesistemebis yofaqcevis cvalebadobaganisazRvreba gardamavali procesebiT, romelTa xangrZliobac(Sesabamisad τ 2da τ 1) SeiZleba Sefasdes utolobiT:τ ~ λ 10 , λ > 0 2mniSvnelobebisTvis gaaCnia erTi trivi-2yaluris ±xs= 0 da ori specifikuri amonaxsni x = ± λ 1λ2. es amonaxsnebierwymian x = 0 , roca λ 0 da ganStovdeba misgan, ro-sca λ > 10 . es iseTi movlenaa, roca icvleba gantolebaTa amonax-1 =38


snebis ricxvi (an mdgradoba) zogierTi mmarTveli parametrebiscvlilebisas. Amas maTematikaSi bifurkacias uwodeben. Mcneba bifurkacia(gaoreba) Semotanilia mecnierul terminologiaSi a.puankaresmier da ganekuTvneba umniSvnelovanes Zireul cnebas ma-Tematikasa da saerTod arawrfivi mecnierebaSi.aq ganxilul amocanaSi mmarTvel parameters warmoadgens λ 1,romlis variaciac iwvevs diferencialuri gantolebis (1.22) yofaqcevisxarisxobriv cvlilebas.nax. 1.5-ze gamosaxulia ,,bifurkaciuli diagrama”, romelicasaxavs stacionaluri amonaxsnebis funqcionalur damokidebulebasλ 2 mmarTveli parametris cvalebadobisas. (1.22) diferencialurigantolebas gaaCnia zusti amonaxsnix2λ1λ2x0( t)±22 −2λ1tx0+ ( λ1λ2− x0) e= . (1.24)nax.1.5 bifurkaciuli diagrama(1.24)-dan gamomdinareobs, rom roca λ < 10 da λ > 0 erT-2aderTi amonaxsnix s= 0aris asimptoturad mdgradi mTelSi, (nax.1.5. xazi a), e.i → ∞ pirobis dros nebismieri sawyisi x pirobe-t039


isaTvis. roca λ > 10 , λ > 0 2warmoiqmneba ukve ori amonaxsnix = + λ 1λ 2da x = − λ 1λ2, romlebic asimptoturad mdgradebia,s ±s ±ara mTelSi, aramed maTi mniSvnelobebis garkveul areSi. sxvanairadrom vTqvaT Stoebi xs±= λ1λ2, (roca λ2= const ) warmoiSoba1±bifurkaciis Sedegad im momentSi roca mdgomareobax s= 0kargavsmdgradobas, xolo da b Stoebi asimptoturad mdgradebia(nax 1.5). (1.22) modelis aRniSnuli Tvisebebi dakavSirebuliastatikuri mdgomareobis (1.23) gantolebasTan, romelic mdoredaris damokidebuliwertilis midamoSib12λ1parametrze ( λ2 = const ) . Tumca bifurkaciisλ1 e= 0 warmoiqmneba gansakuTrebuloba, romelicmdgomareobs imaSi rom λ 0 wertilis SemogarenSi amonaxsni1 e=xs±, ar SeiZleba daSlili iqnes xarisxobriv mwkrivSi λ1parametrismixedviT. sxva sityvebiT, (1..23) gantolebis amonaxsnixs±ar aris damokidebuli parametr λ1-ze analizurad. aseTi saxisgansakuTrebuloba warmoadgens (1.22) modelis axali xarisxobriviyofaqcevis maTematikur asaxvas, romelic ganpirobebulia,,Canglis” tipis bifurkaciis TvisebiT [13]. CavweroT exla(1.22) gantoleba υ potencialuri funqciis saSualebiT∂υ( x1λ1λ2)x&( t)= −∂xmaSin (1.22) gantolebis mdgradobis analizi SegviZlia gavakeToTSemdegi saxis potencialis daxmarebiTυ((x)λ 1= −x1 242 x + . (1.25)4λ2υ (x) funqciis safuZvelze SeiZleba movaxdinoT (1.21) gantolebisamonaxsnebis saerTo klasifikacia stacionaluri mdgomareobebismdgradobis cvlilebis wertilis moZebnis saSualebiT,sadac = 0 . am tipis klasifikacia dakavSirebulia (1.21)∂υ ∂xgantolebis struqturuli mdgradobis dakargvasTan, radgan wer-40


∂υtilebSi = 0∂xxdeba mdgradobis xasiaTis cvlileba. λ qda λ 2parametrebis mniSvnelobebze damokidebulebiT, am wertilebSiSeiZleba amonaxsnebs hqondes ganStoeba an potenciali υ( x , λ , 1λ )aRwevs minimums minimum or wertilSi mainc. sxvagvarad romvTqvaT, am wertilebSi xdeba sistemis dinamikis xarisxobrivicvlileba. (1.25) gamosaxuleba aRwers ,,potenciur ormo”-s, romlisminimumic asaxavs mdgrad mdgomareobas. (1.25) -dan gamomdinareobs,rom roca λ < 10 , λ > 0 2arsebobs erTaderTi mdgradimdgomareoba x = 0 nax 1.6 ( λ < 0)roca λ ≥ , λ 0 potencialis110 2>(1.25) forma icvleba parametris kritikuli mniSvnelobis gavlisasλ = 10 . nax.1.6-ze ( λ > 0 1) gamosaxulia ,,potencialuri ormo “υ(x)SemTxvevisaTvis roca λ 1> 0,λ 2> 0.am dros stacionalurimdgomareobebis ricxvi gaxda samis toli, radganac (1.23) gantolebisyvela sami fesvi namdvilia, Tumca wina ( λ < 0, > 20)1λmdgradi mdgomareoba x = 0 xdeba ukve aramdgradi da wertiligadadgildeba oridan erT Tanabrad SesaZlebel mdgomareobaSix = + λ 1λ 2an x = − λ 1λ2. maSin gantoleba (1.21) SeiZleba warmovidginoTSemdegi saxiT:1x& ( t ) = − x( x − λ1 λ2)( x + λ1λ2) (1.26)λ2± λ 1λ 2fesvebi (1.26)-Si damokidebulni arian λ 1,,mmarTvel parametrze”,rac ganapirobebs or mdgrad mdgomareobas. wertili2x s= 0,romelic imyofeba aramdgrad mdgomareobaSi Zalian mcireagznebebis da fluqtaciebis zemoqmedebis Sedegad, romelic yovelTvisarsebobs realur sistemaSi, gadava oridan erT-erT SesaZlomdgrad mdgomareobaSi (nax. 1.6, λ > 0 1). rac niSnavs rom,marto erTi wrfivi aramdgradobiT ,,pataraSi” sruliad sakmarisia,rom warmoiSvas sistemis araprognozirebadi yofaqceva, romelicSeiZleba davakavSiroT arawrfivi sistemebis stoqastikurobasTan.Uufro meti TvalsaCinoebisaTvis, SeiZleba warmovidginoT41


urTula, romelic Camocurdeba ,,larnakis” υ (x)kedelze, maSinrodesac ,,larnaks” aqvs (1.6) naxazze naCvenebi forma λ < 10 wiriT,maSin burTula aucileblad SeCerdeba x = 0 wertilSi, Tuki,,larnaks roca λ > 0 , maSin burTela SeiZleba SeCerdes oridanerT-erT wertilSi xs= + λ 1λ2an xs= − λ 1λ2.snax. 1.6 struqturuli mdgradobis grafikiam SemTxvevaSi( λ 1> 0)mTavar rols TamaSobs fluqtaciebi. vivaraudoT,rom burTula Tavdapirvelad imyofeboda wertilSix s= 0, maSin marjvena an marcxena ormoebSi burTulis moxvedrisSesaZlebloba damokidebuli iqneba mcire fluqtuciebze, romlebicmoqmedeben sistemaSi.arawrfiv sistemaSi mmarTveli parametrebis Secvlisas SeiZlebawarmoiSvas sxvadasxva gardamavali movlenebi, romlebic analogiuriafazuri gadasvlebisa da damaxasiaTebelia mraval fiziko-qimiurisistemisaTvis. am movlenebs axasiaTebT zogierTisaerTo Tvisebebi, romelTaganac ZiriTadi mdgomareobs SemdegSi.(1 2>parametrebis λ < 0, λ 0)zogierTi mniSvnelobebisaTvis sistemaSiSesaZlebelia arsebobdes mxolod erTaderTi Sto (amonaxsni),romelic xasiaTdeba mniSvnelovani TvisebiT-asimptoturimdgradobiT (wiri a nax.1.5). am SemTxvevaSi sistema aqrobs42


Sinagan fluqtaciebs an SezRudul garegan aRSfoTebebs. xolo1 e( λ 1c=mmarTveli parametrebis λ 0)garkveuli kritikuli mniSvnelobebzegadasvlisas mdgomareoba am Stoze (wiria′ nax. 1.5) xdebaaramdgradi da maSin ar xdeba fluqtaciebis Caqroba. sistemismoqmedebis gaZlierebis analogiurad, daiwyebs gadaxras normalurimdgomareobidan da Semdgom gadava axal reJimSi. naCvenebiori reJimi erwymis erTmaneTs mmarTveli parametrisλ1 c= 0 kritikulimniSvnelobisas da gansxvavdebianλ > λ 1 cmniSvnelobisaTvis,aseT movlena warmoadgens bifurkaciis movlenas, romelicdakavSirebulia sistemis katastrofiul cvlilebebTan. gansazRvrulgansakuTrebuli gadasvlis momentSi, romelic mdebareobsλ1= λ 1 csiaxlovis SemogarenSi sistemas SeuZlia ganaxorcieloskritikuli arCevani [10,20]sistemis aRwerili arCevanis procesebis Sedegad warmoiqmnebaerTgvari mimzidavi struqtura-atraqtori, da Sesabamisadxdeba misi TviTorganizacia.amrigad, evoluciuri erTcvladiani modelis (1.22)-is safuZvelze,romelic aRwers (1.21) gantolebis ,,wesrigis parametris”qcevas, dgindeba rom warmoiSoba TanabradaralbaTuri warmoSobismravali (Cven SemTxvevaSi ori) amonaxsnisa, romlebicerTdroulad floben asimptoturi mdgradobis Tvisebas. naCvenebimovlena uSualod asaxavs arawrfivi sistemebis gadaxrisTvisebas, rac Seesabameba amocanebis amoxsnas. meores mxriv, aRwerilievoluciuri modelis ,,martivi” Tviseba (1.22) miuTiTebsrTul yofaqcevaze. es adasturebs imas rom erTi cvladian sistemaSi,romelic Seicavs ufro maRali rigis xarisxobriv arawrfivobas,SeiZleba warmoiqmnas ufro rTuli gardamavali movlenebi.mTavaria, rom es dabali ganzomilebis ,,martivi” arawrfivimovlenebi unda aRwerdnen mravali klasis mravalganzomilebianfizikuri sistemebis yofaqcevas maTi moZraobis safiniSoO etapebze,anu eqsperimentalurad aRweren rTul dakvirvebad gardamavalmovlenebs.43


sinergetikis evoluciuri gantolebebi andronov-hopfisbifurkacia. zemoT ganxilul SemTxvevaSi x ( t)`wesrigis parametris~yofaqceva aRiwereba pirveli rigis (1.22) diferencialurigantolebiT da Sesabamisad, (1.21) sawyisi sistebis evolucia finaluretapze faqtiurad mimdinareobs erTganzomilebian fazursivrceSi, rac niSnavs, rom am sivrceSi SesaZlebelia arsebobdesmimzidveli invariantuli simravlis–atraqtorebis mxolod erTitipi uZravi wertilebis ( xs= ± λ1⋅ λ2) saxiT, romelsac Seesabamebastacionaluri mdgomareobebi.aseTi saxis disipatiuri sistemebi icvlebian droTa ganmavlobaSi.ganvixiloT organzomilebiani evoluciuri gantolebebistipiuri magaliTi.davuSvaT, rom rTuli sistemis etapze `wesrigis parametrebis~mimarT yofaqceva misi miZraobis finalur aRiwereba puankaresSemdegi evoluciuri gantolebebiT [10,15]:2 22 2x& 1() t = ( λ −x1 −x2)x1−ωx2x2 t = ωx1+ ( λ−x1 −x2)& () x2(1.27)SemovitanoT Semdegi saxis aRniSvna: x 1= rcosϕda x 2= rsinϕe.i. polaruli kordinatebis r da ϕ SemotaniT (1.27) gantolebaCaiwereba Semdegi saxiT:rt &()= λr−r3(1.28)& ϕ()t = ω(1.29)sadac λ da ω - parametrebia. (1.29)- dan uSualod gamomdinareobs:& ϕ()t = ϕ + ωt. (1.30)gantoleba (1.28) zustad emTxveva adre gamokvleul (1.22) gantolebasλ = λ1da λ 2= 1 pirobebis dros. iseve rogorc (1.22) gantolebisaTvis,gvaqvs Semdegi stacionaluri mdgomareobebi rs= 0da r s0= λ (λ>0). am SemTxvevaSi am or amonaxsns r , ϕ sistemis fazursivrceSi (1.28), (1.29) Seesabameba kordinatebis saTave r=0 ϕ=0da egreTwodebul `zRvruli cikli~-atraqtori wrewiris saxiTcentriT kordinatTa saTaveSi radiusiTλ , romelzedac sis-44


tema moZraobs ω siCqariT. naTelia, rom roca λ0stacionaluri mdgomareoba r=0, ϕ=0 xdeba aramdgradi. (1.30) gamosaxulebisgaTvaliswinebiT vpoulobT (1.27) – gantolebis gardamavalamonaxsnsebs:x1() t = r()t cos( ϕ0+ ωt)da x2() t = r()sin( t ϕ0+ ωt)(1.31)sadac funqcia r(t) ganisazRvreba (1.24) tipis damokidebulebidan,roca sruldeba piroba r(t)>0 e.i.rt () =λr202 20+ −0( λ )− tr r e 2λ(1.32)(1.31) da (1.32)–dan gamomdinareobs Semdegi stacionaluriamonaxsnebix () t = λ cos( ϕ + ωt)da x 2() t = λ sin( ϕ + ωt). (1.33)1s0s 0(1.31), (1.32) gardamavali amonaxsnebi gviCveneben, rom nebismieritraeqtoriebi, romlebic gadian fazur sibrtyis (nax. 1.7)nebismier wertilebSi, sistemis fazur sivrceSi garduvalad`garsSemoexvevian~ Signidan Tu garedan `zRvrul cikls~-harmoniulatraqtors. es mtkicdeba (1.33)-is stacionaluri amonaxsnebiT,romlebic aRweren harmoniul rxevebs da Sesabamisad fazur sibrtyeSiSekrul wriul traeqtoriebs.)nax 1.7 zRvruli cikli - harmoniuli atraqtori45


axla ganvixiloT sistemis Sekruli traeqtoriis mdgradobadamyarebul reJimSi pirobiT r s=± λ . amisaTvis traeqtoriasmivaniWoT mcire nazrdi. e.i. davuSvaT romr = λ + Δrda CavsvaTigi (1.28) gantolebaSi. maSin miviRebT:3/2 3 2( ) ( )Δ rt &() = λ λ +Δr − λ +Δ r + 3 λΔ r + 3λΔr(1.34)vinarCunebT ra (1.34) gantolebaSi wrfiv wevrebs Δr vpoulobTdiferencialur gantolebas:3/2 2( ) ( )Δrt &() ≈ λ λ +Δr − λ + 3 λΔ r =−2λ Δr(1.35)romelis amonaxsnsac aqvs saxe2rt () re − λt0Δ =Δ . (1.35)-dan gamomdinareobsrom Sekruli traeqtoriars= λ asimptoturad mdgradiaradganac radiusi r mcirdeba roca Δ r 0> 0 -ze da izrdeba rocaΔ r 0< 0-ze. es niSnavs, rom zRvruli cikli warmoadgens asimptoturadmdgrad wriul traeqtorias romliskenac garduvaladmiizideba sistemis yvela traeqtoria aris gansazRvrul midamo-Si. xazi unda gaesvas, rom mocemuli traeqtoriebi ( rs= λ ) araris damokidebuli (1.28) (1.29) sistemis sawyisi mdgomareobaze.sxvagvarad rom vTqvaT, Sekruli traeqtoriebis moZraobis periodiganisazRvreba mxolod sistemis Sinagani parametrebiT ( λ, ω).amasTan, am dros sistema moZraobs droSi zRvrul ciklze –atraqtorze mxolod erTi mimarTulebiT. SevniSnoT, rom evoluciurigantolebis (1.22) tipiur formamde SegviZlia daviyvanoTdiferencialuri gantolebaTa sistemac (1.28), (1.29). imisaTvis SemoviRoTkompleqsuri cvladi ( z = x1+ ix 2) maSin gantolebebi (1.27),(1.28) da (1.29) miiReben saxes:2zt &() = ( λ+ iω)z−zz(1.36)romelic Tavisi struqturiT warmoadgens (7.22) evoluciuri gantolebiskompleqsur eqvivalents. (1.36)-is gantoleba amonaxsniSegviZlia warmovadginoT formiT()( zt () = rte () iϕt ).46


an zt () rt ()cos [ ϕ() t isin ϕ()t]= + (1.37)sadac funqcia ϕ () t da rt () ganisazRvreba (1.30) da (1.32) gamosaxulebebiT.(1.36) CavsvaT (1.37)-Si da gavaTanabroT namdvili dawarmosaxviTi nawilebi miviRebT kvlav (1.37) gantolebas. stacionalurireJimisaTvis, rodesacgaTvaliswinebiT Rebulobs saxes:rs[ ]0 0= λ , (1.36)-is amonaxsni (1.30)z () t = λ cos( ϕ + ωt) + isin( ϕ + ωt)(1.38)sstacionaluri amonaxsni (1.38) gviCvenebs rom sistema asrulebsdamyarebul harmoniul rxevebs da amiT mtkicdeba, rom sistemisfazur sivrceSi arsebobs zRvruli cikli-harmoniuli atraqtori.amrigad, (1.28), (1.29) disipatiuri sistemas gaaCnia mimzidavisimravle-harmoniuli atraqtorebi. masze moZraoba ar aris damokidebulisawyis pirobebze (sistemis traeqtoria exveva zRvrulcikls Signidan an garedan) da gamovlenilia drois mimarTuleba.zemoT aRwerili movlena warmoadgens axali tipis bifurkaciiswertils (roca λ=0), romelsac maTematikur literaturaSiuwodeben andropov-hopfis bifurkacias. sworedac organzomilebianevoluciur gantolebebSi am dinamikuri Tvisebas mivyevarTperiodul qcevamde. amasTan λ parametris misi bifurkaciuli(λ c =0) mniSvnelobidan gadaxrisas mdored izrdeba perioduliamonaxsnis amplituda. Zalze mniSvnelovan garemoebaswarmoadgens is faqti, rom damyarebuli harmoniuli rxevebis rogorcamplituda, iseve periodi zRvrul ciklze ganisazRvrebasistemis evoluciur gantolebebis λ da ω sakuTrivi parametrebiT.disipatiuri sistemebis rxevis es Tviseba gansxvavdebakonservatiuli sistemebis rxevebis Tvisebebisagan, romelTamaxasiaTeblebi damokidebulia sawyisi pirobebze. konservatiulisistemebi, romelTac miekuTvnebian kvanturi da klasikuri meqanikisZiriTadi gantolebebi, warmoadgenen liapunovis mixedviTmdgradebs, magram ara asimptoturad mdgradebs. e.i. isini mgrZnobiareniarian fluqtaciebisadmi. amave dros disipatiuri siste-47


mebi andronov-hopfis bifurkaciiT asimptoturad mdgradebi arianzRvrul ciklTan mimarTebaSi da Sesabamisad mcired mgrZnobiareniarian fluqtaciisadmi da garegani zemoqmedebisadmi.zogad SemTxvevaSi, rTul mravalganzomilebian sistemebSiSeiZleba iyos ramodenime wesrigis parametri (x 1 , x 2 ......), romelicxSirad mcire ricxvia, arsebiTad mcire vidre sistemis sawyisganzomileba. am koleqtiur cvladebs – wesrigis parametebs (miesadageba)sxva cvladebi, romlebic SeiZleba gamovricxoT makroskopiuliqcevis gamokvlevisas. swored evoluciuri gantolebebismcire ricxvi gvexmareba gamovavlinoT wesrigis parametrebida gamovikvlioT sawyisi mravalganzomilebiani sistemebisTvisebebi. aRwerili Tvisebebis mqone sistemebis ganmasxvavebelTaviseburebas warmoadgens sawyisi pirobebis `daviwyeba~da uwonasworo struqturebis formireba. swored uwonasworobaSeiZleba gaxdes mizezi mowesrigebulobisa e.i dinamikuri arawrfivisistemis TviTorganizebisa.unda aRiniSnos, rom TviTorganizaciis procesebs gaaCniaTspontanur xasiaTi, romelic warmoiqmneba arawrfiv sistemebSimaTi ganzomilebebis moaxloebuli reduqciasTan da mmarTveliprocesebis SemTxveviTi Seecvlisas: TviTorganizaciis mizezobrivixerxi, aRmoCenaa romelmac mogvca saSualeba aRmogveCinaaraCveulebrivi sxvadasxva bunebis kooperatiuli movlenebi.bifurkaciebi da qaosi. arawrfivi sistemebis konservatulobisda disipatiurobis Tvisebebis kombinaciam migviyvana maTiqcevis or gansxvavebul tipamde. pirvel SemTxvevaSi es mowesrigebulida regularuli moZraobebia, romlebsac miekuTvnebianumetesoba procesebisa, romlebic mimdinareoben teqnikur da rigbunebriv sistemebSi. SeiZleba sakmaod safuZvlianad vivaraudoTqcevebi am sistemebSi, Tu viciT maTSi moqmedi kanonebi dazemoqmedebebi.arawrfiv dinamikur sistemebSi sxva farTod gavrcelebuliprocesebi miekuTvneba qaosurs. arc ise didi xnis win mecnierebifiqrobdnen rom rTuli qaosuri moZraoba SeiZleba warmoiqmnas48


mxolod zogierT mravalganzomilebian uamravi cvladebis SemcvelsistemebSi. Tumca, srulad moulodnelad aRmoCnda, romarawrfiv determinirebul mxolod mesame rigis sistemebSi maTfazur sivrceSi SeiZleba vixiloT Zalian rTuli qaosuri moZraobebi.magaliTisaTvis ganvixiloT Semdegi diferencialurigantolebebaTa sistema:x&() t = σ y−σxyt &()= − y+ rx−xz(1.39)zt &()= − bz+xysadac σ, r, b – mudmivi parametrebia(1.39) gantoleba warmoadgens lorencis cnobil models,romelic aRwers mravalferovan bunebriv procesebs. (1.39) sistemasmis fazur sivrcis raime simravleze gaaCnia fraqtaluriganzomilebis `ucnauri~ atraqtori. am simravleze sistemas gaaCnia,agreTve, sawyisi pirobebisadmi gazrdili mgrZnobiaroba,romlis Sedegedac warmoiqmneba moZraobis qaosuri reJimebi.gamovikvlioT lorencis maTematikuri modelis Tvisebebi[10,15,20]. Tavidan ganvixiloT misi stacionaluri mdgomareoba,rodesacx& () t = y& () t = z&() t = 0s s sxssmaSin (1.39) sistemidan gvaqvs= y , y − rs + x z = 0 , xy − bz = 0(1.40)s s s s(7.40)-is gaerTianebiT vpoulobT3xsb r x ss s s+ (1 − ) = 0(1.41)naTelia, rom (1.41)-Si SesaZlebelia Semdegi stacionaluri mdgomareobebia) x = 0, y = 0, z = 0(1.42)s s sb) x = y =± b( r− 1), z = r− 1(1.43)s s s(1.42) da (1.43)-is stacionaluri mdgomareobebis mdgradobisSeswavlisaTvis ganvixiloT lorencis modelis wrfivi miaxloeba.am SemTxvevaSi kvadratuli wevrebi SeiZleba ugulvebelvyoT,maSin (1.42) mdgomareobisaTvis miviRebT gantolebebis sistemas49


x&() t = σ ( y−x)y&() t = rs−y(1.44)zt &()=−bz(1.44)-dan gamomdinareobs rom mesame gantoleba ar arisdakavSirebuli pirvel orTan, xolo komponentizt () = ze −bt0miilevarodesac (z→0) radganac parametri b>0, komponentebis x(t) day(t) gansazRvrisaTvis saWiroa maxasiaTebeli gantolebis2λ ( σ 1) λ σ(1 r)0+ + + − = (1.45)amonaxsna.rodesac r1-ze, maSin fesvebidan erT-erTi xdeba dadebiTi damdgomareoba (1.42) wrfivad aramdgradia. cxadia, rom r c =1 mniSvnelobawarmoadgens wrfivi mdgomareobis sazRvars da atarebsbifurkaciis wertilis saxelwodebas [7.14, 7.20].ase, rom (1.42) wrfivad mdgradia roca 0≤r≤1 da aramdgradiaroca r>1-ze. sinergetikaSi r parametrs hqvia mmarTveli parametri.gamovikvlioT exla (1.43)-is stacionaluri mdgradoba wrfivimdgradoba, romlisaTvisac gawrfivebul sistemas aqvs saxex& () t = σ ( y−x),yt &() = x− y− br ( −1)z(1.46)zt &() = − br ( − 1)( x+ y)−bz.(1.46) sistemis maxasiaTebeli gantolebas eqneba saxe:−σ−1 σ 01 −λ−1 − br ( − 1) = 0br ( −1) br ( −1)−b−λ(1.47)+ ( + b+ 1) + b( + r) + 2 b ( r− 1) =03 2λ σ λ σ λ σroca r>1, (1.47) gantolebis fesvebis namravli arisuaryofiTi ricxvi, es ki rogorc viciT algebridan, niSnavs romsul mcire erTi mainc fesvebidan erTi namdvilia da uaryofiTi,xolo danarCeni ori namdvilia erTi niSniT an kompleqsuradSeuRlebulia. ismis kiTxva: ra mosdis sistemis (1.46)-is staci-50


onalur mdgomareobas (1.43) mmarTveli parametris r>>1 Semdgomizrdisas? cnobilia, rom (1.46) mesame xarisxis wrfivi sistemisaramdgradoba damokidebulia pirobaze, rom (1.47) gantolebiskompleqsur-SeuRlebuli fesvebi arian xSirad warmosaxviTi,radganac 2bσ(r-1)>0. algebridan cnobilia, rom (1.47) gantolebisfesvebis warmosaxviToba SeiZleba uzrunvelyofil iqnes maSin,rodesac misi koeficientebis warmoebulebi toli iqneba Tavisufaliwevrisa (roca λ + da λ). e.i. Tu sruldeba pirobab( σ + b+ 1)( σ + z) = 2 bσ(r− 1) , maSin aqedan vpoulobT' σ ( σ + b + 3)rc=(1.48)σ −b−1gamosaxuleba (1.48) gansazRvravs mmarTveli parametris r-iskritikul mniSvnelobas, romlis drosac (1.43) stacionalurimdgomareoba xdeba wrfivad aramdgradi. cxadia, rom roca σ


sadacμ =−σ + 1σ ( r −1)mcire parametria.(1.50) sistemaSi pirveli gantoleba aRwers arawrfivi oscilatorisrxevas20q −1ω =sixSiriT, romelic mdored icvleba meoregantolebis q(t) amonaxsnis Sesabamisad. mcire parametri μdidi r-ebis dros SeiZleba gavutoloT 0-s, maSin q=const dapirveli gantoleba warmoadgens arawrfiv osciliators. gamokvlevebigviCveneben [15], am SemTxvevaSi (1.39) lorencis modelisfazur sivrceSi warmoiqmneba zRvruli cikli. sur 1.8-ze naCvenebiamisi fazuri portreti roca r=315, σ=10, b=8/3 romlebic amtkicebenzemoT Camoyalibebuli mosazrebebs perioduli moZraobiswarmoSobis Sesaxeb.qaosurobis warmoSobis mizezebis TvalnaTeli warmodgenisaTviskvlav gardavqmnaT (1.39) lorencis gantoleba. CavsvaT1 y = x + x & () t cvladi pirveli gantolebidan da cvladiσmeore gantolebaSi, maSin miviRebT [21]:1 z = ( xy − &z )b21 1+ σ −x1 3 xx() t = F =− x() t + ( r−1) x− x + z()t&& & & & (1.51)σσb bnax. 1.8. zRvruli ciklis funqciis grafiki52


nax. 1.9 potencialuri fazuri portretiSemovitanoT potencialir −1 2 1 4V =− x + x , (1.52)2 4bromelsac gaaCnia sxvadasxva saxe roca r>1-ze da r1 warmoiSoba bifurkacia da warmoiSoba erTiaramdgradi (x s =0) da ori mdgradi mdgomareobaxs= ± b( r− 1)(1.53)potenciali (1.52)-is gamoyenebiT, (7.51)-i gantoleba exlaSegviZlia CavweroT Semdegi saxiT21 1+ σ −() x ∂&& x t =− x& () t − V +x &z()t , (1.54)σσ ∂xbromelic mosaxerxebelia xarisxobrivi analizisaTvis. miRebuli(1.54) gantoleba bolo wevris gareSe warmoadgens materialuriwertilisV ( x)potencialur ormoSi moZraobis gantolebas x(t)xaxunis ZaliT romelsac gaaCnia xaxunis koeficienti, romlis53


niSanic icvleba dadebiTidan uaryofiTisaken, rocax2≥ 1+ σ .(1.54) gantolebis bolo wevrs gaaCnia xisti Zalis forma, romlisdrekadobis koeficienti − 1 zt &()damokidebulia droze. rodesacbwarmoebuli z& () t mcire sididis ar aris, maSin es wevri warmoadgensgarkveul maiZulebel Zalas, romelic damokidebulia yda z cvladebze. Tu kavSirs z da x –s Soris ugulvebelvyofT,maSin bolo wevri SeiZleba gamoiyurebodes rogorc garkveuliSemTxveviTi Zala [21]. sxvanairad rom vTqvaT, materialuriwertili, romelic aRiwereba (1.54) gantolebiT SemTxveviT ZalismoqmedebiT imoZravebs orkuzian potenciur ormoSi (1.52), amasTanxaxunis koeficientma SeiZleba miiRos sxvadasxva niSani.warmodgenili mosazrebebi miuTiTeben lorencis modelisyofaqcevis, rogorc atraqtoris, rTul qaosur xasiaTze. nax. 1.8da nax.1.10-ze mocemulia moZraobis procesebi r parametrissxvadasxva mniSvnelobebisaTvis, rac adasturebs (1.39) lorencismodelis traeqtoriis rTul qaotur xasiaTs, romlebic aRiwerebamesame rigis determinirebuli diferencialuri gantolebebiT.dadgenili faqti adasturebs Tanamedrove arawrfivi dinamikisurTierTgansacvifrebel movlenas.nax 1.10 lorencis modelis fazuri traeqtoria54


ase rom, maRali rigisn ≥ 3determinirebul obieqtebzebifurkaciuli meqanizmebis moqmedebis Sedegad SesaZlebelia warmoiSvasrTuli qaosuri movlenebi. aseTi movlenebis maTematikurimodelebi Seicaven xarisxobriv da kvadratul arawrfivobas,romlebic gavrcelebulia sxvadasxva bunebis obieqtebSi. amasTanwarmoiSveba qaosis marTvis problema, romelsac eqceva sulufro meti yuradReba mecnierul literaturaSi.aqve ganvixilavT marTvis meTodebis gamoyenebi magaliTiarawrfivi obieqtisa bifurkaciiT, romelSic marTvis ararsebobisaswarmoiSvebian katastrofuli procesebi.ganvixiloT amocana romelime biologiuri populaciis optimalurdoneze SenarCunebisa, romelic, sakvebis mopovebis konkurenciisgaTvaliswinebiT, aRiwereba Semdegi saxis gantolebiT[59].2xt &() = α x−βx−μ,(1.55)sadac x populaciis konkretul saxeobaSi warmomadgenelTaraodenobaa; α,β - dadebiTi ricxvebi; μ - mmarTveli parametri.(1.55) gantolebiT SeiZleba aRvweroT, magaliTad, TevzWeris modeli.am SemTxvevisTvis μ warmoadgens TevzWeris kvotas (gegmas).Tavdapirvelad davuSvaT, rom μ=μ 0 winaswar mocemuli sididea.movZebnoT TevzWeris SesaZlo maqsimaluri kvota. amisaTvis (1.55)gantolebis marjvena mxare gavawarmooT x – iT da Semdegi gavutoloTnuliT, miviRebT:xmaxα= da2β2αμ0= (1.56)4βgamovikvlioT (1.55) gantoleba Tvisobrivad. amisaTvis TavdapirveladSeviswavloT misi stacionaluri mdgomareoba:αx− βx− μ = 0 . (1.57)(1.57)-dan ganvsazRvroT damokidebuleba xs( μ ):s2sx s=2α ± α −4β2βμ , (1.58)55


omlis grafikuli saxec, α=β=1 pirobis SemTxvevaSi, warmodgenilianax. 1.11-ze.xxmaxμ 0μnax. 1.11. stacionaluri mdgomareobis grafikixs( μ ) damokidebulebis grafiks gaaCnia ori toti, romlebicerTmaneTs erwymeba μ da x s im mniSvnelobisaTvis, romlebicgansazRvrulia (1.56) pirobiT. am movlenas uwodeben bifurkaciuls,xolo wertils, romlis koordinatebic gamoiTvleba (1.56)gamosaxulebiT, uwodeben sistemis bifurkaciul wertils. gamovikvlioTsistema am wertilis midamoSi amisaTvis SemoviRoT gadaxray = x− q0da CavsvaT potenciuri funqciis gamosaxulebaSi,romelsac Cveni SemTxvevisaTvis aqvs saxe:martivi gardaqmnebis Semdeg miviRebT moZraobis gantolebasVμ −1 1x x2 4β2= +20 0 04yt &() = αq − βq + ( α −2 βq)y−βy 2 −μ (1.59)sadac q 0 wertilis koordinataa.ukanaskneli gantolebis Tviseba damokidebulia μmmarTveli parametris mniSvnelobaze. SevirCioT optimaluri mni-Svnelobebi μ=μ 0 da q 0 =x max , romlebic uzrunvelyofen TevzWerismaqsimalur kvotas. Sedegad miviRebT gantolebas56


