sbvm!lvqsbwb! - საქართველოს ტექნიკური ...gtu.ge/book/ims/RaulKuprava.pdfsignalad. magaliTad, organizmSi sisxlis mimoqcevis Seswavlis dros gamomaval

1

2

3

sbvm!lvqsbwb!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!

nbsUwb!cjptbnfejdjop!tjtufnfcTjnbsUwb!cjptbnfejdjop!tjtufnfcTjnbsUwb!cjptbnfejdjop!tjtufnfcTjnbsUwb!cjptbnfejdjop!tjtufnfcTj!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Ucjmjtj!3124!

4

tbsDfwj!

Tavi Tavi Tavi Tavi IIII

marTvis Teoriis ZiriTadi cnebebi da biosamedicino sistemebis

specifika

6666

avtomaturi marTvis Teoriis MmeTodologia 7777

marTvis ZiriTadi cnebebi da principebi 11111111

avtomaturi marTvis sistemebis klasifikacia 16161616

avtomaturi marTvis gaxsnili da Caketili

sistemebi

18181818

sinTezis amocana 26262626

wrfivi da arawrfivi sistemebi, erTganzomilebiana da

mravalganzomilebiani sistemebi, uwyveti da diskretuli

(cifruli) sistemebi

29292929

biologiuri sistemebis specifika

34343434

Tavi Tavi Tavi Tavi IIIIIIII

49494949

marTvis sistemebis analizis meTodebi 49494949

gawrfiveba mcire gadaxrebis SemTxvevaSi 49494949

klasikuri meTodi 60606060

laplasis gardaqmnebi 66665555

Tavi IIITavi IIITavi IIITavi III

72727272

dinamiuri marTvis sistemebi, modelirebis sqemebi

72727272

Tavi Tavi Tavi Tavi IVIVIVIV

marTvis impulsuri sistemebi

signalis daqvantvis saxeebi

79797979

a.m.s impulsuri sistemebis maTematikuri aRwera 82828282

avtomaturi marTvis impulsuri sistemebis kvlevis meTodebi 84848484

cifroli a.m.s. ganmarteba da Tvisebebani

86868686

cifruli a.m.s. dinamikis Taviseburebani 88888888

5

avtomaturi marTvis adapturi sistemebis cneba da maTi

klasifikacia

90909090

TviTmwyobadi marTvis sistemebi marTvis xarisxis stabilizaciiT 94949494

Tavi Tavi Tavi Tavi VVVV

wrfivi avtomaturi marTvis sistemebis mdgradoba 96969696

mdgradobis raus-hurvicis kriteriumi 106106106106

mixailovis mdgradobis kriteriumebi 109109109109

naikvistis mdgradobis kriteriumi

113113113113

TTTTavi avi avi avi VIVIVIVI

116116116116

marTvis biologiuri sistemebi 116116116116

mdgomareobis gantoleba 119119119119

konveqciuri transporti 126126126126

urTierTqmedeba da kompopentTa

urTierTgardaqmna

129129129129

moqmed masaTa kanoni 131131131131

fermentaciuli reaqciebi 133133133133

ekosistemebSi populaciaTa Soris urTierTqmedeba 136136136136

komponentTa utilizacia

138138138138

TaviTaviTaviTavi VIIVIIVIIVII

142142142142

kompartmentuli modelireba bio-samedicino sistemebSi. 142142142142

cocxali organizmis Tboregulaciis marTvis sistema 153153153153

marTvis sistemis stacionaruli reJimi 158158158158

marTvis sistemis mgrZnobeloba 159159159159

marTvis sistemis dakvirvebadoba 161161161161

rekomendirebuli literatura 163163163163

6

Ubwj!!J!

marTvis Teoriis ZiriTadi cnebebi dmarTvis Teoriis ZiriTadi cnebebi dmarTvis Teoriis ZiriTadi cnebebi dmarTvis Teoriis ZiriTadi cnebebi da biosamedicino a biosamedicino a biosamedicino a biosamedicino

sistemebis specifikasistemebis specifikasistemebis specifikasistemebis specifika

SesavaliSesavaliSesavaliSesavali

marTvis Teoriis meTodebi sul ufro da ufro

farTo gamoyenebas pouloben sxva da sxva saxis

sistemebis analizisaTvis – teqnikur, biologiur,

samedicino, ekonomikur sferoebSi. yovel am sferoSi

mkvlevaris winaSe dgas amocanebi, specifikuri

TaviseburebebiT. bevr teqnikur sistemebs aqvT cvladebisa

da parametrebis SedarebiT mcire ricxvi, ris gamoc maT Soris kavSiri TiTqmis yovelTvis SeiZleba ganvsazRvroT

sistemis eqsperimentuli kvlevis an misi konstruirebis

da damuSavebis dros. biologiur da samedicino amocanebSi

arsebiTi cvladebis ricxvi rogorc wesi Zalian didia,

ris gamoc erTi cvladis meoreze gavlenis xasiaTi da maT

Soris kavSirebi xSirad gaurkvevelia. amave dros marTvis

Teoriis axali meTodebis gamoyeneba sxvadasxva sirTulis

biologiuri obieqtebis gamokvlevis dros efuZvneba maTi

funqcionirebis ZiriTadi kanonebis codnas, cocxali

sistemebis organizaciuli principebis Seswavlas, romelnic uzrunvelyofen funqcionirebis saimedobis

maRal maCveneblebs garemos cvlilebis pirobebSi.

aucilebelia agreTve gaviTvaliswinoT, rom

jandacvaSi did rols TamaSobs mosaxleobis samkurnalo-

profilaqtikuri momsaxureobis avtomatizebuli marTvis

sistemebi, samkurnalo procesebis marTvis axali

meTodebis damuSaveba, avtomatizebuli diagnostikis da

mkurnalobis kompleqsis ageba.

7

aRniSnuli amocanebis gadawyveta SeuZlebelia

avtomatizebuli marTvis sistemebis da marTvis Teoriis

safuZvlebis codnis gareSe. aseve aucilebelia cocxal

organizmSi biologiuri procesebis marTvis principebis

codna, bioobieqtebis mdgomareobis marTvis sxvadasxva

gadawyvetis dros.

avtomaturi marTvis Teoriis MmeTodologiaavtomaturi marTvis Teoriis MmeTodologiaavtomaturi marTvis Teoriis MmeTodologiaavtomaturi marTvis Teoriis MmeTodologia

marTvis Teoriis ganviTarebaSi pirobiTad SeiZleba gamovyoT ori ZiriTadi periodi. marTvis Teoriis

ganviTarebis pirvel etapze – xSirad gamoiyeneba termini

avtomaturi regulirebis Teoria – mkvlevarebma

gaamaxviles yuradReba martiv sistemebze, romlebsac

hqondaT erTi Semavali da erTi gamomavali sidideebi da

funqcionirebdnen martiv pirobebSi.

meore etapze mkvlevarebis yuradReba miipyro

rTulma sistemebma, cvladebisa da parametrebis didi

ricxviT.

pirvel periods pirobiTad uwodeben gadacemis funqciis periods, romlis drosac damuSavda ZiriTadi

8

marTvis Teoriis klasikuri meTodebi – cnebebi

ukukavSirze, cneba sistemis gadacemis funqciaze,

diferencialuri gantolebebis amoxsnisa da gardaqmnis

meTodebi. marTvadi sistemebisaTvis am periodSi mTavari

yuradReba eqceoda sistemebis kvlevebs erTi Seamavali da

erTi gamomavali sidideebiT, Tumca garkveuli

warmatebebiT sargeblobdnen sistemebi ramodenime Semavali

da gamomavali sidideebiTac.

meore periodi xasiaTdeba, rogorc mraval Semaval

da gamomaval sidideebiani marTvis sistemebis Seswavlis

periodad. am etapis ZiriTad miRwevad iTvleba sistemis mdgomareobis sivrcis cnebis formulireba, xolo TviT

periods ki xSirad mdgomareobis cvladebis periods

uwodeben. am periodSi didi ganviTareba hpova sistemis

aRweris da gamokvlevis matriculma meTodma,

mdgomareobis meTodebi liapunovis mixedviT, sistemis

kvlevis meTodebma terminebSi `Semavali-mdgomareoba-

gamomavali~.

gadasvlas pirveli etapidan meoreze daemTxva

marTvis Teoriis iseTi axali nawilebis daarseba,

rogoricaa didi sistemebis Teoria, ierarqiuli struqturebis Teoria, adaptaciis Teoria, arawrfivi

dinamika an sinergetika da sxva.

sinergetika axali samecniero mimarTulebaa. igi

Seiswavlis ierarqiuli struqturebis

sxvadasxvaelementebs Soris kavSirebs. amgvari

struqturebi sistemebSi nivTierebis da energiis

garemosTan intensiuri cvlis Sedegad warmoiqmneba. aseT

sistemebSi calkeuli struqturebis SeTanxmebuli muSaoba

SeimCneva, ierarqulobis principis dacviT, ris Sedegadac

izrdeba sistemis mowesrigebis xarisxi, mcirdeba entropia anu xdeba sistemis TviTorganizacia.

9

sinergetikis kvlevis sagania nebismier fizikur, qimiur,

biologiur, samedicino Tu teqnikur sistemaSi

TviTorganizaciis procesiis kvleva.

marTvis TeoriaSi momuSave bevri specialisti

sinergetikas ganixilavs rogorc marTvis Teoriis nawils,

radganac sinergetikis mTavari cnebaa marTva. sinergetika

sistemis TviTorganizaciis process Seiswavlis, roca

mmarTveli parametrebi metnaklebad SemTxveviTi

cvlilebebiT xasiaTdeba.

amgvarad sinergetika Seiswavlis mravalkomponentian

sistemas, romelic Tavis mxriv, stoqastikur midgomas saWiroebs, rac Zalian mniSvnelovania biologiuri da

samedicino sistemebis Seswavlisas.

bolo dros arsebobs tendencia pirveli

periodisaTvis miRebuli TeoriisaTvis gamoviyenoT

termini marTvis klasikuri Teoria. is gansakuTrebiT

gamoiyeneba biologiuri da samedicino sistemebis marTvis

TeoriaSi arsebul wignebSi, sadac aRwerilia marTvis

Teoriis meTodebi, biologiuri sistemebis kvlevebisaTvis.

avtomaturi marTvis Teoria iZleva pasuxs ori

tipis amocanaze. pirvel tips warmoadgens analizis amocana, romlis mixedviTac unda ganisazRvros sistemis

reaqcia, anu gamomavali sidide ama Tu im Semavali

zemoqmedebis dros da ganisazRvros sistemis qcevis

zogadi Tvisebani.

amocanebis meore tipi, romelSic xSirad vxvdebiT

marTvis TeoriaSi aris mocemuli Tvisebebis mqone

sistemebis sinTezi, romlis arsi mdgomareobs SemdegSi:

ar gvaqvs realuri obieqti maSin sinTezis amocanis

mizania iseTi marTvis mowyobilobis an regulatoris

arCeva, romlis drosac obieqti masTan mierTebul

10

regulatorTan erTad iZleva marTvis sistemas saWiro

maxasiaTeblebiT.

marTvis sistemis sinTezis amocana, romelic

uzrunvelyofs yvela Sesasvlel maxasiaTeblebidan

saukeTesos cnobilia rogorc optimaluri marTvis

amocana. optimaluri marTvis sistemis SeqmnisaTvis

aucilebelia gvqondes amomwuravi informacia sistemaze

da garemoze, romelic faqtiurad ara gvaqvs, Tundac

imitom, rom gamoTvlebisas viyenebT realuri sistemebis

arazust aRweras. amitomac warmoiSva marTvis sistemis

sinTezis amocana, romlebSic procesebi inarCuneben maRal xarisxs miuxedavad imisa, rom marTvis obieqtis

parametrebi da garemos cvladebi SeiZleba iyos ucnobi an

icvlebides gaurkvevlad. aseT sistemebs uwodeben

adaptur sistemebs, romelTa ganasakuTrebuloba is

gaxlavT, rom obieqtis maxasiaTeblebis da garemos

parametrebis cvlilebis sapasuxod xdeba mmarTveli

mowyobilobebis parametrebis cvlileba. ra Tqma unda

adapturoba da optimaloba sruliad gansxvavebuli

cnebebia. adapturi sistema SeiZleba ar iyos optimaluri,

an optimaluri sistema ar iyos adapturi. gadauWarbeblad SeiZleba iTqvas rom, bunebaSi ar

arsebobs ubani, romelsac Sexeba ar hqondes avtomatur

marTvasTan. swored aman ganapiroba is pasuxismgebloba,

romelic ekisrebaT marTvis TeoriaSi momuSave

specialistebis Teoriul momzadebas. Cveni mizania, am

wigniT mcireodeni wvlili SevitanoTYqarTveli

specialistebis aRzrdis saqmeSi, romlebmac gadawyvites

daeuflon marTvis Teorias da mis gamoyenebas bio-

samedicino sistemebis marTvis sferoSi.

11

marTvis ZiriTadi cnebebi da principebimarTvis ZiriTadi cnebebi da principebimarTvis ZiriTadi cnebebi da principebimarTvis ZiriTadi cnebebi da principebi

romelime obieqtis marTva – es aris obieqtze

zemoqmedebis procesi masSi mimdinare procesebis

sasurveli mimdinareobis uzrunvelyofisaTvis an misi

mdgomareobis sasurveli cvlilebisaTvis.

marTvis obieqti SeiZleba iyos rogorc

biologiuri, aseve arabiologiuri. Sesabamisad marTva

SeiZleba xorcieldebodes adamianis mier an teqnikuri

mowyobilobis saSualebiT.

marTvas, romelic xorcieldeba adamianis Carevis gareSe ewodeba avtomaturi marTva, xolo sistemas,

romelSic yvela operacias axorcielebs avtomaturi

mowyobiloba, vuwodebT avtomaturi marTvis sistemas.

sistemas, romelSic avtomatizebulia operaciaTa mxolod

nawili, xolo sxva nawilis marTvas axorcielebs adamiani,

vuwodebT marTvis avtomatizebul sistemas.

marTvis sistemebis analizi moicavs dinamikuri

sistemebis yofaqcevis Seswavlas da mis bazisad

miRebulia Semadgeneli komponentebis (elementebis) mizez-

Sedegobriv kavSirze damyarebuli gantolebani. nebismieri mizez-Sedegobrivi procesi, romliTac xasiaTdeba

biologiuri sistema, ekonomika, socialuri mdgomareoba

da sxva, warmoadgens marTvis obieqts.

obieqtis grafikuli gamosaxuleba mocemulia nax.

1.3.1-ze, sadac y={y 1, y2,….yn} warmoadgens procesis

samarTav sidides (gamomavali sidide) F={F1, F2,….Fk} aris SemSfoTi zemoqmedeba, xolo X={x 1, x2,….xm} mmarTveli

zemoqmedeba, romelic moqmedebs y gamomaval sidideze

(yvela es sidide SeiZleba warmodgenil iqnas veqtorulad

an skalarulad).

12

zemoT aRniSnuli sidideebi Y, X, F erTmaneTTan damokidebuli arian (1) maTematikuri gamosaxulebiT,

romlis saxe ganisazRvreba obieqtis bunebiT.

y=G(F,X)

sadac, G warmoadgens operators, romelic gamoxatavs

cvladebs Soris kavSirs. Tu es kavSiri martivia da

cvladebs Soris Cveulebrivi funqcionaluri

damokidebulebaa, maSin

y=ϕ(F, X) obieqts vuwodebT statikurs (anu uinercios), magram,

rogorc wesi, obieqts axasiaTebs inerciuloba da Tu F SeSfoTebisa da X mmarTveli signalis zemoqmedeba Y samarTav sidideze xdeba aramyisierad, vambobT, rom

nax. 1.

13

obieqti dinamikuria. am SemTxvevaSi cvladebi Y, X, F dakavSirebulni arian diferencialuri, integraluri an

sxvaobiTi gantolebebiT.

rodesac cnobilia marTvis sistemis statikuri da

dinamikuri Tvisebebi, SegviZlia avagoT sistemis

maTematikuri modeli, SevadginoT marTvis iseTi

algoriTmi, romelic uzrunvelyofs mocemuli

funqcionirebis algoriTmis Sesrulebas (cnobili

SemSfoTi zemoqmedebis dros) sistemis marTvis

algoriTmi damokidebulia rogorc funqcionirebis

algoriTmze, aseve obieqtis dinamikur Tvisebebze da is warmoadgens sistemis funqcionirebis raRac wesebis

erTobliobas dakavSirebuls maTematikuri

damokidebulebebiT.

bunebrivad migvaCnia, rom nebismieri biologiuri

procesis maTematikuri modelis Sesaqmnelad saWiroa

kargi maTematikuri aparati da dinamikuri sistemebis

Teoriis codna. miuxedavad imisa, ragind zustad ar unda

SeadginoT maTematikuri modeli, is yovelTvis

gansxvavdeba realuri sistemisagan. am SezRudvas emateba

kidev is, rom SemSfoTi zemoqmedebis cvlilebis kanoni xSirad winaswar cnobili ar aris. swored am mizezebis

gamo sistemis faqtiuri yofaqceva gansazRvruli marTvis

algoriTmiT, gansxvavdeba sistemis sasurveli

yofaqcevisagan gansazRvruli funqcionirebis algoriTmiT.

amrigad, mivediT im daskvnamde, rom sasurveli

Sedegis misaRwevad marTvis algoriTmSi unda aisaxos ara

marto sistemis Tviseba da funqcionirebis algoriTmi,

aramed realuri sistemis yofaqcevac. gansakuTrebiT unda

aRiniSnos, rom sistemebis yofaqceva mniSvnelovan wilad

damokidebulia sistemaSi Semaval signalebze, romlebic rogorc zemoT avRniSneT ori tipisaa – marTvis

14

signalebi da SeSfoTebebi. marTvis signalebi

gansazRvraven moTxovnebs, romelic unda Seasrulos

sistemam (obieqtma). danarCeni signalebi, maT Soris isinic

ki, romelnic gansazRvraven pirobebs, romelSic

funqcionirebs sistema, miekuTvnebian SeSfoTebebs.

erTi da igive realurad arsebuli sidide

mkvlevaris survilis mixedviT da mis mier sistemis

aRweris xerxis mixedviT SeiZleba ganvixiloT sxvadasxva

signalad. magaliTad, organizmSi sisxlis mimoqcevis

Seswavlis dros gamomaval cvladad irCeven arteriul

wnevas. sidideebi, romelnic icvlebian amoxsnis procesSi da axasiaTeben organizmis sxva sistemebs (mag: filtvebis

ventilaciis sidide) ganixilebian am SemTxvevaSi, rogorc

SemSfoTi sidideebi. sunTqvis sistemis reaqciis aRweris

SemTxvevaSi (naxSirorJangis SesunTqvis SemTxvevaSi)

filtvebis ventilacia ukve TamaSobs gamomavali signalis

rols. Tu am gamokvlevebis dros icvleba arteriuli

wneva, maSin is ganixileba rogorc SeSfoTeba.

zemoT moyvanil msjelobaze dayrdnobiT, avtomaturi

marTvis sistemebSi SeiZleba gamovyoT oTxi tipis amocana,

romelic SeiZleba Sexvdes sistemis mkvlevars. kerZod:

1. cnobilia sistemis Semavali sidide, cnobilia

sistemis Tvisebebi da qcevis kanonebi. unda

ganisazRvros sistemis gamomavali sidide da misi

clilebis kanoni. aseT amocanas ewodeba analizis

amocana `pirdapiri amocana~. igi sxva amocanebTan

SedarebiT yvelaze martivia da xSirad gamoiyeneba

teqnikur sistemebSi.

?

15

2. cnoblia sistemis Tvisebebi da qcevis kanoni,

cnobilia sistemis gamomavali sidide da

cvlilebis kanoni. unda ganisazRvros sistemis

Semavali sidide. aseT amocanas `ukuamocana~

ewodeba. is samedicino sistemebSi cnobilia,

rogorc diagnostikis amocana da masTan xSirad

saqme aqvs praqtikos eqims.

3. cnobilia sistemis Semavali da gamomavali

sidideebi, cnobilia sistemis qcevis zogadi kanoni.

saWiroa vipovoT sistemis Tvisebebis ganmsazRvreli

mudmivebis ricxviTi mniSvnelobebi. es amocanac

cnobilia rogorc `ukuamocana~. samedicino praqtikaSi is aseve cnobilia rogorc diagnostikis

amocana. magaliTad, rodesac eqimma unda

ganaxorcielos funqcieri gamokvlevebis Sedegebis

interpretacia. igive amocana teqnikuri

sistemebisTvis cnobilia, rogorc `sinTezis~

amocana.

?