2& =−β(1.60)y()tromlis amonaxsnsac aqvs saxe:0yy0yt () =βq t+1(1.61)(1.61) gamosaxulebidan Cans, rom (1.60) gantolebis amonaxsnimdgradia (roca y=0) sawyisi pirobebisaTvis y 0 =x 0 -x max >0 da aramdgradia(roca y→0) pirobisaTvis y 0


nax. 1.12 sinTezirebuli sistemis gardamavali procesiamrigad, (γx) ukukavSiris SemotaniT marTvis kanonmasaSualeba mogvca gadavsuliyaviT bifurkaciidan (romelic momavalSigamoiwvevda katastrofas) logistikur gantolebaze, romelsacgaaCnia TviTorganizebis Tviseba. amasTan, SesaZlebeliaganvaxorcieloT TevzWeris mocemuli kvota, romelic SeiZlebaiyos maqsimaluric. ukukavSirSi γ koeficientis mcire gadaxra gamoiwvevswarmadobis mcire dawevas da ara katastrofas, rasacadgili hqonda xisti gegmis u=μ 0 =const arCevis SemTxvevaSi. miuxedavadimisa, rom ganxiluli faqti gamovlenil iqna TevzWerismartiv magaliTze, igi SeiZleba gamoviyenoT marTvis yvela imarawrfiv sistemaSic, sadac SesaZlebelia warmoiqmnas bifurkaciulida qaosuri movlenebi. cxadia, (μ) mmarTveli parametrisSerCevis gziT SesaZlebelia sistema mebismieri sawyisi pirobebidangaviyvanoT misi mdgomareobaTa sivrcis sasurvel atraqtorzeda uzrunvelvyoT mimarTuli TviTorganizeba-mizidva invariantulimravalsaxeobisaken (atraqtorisaken).59


1.4 arawrfivi TviTorganizacia da disipatiuri struqturebi.sinergetikis ZiriTad cnebebs warmoadgens: bifurkacia, daqvemdebareba,wesrigis parametri, marTvis parametri da atraqtori.aRmoCnda, rom wonasworobis dakargvas wrfiv miaxloebaSi, wesrigisparametrebis warmoSobasa da daqvemdebarebis principis realizaciasSoris arsebobs mniSvnelovani Sinagani urTierTkav-Siri. mmarTveli parametrebis cvlilebis Sedegad arawrfivmasistemam SeiZleba dakargos wonasworoba wrfiv miaxloebaSi. zemoTganxiluli meore rigis sistemis (1.11) aseT mmarTvel parameterswarmoadgens parametri λ1 , romlis cvlilebis SedegadR ( λ 1) SeiZleba gaxdes Zalian mcire sidide an Seicvalos niSanieda amiT SeiZleba gaxdes sistemis aramdgradobis maCvenebeliwrfiv miaxlovebaSi. aseT SemTxvevebSi gamoiyeneba daqvemdebarebisprincipi.aqedan gamomdnareobs, rom bifurkaciis wertilebSi, romleb-Sic xdeba struqturuli cvlilebebi, sistemis yofaqceva ganisazRvrebamxolod wesrigis parametrebiT. sxvadasxva bunebis arawrfivdinamikuri sistemebSi kavSiri daqvemdebarebis princips,wesrigis parametrebsa da wonasworobis dakargvas Soris wrfivmiaxloebaSi saSualebas gvaZlevs gamovavlinoT saerTo analogiebi.adiabaturi midgoma, romelic gamosaxulia meore rigiswrfivi sistemebis magaliTze (7.11) ar warmoadgens principialursiaxles da ukve sakmaod didi xania gamoiyeneba arawrfiv meqanikaSi,qimiasa da sxva mecnierebebSi. es midgomebi ki warmoadgenendaqvemdebarebis principis, sinergetikis sabazo principis, dasabu-Tebas, am principze agebulia dinamikuri sistemebis arawrfiviTviTorganizaciis Teoria.sistemaSi arawrfivi TviTorganizacia SeiZleba warmoiSvasmisi zogierTi mmarTveli parametrebis cvlilebisas, sistemiskomponentebis ricxvis Secvlisas da agreTve sistemis axal60


mdgomareobaSi gadasvlisas. TviTorganizaciis procesebis magaliTebswarmoadgenen:marTvis TeoriaSi cneba ,,disipatiuri struqturebi” Semotaniliqna i.prigoJinis mier. is aRniSnavda rom ,,rogorc wonasworobidandacileba, agreTve arawrfivoba SeiZleba sistemaSiuwesrigobis mizezi iyos. uwesrigobas, mdgradobasa da disipaciasSoris warmoiSoba umaRlesi xasiaTis aratrivialuri kavSiri.imisaTvis rom gamovyoT es kavSiri Cven vuwodebT mowesrigebulkonfiguraciebs, romlebic warmoiqmnebian Termodinamikuri Stoebismdgradobis aris gareT. aseTi struqturebi SeiZleba arsebobdnenwonasworobidan Sors sakmaod didi energiisa da nivTierebaTamodinebis xarjze. disipatiuri struqturebi warmoadgenenmagaliTs, romelic demonstracias ukeTebs im faqts, romuwonasworobis unari warmoadgens mowesrigebulobis wayros”.Aarawrfiv Ria sistemebSi mowesrigebulobis warmoSobis paradoqsulobasxazi gaesmeva im saerTo cnobili faqtiT rom Cveulebrivwonasworul sistemebSi saxeldobr disipaciis cvlilebasaerTod spobs yovelgvar wesrigs da iq yovelTvis warmoiSobaTermodinamikuri wonasworoba e.i. qaosi. aRmoCnda rom Ria disipaciursistemebSi disipacias mivyavarT Sesabamisi zomisa daformis struqturebis warmoSobamde e.i. warmoiqmneba TviTorganizaciisprocesi. sxva sityvebiT sistemaze zemoqmedebam gare samyarosTanZalian aramdgrad areSi SeiZleba migviyvanos TavisiTvisebebiT axal dinamikur mdgomareobamde, romelic prigoJinismier iwodeba disipatiur struqturebad, aqve xazi gausvaT moulodneladmWidro kavSirs stuqturasa da disipacias Soris e.ikargvebs sistemaSi. aseTma sistemebma SeiZleba migviyvanon principuladaxal movlenebamde sistemis yofaqcevaSi, kerZod, sistemisSemadgeneli uamravi raodenobis nawilakebis da saerTod komponentebisumaRlesad maRali mowesrigebulobis yofaqcevamde.i.prigoJinis mier disipatiuri struqturebis aRmoCena niSnavs materiisaxali dinamikuri mdgomareobis danaxvas, romelic adrecnobili ar iyo klasikuri mecnierebisaTvis [6].61


marTvis TeoriisaTvis cneba ,,disipatiuri mowesrigebulisturqturebi”, romlebic warmoiqmnebian arawrfiv sistemebSi Zlieraramdgrad areebSi, TavisTavad mniSvnelovania sxvadasxvaTvalTaxedvis kuTxidan. saqme imaSia, rom uwesrigoba (qaosi) mii-Reba mravali meTodebiT, xolo wesrigi ki piriqiT SeiZleba uzrunvelyoTZalze mcire raodenobis xerxebis meSveobiT da racmaRalia wesrigis xarisxi, miT naklebia es ricxvi, rac niSnavsoptimalur marTvas. cneba ,,sturqtura“ agreTve dakavSirebuliacneba marTvasTan, radganac niSnavs romeliRac obieqts, romelsacgaaCnia mdgradoba da unari gauwios winaaRmdegoba Sinaganda garegan zemoqmedebebs da darCes sakuTari Tavis msgavsi da arSeicvalos mTlianobaSi. Tumca rogorc aRiniSna Zalian mkacrimowesrigebuli struqturebis warmoSoba Zlierad aramdgrad are-Si SeiZleba gaxdes negatiuri faqtori marTvis TvalsazrisiT damarTvis sistemebis mravalsaxovani yofaqcevis moTxovnis Sesasruleblad.sinergetikul sistemebSi SeiZleba warmoiSves, rogorc mowesrigebuli,aseve qaosuri rxevebi. imisaTvis rom dinamikur sistemaSiwonasworobidan moSorebiT arsebobdnen mowesrigebulistruqturebi am sistemebze mudmivad unda moedinebodes energiis,nivTierebebisa da informaciis uwyveti nakadi. swored TviTorganizaciawarmoadgens zemoT CamoTvlili mravalferovani sistemebissaerTo Tvisebas, romlebic Sedgebian elementebisagan dasxvadasxva bunebis qvesistemebisagan-atomebis, molekulebis, ujredebis,cxovelebis da a.S TviTorganizacia rac ar unda pirveliSexedviT mogveCvenos ucnaurad saSualebas gvaZlevs SeviswavloTTavisi bunebiT sxvadasxvanairi dinamikuri sistemebisTvisebebi, erTiani maTematikuri poziciebidan da erTiani cnebebiT.ase, rom zemoT CamoTvlil yvela mravalferovan sistemebSiSeiZleba warmoiqmnas TviTorganizaciis procesebi, romlebsacmivyavarT wonasworobis mdgomareobidan Sors mowesrgebuli makroskopulistruqturebis Seqmnamde, romlebsac gaaCniaT principuladaxali Tvisebebi.62


Ees struqturebi, romlebic SeiZleba warmoiSvas kogerentulian qaosuri rxevebiT miiRes dasaxeleba atraqtorebi romlebicmiizidaven fazur sivrceSi simravles, e.w. wertilebiserTobliobas, romliskenac miizidebian yvela axlo traeqtoriebida moZraobebi. rogorc cnobilia, arawrfivi diferencialurigantolebebis amonaxsnebi sawyisi pirobebisagan damokidebulebiTSeiZleba moiqcnen principialurad gansxvavebulad, Tu isini moxvdebianatraqtoris mizidvis areSi, da isini aucileblad gaemar-Tebian misken. sxva sityvebiT, es amonaxsnebi SeiZleba daemorCilonromeliRac yofaqcevis mkacr wesebs, romelic cnobilia,,,analogi” superpoziciis prinipisa arawrfiv sistemebSi. atraqtorebiarawrfivi sistemebis fazur sivrceSi SeiZleba iyos Semdegitipis: ZiriTads maT Soris warmoadgens wertili (kerZodmdgradi fokusi), zRvruli cikli (periodulad cvladi mdgomareobaTasimravle), tori da bolos ,,ucnauri atraqtori “.Cveulebrivi atraqtori (wertili, zRvruli cikli antori) gansazRvravs sistemis moZraobis damyarebul reJims, romliskenacmiiswrafian yvela gardamavali reJimebi, romlebicxvdebian misi mizidulobis areSi. am atraqtorebis mniSvnelovaniklasi gamoirCeva Tavisi mravalsaxeobebiT da isini qmnian dinamikurisistemis fazur sivrceSi zogierT maravalsaxeobebs, romlebsacgaaCniaT Tviseba miizidon Tavisken yvela atraqtori Tavisigarkveuli garSemowirulobidan, maT ewodebaT mimzidvelebi,xolo isini, romlebic rCebian ucvlelebi sistemis moZraobisasiwodebian invariantul mravalsaxeobebad.aRmoCnda, rom rig SemTxvevaSi atraqtorebi ar iyvnen mravalsaxovani,Tumca sistemis moZraobis traeqtoria roca xvdebaaseTi atraqtoris zemoqmedebis areSi rCeba iq ramodenime xnisganmavlobaSi. aseTi atraqtoris magaliTs warmoadgens tori, gan-Tavsebuli sistemis fazur samganzomilebian sivrceSi, Tu moZraobaaseT torze xdeba aramdgradi, maSin torze moZraobis traeqtoriebiramdenjerme gadakveTen erTmaneTs bifurkaciis (gaorebiswertilebSi), maSin warmoiqmneba qaosuri moZraoba. amasTan63


erTad traeqtoriis yofaqcevis xasiaTi Zalze mgrZnobiarea sawyisipirobebis cvlilebebisadmi. es niSnavs, rom droTa ganmavlobaSisul mcire cvlilebebi da fluqtaciebi sistemaSi SeiZlebamniSvnelovnad gaZlierdnen, rac gardauvlad migviyvans qaosurdinamikamde. am SemTxvevaSi rodesac bifurkaciebi xdebamravlobiTi traeqtoriebis areSi droTa ganmavlobaSi xdeba imdenadrTuli da gaurkveveli, rom sirTule gadadis uwesrigobasada qaosSi. swored es Tviseba damaxasiaTebelia ucnauriatraqtorebisaTvis da miRebulia determinirebuli qaosis anqaosuri dinamikis saxelwodebiT. aRwerili paradoqsuli Tviseba,romelic SesaZlebelia arawrfiv determinirebul sistemebSi,gviCvenebs, rom ,,mowesrigebulobas” da ,,mouwesrigeblobas”, ubralosada rTuls Soris arcTu ise uzarmazari ufskrulia, rogorsacamis Sesaxeb acxadebda klasikuri mecniereba.mecnierebis sxvadsxva dargebSi Cveni garemomcveli uZravida cocxali samyaro dayofili iyo determinirebul da SemTxveviTprocesebad. aseTi barieri determinirebul da qaosur sistemebsSoris didixania arsebobda klasikur meqanikasa da fizikaSi.es moCvenebiTi da udavo faqti gadavida Tanamedrove mecnierebebSi:kibernetikaSi, informatikaSi, radioteqnikaSi da a.S.amasTan erTad wamoweuli iyo udavoo dasabuTeba, rom mravaliprocesebis stoxastikuri xasiaTi sxvadasxva sistemebSi aixsnebodaamisi Semadgeneli elementebis didi raodenobiT da maTiTavisuflebis xarisxebiT. swored es debuleba edo safuZvladrTuli procesebis axsnas. erTi SexedviT TvalnaTelia, rom mravalganzomilebawarmoadgens sirTulis arss. sinamdvileSi mxoloderTi nawilakis qceva, romlis moZraoba aRiwereba niutoniskanonebiT, SeiZleba aRmoCndes ganusazRvreli da sruliad moulodneli.( n ≥ 3)ganzomilebis martiv determinirebul avtonomiurdinamikur sistemebs SeiZleba hqondeT arsebiTad SemTxveviTi,stoqastikuri moZraobebi yovelgvari garegani zemoqmedebisgareSe. Tanamedrove sinergetikam daamtkica, rom namdvili Sem-TxveviToba Cveni garemomcveli samyarosi principialurad gani-64


sazRvreba swored qaosuri moZraobebis arsebobiT arawrfiv determinirebuldinamikur sistemebSi, romelTac gaaCniaT bifurkaciisTvisebebi da ara sistemis ganzomilebebiT. es sruladgasaocari aRmoCena, romelic Zireulad cvlis bunebismetyvelebisfundamentalur warmodgenebs iTvleba bolo drois did sensaciadda SeuZlia Zireulad Secvalos Cveni mecnieruli msoflmxedveloba.TviTorganizaciis fundamentaluri Tvisebebis dadgena,arawrfiv dinamikur sistemebSi saSualebas gvaZlevs avxsnaTda gamovavlinoT axali moulodneli movlenebi Cvens garemomcvelsamyaroSi da SevqmnaT teqnikuri sistemebi da mowyobilobebiaraCveulebrivi TvisebebiT.zemoT Camoyalibebuli mosazrebebi gvaZlevs saSualebas gavakeToTmniSvnelovani daskvnebi: Camoyalibebuli axali integralurimecniereba sinergetika, romelic swavlobs koleqtiuriTviTorganizaciis procesebs da praqtikulad moicavs Tanamedrovemecnierebis yvela dargs uZravi da cocxali bunebis Sesaxeb,teqnikur da ekonomikur mecnierebebSi. es ganzogadoebuli mecnierebadafuZnebulia Seuqcevadi procesebis arawrfiv dinamikazeda Termodinamikaze, rogorc sabazo mecnierul disciplinebze.1.5 sinergetika da marTvis procesebirogorc ukve avRniSneT, sinergetika - arawonasworuli procesebisTeoria warmoadgens sayovlTao ganviTarebis Teorias,romelsac gaaCnia didi msoflmxedvelobrivi Sedegebi. am axaliintegraluri mecnierebis azri da Sinaarsi mdgomareobs imaSi,rom Ria sistemebSi, romlebic garemomcvel aresTan axdenen energiis,nivTierebebisa da informaciis urTierTgacvlas, warmoiqmnebianTviTorganizaciis procesebi e.i. zogierTi mdgradi mowesrigebulistruqturis fizikuri (biologiuri–qimiuri da a.S) procesebisqaosidan, warmoiSveba sruliad axali Tvisebebis mqonesistemebi. nebismier maRalefeqtur sinergetikul sistemas gaaCniaori fundamentalur Tviseba: garemomcvel samyarosTan energiis,65


nivTierebebis da informaciis aucilebeli gacvla da dauyovnebeliurTierTqmedeba. es niSnavs rom sinergetikasa da sxva fizikur,teqnologiur, kibernetikul, biologiur, ekonomikur mecnierebebsSiris arsebobs Sinagani urTierTkavSiri. amasTan, sinergetikasTiToeul maTganSi Seaqvs Tavisi Taviseburebebi da midgomebi,romlebic ar aris damaxasiaTebeli am mecnierebebis tradiciulimimarTulebebisaTvis [7,10,14].sinergetika - sinTetikuri mecnierebaa, romelic efuZnebasxvadasxva bunebis dinamikuri sistemebis TviTorganizaciis zogadkoncefcias. misi kanonebi zogadi fizikuri ideebisa da maTematikurimeTodebis erToblioba ki ar aris, aramed axali konceptualurixedvaa mecnierebaze. sinergetikuli midgoma mecnierebaSimogvagonebs sistemur midgomas, sinergetikas gaaCnia mniSvnelovaniSexebis wertilebi sistemebis zogad TeoriasTan. sinergetikisaTvisiseve rogorc sistemebis TeoriisaTvis mniSvnelovania,sxvadasxva bunebis mqone movlenebs Soris ara mxolodzedapiruli analogiebi, aramed gansaxilveli sistemis Semadgenelkomponentebs Soris sakmaod mkacri Sesabamisobebi. sinergetikulmidgomaSi zogadi sistemuri midgomisagan gansxvavebiTSeiswavleba bunebrivi da teqnikuri sistemebis TviTkonstruirebiskonkretuli principebi da meqanizmebi. sxvanairadrom vTqvaT, sistemebis zogadi Teoriisagan gansxvavebiT, sinergetikaamaxvilebs yuradRebas rTul arawrfiv sistemebSi warmoSobilkooperatiul, kogerentul da TviTSeTanxmebul procesebze.aucilebelia avRniSnoT, rom rogorc sistemebis zogadi TeoriisaTvis,kibernetikisaTvis, aseve, sinergetikisaTvis gamaerTianebelcnebas warmoadgens sistemebis cneba. Tumca sinergetikulmidgomaSi saerTo sistemuri koncepciis formirebis-TviTorganizaciisgarda aucilebli xdeba konkretuli fizikuri, (biologiuri,qimiuri) movlenebisa procesebis gaTvaliswineba. mecnierebisklasikur gagebas yovelTvis safuZvlad udevs eqsperimentaluriSedegebis erTgvari erToblioba da mecnierTa mier gamoTqmuliprincipebi da hipotezebi. sinergetika ar aris mecniereba66


am sityvis klasikuri gagebiT, aramed arsebiTad axali koncefciaa,romelic bazirebulia sistemis TviTorganizaciis Teoriaze.sinergetikuli midgoma pirvel rigSi miiswrafvis gamoavlinosama Tu im procesis makroskopuli Tviseba mag. populaciiswarmoqmins procesi da a.S. miTiTebuli midgoma ar gamoyofserTi arsebis an nawilakis yofaqcevas, rogorc es keTdeba klasikurmeqanikaSi, misTvis ufro metad mniSvnelovans warmoadgenscalkeuli komponentebis raodenoba, romelic Sedis saerTo sistemaSi.sinergetikul midgomaSi navaraudebia, rom TviT es raodenoba-wesrigisparametri-marTavs sistemis TiToeuli komponentis(arsebis, nawilakis da a.S.) yofaqcevas.TviTorganizebad procesebs safuZvlad udevs daqvemdebarebissinergetikuli principi, romlis Tanaxmadac sawyisi rTulisistema SeiZleba warmodgenil iqnes ierarqiuli sistemebis saxiT,romelic Sedgeba dinamikuri qvesistemebis erToblobisagan,romlebic eqvemdebarebian erTmaneTs da imyofebian erTmaneTTangarkveul dinamikur urTierTkavSirSi.sinergetikas rTul wrfiv dinamikur sistemebSi safuZvladudevs TviTorganizaciis fundamentaluri movlena. Tumca sinergetikaSijer ar SeuqmniaT zogadi da erTiani TviTorganizaciisTeoria, romelic samarTliani iqneba yvela saxis bunebrivi dateqnikuri sistemebisaTvis, amitom, konkretuli Tvisebebidan gamomdinare,sinergetikuli midgoma iZens ganmasxvavebel Taviseburebebsda Sinaars. amasTan dakavSirebiT, Cven SegviZlia vilaparakoTsinergetikul midgomaze, rogorc Sesabamisi mecnierebiserTgvar mmarTvel koncefciaze. sinergetika xdeba im evoluciurbunebismetyvelebis dargad, romelic gvaZlevs saSualebassxvadasxva bunebrivi da teqnikuri movlenebis erTgvar safuZvelzeaigos sxvadasxva bunebrivi da teqnikuri movlenebis erTianimecnieruli koncefciis erTiani ena.sinergetikasTan rogorc mecnierebasTan, romelic Seiswavlisarawrfivi dinamikuri sistemebis qcevas wonasworobis mdgomareobidanmoSorebiT zogierTi mmarTveli parametris cvli-67


lebisas, yvelaze ufro Tavisi ideologiiT axlos aris iseTifundamentaluri mecniereba, rogoricaa marTvis Teoria. amasTandakavSirebiT, marTvis Teoriis ganviTarebisaTvis saWiroa sinergetikulisistemebis ZiriTadi meTodebis gadatana arawrfivi teqnikuriobieqtebis marTvis sistemebis konstruirebaze. Tumca ammecnierebebis midgomebSi arseboben garkveuli gansxvavebebi. asemagaliTad, amtkiceben, rom ,,rogorc kibernetika ise sinergetikapirvelxarisxovan mniSvnelobas aniWebs marTvis cnebas, Tumca Ti-Toeuli isaxavs sxvadasxva mizans. kibernetika awarmoebs algoriTmebisda meTodebis damuSavebas, romelic gvaZlevs saSualebasvmarToT sistema, raTa igi funqcionirebdes winaswar dasaxuliwesiT. sinergatikaSi ki Cven vcvliT mmarTvel parametrebswinaswar ganusazRvreli wesiT da SeviswavliT am procesSi sistemisTviTorganizacias.sabolood, nebismieri mecnierebis, maT Soris sinergetikisarsi mdgomareobs pirvel rigSi, adamianis mier misi garemomcvelisamyarosa da sakuTari Tavis SecnebaSi. miRebuli codniskonstruqciulad gamoyenebaSi. Kkibernetika da Sesabamisad mar-Tvis Teoria, asaxavs Tanamedrove Sexedulebas mecnierebaze, rogorcgarkveul konstruqciul sawyisze, da ara pasiur dakvirvebasbunebrivi procesebsa da movlenebze.marTvis klasikurma TeoriaSi Zalze warmatebulad gamoiyenebasobieqtebze garegani zemoqmedebis meTodebi, Tumca mar-Tvis amocanebis gadawyvetaSi mniSvnelovania Zalovani midgomidansinergetikis TviTorganizaciis ideaze gadasvla. sxva sityvebiTrom vTqvaT, warmoiSva aucilebloba Seiqmnas xerxebi ur-TierTqmedebis Sinagani Zalebis formirebisa da moqmedebisa,romelTa meSveobiTac sistemis fazur sivrceSi warmoiSveba myaridisipatiuri struqturebi, romlebic adeqvaturni iqnebian Sesabamisisistemebis fizikuri (qomiuri, biologiuri) arsisa.gamovyoT sinergetikis Semdegi meTodologiuri debulebebi,romlebic principulad mniSvenlovania Tanamedrove sinergeti-68


kuli gamoyenebiTi marTvis Teoriis sinergetikuli safuZvlebisformirebisaTvis:• sistemis moZraoba rogorc wesi unda mimdinareobdesmisi sivrcis arawrfiv areSi;• sistema unda iyos Ria, rac igivea energiis an nivTierebis(SesaZloa informaciisac) garemomcvel garemos-Tan gacvlisa;• kooperatiuloba, sistemebSi mimdinare procesebiskogerentuloba;• arawonasworuli Termodinamikuri situaciis arseboba,romlis Tanaxmadac energiis modineba sistemisken, undaiyos sakmarisi, aramarto entropiis zrdis SesaCereblad,aramed misi SemcirebisTvisac, rac aZlierebs sistemaSiwesrigs;• moZraobis safiniSoO etapze sistemaSi unda arsebobdesevoluciis ramodenime gza, romelic aRiwereba wesrigisparametrebis mimarT tipiuri gantolebebiT.TviTorganizaciis es niSnebi gviCvenebs, rom sinergetikassaqme aqvs fizikis, maT Soris marTvis Teoriis, araklasikurprocesebTan da movlenebTan.69


2. Caketili optimaluri disipatiuri marTvis sistemebissinTezi sinergetikuli Teoriis gamoyenebiTAoptimaluri regulatorebis analizuri konstruirebis amocanaTanamedrove etapze warmoadgens optimaluri marTvis sistemebismniSvnelovan amocanas. amdenad mniSvnelovania am amocanebisgadawyvetis axali meTodebis SemuSaveba., romlebSic optimaluriregulatorebis analizuri konstruirebis amocana efuZnebaTanamedrove marTvis sinergetikul Teorias.2.1. Caketili optimaluri marTvis sistemebis disipatiurobaganvixiloT optimaluri disipatiuri marTvis sistemebissinTezis sakmaod efeqturi meTodi, romelsac safuZvlad udevsletov-kalmanis meTodi da dinamikuri sistemebis asimptoturimdgradobis aucilebeli piroba [28].Tanamedrove TviTorganizaciis TeoriaSi sistemebis disipatiurobisTvisebis fundamentalur mniSvnelobaze i.prigoJiniaRniSnavda: `disipatiuri sistemebi warmoadgenen gansacvifrebelmagaliTs, romelic demonstrirebas ukeTebs uwonasworobisunars warmoadgendes mowesrigebulobis wyaros~. aq warmodgenilmeTods safuZvlad udevs fizikuri midgoma.pirvel rigSi mas gaaCnia arsebiTi Teoriuli mniSvneloba,romelic dakavSirebulia im faqtTan, rom optimaluri marTvissinTezis procedura saSualebas gvaZlevs davadginoT kavSiriCaketili marTvis optimalur sistemebsa da fundamentalur fizikurprocesebs Soris. am kavSirebs safuZvelad udevs maTematikurimodeli wrfivi diferencialuri gantolebebis an meorerigis kerZo warmoebuliani utolobas saxiT. rogorc cnobilia,farTo klasis fundamentaluri fizikuri procesebis klasifikaciadakavSirebulia aseTi tipis gantolebebis an utolobebisgansazRvrebebTan. marTvis optimalur sistemebSi procesebsa dafizikur procesebs Soris kavSiri saSualebas gvaZlevs movax-70


dinoT marTvis optimaluri disipatiuri sistemebis axali klasifikacia,romlebic dafuZnebulia bunebis fundamentalur kanonebze.optimaluri sistemebis analizisa da sinTezisadmi aseTimidgoma pasuxobs marTvis Teoriis ganviTarebis Tanamedrove dones,romelic dakavSirebulia marTvis fizikuri Teoriis ganviTarebasTan[29]. optimaluri marTvis sinTezis SemoTavazebul me-Tods gaaCnia analitikuri Tvisebebi, romelic ganasxvavebs masarsebuli optimaluri regulatorebis analizuri konstruirebis(orak)-is meTodebisagan.gadavideT orak-is problemebis gadawyvetis axal midgomaze,romelic efuZneba sinTezirebul optimalur sistemebSidisipatiurobis fizikur Tvisebas. es Tviseba ganekuTvnebamarTvis zogad sinergetikul Teoriis erT-erT sakvanZo sakiTxs.qvemoT ganxiluli Teoria eyrdnoba disipatiuri sistemebis Tvisebasda SeiZleba daxasiaTebul iqnas rogorc optimaluri mar-Tvis sinergetikuli Teoriis erTi nawili.dauSvaT marTvis obieqti aRiwereba diferencialuri gantolebebiT,romelic Cawerilia veqtorul-matriculi formiT:x& = ( t) = f( x) + G( x)u(2.1)sadac x=(x 1 ,...,x n ) T , u=(u 1 ,...,u n ) T -Sesabamisad fazuri kordinatebis damarTvebis veqtorebia; f(x,u)=(f 1 (x,u), f 2 (x,u),...f n (x,u)) T –veqtor-funqcia;G(x)=(g ij (x))-n×m ganzomilebis matricaa.marTvis optimaluri sinTezis amocana CamovayaliboT SemdegisaxiT: moiZebnos marTvis kanoni u=u(x), romelsac gadayavs (2.1)obieqti nebismieri sawyisi x(0)=x 0 mdgomareobidan, fazuri sivrcisx=0 kordinatTa saTaveSi, uzrunvelyofs Caketili sistemisasimptotikur mdgradobas da minimums aniWebs funqcionals( 00∞I = ∫ F ( x ) + u , Du ) dt . (2.2)aq F 0 (x) - niSangansazRvruli x-is mimarT dadebiTi funqciaa.D=diag(d ii ) gantolebis diagonaluri matricaa, m×m; .,. - veqtorebisskalaruli warmoebulia.71


optimaluri marTvis sinTezis formulirebuli amocanawarmoadgens optimaluri regulatorebis analizuri konstruirebiscnobil amocanas. optimaluri sistemebis sinTezSi yvelazemeti gavrcelebulia (orak) optimaluri regulatorebis analizurikonstruirebis procedura, romelic dafuZnebulia dinamikuriprogramirebis meTodze. sinTezis mocemuli procedurismTavar nawils warmoadgens funqcionaluri gantolebis amonaxsnismoZebna garkveuli – dadebiTi funqciis saxiT, romelic iqnebaliapunovis optimalur funqcia Caketili optimalurimarTvis sistemebisaTvis regulatorebi romlebic agebulia liapunovisoptimaluri funqciebis safuZvelze uzrunvelyofen asimptoturmdgradobas da konstruirebuli sistemebis xarisxisfunqcionals aniWeben optimalurobis Tvisebas.marTvis wrfivi stacionaluri obieqtebis SemTxvevaSi, liapunovisoptimalur funqcias irCeven gansazRvruli dadebiTikvadratuli formiT v(x)=x T Cx. Tu aseT formas CavsvamT ZiriTadfunqcionalur gantolebaSi da koeficientebis gavutolebT nuls,miviRebT sxvadasxva xarisxis mqone bazuri koordinatebismimarT rikatis tipis arawrfivi algebruli gantolebaTa sistemas.gantolebebis aseTi tipis sistemis amoxsnis meTodebis Sefasebada maTTan dakavSirebuli problemebi aRwerilia [9.14] naSromSi,sadac, aRniSnulia, rom aseTi tipis gantolebaTa sistemisamoxsnis universaluri meTodi jer-jerobiT ar arsebobs.marTvis wrfivi arastacionaluri obieqtebisaTvis liapunovisfunqcias eZeben dadebiTad gansazRvruli kvadratuli formissaxiT v(x)=x T C(t)x. am SemTxvevaSi Rebuloben rikatis tipis Cveulebrivdiferencialur gantolebaTa sistemas-romlis sawyisi pirobaganisazRvreba funqcionalis terminaluri wevris saxis mixedviT.misi amoxsnisaTvis sargebloben Cveulebrivi diferencialurigantolebebis integrirebis ricxviT meTods (eileris,runge-kutas da a.S.) am SemTxvevaSi gvaqvs arawrfivi Cveulebrividiferencialuri gantolebebisaTvis koSis amocana. aseTi sistemebisamoxsnis proceduras yovel konkretul SemTxvevaSi gaaCnia72


Tavisi Taviseburebebi da siZneleebi, romlebic dakavSirebulniarian sizustesTan, mdgradobasTan da a.S.Tu marTvis obieqti aRiwereba rTuli arawrfivi diferencialurigantolebebis sistemiT (2.1), maSin, am SemTxvevaSi ar arsebobsliapunovis optimaluri funqciis gansazRvris regularulimeTodi. es dakavSirebulia ZiriTadi funqcionaluri gantolebisamoxsnis aucileblobasTan, romelic warmoadgens kerZowarmoebulebian arawrfiv diferencialur gantolebas, aseTiklasis gantolebebis amoxsna dakavSirebulia did siZneleebTan.ase rom letov-kalmanis orak-is meTodis gamoyeneba arawrfiviobieqtebis marTvis sinTezisaTvis kriteriumi (2.2)-is mixedviT dakavSirebuliamniSvnelovani winaaRmdegobasTan, ZiriTadi funqcionalurigantolebis ricxviTi da ufro metad analizuramonaxsnebTan ZiebaSi [25].marTvis arawrfivi obieqtebis optimaluri algoriTmebissinTezis problemis gadawyvetam migviyvana orak-is meTodis SeqmnasTan,sadac optimizirebul funqcionals gaaCnia naxevradgansazRvrulisaxe. am SemTxvevaSi sinTezis amocana daiyvaneba ker-Zowarmobulebiani wrfivi gantolebis amoxsnamde, rac saSualebasgvaZlevs zogierTi klasis arawrfivi obieqtebis marTvismosaZebnad avagoT sinTezis ricxviTi procedurebi.optimaluri regulatorebis analizuri konstruirebisletov-kalmanis meTodi dafuZnebulia liapunovis wonasworobisTeoriisa da optimaluri marTvis Teoriis erToblivi gamoyenebiskoncepciaze. aseTi midgoma avtomaturi marTvis Teoriis ori ZiriTadimimarTulebis Serwymam erT SesaZlebeli gaxada liapunovisoptimaluri funqciis Semotanam. fazuri sivrcis wertilix=0, romelSic minimalur mniSvnelobas Rebulobs, rogorc xarisxisfunqcionali (2.2), aseve liapunovis optimaluri funqcia,warmoadgens wonasworobis mdgrad wertils. es niSnavs, rom CaketilimarTvis optimaluri sistemebis traeqtoriebi miizideba mdgradiwonasworobis am wertilisken-atraqtorisken mis fazursivrceSi.73


sinTezirebuli Caketili optimaluri marTvis sistemazogad SemTxvevaSi TviTon warmoadgens dinamikur sistemas, romlebicaRiwereba wrfivi diferencialur veqtorul-matriculigantolebiT.x& () t = ϕ( x), (2.3)sadac ϕ ( x) = f( x) + G( x) u( x);u(x) – marTvis optimaluri kanonebisveqtoria.(2.3) sistemis fazuri traeqtoriebis yofaqceva, xasiaTdebaveqtoruli veliT ϕ ( x) = ( p1( x),..., p1( x)) T . amitom traeqtoriis Sesaswavladmocemuli sistemis fazur sivrceSi SeiZleba SemovitanoTvelis Teoriis zogierTi maCveneblebi. kerZod, radganac Caketilioptimaluri sistema asimptoturad mdgradia, maSin misitraeqtoriis yofaqcevis Sesafaseblad yvelaze ufro misaRebiaiseTi maCveneblis Semotana, rogoricaa veqtoris divergenciaϕ( x)( divϕ ( x)). romelic axasiaTebs fazuri traeqtoriebis klebadobaswonasworobis mdgradi wertilis SemogarenSi. Semotaniliveqtoruli velis divϕ ( x)maCvenebeli saSualebas gvaZlevs gavukeToTformulireba Semdeg mtkicebulebas [28].asimptoturi mdgradobis aucilebel pirobas warmoadgenswonasworobis wertilis x=0-s uaryofiToba zogierT Ω areSi,romlisTvisac wertili x=0 warmoadgens zRvruls.Caketili optimaluri sistema (2.3) sinTezirebuli letovkelmanisoptimaluri regulatorebis analizuri konstruirebismeTodi disipatiuria. e.i. misTvis sruldeba utolobadivϕ( x)=∂ϕ< 0,∑ n1i=1 ∂x1x ∈Ω(2.4)Caketili optimaluri sistema (2.3), romelic (2.4) utolobasakmayofilebas, vuwodebT Caketil optimalur disipatiur sistemas.ase, rom disipatiuri sistemebisaTvis divergencia (2.4) yovelTvisuaryofiTi sididea, romelic axasiaTebs fazuri moculobisSekumSvis siCqares. aseT SemTxvevaSi yvela traeqtoriebiaucileblad miizidebian romeliRac mimzidavi simravlisken-74


atraqtorisken fazur sivrceSi, xolo misi ganzomilebebi yovelTvisnaklebia sawyisi sistemis ganzomilebebTan SedarebiT.dinamikuri sistemebis Tanamedrove TeoriaSi disipatiursistemebs ukaviaT gansakuTrebuli adgili, romelic bolo periodSiswrafad viTareba. ukanaskneli periodSi warmoiSvnen daviTardebian iseTi mecnieruli mimarTulebebi, romlebic dakavSirebuliarian arawrfiv dinamikasTan, aseTebia sinergetika solitonikada katastrofebis Teoria. aucilebelia avRniSnoT romdisipatiuri sistemebi ganekuTvnebian yvelaze naklebad Seswavlilidinamikuri sistemebis klass, amasTan erTad is warmoadgensufro farTo klass konservatiuli da ergodiuli sistemebisklasTan SedarebiT. masTan erTad SegviZlia vamtkicoT,rom bolo ori dinamikuri sistemis klasi warmoadgenen disipatiurisistemebis erTgvar idealizacias. naTqvamidan gamomdinareobs,rom disipatiuri sistemebi da agreTve misi qveklasi-Caketili optimaluri disipatiuri sistemebi, asruleben mniSvnelovanrols rogorc dinamikuri sistemebis zogad TeoriaSi daaseve gansakuTrebiT marTvis sinergetikul TeoriaSi [30].2.2. optimaluri disipatiuri sistemebis klasifikaciaoptimaluri sistemebi maTi eqstremalur-funqcionaluri maxasiaTeblebismixedviT dayofilni arian klasebad, magaliTad,optimalurebi swrafmoqmedebis, energiisa sawvavis xarjis mixedviT,da a.S. optimaluri marTvis sistemebisa da maTi klasifikaciisadmiaseTi midgoma sakmarisia sistemebis momxmareblebisa-Tvis, magram SezRudulia maTi dammuSaveblebisTvis. es dakav-Sirebulia, pirvel rigSi, imasTan rom optimaluri sistemebisSeqmnisas konstruqtors gaaCnia informacia eqstremaluri procesebismaxasiaTeblebze, magram ar floben codnas am procesebiskavSirze bunebis fundamentalur kanonebTan.75


am paragrafSi ganxilulia Caketili optimaluri disipatiurimarTvis sistemebis klasifikaciis sawyisebi, romlebic dafuZnebuliakerZo warmoebuliani (2.5) meore rigis diferencialuriutolobis gamokvlevaze.sadacnn∂f ( x) 1 ∂ ⎛∂v⎞∂x 2 ∂x ⎝∂x⎠∑i−1T− ∑ ⎜G( x) D G ( x) ⎟ < 0.(2.5)t= 1 t t=1 1gadavweroT (2.5) utoloba Semdegi saxiTn∑a ( x) v + ∑ b ( x) v + c( x) > 0, (2.6)ij xix j j x ji, j= 1 j=1i=1nmgik( x) gjk( x)aij( x)= ∑ (2.7)2dkm ⎛ g ( )nijx ∂gik1 ∂ ⎛ ⎞⎞bj( x) = ∑+gi( x) ,⎜k 0 dk xj2dk x⎜∏k ⎟⎟= ⎝ ∂ ∂n+ 1−j ⎝ i=1 ⎠⎠n∂fi( x)c( x) =−∑,∂xi=1 1∂v∂vvx= , v ,j xix= x∈Ωj∂x ∂x ∂xi i j(2.8)ganvixiloT meore rigis operatorida diferencialuri utolobanLv [ ] = ∑ aij( xv )x...ix+(2.9)ji=1Lv> [ ] 0 , (2.10)sadac wertilebi aRniSnaven diferencirebis meore rigis operatorsv -sTan mimarTebiT. operatori, romelic Seicavs mxolodmeore rigis warmoebulebs, iwodebian diferencialuri operatorismTavar nawilad (2.9). am saxis diferencialuri operatorebisklasifikacia ganisazRvreba imis mixedviT, Tu rogor moqmedebsy i =t i (x 1 ,...x n ) (i=1,2,...,n) (2.11)cvladebis gardasaxva diferencialuri operatoris formazeP P= ( 1 )P: ( x p ) x ,..., x nzogierT wertilSi. aRvniSnoT t ik∂ti∂xkmiviRebT,76


vn= ∑tvxiij iji=1,vn∑= t t v + ...,xixs ji is yiyji, j=1sadac wertilebi cvlian wevrebs, romlebic Seicaven v funqciiswarmoebulebs, ara umetes pirveli xarisxisa. (2.11) gamosaxuleba(2.9) operatoris moqmedebiT, dadis Semdeg saxemde:sadac koeficientia pij ( y )L`1 v = ∑ a y vy ...,iy+(2.12)jp[ ] ij ( )gamoisaxeba formuliT:np( ) (a y = ∑ t t a xij ji is lsls , = 1p)(2.13)amgvarad, operatoris mTavari nawilis L[ v ] koeficientebipx wertilSi gardaiqmneba iseve, rogorc kvadratuli formiskoeficientebinQ= ∑ a z zi=1ij i jTu koeficientebi Z i daeqvemdebareba afinur wrfiv gardaqmnaszinitl=1i(2.14)= ∑ t η(2.15)(2.15) tipis kvadratuli forma afinuri gardaqmnisdaxmarebiT SegviZlia miviyvanoT kanonikur saxemde.nQ = ∑ λη(2.16)sadac koeficientebi Rebuloben mxolod mniSvnelobebs +1, -1 an0. uaryofiTi koeficientebis ricxvs, romelsac ewodeba inerciisindeqsi da agreTve koeficientebis raodenobas, romlebic gardaiqmnebiannulis tolad, warmoadgenen formis afinur invariantebs.es ricxvebi axasiaTeben diferencialur operators x p – wertilSi.i=1diferencialur operators ewodeba eliptikuri x p –wertilSi Tu x i yvela mniSvnelobebi marto dadebiTebia, an mxoloduaryofiTebi. diferencialur operators ewodeba hiperbolikuri,Tu λ i -is yvela mniSvnelobas gaaCniaT erTi niSani magaliTaddadebiTi, erTis gamoklebiT, romelic uaryofiTia, dife-ii77


encialur operators ewodeba paraboliuri, Tu Q-s forma singularulia,e.i. erTi an ramodenime λ i koeficientebi gadaiqcevanulad. diferencialuri operatoris (2.9) klasifikacia uSualodgadadis diferencialur utoloba (2.10)-Si. movaxdinoT diferencialuriutoloba (2.6)-is klasifikacia, amisaTvis ganvixiloTskalaruli da veqtoruli marTvis SemTxveva cal-calke.skalaruli marTva. skalaruli marTvisas diferencialuriutolobis (2.6)-is koeficientebi (2.7) Caiwereba Semdegi saxiTp p( ) j( ) ,gix g xppaij( x ) = x ∈Ω (2.17)2dimisaTvis, rom movaxdinoT diferencialuri (2.6) utolobisklasificireba, (2.7) koeficientebiT, ganvsazRvroT koeficientebiλ i . amisaTvis amovxsnaT gantoleba[ A λE]det − = 0 , (2.18)sadac2⎡ g1 g1g2 ... g1g⎤n⎢2⎥1 gg2 1g2 ... gg2 nA = ⎢⎥2 d ⎢ ... ... ... ... ⎥⎢2⎥⎢⎣ggn 1ggn 2... gn⎥⎦g i _ g i (x) funqciis mniSvneloba wertilSi x p , E – erTeulovanimatricaa.gadavweroT (2.18) gantoleba Semdegi saxiT:saidanac gamomdinareobsn ⎛λ ⎜⎝⎞n2∑ gi+ 2λd⎟ = 0(2.19)i=1⎠⎧λi= 0, i = 1,2,..., n−1⎪n⎨ 1 2⎪ λn= − ∑ g1< 0⎩ 2di=1kanonikur formamde dayvanis Semdeg gveqneba⎧λi= 0, i = 1,2,..., n−1⎨⎩ λn= 1(2.20)78


amrigad, skalaruli marTvis SemTxvevaSi diferencialuriutoloba (2.6) miekuTvneba parabolur tips. aqedan gamomdinareobs,rom optimalur disipatiur sistemebSi skalaruli marTviTSesaZlebelia mimdinareobdnen procesebi, romlebic SeiZleba ganvsazRvruliiqnan, rogorc paraboluri. fizikaSi parabolurigantolebebiT aRiwereba difuziuri tipis amocanebi.amgvarad, optimaluri sistemebi skalaruli marTviT, romlebSicmimdinareoben paraboluri tipis movlenebi. SeiZleba mivakuTvnoTdifuziuri tipis optimalur sistemebs.veqtoruli marTva: veqtoruli marTvisa SemTxvevaSi diferencialuriutolobis (2.6) mTavari nawilis koeficientebi ganisazRvrebian(2.7) formuliT romeliRac Ω aris wertilebSi. iseverogorc skalaruli marTvis SemTxvevaSi diferencialuri utolobis(2.6) klasifikaciisaTvis aucilebelia moiZebnos koeficientiλ i . (2.18) gantolebis amoxsniT, sadac matrica A-s aqvs saxem 2 m m⎡ g1 kg1 kg2k g1kg⎢ ∑ ∑ ... ∑⎢k= 1 2dkk= 1 2dkk=1 2dk⎢ g g g g gA d d⎢ ... ... ... ...⎢m m m 2⎢ gnk g1 kgnk g2k gnk∑ ∑ ... ∑⎢⎣k= 1 2dkk= 1 2dkk=1 2dkm m 2m2k 1k 2k 2k nk⎢∑ ∑ ... ∑ ⎥= ⎢ k= 1 2k k= 1 2kk=1 2dk⎥nk⎤⎥⎥⎥⎥⎥⎥⎥⎦(2.21)SemovifargloT SemTxveviT n=2,3. n=2 SemTxvevaSi matricaG(x) veqtoruli marTvis zemoqmedebiT, zogad SemTxvevaSi aqvsstruqtura( )G x⎡g ( x) g ( x)⎤11 12= ⎢g21( x) g22( x)⎥xolo (2.18) gantoleba Rebulobs saxes:⎣2 2 22 1 gij1 1λ − ∑∑ λ+ Δ2 22 i= 1 j=1 dj2d∏i=1i⎦2(2.22)= 0(2.23)romelSic koeficientebi ucnobi λ-s SemTxvevaSi ganisa-pzRvreba yoveli P x Ω-areSi;. Δ-matricis gansazRvrulia (2.22).: ( )79


vietas formulis Tanaxmad gvaqvs2 2 2⎧22 1 gij⎪λ− ∑∑ = ∑λi> 02 i= 1 i= 1 dj i=1⎪⎨2 21 Δ2 2 = ∏λ1>2i=1⎪⎪⎪⎩∏i=10(2.24)saidanac gamomdinareobs rom (2.19) gantolebis fesvebisaTvisunda ganvixiloT ori SemTxveva:1) Tu Δ≠0, maSin λ 1 >0, λ 2 >02) Δ=0, maSin λ 1 =0, λ 2 >0 an λ 1 >0, λ 2 =0pirvel SemTxvevaSi utoloba (2.6) ganekuTvneba elifsurtips da Sesabamisad meore rigis Caketil optimalur disipatiursistemebSi Δ≠0 pirobisaTvis SesaZlebelia procesebi, romlebicSeiZleba ganisazRvros rogorc elifsuri. meore SemTxvevaSiroca Δ=0, sinTezirebul optimalur disipatiur sistemebSi gaaCniaparaboluri Tvisebebi.ganvixiloT exla mesame rigis obieqti veqtoruli gantolebiTda matriciT⎡g ( x) g ( x) g ( x)⎤G x g x g x g x( )11 12 13=⎢21( )22( )23( )⎥⎢ ⎥⎢⎣g31( x) g32( x) g33( x)⎥⎦am SemTxvevaSi gantoleba (2.18) Rebulobs saxes:(2.25)1 1 1 1 1 1λ λ λ3d3 33 2 2 2− ∑ ∑gij+ Δ2 ij−32 i= 1 dij=1 2 ∏ dk2k∈(1,2,3)\i∏i=1i2Δ = 0 (2.26)sadac Δ ij - element g ij -is minoria Δ matriciT ganmsazRvreli (2.25).(2.26) gantolebisaTvis sruldeba vietas formula⎧⎪3 3 31 1 2⎪ ∑ ∑gji= ∑λi> 0⎪ 2 i= 1 dij= 1 i=13 3 3⎪ 1 12⎨02 ∑ ∑Δ ji= ∑ ∏ λj>⎪2i= 1 ∏ dkj= 1 i= 1 k∈(1,2,3)\ik∈(1,2,3)\i⎪⎪3−131 ⎛ ⎞ 2⎪ d02 iΔ = λi>2⎜∏⎟ ∏⎪⎩⎝ i= 1 ⎠i=1(2.27)80


ogorc (2.27) formulidan gamomdinareobs (2.26) gantolebisfesvebisaTvis SesaZlebelia arsebobdes ori SemTxveva:1) Tu Δ≠0, maSin λ 1 >0, λ 2 >0, λ 3 >02) Δ=0, maSin λ i >0, λ j =0, λ k =0, i≠j≠k; i, j, k=1,2,3an λ i >0, λ j =0, λ k =0, i≠j≠k; i, j, k=1,2,3pirvel SemTxvevaSi utoloba (2.6) warmoadgens elifsurs,Sesabamisad mocemul sistemaSi SesaZlebelia elipsuri tipisprocesebi. meore SemTxvevaSi utoloba (2.6)-paraboluria, Sesabamisad,am SemTxvevaSi sistemebSi SeiZleba mimdinareobdes paraboluritipis procesebi.procesebi, romlebic mimdinareoben marTvis Caketil optimalursistemebSi, dafuZnebulni arian (2.6) utolobis tipis klasifikaciazeromlebic safuZvels uyris fiziko-maTematikur midgomasaxal Tanamedrove optimaluri marTvis TeoriaSi.(2.6)-tipis utolobebis klasifikacia udevs safuZvlad agreTveCaketili optimaluri disipatiuri marTvis sistemebissinTezis meTods.2.3. Caketili optimaluri marTvis disipatiurisistemebis sinTeziganvixiloT optimaluri marTvis amocana, romelic mocemuliaSemdegi pirobebiT⎧ ⎛ ⎞J c x du dt,∞ n2 2= ∫ ⎜∑i i+ ⎟⎪ 0 ⎝ i=1 ⎠(2.28)⎨n⎪x&⎪k( t) = ∑ akixi + bku, k = 1, 2,..., n ,(2.29)⎩i=1sadac x=(x 1 ,...x n ) T – obieqtis mdgomareobis kordinatebis veqtoria,U – mmarTveli zemoqmedebaa (marTva) obieqtze; a ki , b k , c i , d –mudmivebia.saWiroa ganisazRvros marTvis kanoni u=u(x 1 ,...x n ) T uwyvetifunqciebis klasSi, romelic uzrunvelyofs (2.29) obieqtis gada-81


yvanas nebismieri sawyisi mdgomareobidan koordinatTa saTaveSida aniWebs minimums (2.28) funqcionals.CavweroT amocana optimaluri regulatoris analizurikonstruirebis mTavari funqcionaluri gantolebis saxiT⎡∂v∂vn n n2 2min ⎢ akixi + bku + ci xi+ du ⎥ 0uk= 1 i= 1 ∂xk∂xki=1⎣∑∑ ∑ ∑ = (2.30)(2.30) gantolebaSi u -s mimarT minimumi miiRweva maSin, rocan1 ∂vu =− ∑ bk. (2.31)2d∂ximisaTvis rom movZebnoT marTva u, aucilebelia moiZebnosfunqcia v=v(x), romelic akmayofilebs kerZo warmoebulian gantolebask = 12∑∑ n n n n∂v1 ⎛ ∂v⎞2akixi − ⎜ ∑ bk ⎟ + ∑ cix10k= 1 i= 1 ∂xk4d k= 1 ∂x= (2.32)k i=1⎝da sasazRvro pirobasv(0)=0 . (2.33)(2.29) sistema (2.31) marTvis gaTvaliswinebiT Rebulobs saxes:nnkk ∑ ik i ∑ ji= 1 2dj=1 ∂xj⎠kb vx & ∂= a x − b , k = 1,2,..., n (2.34)da aris asimptoturad mdgradi. e.i.x(∞)=0. (2.35)klasikuri gagebiT orak-is amocana (2.32) gantolebidandaiyvaneba ψ(x) funqciis moZebnamde sasazRvro pirobiT (2.33).cnobilia, rom Caketili (2.34) optimaluri sistema (2.34)warmoadgens disipatiurs. es gvaZlevs saSualebas SevavsoT (2.32)gantoleba meore rigis Semdegi saxis wrfivi diferencialurigantolebiT.b ∂ v− < 0n n2iaijbji= 1 j=1 2d∂xi∂xj∑ ∑ ∑ , (2.36)romelic warmoadgens (2.5) utolobas, Cawerils (2.34) sistemismarjvena nawilisTvis.⎤⎦82


Caketili optimaluri disipatiuri sistemebis wrfivi obieqtebismarTvis divergencia veqtoruli velisaTvis gansazRvrulia(2.34) gantolebis marjvena nawiliT da warmoadgens mudmiv sididesmTels fazur sivrceSi. (2.36) utolobidan SeiZleba mivi-RoT Semdegi saxis gantoleba:− b ∂ v+ c = 0n n2iaij∑bji= 1 j=1 2d∂xi∂xj∑ ∑ . (2.37)amgvarad, marTvis wrfivi obieqtisTvis orak-is amocanisamoxsna SegviZlia daviyvanoT sistemis amoxsnamde.orak-is amocanis amoxsna daviwyoT (2.38)–(2.40) gantolebaTasistemis pirveli gantolebidan. misi amoxsna saSualebas mogvcemsganvsazRvroT U(x) funqciis struqtura. mocemuli gantolebawarmoadgens mudmivi koeficientebian meore rigis wrfivgantolebas.n n2⎧bi∂ v⎪ ∑aij− ∑ ∑bj+ c=0⎪i= 1 j=1 2d∂xi∂xj2n n n n⎪ ∂v1 ⎛ ∂v⎞2⎨∑∑akixi − ⎜∑bk ⎟ + ∑cix1= 0⎪ k= 1 i= 1 ∂xk4d ⎝ k= 1 ∂xk⎠ i=1⎪⎪⎪v(0) = 0⎩(2.38)(2.39)(2.40)rogorc cnobilia [33], meore rigis gantolebis amoxsnaemyareba am gantolebebis kanonikur saxemde dayvanis SesaZleblobas.(2.38) gantolebis Seesabamis kvadratuli forma.s aqvssaxe:nnQ b tt(2.41)= ∑∑k= 1 i=12sadac bik= biroca i = k; b ik= bb i kroca i ≠ kjamze(2.41) kvadratuli forma SeiZleba daviyvanoT kvadratebisnki i k2Q = ∑ λτ(2.42)i=1cnobili wrfivi gardaqmnebis gamoyenebiTii83


t = c1τ1+ ... + c τ , k = 1,2,..., n.. (2.43)k k nk nTu x i -is nacvlad gardasaxvisy = c1x1 + ... + c x , k = 1, 2,..., n.(2.44)k k kn ngamoyenebiT SemovitanT axal damoukidebel cvladebs y i,, maSin ampirobebis gamoyenebis Sedegad (2.38) gantoleba gardaiqmneba SemdegisaxiTnn21∂ v∑a− ∑λ∑ b + c = 0 . (2.45)ii i ji= 1 2di=1 ∂xi∂xjganvixiloT ufro dawvrilebiT (2.38) gantolebidan (2.45)kanonikur gantolebaze gadasvlis teqnika meore rigis obieqtismagaliTze. am SemTxvevaSi (2.38) gantoleba Rebulobs saxes:∂ v ∂ v ∂ v2 da ( + a ) −b −2bb − b + 2dc=02 2 22 211 22 1 2 1 2 2 2∂x1 ∂x1∂x2 ∂x2(2.46)xolo (2.41) kvadratuli formas, romelic Seesabameba mocemulgantolebis aqvs saxesQ= b t + 2bb tt + b t2 2 2 21 1 1 2 1 2 2 2. (2.47)am formis Sesabamisi matricas aqvs saxe:2⎡ b1 bb ⎤1 2A = ⎢ 2 ⎥ . (2.48)⎣bb2 1b2⎦maxasiaTebel gantolebasA− λE = λ −λb − λb= (2.49)2 2 21 20gaaCnia fesvebi λ 1 =0,2λ = b + b . gantolebidan22 12Ac = λ c ; i = 1,2(2.50)i i i2 2λ 1 =0, λ = b + b , mniSvnelobebisaTvis movZebnoT sakuTriviveqtorebi2 1c=(c21 11 22TT, c ) da c2 = ( c21,c22)).λ 1 =0-Tvis miviRebT gantolebaTa sistemas2⎧ bc1 11+ bbc1 2 12= 0⎨2⎩bbc1 2 12+ b2c12=0(2.51)84