16

4. cnobilia sistemis Semavali da gamomavali

sidideebi. unda ganisazRvros sistemis qcevis

kanoni da sistemis zogierTi parametrebis

mniSvnelobebi. aseT amacanas `induqciis~, an ~Savi

yuTis~ amocana ewodeba. is yvela amocanaTa Soris

yvelaze rTulia. aseT amocanebTan upiratesad saqme

aqvT mecnierebs.

qvemoT mokled ganvixilavT TiToeul maTgans.

avtomaturi marTvis sistemebis klasifikaciaavtomaturi marTvis sistemebis klasifikaciaavtomaturi marTvis sistemebis klasifikaciaavtomaturi marTvis sistemebis klasifikacia

avtomaturi marTvis sistemebis klasifikaciis

saxesxvaoba damokidebulia zemoT ganxiluli oTxi tipis

amocanidan erT-erTze, romlis gadawyvetasac ganvizraxavT.

(klasifikacia saerTod pirobiTia, romelic droTa

ganmavlobaSi icvleba).

magaliTad, analizis an sinTezis meTodebis Sesabamisad, sistemebi SeiZleba warmovadginoT, rogorc

gaxsnili, Caketili an kombinirebuli, wrfivi an

arawrfivi, stacinaruli an arastacinaruli,

?

17

erTganzomilebiani an mravalganzomilebiani, adapturi an

araadapturi. avtomaturi marTvis sistemebSi an mis

calkeul elementebSi gamoyenebuli signalebis saxis

mixedviT SegviZlia davaxarisxoT rogorc uwyveti,

diskretuli (cifruli), modulirebuli an

aramodulirebuli sistemebi.

Tu avtomaturi marTvis sistemebs ganvixilavT

daniSnulebis mixedviT, e. i. imisda mixedviT, Tu rogori

kanoniT unda Seicvalos marTvadi (saregulirebeli)

sididis mocemuli mniSvneloba, ams (avtomaturi marTvis

sistema) SeiZleba davyoT sam ZiriTad saxed:

1. avtomaturi stabilizaciis sistema, romelic

samarTav sidides inarCunebs mudmivad an garkveul

farglebSi (biologiur sistemebSi homeostazis

farglebSi). magaliTad, sisxlSi Saqris

koncentraciis stabiliazacia, an ujredSi

eleqtrolituri balansis SenarCuneba da a. S.

2. programuli marTvis sistema, romelic samarTav

sidides cvlis winaswar mocemuli programiT. (magaliTad, sisxlSi Saqris koncentraciis

cvlileba organizmSi insulinis SeyvaniT garkveuli

programiT).

3. mimyoli sistema, romelSic funqcionirebis

algoriTmi winaswar cnobili ar aris da icvleba

nebismieri kanoniT. rogorc wesi, es kanoni

gansazRvrulia gare movleniT, romelzec Cven

zemoqmedeba ar SegviZlia, magram saWiroa

gaviTvaliswinoT da sistema unda mihyves mas.

18

avtomaturi marTvis gaxsnili da Caketili avtomaturi marTvis gaxsnili da Caketili avtomaturi marTvis gaxsnili da Caketili avtomaturi marTvis gaxsnili da Caketili

sistemebisistemebisistemebisistemebi

avtomaturi marTvis gaxsnili sistemis arsi

mdgomareobs SemdegSi: marTvis algoriTmi gamomuSavdeba

mxolod funqcionirebis mocemuli algoriTmiT da igi ar

aris damokidebuli SeSfoTebasa da gamomaval sidideebze

(nax. 2). mocemul sistemaSi r Semavali sidide (rac igivea

– funqcionirebis algoriTmi, sasurveli sidide)

gamomuSavdeba specialur organoSi (programis gadamwodSi) an TviTmmarTvel mowyobilobaSi da gadaecema

obieqts. obieqtis y gamomavali sidide imarTeba r funqcionaluri algoriTmis saSualebiT da sxvaobis

mospoba r sasurvel sididesa da y realur sidides Soris

ar xdeba, radgan r kontrols ver uwevs F SemSfoT

zemoqmedebas, romelic moqmedebs y-ze.

imis gamo, rom F SemSfoTi zemoqmedebis kontroli ver xorcieldeba da Sesabamisad ar xdeba mis mier gamowveuli

y – gamomavali sididis cvlilebis koreqcia. gaxsnil

nax. 2.

19

marTvis sistemaSi gvaqvs funqcionirebis algoriTmsa da

marTvis realur algoriTms Soris didi gansxvaveba, anu

sistemaSi gvaqvs y – gamomavali sididis didi cdomileba.

sistemaSi y-sa da r-s Soris Sesabamisobis sizuste

mTlianad ganisazRvreba sistemis parametrebisa da

SeSfoTebebis mudmivobiT da ar kontroldeba. amdenad

aseTi sistemebi gamosadegia mxolod zemoT aRniSnuli

pirobebis maRali stabilurobisas da sistemis sizustis

dabali moTxovnebisas.

gaxsnili ams-is meore variantia – ams SeSfoTebiT,

an rogorc kidev mas uwodeben – sistema avtomaturi kompensaciiT. aseTi sistemebi gamoiyeneba, rodesac

sistemis amocanaa obieqtis gamosasvleli sididis y-is mudmivad SenarCuneba.

ams mmarTveli da SemSfoTi zemoqmedebiT

warmoadgens gaxsnili ams-is srulyofil nairsaxeobas.

Caketil ams-Si mmarTveli mowyobilobis

Sesasvlelze miewodeba rogorc mmarTveli r, agreTve

gamomavali y- sidide. aseT ams-Si mmarTveli mowyobiloba

miiswrafis mospos y-is yvela gadaxra im mniSvnelobisagan,

romelic ganisazRvreba r-iT. es gadaxrebi SeiZleba gamowveuli iyos SeSfoTebebiT (Siga da gare

SeSfoTebebi), an sistemis parametrebis cvlilebiT.

rogorc nax. 2.-dan Cans aseTi ams sistema warmoadgens

Caketil ams-s, romelic Seqmnilia mmarveli mowyobilobiT

da obieqtiT.

am dros mmarTveli mowyobiloba qmnis ukukavSirs

obieqtis garSemo, e. i. akavSirebs obieqtis gamomaval da

Semaval sidideebs. amitom Caketil ams-s uwodeben agreTve

sistemas ukukavSiriT an marTvis sistemas gadaxriT.

20

nax. 3.

ukukavSiri SeiZleba iyos uaryofiTi da dadebiTi.

ZiriTadad ams-Si gamoyenebulia uaryofiTi ukukavSirebi.

praqtikaSi, sistemebs, sadac xdeba procesis maCveneblis

cvlilebis mocemuli kanonis avtomaturi marTva –

vuwodebT avtomaturi marTvis sistemebs. zogadad, mmarTveli zemoqmedeba SesaZlebelia proporciuli iyos

ara marto gamomavali sididis gadaxrisa, aramed misi

warmoebulis an integralis. e. i.

21

∫∆∆∆∆=t

ydtyyyfX0

..,..........,,,( &&& )

sadac ∆y=y0-y warmoadgens gadaxras, y0-procesis

maCveneblis sasurveli sididea, y-procesis maCveneblis

realuri sididea.

ukukavSiris principi gansakuTrebiT farTo

gamoyenebas poulobs cocxal organizmebSi (organizmis

temperaturis regulireba, sisxlis mimoqcevis ritmis

regulireba, adamianis wonasworobis regulireba vestibiularuli aparatis meSveobiT da sxva).

rTuli sistemebis ganxilvisas, romelTac aqvT

mravali Semavali da gamomavali sidideebi iseTi termini

rogoricaa dadebiTi da uaryofiTi ukukavSiri xSirad ar

gamoiyeneba. magram termini `ukukavSiri~ praqtikulad

yovelTvis gamoiyeneba martivi, Caketili ams_ebis

aRwerisas, romelTac aqvT cvladebis mcire ricxvi.

uaryofiTi ukukavSiris cneba Zalian

mosaxerxebelia biologiur sistemebSi mimdinare

procesebis gasagebad. biologiur sistemebSi ukukavSiris yvelaze

gavrcelebul saxes warmoadgens parametruli ukukavSiri.

nax. 3. warmodgenilia parametruli ukukavSiri.

nax. 3.

K

X Y

Y2

22

pirdapir arxze X1 Sesasvlelidan Y1-gamosasvlelisaken gvaqvs parametri K, romelic SeiZleba

icvlebodes Y2 cvladis zemoqmedebiT. e. i. K=f(Y 2). maSin Semavali X1 signalis gazrda (an Semcireba) pirdapiri

arxiT, gamoiwvevs Y1 gamomavali signalis gazrdas (an

Semcirebas), magram Semdgom Y1-is gavleniT icvleba Y2, es

ukanaskneli ki iwvevs K-s sapirispiro mimarTulebiT

cvlilebas (Sesabamisad mis Semcirebas an gazrdas). aseTi

uaryofiTi kavSiris moqmedebis efeqtia – Y1 cvladis

cvlilebis Semcireba, Semavali X1 sididis cvlilebiT.

organizmSi parametruli tipis uaryofiTi ukukavSiris tipiuri magaliTia, sisxlis mimoqcevis

sistemis erT-erTi umniSvnelovanesi parametris cvlileba-

qsovilebis JangbadiT momaragebis cvlilebiT gamowveuli

sisxlZarRvebis winaaRmdegobis cvlileba.

ganvixiloT qsovilebis JangbadiT momaragebisa da

utilizaciis procesebi. Jangbadis moxmarebis tempi rO2 CavTvaloT winaswar mocemulad da ω-s tolad. ujredebSi Jangbadis miwodebis tempi qO2 pirvel miaxloebaSi

ganisazRvreba ori faqtoriT: V=pO2 Jangbadis

daZabulobis gradientiT qsovilebsa da arteriul sisxls

Soris da sisxlis θ raodenobis Soris, romelic

gaedineba qsovilebis sisxlZarRvebiT. sisxlis θ raodenoba Tavis mxriv damokidebulia sistemis

parametrebze – arteriuli wnevis p sidideze da

sisxlZarRvebis R winaaRmdegobaze, romlis sidide

SeiZleba davakavSiroT qsovilebSi Jangbadis X1 raodenobasTan, romelime R(X1).

zemoT CamoTvlili cvlilebisa da parametrebis

urTierTqmedeba SeiZleba warmovadginoT sqemiT, romelic

mocemulia nax. 4. (a)-ze:

23

(a)

nax. 4.

naxazis zemo nawilSi warmodgenilia uaryofiTi

ukukavSiris ZiriTadi konturi, xolo qveda konturi

warmoadgens parametrul uaryofiT ukukavSirs.