(2.51) sistemis amoxsniT movnaxoT sakuTrivi C i - veqtoris ko-b1ordinatebi. miviRebT c cb1 1=− , c ( c c ) c c c12 112. davuSvaTC 11 =1, sabolood miviRebTanalogiurad,22 1T ⎛ b ⎞ ⎛ b= , = ⎜ , − ⎟ = ⎜1,−⎝ ⎠ ⎝1 11 12 11 12 11b2 b2T⎛ b ⎞1c1= ⎜1,− ⎟⎝ b2⎠2λ = b + b pirobis SemTxvevaSi vpoulobT sakuTrivc 2 veqtrs, romelsac aqvs saxeTlac:C 1 da C 2TT⎞⎟⎠2T12=⎜ ⎟b2c⎛ b ⎞1, .⎝ ⎠sakuTrivi veqtorebi arian orTogonalurebi, mar-1 2( cc ), = 11 ⋅− b ⋅ b = 0b b1 22 1movaxdinoT sakuTrivi C 1 da C 2 veqtorebis normirebacb + b , b + b .2 2 2 21 2 1 21= c2=b2 b1sabolood miviRebT A matricis orTogonalur sakuTrivveqtorebs, romlebsac aqvs saxe:⎛ b2 ⎞ ⎛ b1⎜ ⎟ ⎜b + b b + b2 2 2 21 2 1 21=⎜ ⎟, c⎜⎜ 2b ⎟=⎜1b2c−⎜ b b ⎟ ⎜⎝ + ⎠ ⎝ b + b2 2 2 21 2 1 2⎞⎟⎟⎟. (2.52)⎟⎠cnobilia, rom bazisSi, romelic Sedgeba matricis orTonormirebulisakuTrivi veqtorebisagan, kvadratuli forma daiyvaneba(2.43) kanonikur saxemde. kanonikur gardasaxvas aqvs saxe:⎧ b b⎪t= τ +⎪⎨⎪ b bt = τ +⎪⎩ + +2 112 212 2b1 + b2 b1 + b21 222 212 2b1 b2 b1 b2τ2. (2.53)τTu (2.53) CavsvamT (2.47)-Si SeiZleba davrwmundeT, rom kvadratuliforma daiyvaneba (2.42) saxemde, e.i.285


2 2( 1 2 )Q= b + b y(2.44)-is gardasaxvebi Rebuloben saxes:22⎧ b2 b1⎪ y1 = x2 21−x2 2⎪ b1 + b2 b1 + b2⎨⎪ b1 b2y2 = x2 21+x⎪ 2 2⎩ b1 + b2 b1 + b222(2.54)Tu gaviTvaliswineT, rom ψ(x 1 ,x 2 ) funqciis meore rigis warmoebulebisx i damoukidebeli cvladis mixedviT axal damoukidebely i – cvladze gadasvlisas, (2.44) wrfivi gardaqmnebis meSveobiTgamoiTvleba formuliT∂ v∂x ∂x2 n2=∑cc∂vki iji j ki , = 1 ∂yk∂yi,(2.55)da miviRebT (2.46) gantolebis (2.45) kanonikur formas. mas eqnebasaxe:b + b ∂ v− a −a − c=0. (2.56)2 2 21 222d∂y211 22movaxdinoT (2.56) gantolebis integrirebada ( + a + c)v= y + q ( y ) y + q ( y ), (2.57)11 222 2b1 + b222 1 1 2 2 1sadac q 1 (y 1 ), q 2 (y 1 ) – garkveuli diferencirebadi funqciebia. x 1 , x 2sawyis kordinatebSi (2.57) funqcia Rebulobs saxes:da (11+ a22 + c)⎛( ) 2 bx2 1−bx⎞1 2bx2 1−bx1 2v= bx2 1 1+ b2x2 + q1+ q2 22 2 2 2( b1 + b⎜2 )b1 b ⎟⎝ +2 ⎠ b1 + b22⎛bx2 1−bx ⎞1 2. (2.58)⎜ 2 2b1 + b ⎟⎝2 ⎠amrigad, miRebulia meore rigi marTvis wrfivi obieqtisTvisu(x 1 ,x 2 ) funqciis struqtura.rogorc gamokvlevebma gviCvena [33] im SemTxvevebSi rodesacmarTvis wrfivi obieqtis rigi n>2, SeiZleba (2.38)-is integrirebisSedegad miRebul iqnes u(x) funqciis struqtura. es dakavSirebuliaagreTve (2.38) gantolebis (2.41) kvadratuli formis erTerTTvisebasTan. amovweroT am formis Sesabamisi matrica86


2⎡ b1 bb1 2... bb ⎤1 n⎢2 ⎥bb1 2b2 ... bb2 nA = ⎢⎥ . (2.59)⎢ ... ... ... ... ⎥⎢2⎥⎢⎣bb1 nbb2 n... bn⎥⎦A – matricis maxasiaTebel gantolebas eqneba saxen 2 n−1 2 n−1 2 n−1A− λE = λ −b1λ −b2 λ −... − b nλ = 0 , (2.60)romelsac aqvs fesvebiλ = λ = ... = λ = 0, λ =∑ b .1 2 n−1Sesabamisad, SeiZleba moinaxos iseTi wrfivi gardaqmna (2.44)romelsac (2.38) gantolebas daiyvans Semdeg saxemde:12dnni=12i∂ vb − a − c=0n 2 n2∑ 1 2 ∑ ii. (2.61)i= 1 ∂yni=1(2.61) gantoleba warmoadgens paraboluri tipis meore rigiskerZo warmoebulebini diferencialuri gantolebas. am gantolebisamonaxsns aqvs saxe:nd ⎛ ⎞ 2v= aii c yn q1( y1,..., yn 1) yn q2( y1,..., yn1)n ⎜∑+ ⎟ +−+−b ⎝ ⎠∑i=12 i=11(2.62)sadac q 1 (y 1 ,...y n-1 ), q 2 (y 1 ,...y n-1 ), argumentebis mixedviT diferencirebadifunqciebia.imisaTvis, rom ganvsazRvroT u=u(y 1 ,...y n ), funqciis srulisaxe, aucilebelia (2.42) gantolebaSi x 1 ,...x n kodrinatebidan gadavideTy 1 ,...y n kodrinatebze da CavsvaT masSi (2.62) gamosaxuleba. SedegadmiviRebT erTgvar diferencialur tolobas, romlisTvisacCavataroT koordinatobrivi CaZirvis procedura.kordinatobrivi Calagebis procedura mdgomareobs ucnobifunqciebis q 1 (y 1 ,...y n-1 ), q 2 (y 2 ,...y n-1 )-ebis mudmiviT c gansazRvraSi: esmeTodi SemdegSi mdgomareobs radganac saZebni funqciebiq 1 (y 1 ,...y n-1 ), q 2 (y 2 ,...y n-1 )-ar arian damokidebuli y n kordinataze, maSin(2.62) gamosaxulebis (2.32)-Si Casmis Sedegad diferencialurgantolebaSi vutolebT 0-s am koordinatebis sxvadasxvaxarisxebis koeficientebs. miviRebT diferencialur gantolebaTasistemas, saidanac ganvsazRvravT funqciis struqturas. q 1 (y 1 ,...y n-87


1), q 2 (y 2 ,...y n-1 ), romelic damokidebulia axali r 1 (y 1 ,...y n-1 ), r 2 (y 2 ,...y n-1 )da mudmiv c-ze. CavsvaT funqciebi q 1 (y 1 ,...y n-1 ), q 2 (y 2 ,...y n-1 ),diferencialur gantolebaSi da radganac am SemTxvevaSifunqciebi r 1 (y 1 ,...y n-1 ), r 2 (y 2 ,...y n-1 ) ar arian damokidebuli y n-1 cvladze.nuls gavutoloT y n-1 koordinatebis sxvadasxva xarisxisas koeficientebi.miviRebT diferencialuri gantolebebis axal sistemas...a.S. kordinatobrivi Calagebis bolo stadiaze miviRebTdiferencialur gantolebebis sistemas ucnobi funqciebis K 1 (y 1 ),K 2 (y 1 ) da mudmiva c-s. am sistemis amoxsniT miviRebT mudmivismniSvnelobas da K 1 (y 1 ), K 2 (y 1 ) funqciis saxes garkveuli romeliRacc 1 mudmivamde sizustiT.c 1 -i mudmivas sidide SeiZleba ganvsazRvroT, Tu gaviTvaliswinebT(2.33) sasazRvro pirobas.kordinatobrivi Calagebis proceduris daxmarebiTv=v(y 1 ,...y n ), funqciis sruli saxis gansazRvris Semdgom aucilebeliay 1 ,...y n kordinatebidan x 1 ,..,x n koordinatebze gadasvla.sabolood miviRebT v=v(x 1 ,...x n ), funqcias romlis Casmac (2.31)-Simogvcems u=u(x 1 ,...x n ) saZebn optimalur marTvis kanons.2.4. optimaluri disipatiuri marTvis sistemebis analizurikonstruirebaganvixiloT rigi magaliTebi, romlebic gvaZleven saSualebasSevafasoT marTvis optimaluri disipatiuri sistemebissinTezis meTodis efeqturoba optimaluri regulatorebis analizurikonstruirebis klasikur meTodTan SedarebiT.magaliTi pirveli. movaxdinoT obieqtis, romelic aRiwerebadiferencialuri gantolebaTa sistemiTx& () t = x , x& () t = u(2.63)1 2 2marTvis optimaluri kanonis sinTezi. xarisxis kriteriums aqvsSemdeg saxe88


∞∫2 2 21 2)dt (2.64)01J = ( x + x + u2CavweroT ZiriTadi funqcionaluri gantoleba Semdegi saxiT⎡ ∂u∂u1 2 2 2⎤min ⎢x2 + u + ( x1 + x2+ u ) ⎥ = 0(2.65)u⎣ ∂x1 ∂x22⎦(2.65) gantolebaSi minimumi u-s mimarT miiRweva maSin, roca(2.65) gantoleba miiRebs saxes∂uu =− . (2.66)∂ x2∂x⎛ ∂x⎞2 22x2 − ⎜ ⎟ + x1+ x 2= 0∂x1 ⎝∂x2⎠(2.63) sistema, (2.66) Sekruli marTviT SeiZleba CavweroT saxiT:22(2.67)∂vx& 1() t = x2, x& 2()t =− (2.68)∂ x(2.78) sistemis disipatiuroba CavweroT Semdegi saxiTan gantolebis saxiT∂x∂x∂x− < 0∂x2 21 22∂ v− c = 0∂x22(2.69)(2.70)(2.70) gantolebas gaaCnia kanonikuri saxe, amitomac ar arisaucilebeli x 1 , x 2 , kordinatebidan gadavideT y 1 , y 2 , kordinatebze.(2.70) gantolebis integrirebiT miviRebT1 2v= cx2 + q1( x1) x2 + q2( x1)(2.71)2(2.71) CavsvaT (2.67)-Si miviRebT diferencialur gantolebas.dq dq 1 1 1 1x + x − c x −cx q − q + x + x = 0 (2.72)2 1 2 2 2 2 2 22 2 2 2 1 1 1 2dx1 dx12 2 2 2(2.72) gantolebis mimarT gamoviyenoT koordinatuli Calagebisproceduradqx :1 1− c + = 0(2.73)2 22 1 22dx1x: dq1 22 1dx1− cq = 0(2.74)89


0 1 2 1 2x2 : − q1 + x1= 0(2.75)2 2amovxsnaT (2.75) gantoleba q 1 -is mimarT, miviRebTq = ± x(2.76)1dauSvaT q 1 =x 1 , maSin (2.73)-dan miviRebT gantolebas c 2 =3,romlis amonaxsnic c = ± 3 . davuSvaT c = 3 , maSin (2.74)-dan mivi-RebT1RebT3 2q2 = x1+ r(2.77)2sadac r – erTgvari konstantaa, q 1 , q 2 -is CasmiT (2.71)-Si mivi-miviRebT3 2 3v = x2 + x1x2 + x 2 1+ r(2.78)2 2sasazRvro pirobidan miviRevT, roca r = 0. maSin saboloodmarTvis optimalur kanons eqneba saxe3 2 3v= x2 + x1x2+ x2(2.79)12 2u 3x2x1= − − (2.80)(2.80) marTvis kanoni emTxveva letovis mier miRebul kanons.magaliTi meore. ganvixiloT obieqtisx& () t = x , x& () t =− x + u(2.81)1 2 2 2marTvis optimizaciis amocana kvadratuli funqcionaliT∫J = ( x + x + u )dt2 2 21 2(2.82)CavweroT ZiriTadi funqcionaluri gantoleba SemdegisaxiT⎡ ∂v ∂v ∂v2 2 2⎤min ⎢x2 − x2 + u + ( x1 + x2+ u ) ⎥ = 0u⎣ ∂x1 ∂x2 ∂x2⎦(2.83)minimumi u-s mixedviT (2.83)-Si miiRweva, maSin rodesacmaSin (2.83) – gantoleba miiRebs saxes1 ∂vu =− (2.84)2 ∂ x290


∂v ∂v 1 ⎛ ∂v⎞x − x − + x + x⎝ ⎠2 22 2 ⎜ ⎟ 1 2∂x1 ∂x2 4 ∂x22= 0 (2.85)sistema (2.81) Caketili (2.84) marTviT, SeiZleba warmovadginoTsaxiT1 ∂vx& 1() t = x2, x& 2()t =−x2− (2.86)2 ∂ x(2.86)-gantolebis disipaturobis piroba CavweroT SemdegisaxiT∂x∂xan Semdegi gantolebis saxiT1 ∂ v−1− < 022212 ∂x21+ − c =02∂ v1 22∂x22(2.87)(2.87) gantoleba aqvs kanonikuri forma. misi integrirebisSemdeg miviRebTv= ( c− 1) x + q ( x ) x + q ( x )22 1 1 2 2 1(2.88)Tu gaviTvaliswinebT (2.85) miviRebT diferencialur gantolobasdq dq 2 2x + x −2( c−1) x −x q −( c−1)x −2 1 222 2 2 2 1dx1 dx11 2 2 2−( c−1) q1x2− q1+ x1+ x2= 0(2.89)4(2.89) gantolobis mimarT gamoviyenoT kordinatuli Calagebisproceduraxdq: − c + 2= 0(2.90)2 1 22dx1x: dq1 22 1dx12− cq = 0(2.91)0 1 2 2x2 : q1 + x1= 0(2.92)4amoxsniT (2.92) gantoleba q 1 -is mimarT, miviRebTq1 2x1= ± (2.93)91


davuSvaT q 1 =2x 1 (2.90)-dan miviRebT gantolebas c 2 =4, romlisamonaxsnic aris c=±2. davuSvaT c=2, maSin (2.91)-dan gvaqvsq = 4x + r(2.94)22 1sadac r warmoadgens konstantas. CavsvaT q 1 q 2 (2.88) gantolebaSi,miviRebTv= x + 2x x + 2x + r(2.95)222 1 2 1v-s sasazRvro pirobebidan gamomdinareobs, rom r = 0, maSinsabolood gvaqvseqneba saxev= x + 2x x + 2x22 1 221(2.96)(2.96)-dan gamomdinareobs rom marTvis optimalur kanonsu = - x 2 - x 1 (2.97)marTvis miRebuli (2.97) kanoni zustad emTxveva [25]samuSaoSi miRebul marTvis kanons.magaliTi mesame. ganvixiloT aerodinamikuro damuxruWebisamocana [25] samuSaodan. samarTavi obieqtis maTematikur modelsaqvs saxe.x& 1()t = x2, x2()t = x3& , x& 3( t)= u(2.98)movaxdinoT avtopilotis optimizireba funqcionalis Tanaxmad∞∫J = ( x + x + 5x + u )dt02 2 2 21 2 3CavweroT ZiriTadi funqcionaluri gantoleba Semdegi saxiT⎡ ∂v ∂v ∂v2 2 2 2⎤min ⎢x2 + x3 + u + ( x1 + x2 + 5 x3+ u ) ⎥ = 0u⎣ ∂x1 ∂x2 ∂x2⎦(2.99)(2.100)gantoleba (2.100)-Si minimumi u marTvis mixedviT miiRweva maSin,rocaamasTan gantoleba (2.100) Rebulobs saxes.1 ∂vu =− . (2.101)2 ∂ x3∂v ∂v 1 ⎛ ∂v⎞2 2 2x2 + x3 − ⎜ ⎟ + x1 + x2 + 5x3=0∂x1 ∂x2 4 ⎝∂x3⎠CavweroT exla sistema (2.98) (2.101) marTviT2(2.102)92


1 ∂vx& 1()t = x2, x& 2()t = x3, x& 3()t =− (2.103)2 ∂ x(2.103) sistemis disipatiurobis piroba SegviZlia warmovadginoTutolobis saxiT3∂x∂x∂x ∂x ∂x22 31 ∂ v+ −


22 dr1 1 3 d r1 dr21 2 2cx2 − x2 + x2 2− ( c −5)xdx1 2 dx1 dx14− 2(2.114)1 2 1 2 2 2− ( c −5)x21 r − r1 + x1 + x2=02 4(2.114) toloba gamoviyenoT koordinatuli CalagebisaTvisx23 12 2dx1: dr2 1 22dx14= 0(2.115)dr 1x : c − c − 5 + 1= 0(2.116)( ) 2dr 1x : − c − 5 r = (2.117)1 2 22dx12( )10(2.92) gantolebis amoxsniT r 1mimarT vipoviT(2.119)-Sir1= 2x1romlis amonaxsni aris0 1 2 2x2 : − r1 + x1= 0(2.118)4r1 2x1= ± (2.119)CasmiT, (2.116)-dan miviRebT gantolebas1 22c− ( c − 5) 2+ 1= 0(2.120)4⎡ 3.235 ⎤⎢1.163⎥c = ⎢⎥⎢− 2.199 + 0.863i⎥⎢⎥⎣−2.199 −0.863i⎦CavsvaT (9.117) –Si da amovxsnaT gantolebar 2(2.121)mimarT miviRebT1r ( 2 )22= c − 5 x1 + k, sadac k erTgvari konstantaa. , (2.111) da2r r 1 2(2.113) dan moyenebiT Sesabamisad miviRebT2( )q = c − 5 x + 2x1 (2.122)1 2⎡c2 ⎤ 2 1 2 2q2 = ⎢ ( c −5) − 1 x2 + 2cx1x2+ ( c − 5)x +2 ⎥1k (2.123)⎣⎦2aviRoT (2.121)-Si c=3,235, maSin (2.106) gamosaxulebaSi (2.122) da(2.123) CasmiT da imis gaTvaliswinebiT rom k = 0 vipoviTliapunovis optimalur funqcias Semdegi saxiT94


v= A x + A 5x x + A x x + A x + A x x + A x2 2233 3 23 2 3 13 1 3 22 2 12 1 2 11 1(2.124)sadac A 11= 2,732; A12= 6, 47; A13= 2; A22= 7,84; A23= 5,465; A33= 3, 235;da marTvis optimaluri kanoniu x1 x2x 3= − −2,735 − 3,235 .(2.125)ganvixiloT atmosferoSi balistikuri Sesvlis amocanaromlis Semdegac SesaZlebelia xelovnuri Tanamgzavris gamoyeneba,rogorc cnobilia dajdomis traeqtoria, romelic mTavrdebadedamiwis zedapiris f wertilze, ganisaxRvreba wriulitraeqtoriis i sawyisi wertiliT.balistikuri Sesvla ewodeba xelovnuri TanamgzavrisSesvlis traeqtorias atmosferos zeda fenebSi. davuSvaT, romxelovnuri Tanamgzavri am traeqtoriaze moZraobisas asrulebsdedamiwis garSemo sakmarisi raodenobis brunvebs.aerodinamikuri damuxruWebis amocana mdgomareobs imaSirom minimumamde unda iqnes dayvanili gadaxra wriulitraeqtoriidan. am pirobebSisizustiT, Semdeg iwyeba dajdomis faza.iwertili ganisazRvreba sakmarisinax. 2.1 aerodinamikuri damuxruWeba95


davuSvaT rom programuli moZraoba warmoadgens wriulmoZraobas radiusiT r+ y da siCqariT v= g( r+y)davuSvaT rom balistikuri Sesvlisas sruldeba Semdegipirobebi:1. programuli kuTxe θ sakmarisad mcireaprogramuli siCqare v ≈ 7800 m/wm.0θ =− 1 , da2. simaRlis sruli cvlileba Δ y < 30 km, rac ufro mcireavidre r .3. wevas ar gaaCnia damatebiTi regulireba.4. amwevi Zala sakmarisad mcirea wonasTan SedarebiT.5. winaRobis regulireba Q xorcieldeba aerodinamikuridamuxruWebis zedapiris cvlilebasTan damokidebulebiT.am daSvebebis safuZvelze SevadginoT marTvis obieqtisgantoleba aRmSfoTi zemoqmedebis gaTvaliswinebiT. amasTan mxedvelobaSiunda miviRoT rom :a) cvladebi m, ω,β ar icvlebian,b) Δ x daΔ w cvladebi maTi spesifikidan gamomdinare gamoiyofawrfivi modelis gantolebidan aRmSfoTi zemoqmedebisgaTvaliswinebiT.g) Δ y cvladi ar axdens mniSvnelovan zegavlenaswevris cvlilebaze.v cosθr+yaseTi daSvebis safuZvelze moZraobis gantoleba miiRebsSemdeg saxes.⎛ Q ⎞Δ v&=−gΔθ−gΔ ⎜ ⎟ ,⎝mg⎠2Δ & θ = Δv,r+yΔ y&= vΔθ,⎛∂f⎞ ⎛∂f⎞Δ w&= k ⎜ ⎟ Δ v+ k ⎜ ⎟ Δy,⎝∂v⎠ ⎝∂y⎠96


⎛ Q ⎞romlebic marTvis funqciis rolSi aris sidide ξ =Δ ⎜ ⎟⎝mg⎠gantoleba iyofa or jgufad, romelTaganac pirvelis integrirebaSesaZlebelia meorisagan damoukideblad. meore sistemasaWiroa siTbos nazrdis SefasebisaTvis korpusis SigniT,avtopilotis mier sistemis stabilizaciis dros, romelic momavalSicunda ganisazRvros. Cven unda davuSvaT rom es daSvebebisruldeba.amrigad gamartivebis Sedegan gvaqvsΔ v&=−gΔθ−gξ,⎫2 ⎪Δ & θ = Δv,⎬r+ y ⎪Δ y&= vΔθ,⎭ ⎪(2.126)romelic warmoadgens marTvis obieqtis maTematikur models,(2.126) gantoleba gardavqmnaT ise rom SevinarCunoT mxolodcvladiΔymiviRebTΔ 2g2vg&&& y =− y .r y Δ &+ − r+y ξSemovitanoT axali marTva da axali cvladebiy y y&y && y yΔ =1, Δ =2,Δ =3⎫ ⎪⎬2g2gv− Δy&− ξ = ζ,r+ y r+ y ⎪⎭(2.127)maSin maTematikuri modeli miiRebs saxe.y&1= y2⎫ ⎪y&2= y3⎬ y&3= ζ ⎪ ⎭(2.128)avtopilotis optimizacia movaxdinoT funqcionalis∞2 2 2 2( 1 2 2 3 3ζ0I = ∫ y + a y + a y + dt.(2.129))mixedviTTvis miviRebT:AaβkoeficientebisaTvis warmoebuli funqciisa-97


0= 1−A2130= a + 2A −A22 12 230= a + 2A −A23 23 330 = A11 −A13A330 = A12 −A13A330 = A + A −A A13 22 23 33(2.130)mniSvnelobaTa ricxviTi monacemebi sxvadasxva winiTikoeficientebis SemTxvevaSi mocemelia cxrilSi 2.1cxrili 2.1SemTxveva 1SemTxveva 2SemTxveva 3a= 1, a = 52 3a2= a3 = 5a= 10, a = 52 3A , 11A22AA1122= 2,8;= 7,75AA1122= 3,5;= 10,75AA1122= 4, 27;= 13,50A33A12AA3312= 3, 2;= 3, 25AA3312= 3, 4;= 3, 45AA3312= 3, 60;= 3, 65A13A23AA1323= 1, 04;= 2,75AA1323= 1, 04;= 3, 45AA1323= 1, 04;= 4,15Caketil sistemas aqvs saxey&y&=y1 2=y2 3y&= p y + p y + p y 33 1 1 2 2 3sadac avtopilotis gaZlierebis koeficientebi toliap = − A , p =− A , p =− A . 31 13 2 23 3 3(2.131)ganxiluli magaliTidan naTlad Cans is upiratesobaniromelsac gvaZlevs optimaluri marTvis sinTezis meTodi orak –is klasikur meTodTan SedarebiT sinamdvileSi, samuSaoSi liapunovisoptimaluri funciqciis koeficientebis gansazRvrisaTvisaucilevelia amovxsnaT rikatis tipis algebrul gantolebaTasistema.98


nax. 2.2 aerodinamikuri damuxruWebis gardamavali procesi99


gardamavali procesis mrudebi warmodgenilia nax 2.22⎧ 0= 1−A13⎪2⎪0 = 1 + 2A12 − A232⎪ 0= 5+ 2A23 −A23⎨⎪0= A11 − A13A33⎪ 0 = A12 − A13A33⎪⎪ ⎩0= A + A −A A13 22 23 33sadacAij- saZebni koefincientebia, aseTi sistemis amoxsnamoiTxovs specialuri ricxviTi meTodebis gamoyenebas, romelsacyovelTvis ar mivyavarT damakmayofelebel Sedegebamde. zemoTCamoyalibebul magaliTSi (2.124) liapunovis optimaluri funqciiskoeficientebis misaZebnad da (2.125) marTvis kanonis Sesabamisadaucilebelia ramodenime umartivesi diferencialuri gantolebebisda (2.120) meoTxe rigis algebruli gantolebis amoxsnarac gacilebiT advilia da gamoTvlebis zust Sedegs iZlevaSedarebisaTvis moviyvanoT (2.124) liapunovis optimalurifunqciis koeficientebis mniSvnelobebi da [25] samuSaoSi miRebuliSedegebi. Sedegebi mocemulia cxrili 2.1.cxrili 2.1A11A12A13A22A23A33samuSao [25] 2,8 3,25 1,04 7,75 2,75 3,2mag. 2 2,732 6,47 2 7,84 5,465 3,235cxrilSi mocemuli monacemebis analizisas, SeiZleba gavakeToddaskvna rom A 11, A22,A 33, koeficientebis mniSvnelobebi, ganxilulsamuSao [25] da magaliT 2-Si mcired gansxvavdebian erTmane-Tisagan, maSin rodesac , , A , koeficientebi mniSvnelovnadA12A1323ganxvavdebian erTmaneTisagan, radganac (2.120) meoTxe rigis al-100


gebruli gantolebis amoxsnis sizuste arsebiTad ufro maRaliavidre eqvsi algebruli gantolebisagan Semdgari sistemisa.amgvarad SeiZleba vamtkicoT rom wrfivi n>3 rogis obieqtebisTvisacki optimaluri disipatiuri sistemebis marTvis sin-Tezis upiratesoba iqneba ufro metad metad maRali orak –isklasikur meTodTan SedarebiT. avRniSnoT rom am TavSi Camoyalibebuliaxali midgoma, romelis eyrdnoba Caketili disipatiurisistemebis fundamentalur fizikir Tvisebas, saSualebasiZleva Zireulad wamovwioT orak –is problemis gadawyveta. rasakvirveliaes midgoma moiTxovs Semdgom ganviTarebas marTvissistemebis sxvadasxva klasebTan mimarTebaSi.3. arawrfivi marTvis sistemebis sinTezi sinergetikuliTeoriis gamoyenebiTwinamdebare TavSi ganxilulia arawrfivi regulatorebisanalizuri konstruirebis amocanis gadawyveta sinergetikuli me-TodiT, romelic emyareba mdgomareobaTa sivrceSi sistemis koordinatebsSoris funqcionaluri damokidebulebis-atraqtorebisSemoRebas, romlebzedac obieqtis bunebrivi Tvisebebi saukeTesodeTanadeba marTvis teqnologiur moTxovnebs. sinergetikulimidgomis mixedviT damyarebulia Sesabamisoba invariantulmravalsaxeobebsa da saoptimizirebel funqcionalebs Soris.invariantuli mravalsaxeobebis Semotana Sekrul sistemasaniWebs zogad globalur Tvisebebs da saSualebas gvaZlevs gamovavlinoTmsgavseba im sxvadasxva fizikur movlenebs Soris,romlebic mimdinareoben sxvadasxva bunebis marTvis obieqtebSi.am movlenebis warmodgena maTematikur enaze - sistemis diferencialurigantolebebis pirveli (kerZo) integralebis erToblioba-asaxavsmarTvis procesebSi zogadi principis Senaxvas.marTvis arawrfivi sistemis sinTezis problemaSi mocemulikoncepcia iyenebs invariantebis da invariantuli damokidebulebebiscnebas. Tumca, invariantebis klasikuri Teoriisagan gansxvavebiTis efuZneba, pirvel rigSi, disipatiuri struqturebis101


Teorias da meore rigSi, invariantebis-atraqtorebis (sinergiebis)miznobriv Semotanas, romlebSic xorcieldeba sistemis mimarTuliTviTorganizacia.sinergetikul meTodSi gaerTianebulia kavSiri invariantulmravalsaxeobebsa da sistemebis saoptimizacio funqcionalebsSoris [35]. sinergetikis TvalsazrisiT optimizaciis meTodieyrdnoba or warmodgenas: funqcionalis sinergetikul interpretaciasda uSualo kavSiris damyarebas orak-is Teoriis xarisxiskvadratul da sxva kriteriumebs da Tanmxleb funqcionalebs.unda avRniSnoT, rom am midgomis gamoyeneba arawrfivi sistemebismarTvis analitikuri konstruirebis amocanebSi efuZnebainvariantul mravalsaxeobebs da ara romelime optimalurobiskriteriums, romlebsac aq gaaCniaT Tanmxlebi, meoradi xasiaTi.mizanSewonilia ganvixiloT axali sinergetikuli midgomisSesaZleblobani, romlebic dafuZnebulia Semotanili invariantulimravalsaxeobebis-atraqtorebis koncepciaze, raTa gamovleniliqnes axali perspeqtiuli mimarTulebebi farTo klasis arawrfivi,mravalganzomilebiani da mravalkavSiriani obieqtebismarTvis sistemebis konstruirebis problemebis gadawvetisaTvis.am problemis arsi mdgomareobs uaryofiTi da dadebiTi arawrfiviukukavSirebis generaciis analitikuri meTodebis damuSavebaSi.3.1 marTvis sistemebis sinTezis amocanamarTvis TeoriaSi sinergetikis ideebis gamoyenebisaTvisaucilebelia gamovavlinoT TviTorganizaciis ZiriTadi Tvisebebis,arawrfivoba-gaxsniloba-kogerentuloba, Sesabamisoba. maTSoris sistemebis gaxsniloba warmoadgens pirvel xarisxovansmarTvis amocanebisaTvis.marTvis standartuli amocanaSi obieqti aRiwereba diferencialurigantolebebiT, romelTa SemadgenlobaSic saZebniU(t) marTva, aRmgznebi q(t) da (SesaZlebelia) damkveTi M(t) zemoqmedebebi.am Zalebis moqmedebiT obieqtma SeiZleba Seasrulos102


Sesabamisi moZraoba. miTiTebuli garegani zemoqmedebis Sedegadsistemas SeiZleba gaaCndes specifikuri funqcionireba an struqtura,romelic mTlianobaSi ar eTanadeba g.hakenis [41] mier CamoyalibeliTviTorganizaciis movlenis gansazRvrebas. aqedan gamomdinareobs,rom marTvis amocanis aseTi formulireba jer kidevarasakmarisia TviTorganizaciis movlenis warmoqmnisaTvis.zemoT aRwerili sqemidan `obieqti-garegani Zalebi~ Tvi-Torganizaciis principze gadasvlisaTvis saWiroa am Zalebis gamoricxva.amisaTvis fazuri sivrcis `SekumSva-gafarToebis~ principisganxorcielebisaTvis, saWiroa gavafarToviT sistemis sawyisigantoleba `obieqti-garegani Zalebi~ ise rom, sistemis gantolebaSiCarTuli garegani Zalebi aRmoCndnen misTvis Sinaganni.Sedegad axali, gafarToebuli sistemisaTvis misi gantolebebiSeiZleba iqcnen TviTorganizaciis gantolebebad e.i. naCvenebi gafarToebisSedegad SeiZleba gadavideT sistemis organizaciidanmis TviTorganizaciaze. am kanonebma, romlebic warmoadgenen regulatorisgantolebebs unda uzrunvelyon Sekruli sistemis`obieqti-marTvis kanoni~ sasurveli dinamikuri Tvisebebi. maSinaxal gafarToebuli sistemebisaTvis `obieqti-regulatori~ mizanSewoniliagamoviyenoT damokidebulebebi, romlebic axasiaTebensinergetikis TviTorganizacias zemoT xsenebul TvisebebisSesabamisad sxva sityvebiT, sawyisi sistema, romelic Sedgebadinamikuri obieqtisagan da masze moqmedi garegani Zalebis (mar-Tveli, damkveTi da aRmgznebi zemoqmedebebi), pirdapiri da ukukavSirebisSedegad gardaiqmnebian axal gafarToebul sistemad.amasTan, pirvelsawyisi zemoqmedebebi gareni Zalebi sawyisi obieqtismimarT, gadaiqcevian gafarToebuli sistemis Sinagan Zalebad.aseTi gafarToebuli sistema namdvilad xdeba Ria (TermodinamikuriazriT) da misiT moxdeba Sesabamisi wyarodan energiisan nivTierebis gadineba.amrigad, marTvis problemebSi, romlebic dafuZnebuli arianTviTorganizaciis kooperatiul procesebze, sinergetikulimidgomis gamoyenebisaTvis aucilebelia marTvis sawyisi amocani-103


dan, romelic moicavs Tavis TavSi obieqtis gantolebas da gare-Se Zalebs gadavideT amocanis gafarToebul formirebaze im azriT,rom zemoT naxsenebi Zalebi gadaiqcevian (Caketili) sistemisSinagan urTierTqmedebebad. amisaTvis saWiroa garegani aRmznebiq(t) da damkveTi zemoqmedeba M(t) warmovidginoT rogorc damatebiTidiferencialuri gantolebebis amonaxsnebi, romlebic aRwereninformaciul models da amiT movaxdinoT maTi `Calageba~gafarToebuli sistemis saerTo struqturaSi. Semdgom TviTmarTvis problema aucilebelia CamovayaloboT rogorc gafar-Toebuli sistemis komponentebs Soris urTierTmoqmedebis kanonebismoZebnis amocana, romlebic uzrunvelyofen masSi TviTorganizaciisprocesebis warmoSobas. konkretulad es problemadaiyvaneba Caketili marTvis Sesabamisi kanonebissinTezamde gafarToebuli sistemis mdgomareobis koordinatebisfunqciaSi. aqux ( ,..., x, w,..., w)1 n 1w1,...,w μinformaciuli modelebis koordinatebiadamkveTi da aRmgznebi zemoqmedebebisas, romlebic Cawerilni ariandamatebiTi diferencialuri gantolebebis saxiT. sistemasTanenergiis an nivTierebis gadadinebiT SegviZlia SevqmnaT arawonasworulisituacia, romelic aucilebelia TviTorganizaciis mimarTuliprocesebis warmoqmnisaTvis. swored sawyisi sistemisgafarToeba da TviTorganizaciis gantolebebis formireba gvaZlevssaSualebas davamyaroT kavSiri sinergetikis ideasa da arawrfivisistemis marTvis sinTezis problemas Soris invariantulidamokidebulebebis safuZvelze. aqedan gamomdinareobs,rom marTvis, sinergetikuli Teoria pirvel rigSi Caketili mar-Tvis sistemebis sinTezis Teoriaa, romelic dafuZnebulia sxvadasxvabunebis sistemebSi TviTSeTanxmebuli, kooperatiuliprocesebis formirebaze. arawrfivi dinamikuri sistemebis sinTezisproblemaSi fazuri moculobis gafarToeba warmoadgens sinergetikulimidgomis ZiriTad ideas-gafarToebuli diferencialurisistemebis formireba, romlebic asaxaven damkveTi zemoq-n104


medebis damuSavebis procesebs, SeSfoTebebis CaxSobas, optimizacias,koordinatebze dakvirvebas da a.S.Camoyalibebuli sinergetikuli meTodis debulebebi analizuradase Caiwereba. obieqtis sawyis diferencialur gantolebasaqvs saxe:sadac x 1,..., x n,marTvebia,x&k( t) = fk( x1,..., xn) + Mk( t); k = 1,2,... m−1;m≤nx&k+ 1( t) = fk+ 1( x1,..., xn) + uk+ 1+Mk+1( t);..................................................................................x&( t) = f ( x ,..., x ) + u + M ( t),n n 1 n n nM1(t),..., Mn( t)- aRmSfoTi zemoqmedebaa.(3.1)– obieqtis mdgomareobis koordinatebia, uk+1,..., un,-(3.1) saxis gantoleba aRwers sxvadasxva bunebis farTo klasisdinamikuri obieqtebis yofaqcevas. (3.1) – gantolebaTa sistemisarCeva masSi wrfivi saxiT Semavali mmrTveli zemoqmedebisasgvaZlevs saSualebas naTlad vaCvenoT agregirebuli regulatorebisanalizuri konstruirebis meTodis konstruqciuloba (arak).aseTi arCevani ar zRudavs gansaxilveli modgomis zogadobas,miT umetes radganac arawrfivi obieqtebis marTvis realuriamocanebis umetesoba Cveulebriv SeiZleba daviyvanoT (3.1) tipissistemamde mdgomareobaTa sivrcis gafarToebis xarjze. arawrfivimarTvidan wrfivi marTvis saxeze gadasvlisaTvis SeiZleba magaliTadmmarTvel zemoqmedebad gamoviyenoT sawyisi amocanismarTvis veqtoris cvlilebis siCqare an sxva meTodebi.Semdgom etapze (3.1)-sistemas emateba μ - raodenobis gantolebebi,romlebic dakavSirebulia progrozirebis da SeSfoTebebisCaxSobis problemasTanw& = g ( w,..., w , x ,..., x ), j = 1,..., μ;(3.2)ji1 μ 1(3.2) - gantolebis agebisas warmoiqmneba ori damoukidebeli damniSvnelovani amocana: pirvel rigSi realuri M ( t),..., M ( t ) aRwerarogorc zogierTi diferencialuri gantolebis kerZo amonaxsnebisa,meore rigSi sawyisi obieqtis (3.1) gantolebebsa da SeS-nkn105


foTebebs Soris kavSiris formireba, ganvixiloT es ori mniSvnelovaniamocana cal-calke.aRSfoTebebis realurad aRwerisaTvis avirCioT SesabamisadtalRuri gantolebebi naxevraddeterminirebuli gamosaxulebebissaxiT.[ ]M ( t) = W M ( t), M ( t),..., M ( t); c ,..., cr(3.3)1 1 1 1sadac Mi(), t i= 1,2,..., n – cnobili funqciebia, Cj, j = 1,2,..., r –ganusazRvreli parametrebia romlebsac SeuZliaT drois nebismiermomentebSi naxtomiseburad cvalon Tavisi uban-uban uwyvetimniSvnelobebi, (3.3) gantolebebSi cnobili funqciebisMi(), t nakrebiunda aRwerdes SeSfoTebebis yvela im talRur formas, romlebicmoqmedeben obieqtze. realuri SeSfoTebebi SeiZleba warmovadginoTwrfivi talRuri saxiT.rM () t = ∑c M () t , j = 1,..., μ;(3.4)1 1k1kk = 1rogorc gamosaxulebidan Cans, SeSfoTeba Sedgeba wrfivi kombinaciebisada drois uwyveti funqciisagan. SeSfoTebebis talRuriwarmodgena iZleva saSualebas ganisazRvros Mi(), t cvlilebis xasiaTibazuri Mik(), t fazuri funqciis arCevis meSveobiT droismokle intervalevSi. amave dros Mi(), t cvlilebis sidide rCebaucnobi, radganac damokidebulia ucnobC ik– koeficientebze,romlebsac gaaCnia uban-uban uwyveti xasiaTi. umetesi informaciaaRSfoTebaze swored drois mokle monakveTebze xSirad sainteresoa.rodesac statikuri aRwera ar aris efeqturi, ufromisaRebi iqneba SeSfoTebebis warmodgena talRuri saxiT. ammniSvnelovan SemTxvevebSi SesaZlebelia martivad SevarCioTMik(), t bazuri funqciebis konkretuli formebi obieqtze moqmedirealuri SeSfoTebebis aproqsimaciisaTvis. bazuri funqciebisSerCevis Semdeg saWiroa gadavideT aRSfoTebis talRuri funqciisstruqturis-dgomareobis modelis formirebaze garkveulidiferencialuri gantolebebis sistemis saxiT. uwyveti funqci-106


ebis aRwera zogierTi diferencialuri gantolebebis amonaxsnissaxiT marTvis TeoriaSi pirvelad wamoayena v.s. kolebakinma. aR-SfoTebis ganzogadoebuli warmodgena diferencialuri gantolebebissaxiT Caiwereba Semdegi wesiT.& ( ) & ( ,..., )(3.5)wjMt = gjmw1w μjjsadac w () t = M () t (3.4) gamosaxulebis wrfivi talRuri aRwerisaTvisSeiZleba warmodgenil iqnes magaliTad kanonikuri saxiTw& 1( t) = w2, w& 2( t) = w3,..., w&r−1( t) = wr,w&( t) =−aw −a w −... −a w .n1 1 2 2(3.5) mdgomareobis modelis safuZvelze SeiZleba gadavideTkavSiris gantolebis (3.2) formirebaze, rac warmoadgens zemoTnaxsenebi meore mTavari amocanis Sinaarss. aRvweroT am amocanismokle Sinaarsi. SeSfoTebebis(3.5)- modelis gantolebidan (3.2)kavSiris gantolebaze gadasvlis ramodenime xerxi arsebobs.Sesabamisi (3.2) gantolebis struqturis SerCeva moqmedebs sin-Tezirebuli dinamikuri regulatoris struqturaze, romelic SeesabamebaSeSfoTebebs. cxadia, rom (3.2)-Si kavSiris gantolebaSimizanSewonilia SevitanoT sawyisi obieqtis is x 1,..., xn,koordinatebi,romelTa warmoebulebic x& 1( t),..., x& ( t)(3.1) gantolebis Tanaxmad,Seicavs marjvena nawilSi Sesabamis M 1( t),..., Mn( t ) aRSfoTebebs.sinergetikuli [35] meTodis mixedviT x 1,..., xn,koordinatebi SeiZlebavaintegroT, rogorc garkveuli `Sinagani~ marTvebi, romelTamier gansazRvruli (mag. nulovan) mniSvnelobis miRwevisas(3.2) kavSiris gantoleba gadadis SeSfoTebebis (3.5) modelze. eski swored niSnavs regulatoris mier moqmedi SeSfoTebebis `STanTqmas~.rasakvirvelia aseTi saxis `Sinagani~ marTvis arCeva, romlisdrosac SeSfoTebebis STanTqma xorcieldeba ramodenimemeTodiT, maT Soris kerZod optimaluri marTvis safuZvelze.(3.2) kavSiris gantolebis arCevis Semdgom vRebulobT diferencialurigantolebebis gafarToebul saxes:nrr107


w&j= gi( w1,..., wμ, x1,..., xn),x&i( t) = fi( x1,..., xn) + wi,x&i+ 1( t) = fi+ 1( x1,..., xn) + wi+1+u i + 1;...................................................x&( t) = f ( x ,..., x ) + w + u .n n 1 n n nj = 1,..., μ;i = μ + 1,..., m−1...................................................(3.6) gantoleba saSualebas gvaZlevs CamovayaliboTmarTvis kanonebis sinTezis amocana, romelic M1( t),..., Mn( t)(3.6)ui+1,..., unzemoqmedebebsaxSobs da uzrunvelyofs Caketili sistemis mocemuldinamikur Tvisebebs. saWiroa movaxdinoT sinTezi iseTiuu (1,..., u m)marTvis veqtorisa, romelic uzrunvelyofs gafarToebuli obieqtisgadasvlas nebismieri sawyisi mdgomareobidan Tavdapirveladmravalsaxeobebis ψ &s( x1,..., xn, w1,..., w μ) = 0 gadakveTaze, xoloSemdgom maT gaswvriv moZraobaze mocemul mdgomareobaSi. ker-Zod, mdgomareobaTa gafarToebuli sivrcis, koordinatTa saTaveSi.amasTan, Caketili sistemis traeqtoriebze SeiZleba miRweuliqnes romeliRac saoptimizirebuli funqcionalis minimumi,an dakmayofilebul iqnes xarisxis pirveladi maCveneblebi. agreTve,garantirebul iqnes moZraobis asimptoturi mdgradobagarkveul areSi an mTlianad.CavTvaloT, sinTezirebuli sistemebis gamomsaxveli wertilebismoZraoba unda akmayofilebdes funqcionalur gantoleba-Ta sistemas:Tsψ&s( t) + ϕs( ψs) = 0, s = 1,2,..., m,Ts> 0,(3.7)sadac ψs( x1,..., xn, w1,..., w μ) -romeliRac agregirebuli makrocvladebia.amasTan ϕ ( ψ ) funqciebi unda akmayofilebdnen Semdeg piro-ssbas: ϕ (0) = 0 da ϕ ( ψ ) ψ > 0 nebismieri ψ ≠ 0 -Tvis e.i. isini xdebians s s0-is toli mxolod ψs= 0 mravalsaxeobebze, romelTa mimarTacsistema (3.7) asimptoturad mdgradia mTlianobaSi. amas garda,ssfunqcia ϕ ( ψ ) SeirCeva imgvarad, rom garda (3.7)-is asimptoturobisa,uzrunvelyofil iqnes gamomsaxveli mimzidavi wertilisxarisxis sasurveli maCveneblebi mimzidvel mravalsaxovnebazeψ ( ,..., , ,..., ) = 0 s = 1,2,..., m,sx1 xnw1w μs108


agreTve uzrunvelyos (n-m) dekompozirebuli marTvis sistemisdinamikuri Tvisebebi. gw-s moZraobisas mravalsaxeobebis ψs= 0gadakveTis gaswvriv mocemuli saboloo mdgomareobaSi.funqcia ϕ ( ψ ) arCevisas (3.7) gantolebebSi gansakuTrebulissSezRudvebi ar aris. mTavaria, rom swored makrocvladebiasaxavdnen mravaldoniani sinTezirebadi sistemebis sinergetikuli(kooperatiul, kogerentul) Tvisebas. aqedan gamomdinareobs,ssrom ϕ ( ψ ) arCevisas SeiZleba metad sasargeblod gamoyenebul iqnesfizikuri, ekologiuri da sxva sistemebis cnobili kanonzomierebebi,romlebSic yvelaze ufro cxadad gamovlindeba moqmedebebiserTianoba da urTierTkavSiri, am sistemaSi energiiskargvis minimizacia da a.S. kerZod aseT kanonzomierebebs ganekuTvnebiangantolebebi xarisxobrivi arawrfivobebiTrksψs ∑ ksψsk = 1ψsT & () t + a = 0, s = 1,2,..., m.(3.8)sinergetikuli TvalTaxedvidan (3.7) funqcionaluri gantolebebi,uSualod dakavSirebulni arian (3.8) evoluciur gantolebebTanda atraqtorebTan ( ψs= 0) , romlebic aRweren sistemismoZraobis safiniSo etapebs. am etapebze Zireulad izrdeba determinaciisTviseba. cnobilia [38-42], rom evoluciuri gantolebebisumetesoba warmoadgens xarisxobriv an eqsponencialur ise-Ti saxis damokidebulebebs, romelic gaaCnia mecnierebaSi far-Tod cnobil bernulis gantolebas.rTsψ& s() t + a1sψs + a1sψs= 0, s = 1,2,..., m,r = 2,3,...,(3.9)romlis amonaxsnir−1 a1sψs() t =,1−ra1s( a1sψs0+ ars)exp( r−1)t−a rsTsroca t → 0 , a1> 0, a > 0, eqsponencialurad miiswrafis ψ = 0 -ken.Tu davuSvebT r = 2 ,srsa1s< 0, ars> 0, maSin bernulis (10.9) kvadratuligantoleba gardaiqmneba logistikur gantolebad, romlissamonaxsnic eqsponencialurad miiswrafvisa1sψs= mniSvnelobiars109


sken. aseTi gantoleba aRwers, kerZod, ekologiur wonasworobas,xolo misi makrocvladebi ganzogadoebulad asaxaven rTul kooperatiulprocesebs.(3.7)-sistemis gansxvavebul Taviseburebas warmoadgens pirvelitrivialuri integralebis sruli ψ s= B s, s = 1,2,..., m,an nawilobiTisψ = 0 erTobliobis arseboba. es niSnavs rom am sistemisaTvisSeiZleba ganisazRvros albaTobis ganawilebis zogadikanoni:t⎡m1 ∂ϕ⎤sP= exp ⎢ψs( ψ1,..., ψm) + ∫ ∑ dt⎥,⎣0 s−1Ts ∂ψs ⎦sadac ψ ( ψ1,..., ψm)– pirveli integralebis nebismieri funqciaa,maSin albaTobis simkvrive mravalsaxeobebis gadakveTis midamoSiψ = 0 iqneba tolist⎛m1 ∂ϕ⎞sP(0,...,0) = P0exp ⎜∫∑dt ⎟.⎝ 0 s=1 Ts ∂ψs ⎠saq miRebuli gamosaxulebebi ϕ ( ψ ) funqciis arCevis Semdgom gvaZlevssaSualebas, gamovTvaloT sistema (3.7)-is Sesabamisi alba-TobaTa simkvrive. cxadia, rom ϕ ( ψ ) mizanSewonilia avirCioTise, rom simkvrive P iyos maqsimalurad SesaZlebeli SesabamisiSezRudvebis gaTvaliswinebisas, romlebic dadebuli arian funqciebisϕ ( ψ ) saxeobaze, zogierTi damatebiTi kriteriumebisa dasswinapirobebisagan gamomdinare.sses debuleba mTlianad eTanxmeba albaTobis maqsimalurisimkvrivis princips, e.i. fazuri sivrcis SekumSvis maqsimaluradSesaZlo siCqares. ase mag. sistema (3.7) bernulis (r=3) kuburigantolebis (3.9) saxiT gaaCnia albaTobaTa simkvrivet⎡m1⎤2P3 = exp ⎢ψψ (1,..., ψm) + ∫∑( a1s + 3 a3sψs) dt ⎥,⎣0 s=1 Ts⎦romelic metia kvadratuli (r=2) bernulis gantolebis simkvrivesTans110


t⎡m1⎤P2 = exp ⎢ψψ (1,..., ψm) + ∫∑( a1s + 2 a2sψs)dt⎥,⎣0 s=1 Ts⎦SedarebiT. es niSnavs, rom kuburi gantolebebisaTvis fazurimoculobis SekumSvis siCqare iqneba meti fazuri sivrcis gansakuTrebiTgare aresaTvis. analogiurad SeiZleba Sevafasod sxvasaxis funqcionaluri gantolebebis (3.7) albaTobaTa simkvrivesinergetikuli sistemebis sinTezis amocanis gadawyvetisas.rasakvirvelia, rom bernulis gantolebis garda, funqcionalurigantolebebis agebisaTvis SeiZleba gamoyenebul iqnes sxva bunebrivisistemebis cnobili evoluciuri gantolebebi.ganxiluli sinergetikuli midgoma, SeiZleba gadmocemuliiqnes optimaluri marTvis Teoriis terminebiT, amisaTvis gamoiyenebastandartuli variaciuli gamoTvlis meTidebi. vuCvenoT,rom ZiriTadi funqcionaluri gantolebebi (3.7) warmoadgenenSemdgomi ganzogadebuli saoptimizirebuli funqcionalis Sesabamisieiler_lagranJis gantolebebs.∞mm⎡ 2 2 2 ⎤J∑= exp ∫ ⎢∑ϕs ( ψs)+ ∑Tsψs( t) ⎥dt,(3.10)0 ⎣ s= 1 s=1 ⎦sadac m – marTvis veqtoris ganzomilebaa. cxadia, rim gantolebebi(3.7)_(3.9) saxiT gamoyofen eqstremalebis qvesimramles,romlebic aniWeben upirobo minimums (3.10) funqcionals. ϕ ( ψ )(3.10) funqcia integralqveSa gamosaxulebaSi unda akmayofilebdesSemdeg pirobebs:1. erTniSnianoba, uwyvetoba da diferencireba ψs–is yvelamniSvnelobisaTvis2. ϕs(0) = 03. ϕ ( ψψ ) > 0 nebismieri ψ ≠ 0 –saTvisssssxvanairad, rom vTqvaT, funqciebi ϕ ( ψ ) am pirobebis Sesrulebisasiqnebian imave niSnis, rogorc aqvs ψs= 0 . ganvsazRvroTn∂ψs∂ψs( x1,..., xn)funqciis sruli warmoebuli = ∑ x&k( t ).dt k = 1 ∂xsskss111


ix&() t –is CavsvaT nacvlad arawrfivi obieqtis sawyisi diferencialurigantolebebis marjvena nawilebi, kerZod skalarulimarTviT (m=1)x&i( t) = fi( x1,..., xn);i = 1, 2,..., n−1;x&( t) = f ( x ,..., x ) + u,n n 1 nn∂ψ∂ψ( x1,..., xn)∂ψmaSin miviRebT: = ∑ fk( x1,..., xn)+ u.dt ∂x dxk = 1k nvariaciuli aRricxvaSi (3.10) cnobilia invariantobis TvisebissaxelwodebiT. bolo gamosaxulebis gaTvaliswinebiT SeiZlebafunqcionali (3.10) SeiZleba Caweril iqnes Semdegi saxiT:∞2⎡m2 2⎛∂ψ∂ψ⎞ ⎤J∑= ∫ ⎢ϕ ( ψ) + T ⎜∑fk+ ⎟ ⎥dt,0 ⎢ ⎝ k = 1 ∂xk ∂xn⎣⎠ ⎥⎦cxadia, rom Tanmxlebi optimizirebuli funqcionalebis esforma (3.10) asaxavs, rogorc sawyisi obieqtis, aseve misi sistemismarTvis zogad Tvisebebs. es niSvavs rom gansaxilvel meTodSisaoptimizirebeli funqcionali winaswar ar aris gansazRvruli,rogorc es xdeba orak-is standartul meTodSi, aramed konstruirdebaSesabamisi funqciebis ϕs( ψs) da ψs( x1,..., xn) arCevis mixedviTobieqtis gantolebebis gaTvaliswinebiT. aseTi midgoma sa-Sualebas gvaZlevs gaviTvaliswinoT sawyisi obieqtis Tvisebebi,radgan xarisxis kriteriumis arCevis etapze garegani postulirebadikriteriumisa `Tavsmoxvevam” da obieqtis Tvisebebis ignorirebamSeiZleba migviyvanos arawrfivi obieqtebSi sawinaaRmdegoan sulac miuRebel gardamavali procesebis mimdinareobamde.am azriT optimizirebuli funqcionalis formireba obieqtisgantolebebis gaTvaliswinebiT eTanxmeba meqanikaSi cnobili gausispincips. rasakvirvelia, rom amasTan obieqti unda gadataniliqnes sawyisi mdgomareobidan mocemul mdgomareobaSi, radganxdeba moZraobis marTvis sinTezi. gansxvaveba mdgomareobs (3.10)funqcionalis makrocvladebisψ smimarT formirebaSi, romlebicwarmoadgenen mdgomareobis koordinatebis erTgvar arCeviT agregatebs.es amocana cnobilia saoptizirebeli (3.10) funqcionale-112


is, agregirebuli makroelementebis gamoyenebiT arak-is amocanad.agregirebuli makrocvladebisψ da ϕ ( ψ ) funqcia SeiZlebaarCeul iqnas sxvadasxva mosazrebiT: romlebic dakavSirebuliasasurvel gardamaval da obieqtis moZraobis damyarebul re-JimTan – atraqtorebTan sistemis fazur sivrceSi.[35,49] naSromebSi dasabuTebuli iyo marTvis amocanebSifazur sivrceSi funqcionalebis gamoyenebis mizanSewoniloba,romlebic saSualebas gvaZlevden uzrunvelvyoT gare areSi asimptoturimdgradi moZraoba da sakmaod efeqturad CaxSobil iqneswarmoSobili gadaxrebi drois mcire monakveTSi. aseTi saxisfunqcionalebs ganekuTnebian saoptimizirebeli funqcionalebi(3.10), romelTa struqturis Secvla SeiZleba ganxorcieldes pirvelrigSi, rogorc ϕ ( ψ ) funqciis struqturis SecvliT-Sesa-ssbamisi raodenoba maRali xarisxis funqciis e.i 3 , 5 rψ ψ ψ ,..., ψ wevrebisraodenobis SenarCuneba (3.8), aseve, meore rigSi, makrocvladebisψs( x1,..., xn) , formis SecvliT, romlebic dakavSirebulebiarian sinTezirebadi sistemis sasurvel atraqtorebTan. pirvelSemTxvevaSi saoptimizirebeli funqcionals (3.10)-s gaaCnia,mciredi gadaxrebis reJimebisaTvis arsebiTad gansxvavebuli saxeebi,rodesac maRali xarisxiani wevrebi axdenen mcire zemoqmedebasψψ3r( = ... = ≅ 0) , xolo didi gadaxrebis reJimebisaTvis igivewevrebi asruleben mniSvnelovan rols gardamaval procesSi.(3.10) funqcionalSi maRali xarisxis wevrebis arseboba niSnavsimas, rom marTvis kanoni, ufro aqtiurad reagirebs ufro didgadaxrebze da maT intensiurad CaaxSobs mcire droSi. amavdro-2ulad funqcionalSi gagvaCnia kvadratuli wevrebi ψ , romlebic3mogvcems saSualebas sakmaod efeqturad gadaamuSaos sistemamsawyisi mdgomareobidan mciredi gadaxrebiT.sinergetikis terminebSi [38,39] makrocvladebissssssψ s-es xarisxisganzodadoebuli parametrebia, romlebic asaxaven sinTezirebadisistemebis koleqtiur Tvisebebs, isini warmoadgenen `infor-s113