24

movaxdinoT R(x1) funqcionaluri damokidebulebis aproqsimacia wrfivi funqciiT R=a+bx1 da movaxdinoT

parametruli uaryofiTi ukukavSiris zemoqmedebis

ilustrireba sistemis Tvisebebze.

stacionarul reJimSi sistemis gantolebas aqvs

Semdegi saxe:

ωθ −−= 111 )( KXVX&

nax. 4.

25

1bxa

P

R

P

+==θ

ωbPK

aV

bPK

PKX

+−

+=

11

1

11

parametruli ukukavSiris intensivoba izrdeba b parametris gazrdisas. rodesac ara gvaqvs parametruli

ukukavSiri b=0 da

ωPK

aVX

111 −=

damokidebuleba x1(v1) am SemTxvevaSi warmoadgens

wrfiv funqcias daxris koeficientiT 1. parametrul

ukukavSiris SemTxvevaSi b>0 da damokidebuleba x1(v1) mainc rCeba wrfivi, mxolod daxris kuTxe V1 RerZis mimarT b-s gazrdiT mcirdeba.

nax. 4 (b)-ze mocemulia x1(v1) funqciis ojaxi b-s sxvadasxva mniSvnelobebis dros. X1-is gazrdas mivyavarT

iqamde, rom cvladebis cvlileba naklebadaa damokidebuli

garemos V1 pirobebze. ukukavSiris principma miiRo

uaRresad didi gavrceleba biologiuri sistemebis

meTodologiur da maTematikur aRwerebSi, rasac Semdgom

TavebSi ufro dawvrilebiT SevexebiT.

26

sinTezis amocanasinTezis amocanasinTezis amocanasinTezis amocana

sinTezis amocanis mizania, iseTi marTvis

mowyobilobis an regulatoris arCeva, romlis drosac

mocemuli obieqti masTan mierTebul regulatorTan erTad

iZleva marTvis sistemas, saWiro maxasiaTeblebiT.

moTxovnebi sasinTezo sistemebTan xSirad aseTia:

1. marTvis sistema unda iyos mdgradi, e.i

sxvadasxva gadaxrebi gamowveuli SeSfoTebis

signalebis zemoqmedebiT unda miilios droTa

ganmavlobaSi.

2. misi reaqciebi Semaval signalebze unda iyos

iseTi Tvisebebis, rac gansazRvravs sistemis

maRal xarisxs, anu gardamavali procesi unda

iyos sakmaod swrafi, Secdomebi damyarebul

mdgomareobaSi unda iyos mcire da a.S.

marTvis sistemis sinTezis amocana, romelic

uzrunvelyofs yvela SesaZlo maxasiaTeblidan

saukeTesos cnobilia rogorc optimaluri marTvis

amocana. optimaluri marTvis sistemis Seqmnistvis

aucilebelia gvqondes amomwuravi informacia

sistemaze da garemoze, romelic faqtiurad arasodes

ar gvaqvs, Tundac imitom, rom gamoTvlebisas viyenebT

realuri sistemis arazust aRweras. amitomac warmoiSva marTvis sistemis sintezis

amocana, romlebsic procesebi inarCuneben maRal

27

xarisxs miuxedavad imisa, rom marTvis obieqtis

parametrebi da garemos cvladebi SeiZleba iyos

ucnobi an icvlebodes gaurkvevlad. aseT sistemebs

uwodeben adaptur sistemebs.

maTi gansakuTrebuloba mdgomareobs imaSi, rom

obieqtis maxasiaTeblebis da garemos parametrebis

cvlilebebis sapasuxod xdeba marTveli mowyobilobis

parametrebis cvlileba. raTqma unda, adapturoba da

optimaluroba-ori sruliad gansxvavebuli cnebaa.

adapturi sistema SeiZleba iyos ara optimaluri,

optimaluri ki araa aucilebeli iyos adapturi, sistemis qceva drosi aRiwereba zogierTi sididis

cvlilebiT, magaliTad, Semavali an gamomavali.

sidideebs romlebic icvlebian sistemis

funqcionirebis processi ewodeba misi cvladebi.

cvladebis reaqcia, romliTac aRiwereba sistema

damokidebulia mudmivi sidideebis garkveul ricxvze,

romelic gansazRvravs sistemis Tvisebebs. am mudmiv

sidideebs uwodeben sistemis parametrebs. gansxvaveba

cvladebsa da parametrebs Soris sistemasi

mdgomareods imaSi, rom cvladebi sistemaSi icvlebian sistemis Sida kavSirebis zemoqmedebiT, xolo

parametrebi Tu icvlebian mxolod gare garemoebaTa

gamo, romelsac araviTari kavSiri ara aqvs Siga

procesebTan.

aseTi ganmarteba cvladebis da parametrebisa ar

SeiZleba CaiTvalos mkacrad, radgan erTi da igive

parametri SeiZleba Seicvalos eqsperimentis seriebis

dros da miiRos yoveli seriis semdeg axali

mnisvnelobebi. garda amisa SeiZleba zogierTi

SemTxvevasi parametrebi Seicvalos droSi damoukideblad winaswar gansazRvruli wesis erTi

28

eqsperimetris farglebsi (aseT SemTxvevasi amboben

cvlad parametrebian sistemebze. amrigad biosistemebis

aRwerebi erTidaigive sidide, SeiZleba warmogvidges

erT SemTxvevaSi rogorc cvladi, xolo meoreSi

rogorc parametri. magaliTad, cxovelis masa,

sisxlismimoqcevis xelovnuri marTvis amocanebSi aris

parametri da cvladi-ontogenetikuri procesebis

gamokvlevisas.

biosistemebis maTematikuri modelebis aRweris

SemTxvevaSi cvladebi da parametrebi warmoadgens

biosistemebis aRmweri gantolebebis cvladebsa da parametrebs.

29

wrfivi da arawrfivi sistemebi, erTganzomilebiana da wrfivi da arawrfivi sistemebi, erTganzomilebiana da wrfivi da arawrfivi sistemebi, erTganzomilebiana da wrfivi da arawrfivi sistemebi, erTganzomilebiana da

mravalganzomilebiani sistemebi, uwyveti da diskretuli mravalganzomilebiani sistemebi, uwyveti da diskretuli mravalganzomilebiani sistemebi, uwyveti da diskretuli mravalganzomilebiani sistemebi, uwyveti da diskretuli

(cifruli) sistemebi(cifruli) sistemebi(cifruli) sistemebi(cifruli) sistemebi

marTvis sistemebi gamoyenebuli elementebisa da

signalebis xasiaTis mixedviT SeiZleba davyoT wrfiv da

arawrfiv, stacionarul da arastacionarul,

erTganzomilebian da mravalganzomilebian, uwyvet da

diskretul (cifrul) sistemebad. marTvis sistemis wrfivoba (nax. 5. a) an

arawrfivoba (nax. 5. b) ganpirobebulia sistemaSi Semavali

elementebis statikuri maxasiaTeblis xasiaTiT.

nax. 5.

30

Tu mkacrad vimsjelebT, bunebaSi wrfivi sistemebi

ar arsebobs, radgan yvela fizikuri sistema xasiaTdeba

arawrfivobiT. maT Soris didi umravlesoba cocxali

sistemebisa Tavis gamovlenaSi xasiaTdebian arawrfivi

TvisebebiT.

arawrfivobis yvelaze gavrcelebul saxes

warmoadgens gajereba, rodesac sistemaSi Semavali

signalis XSem-is cvlilebisas garkveul diapazonSi

icvleba sistemis gamomavali signali Xgam, xolo am diapazonis gareT gamomavali signali gajerebis areSi

xvdeba da ar icvleba Semavali signalis cvlilebiT.

marTvis sistemebis analizisa da sinTezis

gamartivebis mizniT, zogierTi arawrfivi sistema SeiZleba

warmovadginoT wrfivi modeliT, Tu CavTliT, rom

sistemaSi mimdinare procesebis maCveneblebis gadaxra

sakmaod mcirea. am SemTxvevaSi sistemaSi mimdinare

procesebi SegviZlia davaxasiaToT n-uri rigis

Cveulebrivi wrfivi diferencialuri gantolebebiT.

wrfivi sistemebi arawrfiv sistemebTan SedarebiT xasiaTdebian mTeli rigi upiratesobiT da, rac mTavaria,

SegviZlia gamoviyenoT superpoziciis principi. aRniSnuli

principi xasiaTdeba ori TvisebiT: aditiurobiT da

erTgvarovnebiT.

marTvis sistema aditiuria Semavali signalis

mimarT, Tu nulovani sawyisi pirobebis dros r1(t) Semavali

signali iwvevs y1(t) reaqcias gamosasvlelze da r2(t)=>y2(t), maSin r1(t)+ r2(t) Semavali signali gamoiwvevs y1(t)+ y2(t) reaqcias.

marTvis sistema erTgvarovania Semavali signalis

mimarT, Tu nulovani sawyisi pirobebis dros r(t) Semavali

31

signali iwvevs y(t) reaqcias, xolo Kr(t) gamoiwvevs

gamosavalze Ky(t) reaqcias. im dros, rodesac wrfivi sistemebis

dasaxasiaTeblad arsebobs zogadi meTodebi, arawrfivi

sistemebisTvis amas ver vityviT, radgan yvela saxis

arawrfivi sistemebis damaxasiaTebeli zogadi

maTematikuri aparati ar gagvaCnia.

wrfivi sistemebis mniSvnelovan qveklasad

miCneulia wrfivi stacionaruli sistemebi, romlebic

SegviZlia ase davaxasiaToT: Tu nulovani sawyisi

pirobebis dros r(t) Semavali signali iwvevs y(t) reaqcias, maSin t drois gasvlis Semdeg r(t-T) Semavali signali

gamoiwvevs y(t-T) reaqcias. wrfivi stacionaruli sistemebis analizis

saukeTeso aparats warmoadgens laplasis gardaqmnis

meTodi. Tu zemoT aRniSnuli stacionarulobis principi

ara aris daculi, maSin sistemas vuwodebT

arastacionaruls, Tumca Tavisi bunebiT arastacionaruli

sistema (Tu igi ar Seicavs arawrfiv elementebs)

miekuTvneba wrfiv sistemaTa ojaxs.

obieqtis gamomaval sidideTa koordinatTa raodenobaze damokidebulebiT, romelnic qmnian gamomavali

sididis Yr veqtors, ams iyofian erTganzomilebian da

mravalganzomilebian ams-ad.

mravalganzomilebian ams SegviZlia davyoT, rogorc

sistemebi SekavSirebuli da SeukavSirebeli marTviT.