matorebs”-sinergetikuli informaciis matareblebs sistemaSimimdinare procesebis Sesaxeb. swored es xarisxis parametrebigansazRvraven TviTorganizaciis mimarTuli procesebis mimdinareobassinTezirebad sistemaSi. makrocvladebisψ s– ganmartebarogorc xarisxis ganzogadoebuli parametrebisa, axasiaTebenmravaldonian sistemebis saerTo mdgomareibas, saSualebas gvaZlevsganvaxorcieloT (3.10) saxis funqcionalebis Semdegi sinergetikuliinterpretacia. hakenis [40,41]-is Tanaxmad TviTorganizebadisistemebis makroskopiuli moqmedebis zomad SeiZleba gamoyenebuliiqnes xarisxis parametris kvadrati am sazoms pirobiTadSeiZleba davarqvaT sistemis mier Sesrulebuli samuSao.aqedan gamomdinareobs Tanmxleb funqcionalSi (3.9) kvadratulissmdgenelebis & ϕ ( ψ ) Semoyvanis mizanSewoniloba, romlebic asaxavensinTezirebadi sistemebis makroskopuli zemoqmedebis sidides(sazoms). sistemebis efeqturobis sazomad sinergetikaSi miTebuliamakroskopiuli zemoqmedebis sazomis cvlilebis siCqare,rac Cvens SemTxvevaSi aisaxeba Tanmxleb funqcionalSi mdgenelebis& ϕ ( ψ ) SemotaniT. unda avRniSnoT, rom orak-is TeoriaSissxarisxis kvadratul kriteriumebSi woniTi koeficientebis Ser-Cevis cnobili rTuli problema Rebulobs (3.10) funqcionalebismimarT iol gadawyvetas. aq woniTi koeficientebi T sgansazRvravensistemis gw moZraobis dros mravalsaxeobebis gadakveTamde.saoptimizirebeli funqcionalebma (3.10), romlebic gamoiyenebaarak-is meTodSi, SeiZleba mogvces Sinaarsobrivi interpretaciavariaciuli aRricxvis ukuamocanebSic. sinamdvileSi,variaciuli aRricxvidan cnobilia, langraJiani L lwarmoadgensbunebrivi moZraobis kriteriums, romelsac asrulebs Sesabamisiobieqti misi sakuTari, arakoreqtirebuli (U = 0) dinamikuriTvisebebiT. cnobilia, rom bunebrivi moZraobisas kriteriumisrolSi gamodis integralitkJ = ∫ Le( x1,..., xn, t) dt,tn114


omelsac meqanikaSi uwodeben moqmedebas. es moqmedeba mravalSesaZlo moZraobidan gamoyofs obieqtis im realur moZraobas,romelzedac J kriteriums gaaCnia stacionaluri mniSvneloba(Cveulebrivad minimaluri). naTelia, rom aucilebloba SesabamisimarTvisux ( ,..., x)L1nSemotanisa warmoiqmneba im SemTxvevebSi,rodesac obieqtis bunebrivi moZraobis traeqtoria ar gadissasurvel (miznobriv) mdgomareobaze. maSin obieqtis miznobrivimdgomareobidan warmoSobili gadaxrebi SeiZleba CamovaciloTSemdegi saxis axali lagranJianis SemoyvaniT.= uuL∑ = L + Llromelic asaxavs ukve marTvadi obieqtis Tvisebebs. aqedan gamomdinareobsrom arak-is meTodiT marTvis kanonissin-Tezi SeiZleba interpretirebul iqnes rogorc Cveulebrivi lagranJianisdamatebiTi cvlileba. e.i.ux (1,..., x n)ux (1,..., xn)≡ Lu.sxvanairad rom vTqvaT, warmoiSoba variaciuli aRricxviserTgvari ukuamocana, romelSic saWiroa moinaxos axali kriteriumi-funqcoinali,romelic uzrunvelyofs sinTezirebadi sistemisdasaxuli miznis miRwevas. Cveulebriv es optimaluri mar-Tvis Teoriis Znelad gadasawyveti amocanaa, Tumca arak-is me-TodSi is Rebulobs Tavis efeqtur amonaxsns Tanxmlebi funqcionalebissaxiT. aq mTavaria xazi gavusvaT im faqts, rom aseTiinterpretaciisas saoptimizacio funqcionali iZens axal meTodologiurSinaars. kerZod: is ukavSirdeba pirvel rigSi obieqtismarTvis saboloo mizans-mis gadayvanas fazuri sivrcis sasurvelsaboloo mdgomareobaSi nebismieri sawyisi mdgomareobidan(zogierT dasaSveb areSi), da ara gardamavali procesebissasurveli Tvisebebis uzrunvelyofasTan, rogorc es Cveulebrivmocemulia optimaluri marTvis maTematikur TeoriaSi. arak-ismeTodSi aseTi interpretacias eZleva kardinaluri mniSvneloba,xolo marTvis ZiriTadi problema formulirdeba rogorc115


marTvis kanonebisuu (1,..., u m)sinTezis problema, romelic uzrunvelyofessistemis `obieqt-regulatoris” fazuri moculobisSekumSvisas misi gw-s aucilebel moxvedras asimptoturmravalsaxeobebze – atraqtrebze fazur sivrceSi.meore formulireba arak-is meTodSi variaciuli ukuamocanisasaoptimizirebel (3.10) funqcionalTan mdgomareobs SemdegSi.davuSvaT, rom optimaluri marTvis sistemaSi (3.6) SeiZlebamoiZebnos eiler-lagranJis gantolebis (3.7) sistemis funqcionaluraddamoukidebeli ψs= 0 ( S = 1,..., m)pirveli integralebi,romlebic gansazRvraven (3.10) funqcionalisaTvis eqstremalebisvels da arian Tavsebadni (10.6) sawyis gantolebebTan.maSin marTva ui+ 1,..., Un, romlebic warmodgenilia (3.6) da (3.7)-is saxiT.gantolebebis erToblivi amoxsnis Sedegi saoptimizirebelfunqcionals (3.10) aniWebs upirobo eqstremums. uku amocanisorive tipis formulireba variaciuli aRricxvisaTvis amyarebsuSualo kavSirs arak-is meTodsa da analitikuri meqanikis Ziri-Tad cnebebs Soris.imisaTvis, rom davakonkretoT (3.10) saoptimizirebeli funqcionalissaxe da gaviTvaliswinebT sinTezirebadi sistemebisadmidamatebiT moTxovnebs, saWirod SevarCioT rogorc funqciaψ , agreTve, makrocvladebi ϕ ( ψ ). ganvixiloT mokled amsfunqciebis agebis zogierTi xerxi arak-is meTodSi [35]. SezRudvebisx k≤ A kgaTvaliswinebiT. makrocvladebiss k k k 1 nssψ = x + AthF ( x ,..., x ,...),(3.11)gamoyenebiT SeiZleba uzrunvelyoT miTiTebuli SezRudvebi koordinatebzeda marTvaze.arawrfivi sistemebSi wyvetili marTvis sinTezisaTvis SeiZlebagamoyenebul iqnes uban-uban gluvi Semdegi saxis makrocvladebi.n−1ψ = ∑ β x + β s ans k k nk = 1n−12s=kxk +nk = 1ψ ∑ β β s , (3.12)116


sadac s = xn+ϕ( x1,..., x n − 1) . saoptimizirebeli funqcionali (3.10)-issafuZvelze ϕs ( ψs) da ψs( x1,..., xn) funqciebis SerCeviT SeiZlebaavagoT sxvadasxva naxvevebi, romlebic aRweren kriteriumebs.rogorc mcire aseve didi gadaxrebis SemTxvevaSi sistemebis (3.12)s s soptimizirebul reJimebs (3.11) saxis ϕ ( ψ ) = ψ – makrocvladebisaTvissaoptimizirebeli funqcionals aqvs saxe:dan gvaqvst k sup2⎡2 2sup ( ±k+k)+k k()⎣t0supJ A x T x t ⎤∫& dt(3.13)⎦vivaraudoT, rom obieqtis gantolebebSi ϕn= 0 , maSin (3.13)-t k sup2⎡2 2sup ( ±k+k)+k k()⎣t0supJ A x T U t ⎤∫ dt. (3.14)⎦miRebuli (3.14) kriteriumi asaxavs marTvaze daxarjulienergiis minimums koordinatze xk ≤ ArSezRudvis gaTvaliswinebiT.aseTi tipis amocanebs xSirad ganvixilavT marTvis gamoyenebiTamocanebSi. analogiurad ϕs ( ψs)= thψsda ψ s(3.11) didi gadaxrebisreJimSi ψk= 0 dan, roca thssup≅ ± 1, saoptimizirebeli funqcionali(3.10) Rebulobs saxessupt k sup2 2∫⎡⎣1 +k sup( ) ⎤⎦(3.15)t0supJ T ψ t dtTu obieqtis gantolebebSi fn= 0vRebulobT saoptimizirebel funqcionals.t ksup2 2∫ ( + TkUk)Jsup 1 ( t)dtt0sup, maSin (10.15)-is safuZvelze(3.16)romelic Tavis TavSi moicavs swrafmoqmedebis da energodanaxarjebiskriteriumebs, romlebsac gaaCniaT mniSvnelovani gamoyenebiTiazri marTvis sxvadasxva amocanebSi.didi gadaxrebis reJimebSi agebuli kriteriumebi (3.13)-(3.16)uzrunvelyofen mxolod suboptimalur procesebs. Tumca aqmniSvnelovaniaψ (3.11), (3.12) da ϕ ( ψ ) Sesabamisi arCeva, riTacsss117


SegviZlia davakmayofiloT moTxovnebi sistemebis dinamikuri Tvisebebisadmi.funqcionaluri gantolebebis agebisaTvis SeiZleba miRebuliqnes agreTve optimaluri marTvis sxva kriteriumebic. asemagaliTad swrafmoqmedebis kriteriumi [10.19]-is gamoyenebiT Semdegigantolebebi – orarxiani marTvisas (m=2)&1()t = ψ2, ψ21t =−Uψmaxsignμψ1 ψ2ψ0,5& () ( , ), μ( ψ1, ψ2)= ψ1+ ψ2ψ2;samarxiani marTvis SemTxvevaSi (m=3)&1()t = ψ2, ψ2()t = ψ32ψU ψ max& , ψ& 3() t = −U maxsignμψ (1, ψ2, ψ ),ψ 31μ( ψ , ψ , ψ ) = ψ + ψ (2 ψ + ψ ψ ) + ( ψ + 0,5 ψ ψ ) ψ + 0,5ψ ψ ;3am SemTxvevaSi sistemis aRmweri wertili minimalur droSisadac1 2 3 1 3 2 3 3 2 3 3 2 3 3xvdeba Sesabamisi μ = 0 mravalsaxeobebis gadakveTis wertilSi.Semdgom imoZravebs mis gaswvriv sasurvel safiniSo mravalsaxeobazemoxvedramde.analogiuri gantolebebi SeiZleba avagoT (3.7) formis safuZvelze.iseTi arawrfivi funqciebis ϕ ( ψ ) gamoyenebiT romelicaradiferencirebadia nulovan wertilSi. kerZod, SeoZlebaamovirCioT Semdegi gantolebebi:12p1sψ st ψ +ssT & () + & = 0 p = 1,2,... . (3.17)roca p→∞ gantoleba (3.19) Caiwereba Semdegi saxiT.Tψ& ( t) + signψ= 0 . (3.18)s s sam dros funqcionali (3.1) gardaiqmna Semdeg formaSisJ∞∑= ∫0ψ dtrodesac gardauvalia moZraoba mravalsaxeobis ψs= 0 gaswvriv.rs(3.17), (3.18)-dan gamomdinareobs, rom drois gansazRvrul momentSia.w. maTematikurad zustad xvdeba ψs= 0 mravalsaxeobebisgadakveTaze, Semdgom SeiZleba warmoiSvas moZraobis mcocavi re-118


Jimi, analogiurad sxva xarisxis kriteriumebis gamoyenebisas, daSeiZleba agebuli iqnas Sesabamisi funqcionaluri gantolebebi.sabolood, saoptimizirebeli funqcionalis (3.10) forma yvelazeufro mosaxerxebelia iseTi arawrfivi obieqtebis marTvisamocanebisaTvis, romelTa maTematikuri modelebic warmodgeniliapirveli rigis diferencialuri gantolebebiT. warmodgenilimeTodi SeiZleba misaRebi aRmoCndes sxva cnobili modelebisTvisac.kerZod, meore rigis diferencialuri gantolebebissistemebisTvisac. magaliTad, meqanikuri sistemebis maTematikurimodelebi, romlebic Cveulebriv Caiwereba niutonis meorekanonis an lagranJis formulis mixedviT. am SemTxvevaSi mizanSewoniliamovaxdinoT funqcionalis (3.7) modificireba arak-ismeTodis gamoyenebiT, Tu obieqtis gantolebebs Tu warmovadgenTSemdegi saxiT:2T &&sψs() t + ϕs( ψ& s) + fs( ψ& s) = 0 s = 1,2,..., m(3.19)sadac m - marTvis veqtoruli ganzomilebaa.am gantolebebs SeiZleba mieces energomeqanikuri interpretacia[47]. CavTvaloT, rom (3.19) gantolebebi aRweren im materialuriwertilebis erTobliobis moZraobas, romlebzedac moqmedebenarawrfivi aRmdgeni Zalebi ϕ ( ψ& ). amasTan erTad T 2 parametriSeiZleba CavTvaloT materialuri wertilebis masis analogad.maSin [47]-is Tanaxmad sistemis sruli energia CavweroTSemdegi saxiT:ψ s2 1s() = 0,5 &s+ ( )2 sT∫s 0v t ψ f ψ dψ sam gamosaxulebaSi marjvena mxaris pirveli wevri aRniSnavskinetikur energias, meore – sistemis potencialur energias. garemoswinaRobis ararsebobis SemTxvevaSi ϕ ( ψ & ) = 0 da, Sesabamisadenergiis Senaxvis kanonis Tanaxmad, (3.19) sistemas – eqneba pirveliintegraliqcia v s saWiroebis SemTxvevaSi klebulobs (3.19) sistemis moZra-vssss= const . Tumca realur pirobebSi meqanikurienergia sistemis moZraobis procesSi garemos winaRobis gamo,rogorc cnobilia, gadadis Tbur energiaSi. es niSnavs rom fun-ss119


obis traeqtoriis gaswvriv. amis sailustraciod gavawarmovoTvs–gamosaqxuleba drois mixedviT:[ && ]v&() t = ψ () t + f ( ψ ) ψ&() ts s s s se.i. (3.19) gantolebis gaTvaliswinebiT gvaqvs1v & s( t) =− ( ) ( )2 stT ϕψ& ψ&s.saqedan uSualod gamomdinareobs, rom ϕ ( ψ& ) ψ & ( ) > 0 warmoebulis s stu&() t ≤ 0se.i. am SemTxvevaSi sistemebis sruli energia (3.10) mcirdeba.cxadia romv s–funqcia, romelic asaxavs sistemis energiasiyo gansazRvrulad dadebiTi, amisaTvis aucilebelia Sesruldesutoloba fs( ψs)ψs> 0. Tu vivaraudebT, rom ϕs( ψ &s) = 0, maSin u&s() t = 0da Sesabamisade.i. garemos winaRobis ararsebobis Sem-TxvevaSi sistemas (3.19)-s gaaCnia pirveli integrali, romelicSeesabameba energiis Senaxvis kanons izolirebul meqanikur sistemaSi.us= constase, rom [47]-is Tanaxmad, (3.19)-sistemis asimptoturi mdgradobispirobas aqvs Semdegi saxe:a) f ( ψ ) ψ > 0 ψ ≠ 0 ; b) ψ ( ψ& ) ψ & ( ) > 0 ψ () t ≠ 0ssg)ψ0s∫sf ( ψ ) dψ →∞s s ss s stψs→∞arak-is meTodis Sesabamisad am pirobebis Sesruleba uzrunvelyofsmarTvis sistemis gamomsaxveli wertilis (gw) invariantulimravalsaxovnebebis ψs( x1,..., xn) = 0 da ψ &s( x1,..., xn, x& 1,..., x& n) = 0 gadakve-Taze gadayvanas. cxadia, rom funqcionaluri gantolebebi (3.19)saoptimizirebel funqcionals aniWeben upirobo eqstremums:sadacks∞2 2 2∫⎡⎣ s( ψs) λ1sϕs( ψ& s) λ &&2sψs (s)⎤⎦dt(3.20)0J = f + + t∑λ -erTgvari woniTi koeficientebia, romlebic uSualoddakavSirebulni arian (3.19)-is gantolebis koeficientebTan.wrfivi SemTxvevisaTvis (3.19)-gantoleba da saoptimizirebelfunqcionali (3.20) Rebulobs Semdeg saxes:&& ψ () t + a ψ&() t + a ψ = 0s 1s s 1s ss120


dasadac koeficientebiλksdakavSirebulia erTmaneTTan SemdegidamokidebulebiT:x2 2 2JΣ = ∫⎡⎣ψ s+ λ1sψ& s() t + λ &&2sψs() t ⎤⎦ dtaks0da22λ1 s= a1 s−2a0 s; λ2s= a2savRniSnoT, rom funqciis asimptoturi Tvisebebis ZaliTψ () t → 0 da ψ& () t → 0 roca t→0 kvadratuli saoptimizirebelissfunqcionali SeiZleba CavweroT Semdegi eqvivalenturi formiTxJΣ = ∫⎡⎣&& ψ () t + λ ψ & + λ ψ ⎤⎦dt+ c0s 1s s 2s s ssadac C s - erTgvari mudmivebia, romlebic ar moqmedeben Tof-iseqstremumze.saoptimizirebel funqcionalis minimizaciis (3.10) pirobaanalogiurad tolfasia funqcionalis minimizaciisx2JΣ = ∫ ⎣⎡T &&sψs()t + ψ ⎤s ⎦ dt+cs0J ∑J ∑es niSnavs, rom wrfiv SemTxvevaSi saoptimazirebeli funqcionalebida (3.20) SeiZleba sistemis dinamikaSi warmodgeniliqnen cnobili kvadratuli Sefasebebis saxiT.amrigad, (3.19) funqcionaluri saxis gantolebebs SeiZlebamivcemT fizikuri ganmarteba. (3.7) gantolebac warmoadgens klasikurimeqanikis invariantuli damokidebulebas. rasakvirvelia,rom rTuli, magaliTad eleqtromeqanikuri sistemebisaTvis zogadiSemTxvevaSi SeiZleba aRmoCndes mizanSewonili gamoviyenoTpirveli (3.7) da meore (3.19) rigis funqcionaluri gantolebebiskombinaciebi da Sesabamisi optimizirebadi funqcionalebi.zemoT ganxiluli funqcionalebi (3.10) da (3.20) sinergetikulmidgomaSi, zogadad rom vTqvaT, ar TamaSoben ganmsazRvrulrols. Tumca aq midgomaSi, romelic dafuZnebulia mimzidaviinvariantuli mravalsaxovnebebis SemotanasTan, saSualebasgvaZlevs gamovavlinoT optimaluri marTvis amocanebSiaxali Taviseburebebi.121


3.2. arawrfivi agregirebuli regulatorebis analizurikonstruirebis ganzogadoebuli meTodiagregirebuli regulatorebis analizuri konstruirebisganzogadebuli meTodi (arak) efuZneba marTvis TeoriaSi sinergetikulimidgomis konceptualur debulebebs. am midgomis arsimdgomareobs SemdegSi.funqcebis ψ s da ψ s (ψ s ) –arCevis Semdeg funqcionalurigantolebebis (3.7), (3.8), (3.9) safuZvelze SesaZlebelia marTviskanonebis sinTezi. gansakuTrebiT unda aRvniSnoT, rom es kanonebi,marTvis Teoriis klasikuri meTodebisagan gansxvavebiT,pirvel rigSi gansazRvraven, ara marto erTeuli cvladebis,aramed sistemebis TviTorganizaciis koleqtiuri procesebis mar-Tvis strategias. swored amaSi mdgomareobs dinamikuri sistemebismizanmimarTuli TviTorganizacia. marTvis sinTezis sinergetikulimidgomis arsi mdgomareobs sasurveli gare – da Sidasistemuri invariantebis SenarCunebaSi ψ s =0 fazuri sivrcisstruqturaSi. konkretulad gare invariantebis ricxvi (m), romelicparalelurad SehyavT sistemis struqturaSi, ganisazRvrebamarTvis arxebis ricxviT: ψ 1 =0,... ψ m =0 m≤n. xolo mimdevrobiT SeyvaniliSidasistemuri invariantebis ricxvi SemosazRvrulia gafarToebulisistemis xarisxis ψ m-H =0,... ψ r =0 r≤n-m mixedviT. amisgandamokidebulebiT, sistemis winaSe dasaxuli miznebi mudmiviaTu cvalebadi, SeiZleba Seicvalos Semosatani gare da Sida invariantebis`koleqtivi~. sxva sityvebiT, fizikuri (qimiuri, biologiuri)marTvis sistemaSi SegviZlia ganvaxorcieloT dinamikuriinvariantebis Sesabamisi amonakrebis tipebis amorCeva da amiTrealizeba gavukeToT misi mimarTuli TviTorganizaciis unars.gadavideT agregirebuli dinamikuri regulatorebis sinTeziskonkretuli procedurebis ganxilvaze, romlebic emyarebianarak–is meTodis saerTo idealogias.122


am meTodis Tanaxmad sistemis (3.6) gamomsaxveli wertilebigafarToebuli `gare~ marTvebis u i+1 ,...u n zemoqmedebiT xvdebaψ 1 =0,...ψ m =0 mravalsaxeobebis gadakveTis areSi, romlis gaswvrivacmoZraoba aRiwereba dekompozirebuli sistemis `Sinagani~dinamikis gantolebebiT:w&( t) = g ( w ,..., w , v ,..., v , x ,..., x ), j = 1,..., μ;jψ j 1ψ μψ i+ 1 n 1ψ m−1ψx&( t) = f ( x ,..., x , v ,..., v ), i = μ + 1,..., m−1iψ i 1ψ m− 1ψi+1 n(3.21)sadac v i+1 ,... v n – Sinagani marTvebiamarTvis v i+1 ,...,v n sinTezi warmoadgens (3.21) qveobieqtis mar-Tvis damoukidebel Sida amocanas. amisaTvis gamoiyeneba invariantulimravalsaxeobebis mimdevrobi-paraleluri erToblioba.marTvis Senaxvis principis Tanaxmad Sinagan marTvas v k -s gaaCniaucvleli ganzomileba dimv k =m, romelic emTxveva gare mar-Tvebis ganzomilebs. Sinagani marTvebi v k moqmedebs (3.21) qveobieqtsdekompozirebas ukeTebs mas Semdgom n-m ganzomilebis qveobieqtamde.naCvenebi mimdevrobiTi dekompoziciis procesi grZeldebagw-is moxvedramde arCeul safiniSo mravalsaxeobebamde-atraqtoramde,romlis ganzomilebac ganisazRvreba damokidebulebiTdimA=n-rmsadac n (3.6) gafarToebuli sistemis ganzomilebaa, m – mar-Tvis veqtoris ganzomilebaa, r - mimdevrobiT Semotanili atraqtorebisricxvia.aRwerili proceduris Sedegad Sinagani marTvebi aRmoCndebianrekurentulad erTmaneTTan dakavSirebuli vici rav i+1 ,...,v n marTvebi, SeiZleba SemovitanoT sasurveli makrocvladebi,magaliTad wrfivi saxisψ = γ ( x − v ) + ... + γ ( x − v ), s = 1,..., m (3.22)s s1 i+1 1sm n n(3.7) saxis funqcionaluri gantolebebis da aseve sasurvelimakrocvladebisψ s(3.22) da (3.6)-s gafarToebuli sistemis safuZvelzearak meTodis Sesabamisad moiZebneba `gareSe~ marTvebi.123


sadacu f x x w1i+ 1=−i+ 1( 1,..., n) −i+1−;...................................................................................Dnun =−fn( x1,..., xn) −wn− ,DDD(3.23)γ γ ... γ Φ γ ... γ11 12 1m1 12 1mγ21 γ22 ... γ2mΦ2 γ22 ... γ2mD = ≠ 0, D1= ≠0... ... ... ... ... ... ... ...γ γ ... γ Φ γ ... γm1 m2 mm m m2mmroca Φ = 0sDnγ11 γ12 ... γ1, m−1Φ1γ21 γ22 ... γ2, m−1Φ2=... ... ... ... ...γm 1γm2... γ Φmm , −1m≠ 0 roca Φ ≠ 0(3.24)s1Φ = γ v& () t + γ v& () t + ... + γ v& () t − ϕ ψ (3.25)s s1 1 s2 2sn n sTs( )aq miRebuli Sefardebebi (3.23)-(3.25) gvaZlevs saSualebasmovnaxoT (3.23) veqtoruli marTvis kanonebi, romlebsac gadayavTgw invariantuli mravalsaxeobebis ψ 1= 0,..., ψ m= 0latoris gantolebebis romelic uzrunvelyofs Caketili sistemis(3.6)–(3.23) seleqtiur invariantulobas M1( t),..., Mn( t ) SeSfoTebebismimarT, misi moZraobis asimptotur mdgradobas da sasur-sgadakveTis midamoSi.gw-is moZraoba am gadakveTebis gaswvriv ganisazRvrebaSinagani dinamikis gantolebebiT (3.21). (3.23) marTvis kanonenukavSiris gantolebebTan (3.2) qmnian dinamikur agregirebuli regu-vel dinamikur Tvisebebs.marTvis TeoriaSi axali meTodis princialur gansxvavebaswarmoadgens Semdegi mosazrebiT:pirvel rigSi: mTavari yuradRebis koncentrireba sinTezirebadisistemebis yofaqcevaze mimzidav invariantul mravalsaxeobebze-atraqtorebze,rasac mivyevarT marTvis sistemis dinamikurdekompoziciamde da Sesabamisad, misi yofaqcevis arsebiT gamartivebamde,radganac am dros SesaZlebelia sasurveli evolu-124


ciuri dabali ganzomilebani gantolebebis formireba, romlebicaRweren moZraobis mdgrad asimptotiur reJimebs.meore rigSi: sasurveli dabali rigis ganzomilebiani evoluciurigantolebebis formirebis SesaZlebloba, romlebic aRwerenmoZraobis mdgrad asimptotur reJimebs da warmoadgenenmravalsaxeobebze sinTezirebad sistemebs, dinamikuri mdgomareobisgantolebebs.mesame rigSi: `Sinagani~ marTvebis paralelur-mimdevrobiTierTobliobis kaskaduri sinTezi, romlebic dinamikurad dakavSirebulniarian erTmaneTTan da uzrunvelyofen atraqtorebze dekompozirebulisistemebis sasurvel yofaqcevas.agregirebuli regulatorebis analizuri konstruirebis(arak)-is meTodSi, romelic dafuZnebulia agregirebis _ dekompoziciisproceduraze, maRali ganzomilebis sinTezirebadi arawrfivisistemebis asimptoturi mdgradobis uzrunvelsayofad,gamoiyeneba liapunovis paralelur-mimdevrobiTi funqciaTa er-Toblioba. amasTan erTad Tavdapirvelad Semogvaqvs (3.7) gantolebisaTvisliapunovis umartivesi funqciebiV2s= 0,5ψs, ψsx1( ,..., x )makrocvladebis mimarT, da Semdeg saboloo mravalsaxeobebzerψ = 0 gamoikvleva moZraobis mdgradoba (n-rm) koordinatTa nawilismimarT, romlebic aRweren dekompozirebuli sistemis gamomsaxveliwertilis yofaqcevas moZraobis damamTavrebel etapze.zemoT moyvanili liapunovis funqciaTa erToblioba warmoadgensTavisebur analogs liapunovis veqtoruli funqciisa,da gamoyenebulia arawrfivi sistemebis sinergetikul TaoriaSiagregirebuli regulatorebis analizuri konstruirebis liapunovisveqtorul funqciaTa meTodi gamoirCeva sawyisi sistemebisasimptoturad zusti dinamikuri dekompoziciiT. garda amisaarak-is meTodSi ganixileba marTvis dinamikuri sistemebismdgradobis amocanebi. sxvanairad rom vTqvaT sinergetikulmidgomaSi liapunovis veqtoruli funqciis meTodi dakavSirebuliaagregirebuli regulatorebis analizuri konstruirebisprocedurasTan. swored marTvis struqturuli sinTezisn125


proceduras. romelsac swored marTvis gamomsaxveli wertiligadahyavs erTi mravalsaxeobidan meore dabali ganzomilebismravalsaxeobaSi, saSualebas iZleva arak-is meTodSi ganxorcieldesliapunovis veqtoruli funqciis analitikuri agebismkacri procedura sinTezirebadi arawrfivi sistemebis asimptoturimdgradobis analizisaTvis.arak-is meTodis es Taviseburebebi saSualebas aRvWurvoTsinTezirebad sistemebis gardamavali procesebisTvis, damaxasia-Tebeli robastikuli TvisebebiT struqturuli variaciebiT parametruliSeSfoTebebi. cnobilia, rom fazuri sivrcis garkveulareSi sistemis asimptoturi mdgradoba warmoadgens uxeS Tvisebas,romelic icvleba da Zlierdeba sistemis eqsponencialurimdgradobisas. marTvis sistemebi, romelTa sinTezic xorcieldebasinergetikuli principebiT warmoadgenen rogorc asimptoturadmdgradebs mTelSi (e.i. fazuri sivrcis mTel areSi), agreTveeqsponencialurad mdgradebs Semoyvanili invariantuli mravalsaxeobebisψs= 0 mimarT. es niSnavs rom aseT sistemebs gaaCniaTgardamavali procesebis robastulobis ganmasxvavebeli Tviseba.3.3. agregirebuli regulatorebis analizuri konstruirebismeTodi mimdevrobiT CarTuli invariantuli mravalsaxovnebismixedviTagregirebuli regulatorebis analizuri konstruirebismeTodis Tanaxmad, dinamikuri sistemebis fazur sivrceSi SeiZlebaarsebobdnen mravalsaxeobebi, romlebiskenac miizidebianfazuri traeqtoriebi. zogad SemTxvevaSi SeiZleba ramodenimeiseTi mravalsaxeobis ageba romlebic moicaven mizidulobisgarkveul zedapirebs. amitomac warmoiqmneba iseTi mizidulobiszedapirebis erTibliobis konstruirebis idea, rom gamomsaxveliwertilebi daiwyeben ra moZraobas nebismieri sawyisi mdgomareobidan,mimdevrobiT gadaadgildebian mizidulobis erTi zedapi-126


idan meoreze vidre ar moxvdebian zedapirze, romelsacmivyavarT fazuri sivrcis koordinatTa saTavesTan. am SemTxvevaSigamomsaxveli wertilebi uaxlovdebian mravalsaxeobebsψ1= 0 , Semdeg ψ2= 0 da a.S. saboloo mravalsaxeobasTan ψ = 0miaxloebis Semdeg Camoyalibdeba gansazRvruli mdgomareoba,kerZod, koordinatTa saTavemde mdgradi moZraobis procesi.zogad SemTxvevaSi r mimzidveli mravalsaxeobebis gamoyenebisasyoveli i -uri mravalsaxeobis ganzomileba iqneba erTiTnaklebi winamdebaresTan SedarebiT, e.i. gamomsaxveli wertiliTavidan uaxlovdeba ( n − 1) ganzomilebis mravalsaxeobas, Semdgom( n − 2) da a.S. ( n−r)gamzomilebis mravalsaxeobamde. moZraobisbolo etapze moZraobs koordinatTa saTavisaken aRiwereba ( n−r)rigis diferencialuri gantolebebiT. mimzidveli mravalsaxeobebisSerCeulirraodenobis mixedviT SeiZleba miviRoT arawrfivisistemebis gansxvavebuli Tvisebebi.aRvweroT arawrfivi sistemebis sinTezis procesebi maTematikurad.davuSvaT, rom obieqtis SeSfoTebuli moZraobaaRiwereba Semdegi diferencialuri gantolebebis sistemiT:x&i( t) = fi( x1,..., xn),i = 1,2,..., p;x&j( t) = fj( x1,..., xj) + aj+ 1xj+1,j = p+ 1, p+ 2,..., n− 1;(3.26)x&( t) = f ( x ,..., x ) + u,s n 1 nsadac x , x - obieqtis mdgomareobis koordinatebia;i- mmarTvelizemoqmedebaa,jf ( x x ) uwyveti funqciebia ( 0,..., )i1 ,...,numf x1,...,x -fi0 ;j ( j )uwyveti funqciebia, romlebic diferencirdeba Tavisi cvladebismixedviT:( )fi0,...,0 = 0; = d .dtmiuxedavad gansaxilveli (3.26) diferencialuri gantolebebisspecifikurobisa, am saxis maTematikuri modeliT SeiZlebaaRwerili iqnes sxvadasxva mniSvnelovani obieqtebis klasebi. asemagaliTad, (3.26) gantolebis kerZo SemTxvevas warmoadgens diferencialurigantolebebis sistema ( p= 0)127


x&1() t = f1( x1) + a2x2;x&() t = f ( x ) + a x ;2 2 2 3 3..................................x&n−1() t = fn− 1( xn−1) + anx n;x&() t = f ( x ) + u,n n naRwers teqnikis mraval dargSi farTod gavrcelebuli obieqtebisklass, romelic Sedgeba mimdevrobiT SeerTebuli arawrfividinamikuri rgolebisagan. aseT obieqtebs ganekuTvnebianqimiuri reaqtorebis erToblioba, sxvadasxva gamaTbobeli xelaswyoebi,satransporto da gamamdidrebeli manqanebi, eleqtro,hidravlikuri da pnevmaturi amZravebi, mZlavri manqanebi da sxva.gadavideT agregirebuli regulatorebis analizuri konstruirebis(3.26) meTodis aRweraze arawrfivi obieqtebisaTvismimdevrobiT CarTuli invariantuli (mimzidavi) mravalsaxeobebiserTobliobis gamoyenebiT. gansaxilvelad SemoviyvanoT pirveliagregirebuli makrocvladiamocana mdgomareobs ( )n1=1k k+1 1 nk = 1ψ ∑ β x ϕ ( x ,..., x− 1).(3.27)u ,...,sx1 xniseTi marTvis sinTezSi, romelsacgadayavs (3.26) obieqti nebismieri sawyisi mdgomareobidan (romeliRacaaris) (3.27) mravalsaxeobis midamoSi. am amocanisamoxsnisaTvis gamoviyenoT funqcionaluri gantolebaTψ() t + ψ = 0(3.28)1 1 1maSin arak-is meTodis Sesabamisad (3.27) da (3.28)-is TanaxmadmarTvas eqneba saxepn−11 ⎛ ∂ϕ⎞11 ⎛ ∂ϕ⎞11u1 =− ∑ ⎜β2n + ⎟ fi − ∑ βij + ( fj+ aj+ 1xj+1)− −f β1 n i− 1 xi β ⎜1n j= p+1 x ⎟⎝ ∂ ⎠ ⎝ ∂j ⎠β1 n.1nT ψ (3.29)1u1marTvas gamomsaxveli wertili gadayavs ψ1= 0 mravalsaxeobebze,romlis gaswvrivac moZraoba aRiwereba gantolebebiTx&i( t) = fi( x1,..., xn), i = 1,2,..., p;x&( t) = f ( x ,..., x ) + a x , j = p+ 1,..., n−2;(3.30)j j 1 j j+ 1 j+1n−1ananx& n−1 t = fn− 1x1 xn−1− ∑ β1 kxk − ϕ1 x1xn−1β1n k = 1 β( ) ( ,..., ) ( ,..., ).128