SeukavSirebel marTvian sistemas gaaCnia ramodenime

mmarTveli mowyobiloba, romelTagan TiToeuli

axorcielebs obieqtis Sesabamisi gamomavali sidideebis marTvas. am dros ar gamoiricxeba, rom mmarTveli

mowyobilobebi dakavSirebulni arian erTmaneTTan gare

kavSirebiT.

32

mravalganzomilebian sistemaSi Semaval calkeul

marTvis sistemas ewodeba avtonomiuri, Tu mis mier

marTvadi obieqtis gamomavali koordinata ar aris

damokidebuli sxva koordinatTa mniSvnelobaze.

Tu marTvis sitemaSi mimdinare procesebi

warmoadgens drois uwyvet funqcias, maSin aseT sistemebs

mivakuTvnebT uwyveti sistemebis klass, (nax. 6 a), xolo

Tu marTvis sistemis erT an ramdenime elementSi

mimdinare procesi diskretulia (nax. 6. b) an cifrulia,

sesabamisad marTvis sistemebs ewodebaT diskretuli an

cifruli.

a)

y

t

33

b)

nax. 1.6.2

amrigad, marTvis sistemebis mravalsaxeobam

gansazRvra maTi analizis maTematikuri meTodebi. ase magaliTad, wrfivi sistemebis analizis dros farTod

gamoiyeneba gadacemis funqciebi, impulsuri funqciebi,

sixSiruli funqciebi, blok-diagramebi, grafebis meTodebi,

xolo arawrfivi sistemebis dasaxasiaTeblad miRebulia

fazuri portretis, harmoniuli balansis, wertilovan

gardaqmnaTa da sxva meTodebi, romlebsac zogadad

vuwodebT klasikur meTodebs. amJamad, marTvis sistemebis

analizisa da sinTezis gansaxorcieleblad warmoiSva

Tanamedrove meTodi – mdgomareobaTa sivrcis da

mdgomareobaTa parametrebis meTodi, romeli gamoTvliT manqanasTan erTad kvlevis Zalze mZlavri iaraRia da is

y

t

nax. 6.

34

yvelaze metad fexs ikidebs bio-samedicino marTvis

sistemebis analizsa da sinTezSi.

biologiuri sistemebis specifikabiologiuri sistemebis specifikabiologiuri sistemebis specifikabiologiuri sistemebis specifika

biologiuri sistemis qveS Cveulebriv igulisxmeba

urTierTmoqmed elementTa zogierTi erToblioba,

romelic qmnis erTian biologiur obieqts.

biologiuri obieqtebi, romlebic SeiZleba

ganvixiloT, rogorc cocxali sistemebi Semdegia: -

ujredebi, qsovilebi, organoebi, organizmebis populaciebi, ekosistemebi, biosfero da sxva.

`sicocxle es aris procesebis rTuli erToblioba,

romelic mimdinareobs organizebul materialur sistemaSi

da xasiaTdeba sistemis specifierobis SenarCunebiT

nivTierebaTa cvlis nakadSi da garemos sxvadasxva

35

zemoqmedebisas, Tu es moqmedebebi ar aRemateba sistemis

regulatiur SesaZleblobebs~.

ra Tqma unda sicocxlis es ganmarteba wminda

materialuria da masSi mocemuli ar aris umTavresi, rac

sulier substanciasTan aris dakavSirebuli da romlis

gareSe sicocxle warmoudgenelia. magram unda

gaviTvaliswinoT is garemoeba, rom marTvis Teoria

aRwers materialuri obieqtebis marTvas, maT Soris

biologiurisac, amitom zemoT moyvanili sicocxlis

ganmarteba ufro damkvidrebulia mecnierebaSi.

Cveulebriv msjeloba moTxovnebze, romlebsac unda

akmayofilebdes cocxali sistema daiyvaneba sam ZiriTad

sakiTxamde.

1. cocxali sistemebi warmoadgenen Ria sistemebs. Ria

sistemas uwodeben iseT sistemas, sadac xdeba

energiisa da nivTierebaTa cvla garemosTan.

nivTierebaTa cvlis procesebi (metabolizmi)

cocxali sistemebis Seswavlis dros umTavresia.

2. cocxali sistema warmoadgens TviTmarTvads,

TviTregulirebads, e. i. maT SeuZliaT garemosTan

informaciis mimocvla da am informaciis

gadamuSaveba sakuTari struqturis SesanarCuneblad

da metaboluri procesebis samarTavad.

3. cocxali sistemebi unda iyvnen TviTaRdgenadi. ra

Tqma unda es ar exeba yvela biologiur cocxal

sistemas; magaliTad Tirkmeli, romelic elodeba

gadanergvas da sicocxle SenarCunebulia xelovnurad, TviTaRdgenis procesi gamoricxulia.

36

magram TviTaRdgena ujredis an organizmis doneze

savsebiT SesaZlebelia.

cocxali sistemebis mravali ZiriTadi

fundamenturi Tvisebebi dakavSirebulia marTvis

procesebTan (TviTregulireba, TviTmarTva) romlebic

miedinebian Ria sistemebSi.

cneba Ria sistemis Sesaxeb pirvelad Semotanili

iqna biologiuri mecnierebidan. am sistemisaTvis

damaxasiaTebelia is, rom masSi garedan mudmivad Semodis

nivTiereba, romelic sistemis SigniT sxvadasxva reaqciaSi Sedis. anabolizmis procesebis Sedegad sistemaSi

warmoiqmneba ufro maRali sirTulis komponentebi,

romlis utilizacia xdeba organizmis mier. erTdroulad

mimdinareobs katabolizmis procesebi, romlis saboloo

produqtebi gamodis sistemidan. anabolizmisa da

katabolizmis procesebis erTobliobas ewodeba

metabolizmi.

Ria sistemebis erT-erT ZiriTad damaxasiaTebel

Tvisebas warmoadgens is, rom masSi miiRweva dinamikuri

wonasworoba, anu moZravi wonasworuli mdgomareoba. amasTan sistemis struqtura rCeba mudmivi, magram es

mudmivoba SenarCunebulia nivTierebaTa cvlis pirobebSi

garemosa da sistemas Soris.

nax. 1.7.1 warmodgenilia Ria sistemis martivi sqema.

A B

y y

nax. 7.

37

mis Sesasvlelze gvaqvs Semavali nivTierebaTa

nakadi – substratebi y1, gamosavalze – katabolizmis

produqtTa nakadi – y2. sistemis SigniT A nivTiereba Seqcevadi reaqciebis msvlelobisas gardaiqmneba B nivTierebad.

cxadia, nebismier biologiur sistemaSi,

metaboluri reaqciebis gza gacilebiT rTulia: isini

miemarTebian paralelurad, Semdgom an ikribebian an

ifantebian. metaboluri reaqciebis sistemas axasiaTebs autoregulirebis Tvisebebi, romelic ganpirobebulia

maTSi pirdapiri da ukukavSiris wredebis arsebobiT.

rTuli bioqimiuri sistema warmodgenilia nax. 7-ze.

ukukavSiri warmoadgens E1 fermentis daTrgunvas Sn saboloo produqtiT, romelic miiReba S0 sawyisi

substanciidan S1,………Sn-1 saSualedo formebis

saSualebiT. fermentebi, romlebic ukeTeben katalizirebas

Sualedur reaqciebs aRiniSneba E1, E2,……….,En-iT.

M1 aris nivTiereba, romelsac aqvs unari

daTrgunos an gaaqtiuros fermenti E1. Mk iseTive nivTierebaa Ek fermentisaTvis. pirdapiri kavSiri

warmodgenilia fermentis aqtivaciiT nivTierebis mier.

38

(nax. 8.)

pirdapiri da ukukavSiris meqanizmis didi

raodenobis erToblivi moqmedeba, romelic aregulirebs fermentebis sinTezis aqtiurobas da siCqares, gvaZlevs

saSualebas SevinarCunoT mudmivad saboloo produqtebis

koncentracia, garemo SeSfoTebis sakmaod farTo

intervalSi cvlilebisas. ganxiluli tipis meqanizmebi

TiTqos CaSenebulia saregulirebel sistemaSi, amitom

maTTvis gamoiyeneba termini `Sida marTva~.

Sida marTva aris pasiuri. es niSnavs, rom

realurad arsebul sistemaSi stacionaruli

wonasworuli mdgomareobis SenarCuneba ar saWiroebs

raRac metaboluri samuSaos Sesrulebas. pasiuri marTva sistemaSi warmoiqmneba, rogorc

Sedegi elementTa urTierTqmedebisa romlebic qmnian am

sistemas. pasiuri marTvis tipiur magaliTs warmoadgens

sistemaSi myof nivTierebaTa urTierTqmedeba da am

39

nivTierebaTa nakadebi, romlebic cirkulireben sistemis

SigniT da kveTen sistemis sazRvrebs garemosTan.

pasiuri marTvis meqanizmebi Zevs sazRvarze, marTvad

da umarTav sistemebs Soris.

marTvis TeoriaSi gadawyvetileba imis Sesaxeb

mivakuTvnoT Tu ara sistema marTvadi sistemebis klass

miiReba Tu sistemaSi gvaqvs ukukavSiris arxebi. marTvis

TeoriaSi pasiuri sistemebi Cveulebrivad ganekuTvneba

aramarTvad sistemebs.

biologiur sistemebSi teqnikuri sistemebisagan

gansxvavebiT pasiuri sistemebi miekuTvneba marTvadi sistemebis klass. cocxal sistemebSi marTvis pasiuri

xasiaTi mdgomareobs SemdegSi: Tu mocemulia zogadi

fizikuri da qimiuri kanonebi, aseve cocxali sistemebis

organizacia da masSi Semavali komponentebi, maSin

stacionaruli mdgomareobis SesanarCuneblad ar aris

saWiro metaboluri samuSaoebis Sesruleba.

aqtiuri marTva biosistemebisagan Txoulobs

metaboluri samuSaoebis Sesrulebas. amdenad aqtiuri

marTva aris garegani, xolo pasiuri marTva Sinagani.

gare marTvis meqanizmi zed edeba Sinagans, ris xarjze iqmneba erTiani sistema marTvis Caketili

konturebiT. iqmneba pirdapir da ukukavSirebis

kompleqsebi.

cocxal sistemebSi, gansxvavebiT teqnikuri

sistemebisagan, pasiuri marTva SeiZleba warmodgenil iqnas

ukukavSiris saxiT.