Tu aRvniSnavT β1 nu2 anϕ1( x1,..., xn−1),= 1ϕ - vuwodoT SinaganimarTva, romelsac gadayavs sistemis moZraobis traeqtoriisaRmweri wertili meore mravalsaxeobazen−1ψ = ∑ β x + ϕ ( x ,..., x ) = 0. (3.31)2 2k k 2 1 n−2k = 1ψ2= 0 ganzomileba romelic Cawerilia (3.31) gamosaxulebiTerTi erTeuliT naklebia ψ1= 0 mravalsaxeobis ganzomilebaze.2 1(3.30) qveobieqtis marTvas u ( x ,..., 1) , romlis sinTezic xorcieldebaagregirebuli cvladisxn−ψ2-is bazaze, aqvs saxepn−21 ⎛ ∂ϕ⎞21 ⎛ ∂ϕ⎞2u2 = ∑⎜β2i + ⎟ fi + ∑β2 j+ ( fj+ aj+ 1xj+)β2, n−1 i= 1 ∂xi β ⎜2, n−1j= p+1 ∂ϕ⎟1−⎝ ⎠ ⎝ j ⎠a1− + +βn−1n∑ β1 kxk ψ2 fn−11nk = 1 β2, n−1T2.(3.32)u2- (3.32) marTva uzrunvelyofs sistemis moZraobis traeqtoriisaRmweri wertilis gadayvanas (3.31)-is mravalsaxeobis SemogarenSi.mis gaswvriv moZraoba aRiwereba diferencialuri gantolebebissistemiTx&i( t) = fi( x1,..., xn), i = 1,2,..., p;x&( t) = f ( x ,..., x ) + a x , j = p+ 1, p+2,..., n−3;j j 1 j j+ 1 j+1..................aa( ) ( ,..., ) ( ,..., ).n−2n−1 n−1x& n−2 t = fn−2 x1 xn−2 − ∑ β2kxk − ϕ2 x1 xn−2β2, n−1 k = 1 β2, n−1SemovitanoT aRvniSvna β2, n−1u3 = an− 1ϕ2( x1,..., xn−2),Tavis mxrivSeiZleba moinaxos marTvaromelic uzrunvelyofssistemis moZraobis traeqtoriis aRmweri wertilis gadayvanasmravalsaxeobebis midamoSi.n − 2u3( x1,..., xn− 2)ψ = ∑ β x + ϕ ( x ,..., x ) = 0 .3 3k k 3 1 n − 3k = 1analogiurad zogad SemTxvevaSi SeiZleba ganxorcieldesgamomsaxveli wertilis gadatana mravalsaxeobebis ψ4= 0 , ψ5= 0129


midamoSi da a.S. ψ 1= 0 mravalsaxeobebamde. amasTan Sinagani mar-Tvebis simravle gamoisaxeba gamosaxulebiTaux ( ,..., x ) = ϕ ( x,..., x ), i=1,2,... r;ni −+ 1i 1 n−+ i 1 i−1 1 n−+i 1βi−1,n−+i 2pn−11 ⎛ ∂ϕ⎞ 1 ⎛i∂ϕ⎞iui= ∑⎜β1i+ ⎟fi+ fn −+ i 1+ ∑βij+ ( fj+ aj+ 1xj+1)−βin , −+ i 1 i=1⎝∂xi⎠ β ⎜in , −+ i 1 j= p+1 x ⎟⎝ ∂j⎠a1− + ψi β.ni −+ 1ni −+ 2∑βi−1, kxki−1, n−+i 2 k=1 T1 βi, n−+i 1(3.33)gamomsaxveli wertilis moZraoban−+i 1i=ikxk +ix1 xn−1k = 1ψ ∑ β ϕ ( ,..., ) = 0(3.34)Sesabamisi mravalsaxeobebis gaswvriv aRiwereba diferencialurigantolebaTa sistemiTx&i( t) = fi( x1,..., xn), i = 1,2,..., p;x& ( t) = f ( x ,..., x ) + a x , j = p+ 1, p+ 2,..., n−i+1;u1( x1,..., x n)j j 1 j j+ 1 j+1.................. (3.35)aax&( t) f ( x ,..., x ) x ( x ,..., x ),n−in... i+ 1 n... i+1n−i =n−1 1 n−i − ∑ βik k−ϕi 1 n−iβin , ... i+ 1 k = 1 βin , ... i+1i = 1,2,..., r, r ≤n−1Camoyalibebuli proceduris mixedviT (3.29) marTvassistemis moZraobis traeqtoriis aRmweri wertili jergadayavs mravalsaxeobebis SemogarenSi, Semdgom (3.33) SinaganmarTvas gadayavs isiψ = 0 (3.34), ( l = 2,..., r)mravalsaxeobebis midamoSiromlis gaswvrivac moZraoba aRiwereba (3.35)-saxis diferencialurigantolebebiT. (3.33) marTvisuix1xn−+1 i( ,..., )sinTezi ganxorcieldayovel etapze funqcionaluri gantolebebis1 1Tψ& () t + ψ = 0, l = 1,2.(10.36)isafuZvelze. (3.33) marTvisu ikanonebi uzrunvelyofen sistemismoZraobis traeqtoriis aRmweri wertili gadasvlas i-urimravalsaxeobebidan( i + 1) mravalsaxeobaze, romelTa gaswvrivacmoZraoba aRiwereba (3.35) saxis gantolebebiT. (3.34) ψ 1,..., ψrgamo-130


saxulebidan gamomdinareobs rom i-uri mravalsaxeobebis ψi= 0ganzomileba erTiT naklebia (i-1) mravalsaxeobis ganzomilebas-Tan SedarebiT. fazuri sivrcis ganzomilebis aRwerili TanmimdevruladSemcireba, sistemis moZraobis traeqtoriis aRmweriwertilis moZraobisas ( i − 1) mravalsaxeobebis gaswvrivanalogiuria procesebis optimalurebisa swrafmoqmedebis mixedviT,romelTa Taviseburebas warmoadgens gadarTvis hiperzedapirebisganzomilebebis Tanmimdevruli Semcireba.amrigad mimzidavi mravalsaxeobebis ψi= 0 garkveuli TanmimdevrobebisSemoReba saSualebas iZleva daCqardes gardamavaliprocesebis mimdinareobis dro. SemoTavazebuli meTodisTanaxmad sinTezis procedura mdgomareobs marTvis u ( x ,..., )(3.29) kanonis formirebaSi. amasTan mTavar amocanas warmoadgensϕ1( x1,..., x n) funqciis da misi (n-1) warmoebulebis 1 ∂xj∂ϕ1 1x ngansazRvra.funqcia ϕ 1( x 1,..., x n − 1)SeiZleba moZebnili iqnes damxmare funqciebisϕ2 x1x ( x ,..., x n − r)( ,...,n−2),...,ϕr1Tanmimdevruli gansazRvris Sedegad,romlebic Sedian Sesabamis Sinagan (3.33) marTvaSiur( x1,...,xn −+ i 1) konkretulad es amocana ixsneba ukuTanmimdevrobiT:jer irCeva funqcia ϕr( x1,..., xn− r)asimptoturi mdgradobis pirobidanda moZraobis damamTavrebel rlebidan gamomdinare, Semdgom iZebnebaetapze xarisxis moTxovni-ϕr − 1( x1,...,xn − r + 1)funqcia, amisSemdgom ϕr − 2( x1,...,xn − r + 2) funqcia da a.S. da im funqciamde romelicuSualod daamTavrebs marTvis sistemis kanonis ui(x 1,..., xn−+ i 1)(3.33)sinTezis proceduras. funqciis gansazRvris mocemuli TanmimdevrobisrealizaciisaTvis daSinagani marTvebis gamoTvlisaTvis,romlebic moZraoben Sesabamisi mravalsaxeobebisψ = 0 gaswvriv, zemoT moyvanilia saWiro Tanafardobebi (3.29)-i(3.36). maT safuZvelze SegviZlia movnaxoT Sesabamisi konkretuligamosaxulebebi ui,ψida ψida jamSi movaxdinoT marTvisu 1131


mastabilizebeli kanonebis sinTezi. sinTezis damamTavrebeletapze agregirebuli cvladis safuZvelze ganisazRvreba marTvaSemdegi gamosaxulebis safuZvelze:n−+i 1∑ψ = β x + ϕ ( x ,..., x ) = 0r rk k i 1 n−rk = 1ϕr( x1,..., xn−r)funqciis daiβ koeficientebis SerCeviTSeiZleba dasruldes arawrfivi sistemis sinTezis amocanis amoxsna,arawrfivi sistemebis sinTezis warmodgenili midgoma dafuZnebuliafazur sivrceSi mimzidavi mravalsaxovnebis simravleTaSemotanaze, da mTlianad Seesabameba sistemebis Tanmimdevrul optimizaciasradganac gamomsaxveli wertilebi gare aredan xvdebapirvel mravalsaxovnebaze, Semdgom meoreze da a.S. r-mde. amasTanyoveli i-uri mravalsaxeobebis gaswvriv moZraoba aris asimptoturadmdgradebi mTelSi. amgvarad, xorcieldeba moZraobisTanmimdevruli optimizacia gamomsaxveli wertilebis fazursivrceSi saboloo mravalsaxeobebze, moxvedris procesSi. sistemebisTvis,romelTa sinTezic xorcieldeba aq CamoyalibebulimeTodiT, saSualebas gvaZlevs gamovavlinoT rigi saerTo kanonzomierebanimaT dinamikur TvisebebSi da xarisxis maCveneblebSi.ganvixiloT magaliTad gardamavali procesebis milevisdrois Sefasebis amocana. gansaxilvel sistemebSi regulirebisdro ganisazRvreba gamomsaxveli wertilebis r mravalsaxeobeb-Tan miaxloebis droisa da maTi ψr= 0 saboloo mravalsaxeobebisgaswvriv fazuri sivrcis koordinatTa saTavemde moZraobisdrois jamiT. amasTan yoveli moZraoba akmayofilebs gantolebebsTψ& ( t) + ψ = 0, l = 1, 2,...(3.37)i i rmaSin (3.37)-is Tanaxmad regulirebis drois jamuri SefasebamiiRebs saxes:r=1t ≤ (4...5) T + tψ,(3.38)∑ p∑i=1ir132


sadact ψ r - safiniSo mravalsaxeobebis ψr=saTavisken moZraobis droa.0gaswvriv kordinatTaamgvarad, arawrfivi agregirebuli regulatorebis analizurikonstruirebis SemoTavazebuli meTodi, romelic dafuZnebuliafazur sivrceSi mimzidavi mravalsaxeobebis TanmimdevrobisSemotanaze, warmoadgens sinTezirebuli sistemebis SeSfoTebulimoZraobis asimptoturi mdgradobis garantias e.i. gadawyvetiliaarawrfivi sistemebis stabilizaciis amocana. uzrunvelvyofiliagardamavali procesebis milevadobis saWiro dro da a.S.SemoTavazebuli sinTezis meTodi Tavisi arsiT efuZnebasawyisi n rigis amocanis ganzomilebaTa klebadi r raodenobisqveamocanis(n− r)rigis qveamocanaze mimdevrobiT dekompozicias.amasTan, sawyis fazur sivrceSi saerTo moZraobis dayofa parcialurmoZraobebad dafuZnebulia gansaxileveli obieqtebisara romelime miaxlovebul Tvisebaze, aramed gamomsaxveli wertiliserTi mravalsaxeobebidan sxvanaklebi ganzomilebis mravalsaxeobazedroSi mimdevrobiT gadayvanaze3.4. invariatuli mravalsaxeobebis paralelur – mimdevrobiTierToblioba da mravalkavSiriani sistemebis sinTezizemoT aRwerili mimzidveli mravalsaxeobebis simravleTaparalelurad da mimdevrobiT Semotanis meTodebis gaerTianebisamocana gavaerTianoT erT ganzogadoebul meTodad, romelsacSenarCunebuli eqneba orive meTodis upiratesobani.davuSvaT, rom obieqtis moZraoba aRiwereba Semdegi saxisdiferencialuri gantolebebis sistemiT:sadacx&( t) = f ( x ,..., x ),i i 1 nx&( t) = f ( x ,..., x ) + a x ,j j 1 j j+ 1 j+1x&( t) = f ( x ,..., x ) + bu ,s s 1 j s si = 1,2,..., p;j = p+1,..., μ;(3.39)s = μ + 1,..., n,1 nx ,...,x obieqtis mdgomareobis koordinatebia; f () ⋅ uwyvetaddiferencirebadi funqciebia.133


gadavideT marTvis kanonebis ( )u ,...,sx1 xnsinTezis procesisaRweraze. jer Tavdapirvelad vaxorcielebT `gare~ u ( x x )marTvebis sinTezs adre ganxiluli meTodis mixedviT. amasTanerTad saWiroa amovirCioT agregirebili cvaladebidan erT-erTi,Semdegi saxiT:maSin marTva ( 1,..., )snns= ∑ skxk +sx1x μk = 1ψ β ϕ ( ,..., ).s1 ,...,u x x uzrunvelyufs fazur sivrceSi gamomsaxveliwertilis gadayvanas nebismieri sawyisi mdgomareobidanmravalsaxeobebis gadakveTis midamoSin∑ψ = ξ x + b μ ( x ,..., x ) = 0,1,2... μ sk k sk k 1k = 1romlis gaswvrivac moZraoba aRiwereba gantolebaTa sistemiTx& ( t) = f ( x ,..., x ), i = 1,2,..., p;i i 1 nab ax& ( t) f ( x ,..., x ) x ( x ,..., x ). (3.40)μμ+ 1 sμ μ+1i=i 1 n− ∑ξsk k− ϕs1ξμ+ 1 k = 1 ξμ+1(3.40) sistemis ganzomileba tolia μ . pirveli Siga marTvaganisaZRvreba Semdegi gantolebidan:ub a= ϕ ( x ,..., x ),sμμ+1μ1 s 1ξμ+ 1movaxdinoT im Siga marTvis kanonis sinTezi, romelsac gamomsaxveliwertili gadahyavs pirveli mimzidveli qvemravalsaxeobebisψμ −1∑= a x + ϕ ( x ,..., x ) = 0.μ1 sk k s 1k = 1midamoSi. ψμ1- mravalsaxeobebis ganzomileba erTiT naklebia gadakveTisψ12μmravalsaxeobebis ganzomilebisa. Semdgom avRniSnoTmeore Siga marTvauaμμ−1μμ2 =μ1 x1x μ −1aμ−1μϕ ( ,..., ),μn134


SegviZlia Tavis mxriv movaxdinoT marTvisu ( x ,..., x )μ2 1 μ−1kanonissinTezi romelic uzrunvelyofs gamomsaxveli wertilis gadayvanasSemdgom mravalsaxeobazeψμ −2∑= a x + ϕ ( x ,..., x ) = 0.μ2 2kk μ2 1 μ−2k = 1aRniSnuli procesi gagrZeldeba adre aRwerili procedurisanalogiurad. amasTan erTad xorcieldeba klebadi ganzomilebianimimzidveli mravalsaxeobebis Semotana. moZraoba boloetapze xorcieldeba Semdegi mravalsaxeobebis gaswvriv.ψμ−p∑= a x + ϕ ( x ,..., x ) = 0.μp pk k μp 1 μ−pk = 1Tu CavsvamT ψ 1... μ= 0,...,ψμp= 0 gantolebebidan xμ, xμ−1,...koordinatebsobieqtis (3.39) pirvelp gantolebaSi, miviRebT diferencialurigantolebaTa sistemasi i 1 nx& ( t) = f ( x ,..., x ), i = 1,2,..., p;romelic aRwers moZraobas bolo mravalsaxeobebis ψ − + 1= 0 gaswvrivfazuri sivrcis koordinatTa saTavsaken. Sesabamisi funqciebisϕsϕμ1,...,ϕμ pda koeficientebis βsk, a1k,...,apkarCeviT SeiZlebauzrunvelyoT asimptoturi mdgradoba da sistemebis moTxovnilidinamikuri Tvisebebi.ganzogadoebuli analizuri konstruirebis mocemuli meTodidafuZnebulia or ZiriTad proceduraze. pirvel rigSimarTvis ( )u ,...,sx1 xnsinTezi, romelic uzrunvelyofs gamomsaxveliwertilis dadaadgilebas mravasaxeobebis ψ 12 μ= 0 gadakveTazeda meore rigSi Siga marTvebisu ( x ,..., x ),..., u ( x ,..., x ),μ 1 μ μ− p+ 1 1 μ− p+1sinTezi romelsac gadahyavs gamomsaxveli wertili mravalsaxeobebisψ 12 μ= 0 gadakveTidan ψμ 1= 0 pirvel mimzidvel qvesimravleze,Semdeg meoreze ^ψ μ 2= 0 a.S. TviT bolo qvemravalsaxeobamdeψμ p= 0 , miTiTebuli procedurebis Sesrulebis Semdeg jerμ p135


xorcieldebasmravalsaxeobis paraleluri Semotana, xolo Semdegxdeba sistemis fazur sivrceSiμ − p mimzidveli qvemravalsaxeobebiserTobliobis mimdevrobiTi Seyvana.(3.39) obieqtebisaTvis roca p = 0, aRiwereba arawrfivi diferencialurigantolebaTa sistemiT&xj( t) = fj( x1,..., xj)+ a , j+ 1x j + 1x&( t) = f ( x ,..., x ) + bu ,s s 1 j s sj = p+1,..., μ;s = μ + 1,..., n,(3.41)romelsac gaaCnia samkuTxa funqcionaleri matrica μ koordinatamde,paralelur-mimdevrobiTi mimzidveli mravalsaxeobebissimravlis SemoyvaniT SeiZleba analizurad movaxdinoTs( 1,..., )u x xnsinTezi romelic uzrunvelyofs gardamavali procesebisada Caketili sistemebis Tvisebebis garantirebul asimptoturmdgradobas. am SemTxvevaSi Tavidan paralelurad Semoitanebas mravalsaxeoba ψ = 0 da arawrfivi regulatorebis ana-slizuri konstruirebis zemoT mocemuli meTodis TanaxmadusmarTvebi. amasTan erTad erT-erTi makrocvladiCioT Semdegi saxiT:maSin ( 1,..., x )snn∑ψs= βskxk+ ϕs( x1,...,x μ).k = 1ψ ssaWiroa avir-u x marTvebi uzrunvelyofen gamomsaxveli wertilisgadayvanas nebismieri sawyisi mdgomareobidan mravalsaxeobebisgadakveTazeμ∑ψ = a x + b1ϕ s( x ,..., x μ) = 0,1− p sk k p+1k = 1romlis gaswvrivac moZraoba aRiwereba gantolebebis sistemiT:aβ a∑ (3.42)μj( ) =j( 1,..., μ + 1p+ 1 μ + 1j) − askxk − ϕs.aμ+ 1 k = 1 aμ+1x&t f x x(3.42) sistemis ganzomileba tolia μ . (3.42) qveobieqtebis SigamarTvebis sinTezisaTvis SeiZleba SemovitanoT klebad ganzomilebianimimzidveli mravalsaxeobebis mimdevrobiTi erToblioba.136


amisaTvis jer SemovitanoT pirveli mimzidveli mravasaxeoba μfazur qvesimravleSiμψ = ∑ γ x + ϕ ( x ,..., x−) = 0 . (3.43)μ1 1kk μ1 1 μ 1k = 1(3.39) - (3.35) gamosaxulebebis Sesabamisad movaxdinoT Siga marTvisϕ ( x ,..., x ) sinTezi, romelsac gadayavs gamomsaxveli wertilis1nψp1= 0 (3.43) mravalsaxeobis midamoSi, romlis gaswvrivac moZraobaukve aRiwereba μ − 1 ganzomilebis diferencialuri gantolebebiT,Semdeg unda moinaxos momdevno Siga marTva da a.S. amasTanerTad xorcieldeba klebadi ganzomilebis mimzidvel qvesimravleTamimdevrobiTi Semotana. arawrfivi qveobieqtebis (3.43) SesabamisimarTvis arCeviT da (3.41)-obieqtebisaTvis SeiZleba yovel-Tvis garantirebul iqnes moZraobis asimptoturi mdgradoba mTelSi,gardamavali procesebis aperioduli xasiaTi da arawrfivisinTezirebadi procesebis rigi mTavari Tvisebebi.konkretulobisaTvis moviyvanoT (3.41) – obieqtebisaTvis ZiriTadidamokidebulebebi ori marTviT:x&i( t) = fi( x1,..., xn) + aj+ 1xj+1,x&j( t) = fj( x1,..., xj) + b2un−1,x&( t) = f ( x ,..., x ) + bu .n n 1 n i nSemoviyvanoT paralelurad ori mravalsaxeobai = 1,2,..., μ = n− 2; (3.44)n∑ψ = β x + ϕ ( x ,..., x ) = 0;1 1k k 1 1 n−2k = 1n∑ψ = β x = 0;2 2kk = 1maSin miviRebT marTvebs]kn−2⎛∂ϕ⎞1Bb1u n=−∑⎜β2nβ1 k− β1 nβ2k + β2n fn( x1,...,x )k 1ϕ⎟⎣⎡k+− ⎝∂k ⎠β β+ a x −Bf− ψ + ψ ;2n1nk+ 1 k+ 1 n−1 1 1T1 T2(3.45)(3.46)137


n−2⎛∂ϕ⎞1Bb2un− 1=−∑⎜β2, n−1β1k − β1, n−1β2k + β2n fn( x1,...,x )k 1x⎟⎣⎡k+− ⎝∂k ⎠β β+ a x −Bf− ψ + ψ ,]2, n−1 1, n−1k+ 1 k+ 1 n−1 1 1T1 T2(3.47)sadac B = β1, 1β2 −β1 β2, − 1, romelsac gadahyavs gamomsaxvelin−n n nwertilebi (3.45) mravalsaxeobebis gadakveTazen−1∑ ( n k n k)xk n( x ,..., xn−) = 0. (3.48)ψ = β β − β β + β ϕ(3.48)-dan koordinatis moZebniT1−2 1 2 2 1 2 1 1 2k = 1x21 n −2nn−1 = ∑( β1 nβ2k −β2nβ1k)xk−B k = 1Bda misi CasmiT (3.44) miviRebT gantolebaTa sistemasx&( t) = f ( x ,..., x ) + a x , j = 1,2,..., n−3;j j 1 j j j+1aax&( t) f ( x ,..., x ) β β β β x ϕ ,n−2n−1 β2nn−1n−2 =n−2 1 n−2 + ∑( 1n 2k −2n 1k)k−B k = 1Bβϕ11(3.49)romelic aRwers gamomsaxveli wertilis moZraobas ψ 1− 2= 0 (3.48)gaswvriv. SemovitanoT mimdevrobiT qvemravalsaxeoban−2∑ψ = γ x + ϕ ( x ,..., x ) = 0,...,μ1 1k k 2 1 n−3k = 1,..., ψ = γ x + γ x + ϕ ( x ) = 0μ, n−3 n−3,1 1 n−3,2 2 n−1 1(3.50)da arawrfivi regulatorebis analizuri konstruirebis meTodebisSesabamisad, movnaxavT qveobiqtebis ϕ1( x1,..., x n − 2)(3.49) mar-Tvas, Semdeg ϕ 2( x 1,..., x n − 2) marTvas da a.S. ϕ n−1( x 1)- marTvamde, romelicuzrunvelyofs gamosaxveli wertilis moZraobis sasurveliTvisebebis fazuri sivrcis koordinatTa saTavisaken bolo intervalSi.CavsvaT ϕ 1,..., ϕmarTvasun−1(x1,..., xn)n−1(3.50)-dan (3.46) da (3.47)-Si movnaxavTda vuzrunvelyobT moZraobis asimptoturmdgradobas mTelSi drois gansazRvrul periodSi da gardamavaliprocesebis (aperioduli) milevadobas marTvis sinTezirebulsistemaSi. analogiuri saxiT SegviZlia movnaxoT (3.41)obieqtebis Sesabamisi marTvebi sami da meti raodenobis marTvisSemTxvevaSi.138


amgvarad, aq Camoyalibebuli arawrfivi regulatorebisanalizuri konstruirebis ganzogadoebuli meTodi raomodenimemarTvis SemTxvevaSi emyareba or ZiriTad proceduras: pirvelrigSi us( x1,..., xn)marTvis sinTezs, romelic uzrunvelyufs gamomsaxveliwertilis gasdayvanas mravalsaxeoba ψs= 0ϕ ( x ,..., x−),..., ϕ−( x ) sinTezs,Taze da meore rigSi Siga marTvis 1 1 n 2 n 1 1-is gadakve-romelTac mimdevrobiT gadahyavs gamomsaxveli wertili pirvelimravasaxeobis ϕμ1 = 0 midamoSi, Semdeg meoris ϕμ 2= 0 da a.S. mismoxvedramde fazuri sivrcis koordinatTa saTaveSi.3.5. arawrfivi dinamikuri obieqtebis skalaruliregulatorebis analizuri konstruirebaganvixiloT arawrfivi dinamikuri obieqtebis skalaruliregulatorebis analizuri konstruirebis meTodis Taviseburebebisxvadasxva bunebis arawrfivi obieqtebis skalaruli marTvissistemebis sinTezis amocanis gadawyvetisas. kompiuteruli modelirebaSesrulda programuli uzrunvelyofa Maple-is gamoyenebiT.magaliTi 1. vivaraodoT, rom obieqti aRiwerebadiferencialuri gantolebiTx& () t = ax + x ; x& () t = u. (3.51)31 1 2 2(3.51) obieqtis Taviseburebas roca a > 0 , warmoadgens misiarsebiTi aramdgradoba, radgan roca x () t → 0 koordinatax () t →∞ 1, rac uyenebs marTvis kanonebs ux (1,x ) 2damatebiTmoTxovnebs romlebmac unda uzrunvelyon0)sistemisstabilizacia nebismieri sawyisi pirobebisas. gamoviyenoT arak-ismeTodi aseTi marTvis sistemebis sinTezisaTvis. amisaTvisamovirCioT ψ funqcia Tavidan Semdegi saxiT31 2 1 12( x →0,x12 →ψ = x + β x + bx(3.52)139