marTvis garegani meqanizmebi organizebulia

imgvarad, rom nebismieri SeSfoTeba F sistemaze, romelic

cvlis mis mdgomareobas, SeiZleba aRmoCenil iqnas

specialuri mgrZnobiare elementebiT. mgrZnobiare elementebidan signali miewodeba konturis gare

40

elementebs, sadac xdeba misi damuSaveba da gamomuSavdeba

samarTavi signali, romelic xels uSlis sistemaze moqmed

SeSfoTebebs.

xSirad, rodesac cocxali sistemebis

kvlevebisaTvis gamoiyeneba avtomaturi marTvis Teoriis

meTodologia biolog-specialistebs es meTodologia ar

akmayofilebT. es upirveles yovlisa ganpirobebulia imiT,

rom marTvis Teoria gansakuTrebul aqcentebs akeTebs

sinTezis amocanebze, maSin roca biolofiuri amocanebi

ufro analizis amocanebs ganekuTvnebian.

amis gamo xSirad ismis bunebrivi kiTxva. ra zomiT SeiZleba gamoviyenoT teqnikuri sistemebis sinTezis dros

miRebuli modelebi biologiuri sistemebis

analizisaTvis? yovel SemTxvevaSi Zalian mkafiod unda

gvesmodes, rom avtomaturi marTvis Teoriis yvela meTodi

da cneba sufTa saxiT ar SeiZleba gamoviyenoT

biologiaSi. cnebebi, meTodebi da marTvis Teoriis ideebi

– es agurebia, romlebic cocxali obieqtebis Senobis

asaSenebladaa saWiro, magram xSirad misi xelovnuri

CarTva biologiuri sistemebis analizisaTvis yovlad

dauSvebelia. bolo periodSi xSirad vxvdebiT naSromebs marTvis

TeoriaSi, romlebic dawerilia biologebisaTvis da

mTlianad agebulia masalisagan cocxal sistemebze. magram

aseT naSromebSi masalis gadmocema rogorc wesi iwyeba

marTvis Teoriis umartivesi meTodebiT da ar

iTvaliswinebs Tanamedrove meTodebs, romlebic saWiroa

bilogiuri sistemebis analizisaTvis.

ukanasknel wlebSi, optimaluri sistemebis

terminologia da ZiriTadi ideebi farTod gamoiyeneba

teqnikaSi da biologiuri sistemebis analizisas. marTvis optimaluri procesebi, optimaluri konstruqciebi da

41

organizmis maxasiaTeblebi, bunebrivi obieqtebisa da

sistemebis eqspluataciis optimaluri meTodebi – es

gansazRvrebebi biologiur mecnierebaSi Semovida

teqnikuri disciplinebidan.

idea biosistemebis optimaluri konstruqciebis

Sesaxeb mTlianad emyareba optimalurobis warmodgenebs,

romlebic marTvis Teoriisgan aris nasesxebi. xolo

termini `adaptacia~ marTvis TeoriaSi aRebulia

biologiidan. biologiaSi es termini aRniSnavs organizmis

Semgueblobas garemo pirobebTan. amitom aseTi terminis

uSualo gamoyeneba marTvis sistemebSi uxerxulia, vinaidan nebismier, Tundac yvelaze martiv sistemasac ki,

ukukavSiriT gaaCnia unari avtomaturad Seeguos garemo

pirobebis cvalebadobas.

saboloo jamSi, marTvis TeoriaSi adaptur

sistemebs miakuTvneben mxolod imaT, romlebic

avtomaturad eguebian obieqtis ganzomilebisa da garemo

faqtorebis winaswar ganusazRvrel cvlilebas.

marTvis sistemebSi adaptaciis yvelaze martiv

SemTxvevas, garemo pirobebisa an parametrebis cvlilebaze

warmoadgens is, rom am cvlilebebis sapasuxod mimdinareobs regulatoris parametrebis cvlilebani,

romlebic mimarTulia sistemis muSa mdgomareobisa da

xarisxis SenarCunebisaken. amisaTvis, garda ZiriTadi

konturisa, marTvis sistemaSi Segvyavs agreTve adaptaciis

damatebiTi jaWvi (nax. 8.).

42

nax. 8.

garda ZiriTadi konturisa, romelic regulatorsa

da obieqts Sorisaa, sistemas gaaCnia adaptaciis

damatebiTi jaWvi. adapturi mowyobiloba zomavs sistemis

Semaval da gamomaval cvladebs da maTi analizis

safuZvelze cvlis regulatoris parametrebs, sistemis

muSaobis unarianobisa da xarisxis SenarCunebis mizniT.

ZiriTadi konturisa da adaptaciis damatebiTi

jaWvis cvalebadi nawilis erTobliobas ewodeba, adaptu ri regulatori. Cveulebriv, adaptaciis jaWvi

xasiaTdeba procesebis ufro neli dinebiT, vidre marTvis

ZiriTadi konturi. parametrul ukukavSirs gaaCnia

moyvanili sqemis analogiuri struqtura, amdenad

biosistemis nebismieri modeli, romelic Seicavs Tundac

erT parametrul ukukavSirs, marTvis Teoriis terminebiT

miekuTvneba adaptur sistemas.

publikaciebSi biosistemebSi marTvis Teoriis

gamoyenebis Sesaxeb arsebobs tendencia, miCneul iqnas

adapturi sistemebi srulyofil optimalur sistemebad. Tumca adapturoba, aranairad ar aris dakavSirebuli

optimalurobasTan; es marTvis sistemis ori

43

damoukidebeli maxasiaTebelia. adaptacia aumjobesebs

sistemis xarisxs, Tumca ar mihyavs ~idealamde~ -

optimalur sistemamde. adaptacia marTvis TeoriaSi,

xSirad gamoiyeneba, rogorc saSualeba optimalurobis

misaRwevad winaswar ucnob pirobebSi.

sistemaSi procesebis kargi xarisxis (an

optimalurobis) miRweva adaptaciis xarjze, Cveulebriv,

dakavSirebulia sistemaSi damatebiT metnaklebad rTuli

informaciis gadamamuSavebeli mowyobilobis SeyvanasTan.

amitom adaptur sistemebs SeuZliaT Seasrulon

adekvaturi modelis roli sakmaod maRalganviTarebuli cocxali organizmebis adapturi qcevisa, romlebsac

informaciis gadamamuSavebeli kargad ganviTarebuli

aparatis meSveobiT SeuZliaT moaxdinon Sefaseba garemos

savaraudo maxasiaTeblebisa da moaxdinon reagireba am

Sefasebis mixedviT.

cocxal organizmebSi zeorganizebul da

suborganizebul doneze adaptaciis procesi saWiroa

mimdinareobdes, SedarebiT ubralo gagebiT, algoriTmuli

uzrunvelyofis doneze. adaptaciis yvelaze ubralo

SesaZlo meqanizmad gvevlineba parametruli ukukavSiris gamoyeneba. cocxali organizmebis adaptiuri qceva,

ganxorcielebuli parametruli ukukavSiris saSualebiT,

nawilobriv ganixileba r. rozenis wignSi.

amasTan sistemis qcevis adapturi xasiaTi niSnavs,

rom Tu misi reaqcia mocemul zemoqmedebaze ar aris

sasurveli, maSin sistemaSi mimdinareobs TandaTanobiT

iseTi cvlilebebi, romelic mas sasurvel mdgomareobamde

miiyvans. arawrfiv biologiur sistemebSi adaptacia

xorcieldeba imis xarjze, rom sistemis calkeuli

rgolebis maxasiaTeblebi iwyeben cvlilebas sistemis

44

gamomavali signalebis (an raime Sinagani signalebis)

mniSvnelobaze damokidebulebiT.

sainteresoa, rom r. rozeni anviTarebs ra

optimaluri sistemebis Teorias, mkiTxveli, uneburad,

mihyavs `eretikul~ azramde – imis magivrad, rom SevqmnaT

rTuli optimaluri konstruqciebi da algoriTmebi,

biosistemebisaTvis sakmarisia miviRoT parametruli

ukukavSiris tipis martivi da saimedo meqanizmebi garemos

pirobebTan SeguebiT, Tumca ki ufro energo tevadi

saSualebebiT. optimaluroba da adaptacia gvevlinebian

or gansxvavebul da Tanac konkurirebad variantad erTi saerTo miznis misaRwevad – biosistemis sasurveli

mdgomareobis miReba.

boilogiaSi kibernetikuli warmodgenebisa da

marTvis Teoriis meTodebis gamoyenebamde terminebi

optimumi, optimaluroba gamoiyeneoboda sxvadasxva

biologiur disciplinebSi metadre ukeTesi saarsebo

pirobebis aRniSvnisaTvis, romlebic Seesabameba

cxovelmoqmedebis procesis mimdinareobis sasurvel

pirobebs, anu ~yvela fiziologiuri da bioqimiuri

procesebis garemo pirobebTan damokidebulebis faqtis aRwerisaTvis~.

Tumca, bolo dros, biosistemebis aRwerisas

termini ~optimaluroba~ farTod gamoiyeneba sxva gagebiT,

rodesac optimalurs warmoadgens TviTon am sistemebis

Tavisebureba – maTi yvelaze saukeTeso funqcionireba

(optimaluri procesebi, optimaluri algoriTmebi) an

misi Semadgenloba (optimaluri struqtura).

biologebisaTvis cocxal bunebaSi mimdinare

procesebis siRrmis dauflebulisTvis, damaxasiaTebelia

ara marto codnis miReba da gageba cxovrebiseuli movlenebis kanonebisa, aramed Semadgeneli ena, romelsac

45

isini iyeneben biosistemis aRsawerad. isini ar gaurbian

emociurad Selamazebul terminebsac. arseboba,

araCveulebrivi mudmovoba maTi Sinagani garemosi –

terminebia, romlebsac biologiuri mecniereba siamovnebiT

iyenebs, raTa daaxasiaTon biosistemebSi regulaciis

procesebi.

optimaluroba aris biosistemebis saukeTeso

Semguebloba garemo pirobebTan da misi fesvebi

eqstremaluri procesebis xasiaTsa da cocxal sistemebSi

movlenebze unda veZeboT aracocxali bunebis

eqstremalur principebSi. pirvelad istoriaSi, am principebis formulireba

ekuTvnis leonardo eilers: ~ise rogorc yoveli Senoba

msoflioSi srulyofilia da aSenebulia brZeni

Semoqmedebisgan, arc araferi xdeba qveynad ise, rom masSi

Candes azri raime maqsimumisa, an minimumis.~ es mtkiceba,

romelsac SeiZleba davarqvaT eqstremalurobis principi,

an Tu gnebavT, - optimalurobis principi, devs

sabunebismetyvelo fundamenturi kanonebis safuZvelSi. am

saxis principebi cnobilia optikaSi, meqanikaSi,

eleqtrodinamikaSi da albaTobis TeoriaSi. analogiuri saxiT xdeba daSvebebi cocxal

sistemebSi eqstremaluri xasiaTis procesebze. ase

magaliTad, ~umciresi urTierTqmedebis principi~, am

principis arsi mdgomareobs SemdegSi. rTuli marTvadi

sistemebisTvis tipiuria struqtura, avtonomiuri

qvesistemebiT. qvesistemebis moqmedebis arsi gamovlindeba

imaSi, rom mdgrad mdgomareobaSi yvela isini miiswrafian,

damoukidebeli, avtonomiuri funqcionirebisken. sistemas

hqvia gare samyaroSi mizanmimarTulad momuSave, Tu mas

minimumamde aqves dayvanili urTierToba garemosTan. Tumca SesaZlebelia, rom aseT saerTo formaSi es principi iyos

46

arasakmarisad konstruqciuli da saWiro iqneba misi

dazusteba zogierTi konkretuli cocxali sistemebisTvis.

radgan aracocxali bunebisTvis eqstremaluri principebi

formulirdeba mxolod calkeuli ~klasebis~

gamomJRavnebisTvis.

warmodgena imaze, rom yvela cocxali sistema

SeiZleba ganixilebodes, rogorc marTvis optimaluri

sistemebi, ganicdis kritikul ganxilvas sxvadasxva

sawyisi poziciidan. Cven SevecdebiT gavarCioT sami aseTi

midgoma, romlis arsi miaxloebiT SeiZleba gamoixatos

Semdegi saxiT:

1. evoluciasa da bunebriv gadarCevas ar mivyavarT

optimaluri biosistemebis Seqmnamde.

2. biosistemebs ar sWirdebaT optimaluroba im

gagebiT, rogorc es gaigeba marTvis TeoriaSi.

3. biosistemebis optimalurobis gageba ar Seesabameba

optimalur sistemebze amocanis dayenebas marTvis

TeoriaSi.

evolucias da bunebriv gadarCevas ar mivyavarT

optimaluri biosistemebis Seqmnamde. sanam ganvixilavdeT

arsobriv sakiTxs, SevniSnoT filosofiuri xasiaTis erTi

garemoeba, romelic biosistemis optimalurobis

ganxilvisas, Cveulebriv, ar miiReba angariSSi.

saqme is gaxlavT, rom aracocxali bunebis

eqstremaluri principebi da cocxali sistemebis

evoluciis principebi sxvadasxva rigis kategoriebia.

ucnobia, aracocxali bunebis eqstremaluri kanonebi warmoadgenen Tu ara materiis evoluciis raRac procesis

47

Sedegs. yovel SemTxvevaSi kacobriobisaTvis xelmisawvdom

drois masStabebSi es kanonebi ganixileba rogorc

ucvleli monacemebi – samyaros stacionaruli Tvisebebi.

cocxali sistemebis evolucia, piriqiT,

mimdinareobs sakmaod swrafad. evoluciis yvelaze neli

procesebi mimdinareoben drois geologiur masStabebSi,

yvelaze swrafi ki Cvens Tvalwin mikroorganizmebisa da

virusebis swrafi evolucia (maT Soris xSirad

adamianisaTvis sasargeblo).

evoluciis procesebi midis Taviseburi

eqstremaluri traeqtoriebis mixedviT, magram aRweven ki es traeqtoriebi stacionarul wertilebs? biosistemebi

amJRavneben tendencias Tavisi struqturis, evoluciis

procesSi Tavisi funqcionirebis gaumjobebisaken, magram

miswrafebis ra tempze imyofebian isini evoluciuri

procesis yovel mocemul momentSi garemosTan optimalur

srulyofasTan? gauzes gamoricxvis principis Tanaxmad

erT niSSi erT-erTi maTgani bolos da bolos unda

ganidevnos sruliad. gamodevnis procesi ki SeiZleba

sakmaod didxans gagrZeldes.

paleontologiaSi eWvs ar iwvevs erTsa da imave arealSi saxeobis-aborigenisa da metad

maRalorganizebuli saxeobis-mosulebis xangrZlivi

Tanacxovrebis faqti, romlebic TandaTan aZeveben

pirvelyofil mosaxleobas: ~sicocxlisaTvis

sabrZolvelad ukeTesad SeiaraRebuli emigrantebis mier

organizmebis gaZeveba cnobili dakavebuli

adgilsamyofelidan moiTxovs cotad Tu bevrad

mniSvnelovan dros. SegaxsenebT, rom aq laparakia

geologiur droze. naTelia agreTve, rom drois iseTi

xangrZlivi periodebis ganmavlobaSi SeiZleba ganicados mniSvnelovani cvlilebebi garemomcveli garemos pirobebma.

48

es garemoeba miRebulia mxedvelobaSi SerCevisa da

evoluciis maTematikur TeoriaSi.

garemo, romelSic evoluciurad viTardeba

biosistema, droSi icvleba iseTive siCqariT, rogorc

TviT biosistema. inJinrebma ician, rom aseT pirobebSi

optimumis miReba Znelia. ase magaliTad, maqsimumis

miRwevisas saxeoba TandaTan kargavs ~maqsimaluri

Seguebis~ Tvisebas da rogorc es SesaZlebelia

garemomcvel garemosTan amyarebs yvelaze xelsayrel

Tanafardobas.

49

Ubwj!!JJ!

marTvis sistemebis analizis meTodebimarTvis sistemebis analizis meTodebimarTvis sistemebis analizis meTodebimarTvis sistemebis analizis meTodebi

SesavaliSesavaliSesavaliSesavali

imisaTvis, rom SevisawvloT sxvadasxva saxis

sistemebi unda gamoviyenoT saWiroebis da mixedviT Cvens

xelT arsebuli logikuri, maTematikuri da grafikuli

saSulebani, radganac Cven ar SegviZlia yvela saxis

sistemas erTidaigive aRmweri saSualeba mivusadagoT.

wrfivi uwyveti ukukavSiriani sistemebis analizisa

da sinTezis karg iaraRs warmoadgens laplasis

gardaqmnebi, grafebis Teoria, mdgobareobaTa sivrcis meTodi.

iq sadac saqme exeba wrfivi-diskretuli (cifruli)

sistemebis gamoyenebas SegviZlia gamoviyenoT z gardaqmnebi, informaciis Teoriis zogierTi meTodebi, im SemTxvevaSi,

Tu sistemis Semavali signali aris SemTxveviT-kanoniT

gansazRvruli, SegviZlia gamoviyenoT albaTobisa da

statistikis meTodebi.

gawrfiveba mcire gadaxrebis SemTxvevaSigawrfiveba mcire gadaxrebis SemTxvevaSigawrfiveba mcire gadaxrebis SemTxvevaSigawrfiveba mcire gadaxrebis SemTxvevaSi

Cvens garSemo arsebuli sistemebi da elementebi

Tavisi bunebiT arian arawrfivi, romelTa analizuri

daxasiaTeba Zalian rTulia. wrfivi sistemebis analizis

simartivisa da moxerxebulobis gamo mizanSewonilia

movaxdinoT arawrfivi sistemebis gawrfiveba. es TiTqmis

yovelTvis SesaZlebelia, Tu CavTliT, rom nebismieri

marTvis sistemaSi cvladi sidideebis moqmedebis are

50

SemosazRvrulia da maTi gadaxrebi sistemis nominaluri

muSaobis donidan sakmaod mcirea. am gadaxrebis

mniSvnelobis dakanoneba SeuZlebelia – is yovelTvis

damokidebulia konkretul arawrfivobaze da mkvlevaris

intuiciaze.

wrfivi sistemebisaTvis samarTliania superpoziciis

meTodi, romlis sailustraciod mocemulia grafikuli

magaliTi (nax. 9.). rogorc naxazidan Cans, arawrfiv

maxasiaTebelze cvladebis didi gadaxris SemTxvevaSi

superpoziciis meTodi darRveulia.

gantoleba iTvleba wrfivad, Tu damokidebuli

cvladebi an maTi warmoebulebi arian pirvel xarisxSi

(da ar gvaqvs maTi namravlebi). unda avRniSnoT, rom am

nax 9.

51

mxriv damoukidebel cvladebze araviTari SezRudva ar

aris gaTvaliswinebuli.

qvemoT mocemulia wrfiv da arawrfiv gantolebaTa

magaliTebi, sadac x da y warmoadgenen damokidebul

cvladebs da arian t damoukidebeli cvladis funqciebi:

txdt

dyx

dt

yd324 2

2

2

=++

tydt

dyt

dt

ydsin44

2

2

=++

gantolebaSi meore da mesame wevri aris arawrfivi da

amitom aRniSnuli gantoleba arawrfivia. xolo, rac

Seexeba meore gantolebas, miuxedavad wevrebisa dt

dy da

sint, igi warmoadgens wrfivs (radgan t damoukidebeli

cvladia da SezRudva masze ar vrceldeba).

arawrfivi sistema an gantoleba SesaZlebelia

gawrfivebuli iqnas, Tu davuSvebT, rom damoukidebeli

cvladebis gadaxra nebismieri wonasworobis wertilidan (anu muSaobis nominaluri mdgomareobidan) aris `mcire~,

ise rom am gadaxraTa da maT warmoebulTa namravli da

xarisxebi SeiZleba ugulebelvyoT. mocemulia arawrfivi

gantoleba:

032 2 =−+ xydt

dy

gantolebaSi mocemuli damokidebuli x da y cvladebi

gamovsaxoT Semdegnairad:

52

)()( 0 txxtx ∆+= da )()( 0 tyyty ∆+=

sadac x0 da y0 warmoadgens wonasworobis wertilebs da

arian mudmivebi. )(tx∆ da )(ty∆ gamosaxaven x(t) da y(t) damokidebuli cvladebis wonasworuli mdgomareobidan

gadaxrebs (nax. 10 a, b).