Tu (3.52)-s CavsvavT gantolebaSiTψ&+ ϕψ ( ) = 01 1 1miviRebT Semdeg zogad gamosaxulebas2 3 1u1( x1, x2) = − (3 bx1 + β)( ax1 + x2) − ϕ( ψ), T > 0 (3.53)Tromelic SerCeuli ( ) ϕ ψ 1funqciis mixedviT gvaZlevs saSualebasmiviRoT marTvis sxvadasxva kanonebi. es kanonebi uzrunvelyofengamomsaxveli wertilis moZraobis asimptotur mdgradobas mravalsaxeobis( ) 1amoirCeva ise, rom ( )ϕψ = 0 (3.52) SemogarenSi, radganac funqcia ϕψ ( )ϕψ 1⋅ ψ 1> 0 . diferencialuri gantoleba, romelicaRwers moZraobas ψ 1= 0 -is gaswvriv aqvs saxesx& () t =−βx −( b−a)x . (3.54)ψ31ψ1 1ψ1 1 111(3.54) gantolebis mdgradobis SefasebisaTvis viyenebT liapunovisfunqciasV= 0,5 , maSin misi warmoebuli drois mixedviT2x1 ψ 1(3.54) miiRebs saxes:Vt & () = −βx −( b− ax ) < 0 .2 41ψ1 1ψ1aqedan gamomdinareobs rom utolobebi β > 0, b≥ a, T1> 0 warmoadgenensinTzirebul Caketil sistemis (3.51), (3.53) mTelSi asimptoturimdgradobis pirobas1( ) =− (3 + β )( + ) − ϕψ ( 1). (3.55)T32 3x& 1() t = ax1+ x 2; x2 t bx1 ax1 x2ganvsazRvroT (3.55) sistemis pirveli integralebi, risTvisacigi warmovadginoT Semdegi simetriuli formiT:dxT dxdt.ax x = − T (3 bx )( ax x ) ( )=1 1 23 2 31+2 1 1+ β1+2+ ϕ ψ11(3.56)Tu (3.56) funqciaSi CavsvamT ψ1= 0 da Sesabamisad, ϕ (0) = 0 ,3integrirebis Semdeg vpoulobT pirvel integrals ax + bx =−x2,romelic emTxveva gamosaxulebas1 1ψ1= 0 (3.52). Cven davrwmundiTrom mocemeli integraluri mravalsaxeoba ψ 1= 0 (3.52) namdvilad140


warmoadgens arawrfivi sistemis sasurvel mimzidvel mravalsaxeobispretendents.saxes:rocaϕ = ψ = x + βx + ax , marTvis kanoni (3.53) Rebulobs31 2 1 1β 1 au =− x − x − x − (3 ax + β )( x + x ). (3.57)3 2 31 1 2 1 1 1T1 T1 T12nax.3.1-ze am kanonisaTvis da parametrebisaTvis β = 1, a = 1, T1= 1gamosaxulia Caketili sistemis moZraobis traeqtoriebi (danarTisistema 1). rogorc nax. 3.1 Cans fazuri traeqtoriebi `exvevian~ψ1= 0 (3.52) mravalsaxeobas, ikribebian misken koordinatTasaTaveSi. amasTan sistema warmoadgens asimptoturad mdgradsgardamavali procesebis milevadobis aperioduli xasiaTiT.nax. 3.1 gardamavali procesi Caketili sistemis moZraobis traeqtoriebiparametrebesaTvis β = 1, a = 1, T1= 1axla vivaraudoT, rom x2koordinataze dadebulia SezRudvax2≤ A , maSin Tu SemovitanT funqcias3( β )ψ = + + (3.58)2x2 Ath x1 bx1 ,marTvis kanonisaTvis vRebulobT gamosaxulebasuA( β + 3 bx )( ax + x ) 1=− − ( ),T2 > 0 (3.59)2 31 1 22ϕψ2 32ch ( β x1+bx1)T1141


a)b)g)nax. 3.2 gardamavali procesi Caketili sistemis moZraobistraeqtoriebi ( b= 1, T = 1, A= 1 ).142


omelsac gadahyavs gamomsaxveli wertili ψ2= 0 (3.58)mravalsaxeobebis midamoSi arCeuli ϕ( ψ2)_ funqciis daparametrebis β da a -ze damokidebulebiT uzrunvelyofsgardamavali procesebis Sesabamis xarisxs.gaswvriv moZraobis gantolebas aqvs Semdegi saxe:3( β )31ψ 2 1ψ 2 1ψ 2 1ψ2ψ2= 0 mravalsaxeobisψ& () t = x − Ath x + bx .(3.60)moZraobis (3.60) gantolebidan gamomdinareobs rom pirobebiβ > 0, b ≥ 1 uzrunvelyofes mis asimptotur mdgradobas mxolodgarkveul areSi. es niSnavs rom x 2≤ A SezRudvis da Sesabamisadψ2(3.58) funqciis SemotaniT mcirdeba Caketili sistemebis (3.51),(3.59) asimptoturad mdgradobis are.nax. 3.2-ze gamosaxulia moZraobis traeqtoriebia , б daSesabamisadϕ ψ ϕ ψ=2, = th2da si gn2ϕ = ψ funqciebisaTvis, parametrebisaTvisb= 1, T = 1, A=1 , romlebic amtkiceben sinTezirebulsistemebSi aperioduli gardamavali procesebis asimptoturadmdgradi areebis arsebobas.mcire gadaxrebis reJimSi, rodesac ψ2inf= x2 + Aβx1, u 2(3.59)da u (3.53) marTvis kanonebi ( A = 1)iqnebian optimalurebi1kvadratuli kriteriumebi mixedviT:x2 2 2 2 2 2 2 2 2inf= ⎡⎣ 1+ (1 + )2+0J ∫ β A x β A T x T u ⎤⎦dt. (3.61)(3.61) kriteriumSi woniTi koeficientebis arCeva damokidebuliasasurveli gardamavali procesebis xarisxze. ase rom,agregirebuli regulatorebis analizuri konstruirebis marTvisu12kanonebi (3.53) da u (3.59) uzrunvelyofen asimtotur mdgradobasmTelSi anx2≤ A areSi da uzrunvelyofen Caketilisistemis moTxovnil Tvisebebs.magaliTi 2. movaxdinoT obieqtis marTvis kanonis sinTezix& () t = x + x , x& () t = u(3.62)21 1 2 2romelsac gaaCnia eqstremaluri xasiaTis arawrfivoba. funqciis143


ψ = x + β x + ax x(3.63)2 1 1 1SemotaniT da misi CasmiT funqcionalur gantolebaSiTψ&ψ1+ = 0obieqtis (3.62) gantolebidan vipoviT marTvis Semdeg kanons:β a 12u1 =− x1− x1 x1 − x2 − (2 a x1 + β )( x2+ x2).(3.64)T T T(3.64) kanons gadayavs gamomsaxveli wertili ψ = 0 (3.63)mravalsaxeobis areSi, romlis gaswvriv moZraoba aRiwerebadiferencialuri gantolebiTx & () t = x −βx −a x x21ψ 1ψ 1ψ 1ψ1ψgamovikvlioT bolo gantolebis mdgradoba x&ψ= -is10mimarT. amisaTvis SemovitanoT liapunovis funqcia3 2ganvixiloT misi warmoebuli: Vt &()= x1 x1 ax1x 2 ψ−βψ−ψ 1ψV= 0,5 da2x1ψnaTelia, rom x& 1 ψ< 0 -Tvis warmoebuli Vt &() < 0 rocaβ > 0; a > 0 , xolo x& ψ> 0 warmoebuli Vt &() < 0 roca β > 0; a ≥ 1. es1niSnavs rom roca sruldeba piroba β > 0; a ≥ 1, es gantoleba daSesabamisad sinTezirebad sistema asimptoturad mdgradiamTelSix1= x2 = 0mdgomareobasTan mimarTebaSi.nax. 3.3-ze gamosaxulia Caketilisistemis moZraobis traeqtoria roca β = 1; T = 1, a=2.144


nax. 3.3 mocemulia meore rigis sistemebis analogiuri fazuriportretebis SesamCnevi gansxvaveba mdgomareobis traeqtoriisyofaqcevis mesame meoTxedSi (danarTi sistema 2). CveulebrivimarTvis kanonebi warmoadgenen erTgvar mravalsaxeobebs romlebicgaivlian fazuri sibrtyis meore da meoTxe meoTxedebs.magaliTi 3. ganvixiloT moZravi obieqtis masis centrismoZraobis marTvis sistemis analizuri konstruirebis amocana,romlis moZraobac aRiwereba [46] diferecialuri gantolebebissistemiT:sadacmΔ h&&() t = aδ + bδ ; T & δ()t + δ = cu31 1 1Δh-masis centris kordinataa,(3.65)δ - mmarTveli organosgadaxraa, ma , , b, c,T –mudmivi koeficientebia CavweroT 3.65-1 1 1gantoleba Semdegi formiT:x&t x x&t ax bx 3x&() t =− ωx + cu,31() =2; 2() =3+;3 3(3.66)a1 b11sadac x1 =Δ h, x2 =Δ h& c1( t), x3= δ, a = , b= , ω = , c=m m T TsaWiroa moiZebnos–marTvis kanoni, romelic uzrunvelyofs(3.66) obieqtis gadaadgilebis sivrcis nebismieri wertilidanwonasworobisux ( , x, x),1 2 3xk(0,0,0)wertilSi. amasTan, gaTvaliswinebuliunda iqnes moTxovnebi sistemis dinamikuri Tvisebebisadmi,gamosaxuli, magaliTad zogierTi kvadratuli xarisxiskriteriumis minimizaciis formiT.Tavidan jer gamoviyenoT arak-is meTodi (3.66) obieqtisaTvis,romelic warmodgenilia diferencialuri gantolebiTkanonikur formaSiy& () t = y , y& () t = y , y& () t = u(3.67)1 2 2 2 3sadac y = x , y = x , y = x&( t) = ax + bx 3 , u = ( a+ 3bx 3)( − ωx + cu )1 1 2 2 3 2 3 3 0 3 3 1amovirCioT Semdegi wrfivi makrocvladi0ψ1= p1y1+ p2y2 + y3(3.68)145


Tu ukansknel gamosaxulebas CavsvamT (3.68) funqcionalurgantolebaSiTψ& () t + ψ = 0(3.69)1 1 1maSin (3.67) obieqtis gantolebidan vRebulobT marTvis kanonsp ⎛1p ⎞ ⎛21 ⎞u0 =− y1− ⎜ p1+ ⎟y2 − ⎜ p2+ ⎟y .(3.70)T1 ⎝ T1⎠ ⎝ T ⎠ 31sawyis kordinatebze gadasvlis Sedegad miviRebT:1 ⎡ p ⎛1p ⎞ ⎛21 ⎞ ⎤2cu1 =− ⎢ x1+ ⎜ p1+ ⎟x2 + ⎜ p2 + ⎟( a + bx3)x ⎥+ωx+ ⎣T1 ⎝ T1 ⎠ ⎝ T1⎠ ⎦2( a 3bx3)3 3(3.71)ganvixiloT sinTezirebuli sistemis Tviseba arak-is meTodispoziciidan. marTvis kanonsu 0(3.70)-s gadayavs obieqtis aRmweriwertili nebismieri sawyisi mdgomareobidanψ1= py1 1+ py2 2+ y3 = 0(3.72)mravalsaxovnebis areSi, romlis gaswvrivac moZraoba aRiwerebaSemdegi diferencialuri gantolebaTa sistemiT:y& () t = y , y () t =−p y −p y .(3.73)1ψ1 2ψ1 2ψ1 1 1ψ1 2 2ψ1(3.73) gantolebas da Sesabamisad, (3.66), (3.71) Caketili sistemebimTelSi asimtoturi mdgradobis pirobebs eqneba martiviutolobis saxe:p > 0; p > 0; T > 01 2 1(3.66) - obieqtis (3.71) regulatoriT marTvis Caketili sistemismodelirebis Sedegebi (danarTi sistema 3) warmodgenilia nax. 3.4da nax. 3.5-ze146


nax. 3.4 gardamavali procesis mrudinax. 3.5 fazuri portretigadavideT (3.66) - obieqtis marTvis sxva kanonebis sinTezisSesaZleblobebis ganxilvaze. arak-is meTodis gamoyenebiT amkanonis struqtura damokidebulia arCeuli invariatiuli147


mravalsaxeobis formisa da funqcionaluri gantolebis saxezeamasTan dakavSirebiT SemoviyvanoT gansaxilvelaT Semdegimakrocvladida funqcionaluri gantoleba( )sadac F ψ ψ > .2 20arak-is meTodebis Sesabamisadψ = x + ϕ( x , x )(3.74)2 3 1 2Tψ& () t + F( ψ ) = 0(3.75)2 2 2u2( x1, x2, x3)sinTezirebadi marTviskanonebi uzrunvelyofen obieqtis aRmweri wertilis gadayvanas(3.74) ψ2= 0 mravalsaxeobebis midamoSi, romlis gaswvrivacmoZraoba aRiwereba (3.66) Semdegi diferencialuri gantolebisTanaxmad3( ) ϕ ( )x& () t = x ; x& () t =−aϕx , x −b x , xψ(3.76)1ψ 2 2ψ 2 2ψ 2 1ψ 2 2ψ 2 1ψ2 2 2Sinagani marTvis ϕ( x1, x2)Sesabamisi sinTeziT SesaZlebelia uzrunvelvyoTmravalsaxeobebis ψ 2= 0 gaswvriv moZraobis saWirodinamikuri Tvisebebi. ψ2= 0 CasmiT (3.75)-Si obieqtis marTvisgantolebisa (3.66)-is ZaliT vipoviT gamosaxulebas:∂ϕ∂ϕ1cu =− x − a + bx x − F( ψ ) + ωx,2( )2 2 3 3 2∂x1 ∂x2 T23(3.77)romelic moicavs marTvis dasaSvebuli kanonebis gansazRvrulerTobliobas. avirCioT Tavidan ubralo wrfivi funqcia( , )ϕ x x = β x + β x(3.78)1 2 1 1 2 2maSin (3.78) gamosaxulebis CasmiT (3.77)-Si, ψ2-gaTvaliswinebiT(3.71)-Si, miviRebT marTvis Semdeg kanons roca F( ψ2)= ψ2;β ⎛1β ⎞ ⎛22 1 ⎞cu2 =− x1− ⎜β1+ ⎟x2 − ⎜β2a + β2bx3+ −ω ⎟x, (3.79)T2 ⎝ T2 ⎠ ⎝ T2⎠ 3amasTan (3.76) gantolebas eqneba saxe( ) 3x& () t = x ; x& () t =−β ax −β ax − b β x + β x (3.80)1ψ 2 2ψ 2 2ψ 2 1 1ψ 2 2 2ψ 2 1 1ψ 2 2 2ψ2148


(3.80)-gantolebis asimtoturi mdgradobis pirobebi mTelSi daSesabamisad Caketili sistemebisaTvis (3.66), (3.79) Rebulobs martiviutolobis saxesmarTvis kanoniu 2β > 0; β > 0; T > 0;(3.81)1 2 2(3.79), (3.81) utolobebis Sesrulebisas uzrunvelyofsaRmweri wertilis moZraobis asimtotur moZraobasψ2= 0 (3.74) mravalsaxeoba gaswvriv moZraobisas, is ufro martiviau 1(3.71) kanonze. es aixsneba imiT, rom aq gamoyenebuli iyo(3.66) - obieqtis sawyisi gantoleba, xolo (3.79)-is kanonis sin-Tezi dafuZvnebulia Sesabamisi Sinagani marTvis kanonisu 2ϕ( x , x )arCevaze, sasurveli moZraobis uzrunvelyofisaTvis mravalsaxeobaψ2= 0 (3.74)-is gaswvriv moZraobisas, gansazRvrulia (3.80)diferencialuri gantolebiT. parametrebi β 1, β 2da T 2(3.79) kanoniunda akmayofilebes (3.81)-is pirobebs da SeiZleba, gansazRvruliiqnes mcire gadaxrebis reJimSi Caketili dinamikuri1 2sistemis sasurveli dinamikuri Tvisebebidan gamomdinare. Caketilisistemis modelirebis Sedegebi (3.79)gamosaxulia3.6 da 3.7 naxazebze.u 2marTvis kanoniTnax. 3.6 gardamavali procesis mrudi149


nax.3.7 fazuri portretisistemis traeqtoriis aRmweri wertilis moZraobisasmravalsaxeobis ψ2= 0 (3.74) swrafmoqmedebis gasazrdeladSeiZleba Semoyvanili iqnes arawrfivi funqcia3( x , x ) = x + x + xϕ β β β1 2 1 1 2 2 2 2maSin kanoni (3.77) roca F( ψ2)= ψ2Rebulobs saxes1cu =−β x − β + 3 β x a + bx x − ψ + ωx. (3.82)A2 2( )( )2 1 2 2 3 3 3 3 2T23Tu avirCevT funqciebs ϕ ( x , x ) da ( )1 2F ψ , maSin (3.79), (3.82)analogiurad miviRebT marTvis Sesabamis kanonebs. Caketilsistemis u = (3.82) marTvis kanoniT modelirebis Sedegebi20mocemulia nax. 3.8 da 3.9-ze2150


nax. 3.8 gardamavali procesis mrudinax.3.9 fazuri portretivivaraodoT, rom masis cvlilebis siCqareze dadebuliaSezRudva x2 ≤ A/β 2. maSin, misi aRricxvisaTvis SeiZlebagamoyenebul iqnes Semdegi makrocvladi;ψ = β x + Ath( x + β x )2 2 2 3 1 1151


Tu ψ 3CavsvamT funqcionalur gantolebaSiTψ&() t + ψ = 03 3 3(3.66) obieqtis gantolebis ZaliT vRebulobT marTvis kanons1 2 3 1cu ω 3x β 3 1x2 ch ( x3 1x1)2ax3 2bx3 3.Aβ ⎛β β Tψ ⎞= − + ⎜ + + ⎟⎝3 ⎠(3.83)es kanoni uzrunvelyofs x 2≤ A SezRudvas da gadayavs aRmweriwertili obieqtis nebismieri sawyisi mdgomareobidan x1da x3koordinatebis mixedviT mravalsaxeoba ψ 3= 0 midamoSi, romlisgaswvrivac moZraoba, aRiwereba diferencialuri gantolebis sistemiT:x&() t = x ,1ψ3 2ψ3β() ⎛β⎜⎞⎟ .A ⎝A ⎠2 2x& 2ψ3t =−β1x1 ψ 2−aArth x2 ψ3− b β1x1 ψ3+Arth x2 ψ 3am gantolebebis mdgradobis pirobebi daiyvaneba utolobebzeβ 1> 0, β 2> 0 , romlebic gansazRvraven Caketili (3.66), (3.83)sistemisaTvis x kordinatebze moZraobis asimtotur mdgradobasx2 22≤ A/β SualedSi, xolo x 1da x 3kordinatze – mTelSi (3.66)marTvis Caketili sistemis (3.83) marTviT modelirebis Sedegebimocemulia nax. 3.10 da 3.11-ze.zemoT Camoyalibebuli moZravi obieqtis (3.66) masis centrismoZraobis marTvis amocanebi gviCveneben arak-is meTodis Rirsebebs,romlebic gvaZleven saSualebas martivi analitikuri procedurebsmeSveobiT miviRoT marTvis kanonebis erToblioba [35],romlebic uzrunvelyofs Caketili sistemebis dinamikur Tvisebebsmocemulia xarisxis Sesabamisi kriteriumebi, romelTa meSveobiTacxorcieldeba marTvis kanonis optimizacia.3152


nax. 3.10 gardamavali procesis mrudinax. 3.11 fazuri portretizemoT ganxilul magaliTSi ar moqmedemen garegani SeSfo-Tebebi. exla ganvixiloT dinamikuri regulatoris sinTezis maga-153


liTi, romelic gansazRvravs da STanTqavs obieqtze struqturuladgansazRvrul zemoqmedebebs.magaliTi 4 movaxdinoT arawrfivi obieqtis seleqciurinvariatiulisistemis sinTezi.x& () t = x , x& () t = sin x + x , x& () t = u+f(3.85)1 2 2 1 3 3masze harmoniuli SeSfoTebis f = Bsin(2 t)zemoqmedebiT, romelsacgaaCnia ucnobi magram SezRuduli amplituda, (3.84) gantolebebiTaRiwereba maTematikuri qanqaris moZraoba zeda aramdgradmdgomareobaSi. amasTan x 1_ qanqaris vertikalidan gadaxriskuTxea, x 2_ gadaxris siswrafea, x 3_ qanqaraze modebuli momentia.[34]-ze. avRniSnoT, rom maTematikuri qanqaris gantolebiTaRiwereba erTgvari mravali eleqtromeqanikuri obieqti, kerZodsxvadasxva saxis fazuri sistemebi, sinqronuli generatorebi daasinqronuli gaSvebis Zravebi da a.S. aseT obieqtebs gaaCniaTcilindruli fazuri sivrce. ismeba amocana qanqaris stabilizaciisamisi sakidelis RerZze modebuli momentis mixedviTaseTi momenti warmoiqmneba Semsrulebeli meqanizmis meSveobiT,romelic warmodgenilia maintegrirebeli rgolis saxiT. saWiroamoinaxos marTva Semsrulebeli meqanizmis Sesasvlelze, romelicastabilurebs qanqaris wonasworobas zeda mdgomareobaSi e.i.uzrunvelyofs sistemis asimtotur mdgomareobas.sistemaze moqmedi harmoniuli aRmSfoTi zemoqmedebiTaRwerisaTvis amovirCioT talRuri gamosaxuleba Semdegi saxiT& ω1() t = ω2;& ω () t =−4 ω ;(3.85)2f = Bω;(3.84)-sistema (3.85)-is gamoyenebiT warmovadginoT gafarTovebulisaxiT13 11& ω1() t = ω2 + v1( x1, x2, x3);& ω2() t =− 4 ω1+v2( x1, x2, x 3);x&1() t = x2;x&2() t = sin x1+x3;x () t = u+ω ,(3.86)154


sadac ω1,ω2_ w,w mdgomareobis cvladebis Sefasebebia,1 2v1( x1, x2,x3),v2( x1, x2,x3)-kavSiris funqciebia, rocav ( x , x , x ) = v ( x , x , x ) = 0, sistemis pirveli ori gantoleba1 12 3 212 3warmoadgens (3.85) SeSfoTebis models (danarTi sistema 4).dinamikuri regulatoris, romelic SeSfoTebis sawinaaRmdegodmoqmedeben, SemovitanoT invariaciuli mravalsaxeobaψ = x + sin x + β x + β x = 0 . (3.87)3 1 1 1 2 2maSin funqcionaluri gantolebis safuZvelzeTψ&() t + ψ = 0kerZod v1( x1, x2, x3)= aψ1; v2( x1, x2, x3)= aψ2kavSiris funqciisaTvismiviRebT marTvis Semdeg kanons⎛ β2 ⎞ β1⎛ 1 ⎞u = −x2cos x1− ⎜β1+ ⎟x2 − x1− ⎜ + β2⎟( x3+ sin x1)−ω 1. (3.88)⎝ T ⎠ T ⎝T⎠garegani gauzomeli SeSfoTebuli zemoqmedebis SefasebisaTvissaWiro gantolebas aqvs saxe( )( x )& ω () t = ω + a x + sin x + β x + β x ;1 2 1 3 1 1 1 2 2&2t =−1+ a2 x3 + x1+ 1x1+2 2ω () 4ω sin β β .(3.89)nax 3.12 Caketili marTvis sistemis gardamavali procesis mrudi155


nax 3.13 Caketili marTvis sistemis gardamavali procesis mrudi(3.88) marTvis kanons Tanmimdevrulad gadayavs aRmweri wertilimravalsaxeoba (3.87)-is midamoSi. es kanoni SeSfoTebis zemoqmedebisSefasebis (3.89) gantolebasTan erTad warmoqmnis dinamikurregulators romelic STanTqavs harmonol zemoqmedebas(3.85)-s. 3.12 da 3.13 naxazze warmodgenilia Caketili marTvis sistemebis(3.84), (3.88), (3.89) modelirebis Sedegebi. modelireba Catardaregulatoris Semdegi parametrebisaTvisT = β1 = β2 = 1; a2= − 1; a1= − 3.sinTezirebuli dinamiuri regulatori Tavisi struqturiTganisazRvreba miRebuli kavSiris gantolebebiT e. i. mocemulSemTxvevaSi damokidebulia kavSiris funqciis v1( x1, x2)arCevisagan,romelic SeiZleba interpretirebuli iqna rogorc erTgvari`Sinagani” marTvebi, romlebic moqmedeben SeSfoTebis modelze,rasakvirvelia am marTvebis sinTezi SesaZlebelia ganxorcieldesoptimaluri marTvis Teoriis meTodebis safuZvelze.kavSiris gantolebebis arCevisagan damokidebulebiT, arak-is ganzogadoebulmeTodSi SeiZleba avagoT sxvadasxva dinamiuri regulatorebiromlebic ukuqmedeben SeSfoTebebze.156


d a s k v n aCatarebuli gamokvlevebis safuZvelze SeiZleba gavakeToTSemdegi daskvna:1. sadisertacio naSromSi Seswavlilia axali integrirebulimecnierebis - sinergetikis safuZvlebi, romelicSeiswavlis koleqtiuri TviTorganizaciis procesebs dapraqtikulad moicavs Tanamedrove mecnierebis yveladargs. ganmartebulia sinergetikis ZiriTadi cnebebi;2. Seswavlilia Caketili optimaluri disipatiuri marTvissistemebis sinTezis problema;3. optimaluri regulatorebis analizuri konstruirebisamocana gadawyvetilia sinergetikuli meTodebis gamoyenebiT.4. ganxilulia optimaluri disipatiuri marTvis sistemebisklasifikacia - skalaruli da veqtoruli marTvis Sem-Txvevebi cal-calke. sailuatraciod motanilia rigimagaliTebisa, romlebSic naCvenebia marTvis optimaluridisipatiuri sistemebis sinTezis meTodis efeqturobaoptimaluri regulatorebis analizuri konstruirebisklasikur meTodTan SedarebiT.5. Seswavlilia arawrfivi agregirebuli regulatorebis analizurikonstruirebis ganzogadoebuli meTodi, romelicdafuZnebulia fazur sivrceSi mimzidavi mravalsaxeobaTasimravlis Semotanaze da warmoadgens sinTezirebuli sistemebisSeSfoTebuli moZraobis asimptoturi mdgradobisgarantias.6. gadawyvetilia sxvadasxva saxis arawrfivi dinamikuriobieqtebisaTvis skalaruli regulatorebis analizurikonstruirebis problema sinergetikuli meTodebis gamoyenebiT.157


danarTisistema 1> with(DEtools):> u(x1(t),x2(t))=-(3*x1(t)^2+1)*(x1(t)^3+x2(t))-x2(t)-x1(t)-x1(t)^3;>u ( x1( t ),x2( t ))= − ( 3 x1( t )2 + 1 ) ( x1( t )3 + x2( t ))− x2( t ) − x1( t ) − x1( t )3>phaseportrait([D(x1)(t)=x1(t)^3+x2(t),D(x2)(t)=u(x1(t),x2(t))],[x1(t),x2(t)],t=130..150,linecolor=black,method=classical[foreuler],stepsize=.2);sistema 2> with(DEtools):> u(x1(t),x2(t))=-x1(t)-x1(t)*abs(x1(t))-x2(t)-(2*abs(x1(t))+1)*(x2(t)^2+x2(t));u ( x1( t ),x2( t ))= − x1( t ) − x1( t ) x1( t ) − x2( t ) − ( 2 x1( t ) + 1 ) ( x2( t )2 + x2( t ))>phaseportrait([D(x1)(t)=x1(t)^2+x2(t),D(x2)(t)=u(x1(t),x2(t))],[x1(t),x2(t)],t=130..150,linecolor=black,method=classical[foreuler],stepsize=.2);158


sistema 3> with(DEtools):> u(x1(t),x2(t),x3(t))=-1/(1+3*x3(t)^3)*[x1(t)+2*x2(t)+2*(3+x3(t)^2)*x3(t)]+x3(t);[ x1( t ) + 2 x2( t ) + 2( 3 + x3( t )2 ) x3( t )]u ( x1( t ), x2( t ),x3( t ))= − + x3( t )1 + 3 x3( t )3> phaseportrait([D(x1)(t)=x2(t),D(x2)(t)=x3(t)+x3(t)^2,D(x3)(t)=x3(t)+u(x1(t),x2(t),x3(t))],[x1(t),x2(t),x3(t)],t=-130..150,linecolor=black,method=classical[foreuler],stepsize=.2);sistema 4> with(DEtools):> u(x1(t),x2(t),x2(t))=-x2(t)*(cos(x1(t))+2)-x1(t)-2*x3(t)-2*sin(x1(t));u ( x1( t ), x2( t ),x2( t ))= − x2( t ) ( cos ( x1( t ) ) + 2 ) − x1( t ) − 2 x3( t ) − 2 sin ( x1( t ) )> dif1:=diff(x1(t),t)=x2(t);>ddif1 := x1( t ) = x2( t )dt> dif2:=diff(x2(t),t)=sin(x(t))+x3(t);ddif2 := x2( t ) = sin ( x( t ) ) + x3( t )dt> dif3:=diff(x3(t),t)=u(x1(t),x2(t),x3(t))+sin(2*t);ddif3 := x3( t ) = u ( x1( t ), x2( t ),x3( t ))+ sin( 2 t )dt> autonomous({dif1,dif2,dif3},[x1(t),x2(t),x3(t)],t);true> DEplot({dif1,dif2,d},[x(t),y(t)],t=-4..4,[[x(1)=1,x2(1)=1,x3(1)=1],[x(1)=2,x2(1)=2,x3(1)=2],[x1(1)=3,x2(1)=3,x3(1)=3],x1(t)=-20..20,x2(t)=-20..20,x3(t)=-20..20,linecolor=black,scene=[x(t),y(t)],color=black,stepsize=0.1);159


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