(a)

nax. 10

53

(b)

nax. 10.

radgan x0 da y0 warmoadgens mudmivebs, amitom maTi

warmoebulebi udris nuls da gveqneba

[ ])(0 ty

dt

yd

dt

yyd

dt

dy&∆=∆=

∆+=

[ ])(0 tx

dt

xd

dt

xyxd

dt

dx&∆=∆=

+=

CavsvaT gamosaxulebebi gantolebaSi, gveqneba

[ ] 0)(3)(63)(22)()(3)(22)( 20

200

200 =∆−∆−−∆++∆=∆+−∆++∆ txtxxxtyytytxxtyyty &&

54

Cven davuSviT, rom gadaxra wonasworuli mdgomareobidan

aris Zalze mcire, amitom )(2 tx∆ SegviZlia

ugulebelvyoT da miviRebT arawrfivi gantolebis wrfiv

aproqsimacias:

)(63)(22)( 02

00 txxxtyyty ∆−−∆++∆& =0

wonasworuli (anu damyarebuli) mdgomareobis gamomsaxveli gantoleba SesaZlebelia uSualod

ganisazRvros gantolebidan im daSvebiT, rom yvela

gadaxra da maTi warmoebulebi wonasworobis wertilSi

udris nuls e.i.

,0)( =∆ ty 0)( =∆ ty& da 0)( =∆ tx

wonasworobis gantoleba miiRebs Semdeg saxes:

032 200 =− xy

es gantoleba SegveZlo migveRo sawyisi

gantolebidanac im pirobiT, rom damyarebul reJimSi

yvela damokidebuli cvladis warmoebuli udris nuls.

damyarebuli mdgomareobis gantoleba gamovricxoT

gawrfivebuli gantolebidan. miviRebT mudmivkoeficientebian diferencialur gantolebas,

romelic gamoxatavs sistemis yofaqcevas wonasworuli

mdgomareobis mimarT mcire gadaxrebSi.

062 0 =∆−∆+∆ xxyy&

55

es gantoleba cnobilia, rogorc dinamikis gantoleba.

gawrfivebis es meTodi sakmaod Sromatevadia,

rodesac arawrfivi gantoleba rTulia. amitom aseT

SemTxvevebSi unda gamoviyenoT teiloris mwkrivebi,

romlis gaSlac xdeba wertilis mimarT.

vTqvaT, F aris x da y cvladebis arawrfivi funqcia

da akmayofilebs Semdeg pirobebs:

a) x da y cvladebis yvela wonasworuli

mdgomareobidan gadaxra x∆ da y∆ sakmaod mcirea.

b) F funqcias wonasworobis wertilis areSi gaaCnia

yvela cvladis mimarT kerZo warmoebulebi. F funqciis teiloris mwkrivad daSlas F0

wonasworobis wertilis mimarT eqneba Semdegi saxe:

......!2!2

),(),(2

2

22

2

2

00

0

0

0

0

+∆⋅∂

∂+∆⋅∂

∂+∆⋅∂∂+∆⋅

∂∂+=

==

==

y

y

Fx

x

Fy

y

Fx

x

FyxFyxF

yyxx

yyxx

zemoT mocemuli a) pirobis Tanaxmad, teiloris

mwkrivis is wevrebi, romlebic Seicavs gadaxrebis

xarisxebs da maT namravlebs, SegviZlia ugulebelvyoT.

garda amisa, F(x0, y0) wevri gansazRvravs damyarebul

reJims da misi gamoklebiT dagvrCeba dinamikis

gawrfivebuli gantoleba:

56

0

00

=∆⋅∂∂+∆⋅

∂∂

==

yy

Fx

x

F

yyxx

am gantolebis daxasiaTebisas mxedvelobaSi unda

miviRoT Semdegi garemoebebi:

1) gantoleba aRwers igive sistemas garkveuli miaxloebiT, radganac ugulebelvyaviT

maRali rigis mcireebi.

2) cvladebia ara x da y, aramed maTi gardaqmnebi x∆ da y∆ .

3) miRebuli gantoleba aris wrfivi x∆ da y∆

gardaqmnebisa da maTi warmoebulebis

mimarT.

4) F funqciis warmoebulebi warmoadgenen

mudmiv sidideebs da ganisazRvrebian wonasworobis wertilSi gavlebuli mxebis

tangensiT e.i. gadaxris koeficientiT

10

1αtg

x

FK =

∂∂= ;

20

2αtg

y

FK =

∂∂=

(geometriuli interpretacia ixileT nax 11. a, b)

geometriulad gawrfiveba niSnavs (nax 11. a, b) mocemuli

arawrfivi maxasiaTeblebis Secvlas wonasworobis

wertilSi gavlebuli mxebiT da koordinatTa saTavis O′

57

wertilSi paraleluri gadataniT (imis da mixedviT t dro cxadi saxiT Sedis Tu ara F funqciaSi-Sesabamisad gveqneba cvladi an mudmivi koeficientebi da marTvis

sistemac iqneba Sesabamisad stacionaruli an

arastacionaruli).

58

nax. 11.

59

gamoviyenoT arawrfivi gantolebis teiloris

mwkrivis saSualebiT gawrfivebis meTodi da ganvixiloT

zemoT mocemuli gantoleba (2.2.3). winaswar es gantoleba

ganvsazRvroT maRali rigis warmoebulis mimarT:

232 xydt

dy +−=

es gantoleba CavweroT ase:

232),( xyyxF +−=

gavSaloT F(x,y), anu dt

dy, teiloris gawrfivebul

mwkrivad, miviRebT: [ ]

yy

Fx

x

F

dt

yyd∆−

∂∂+∆

∂∂=

∆+

00

0

sadac

[ ]00

2

0632 xxy

xx

F =+−∂∂=

∂∂

[ ] 232 02

0−=+−

∂∂=

∂∂

xyyy

F

CavsvaT gamosaxulebebi gawrfivebul gantolebaSi,

miviRebT:

60

yxxdt

yd ∆−∆=∆26 0

anu

062 0 =∆−∆+∆ xxyy&

romelic wina gantolebis analogiuria.

amdenad gawrfivebuli diferencialuri gantolebebi

SeiZleba amoixsnas ori: klasikuri da laplasis gardaqmnis meTodiT. ganvixiloT orive meTodi.

klasikuri meTodiklasikuri meTodiklasikuri meTodiklasikuri meTodi

avtomaturi marTvis sistemis qcevis aRmwer zogad

gantolebas aqvs saxe:

)(.......... 11

1

1 tFydt

dya

dt

da

dt

yda

n

n

nn

n

n =++++ −

−−

sadac F(t) – sistemis Semavali signalia (damokidebuli

cvladi)

y(t) – gamomavali signali (damokidebuli cvladi)

t - dro (damoukidebeli cvladi)

naaa .......,, 21 mudmivebia (sistemis parametrebi)

gantoleba aris araerTgvarovani n-uri xarisxis

mudmiv koeficientebiani wrfivi diferencialuri

gantoleba. gantolebis amoxsna warmoadgens

funqcionalur kavSiris y(t) damokidebul cvladsa da t

61

damoukidebel cvlads Soris, romelic ar Seicavs y-is warmoebuls t-Ti.

zogadad gantolebas akmayofilebs mravali F(t) funqcia. e.i mas aqvs mravali sxvadasxva amonaxsni. am

gantolebis zogadi amonaxsni ewodeba iseT amonaxsns,

romlidanac SeiZleba miviRoT yvela danarCeni amonaxsni

(romlebsac ewodebaT kerZo amonaxsnebi). Tavisufali

mudmivebis mniSvnelobis Sesabamisi SerCeviT, ganvixiloT

aseTi amonaxsnebis moZebnis meTodi.

gantolebis amoxsna klasikuri meTodiT iwyeba

zogadi amonaxsnis y(t)–s ori Sesakrebis yTav(t) da yiZ(t) jamis warmodgeniT.

y(t)= yTav(t)+ yiZ(t)

am mdgenelTa dasaxeleba damokidebulia imaze, Tu vin

xsnis gantolebas: maTematikosi Tu specialisti marTvis

TeoriaSi. maTematikosi maT uwodebs gantolebis zogad

amonaxsns da kerZo integrals, xolo specialisti

marTvis TeoriaSi Sesabamisad gardamavali procesis

mdgenels (anu Tavisufali moZraobis mdgenels) da

iZulebiTi mdgenels (anu sistemis reaqcias). e.i. yTav(t) warmoadgens gantolebis zogad amonaxsns, rodesac F(t) =0

0.......... 11

1

1 =++++ −

−− y

dt

dya

dt

da

dt

yda

n

n

nn

n

n

zogadi amonaxsni warmoadgens eqsponencialuri wevrebis

jams:

tn

tt neCeCeCty ααα +++= ........)( 2121Tav

62

sadac C1, C2, ……., Cn nebismieri mudmivebia 1α , 2α , ……., nα …maxasiaTebeli gantolebis

sxvadasxva fesvebia.

maxasiaTebeli gantoleba miiReba y-is da misi

warmoebulis Sesabamisi α _iT SecvliT.

;0α=y 1α=dt

dy

22

2α=

dt

yd;…….…, n

n

n

dt

yd α=

gantoleba miiRebs Semdeg saxes:

01.......... 11

1 =++++ −− ααα aaa n

nn

n

gantolebis zogadi amonaxsnis meore wevri yiZ(t), warmoadgens am gantolebis nebismier kerZo amonaxsns.

cnobilia yiZ(t) moZebnis sakmaod zogadi meTodebi

nebismieri F(t)-Tvis. xSir praqtikul SemTxvevaSi F(t) SeiZleba warmovadginoT sxvadasxva saxiT. ase magaliTad,

rogorc mudmivi sidide, t-s dadebiTi mTliani xarisxebis

(e.i. tn), eqsponentis da sinusoidebis saxiT. am SemTxvevaSi yiZ(t)-s mosaZebnad yvelaze mosaxerxebelia gamoviyenoT ganusazRvreli koeficientebis meTodi. aseTi midgomis

dros Tavidanve vuSvebT, rom yiZ(t) saxe Seesabameba F(t)-s saxes. magaliTebi naCvenebia cxril 1-Si.

63

cxrili (1) F(t) yiZ(t)

mudmivi b xarisxi btn

eqsponenta bekt sinusoida bsinω f t kosinusoida bcosω f

t

B

nnnn BtBtBtB ++++ −

−1

110 ........

ktBe tBtB ff ωω sincos 10 ++

tBtB ff ωω sincos 10 ++

yiZ(t) saxis SerCevis Semdeg CavsvaT igi da misi

warmoebuli sawyis gantolebaSi da vipovoT mudmivebis

mniSvnelobebi. amis Semdeg SegviZlia CavweroT gantolebis

zogadi amonaxsni:

++++= tn

tt neCeCeCty ααα ..........)( 2121 yiZ(t)

amiT mTavrdeba gantolebis amoxsna klasikuri meTodiT. aRsaniSnavia, rom amoxsnisaTvis saWiroa n-uri xarisxis algebruli sistemis amoxsna, rac sakmaod

Sromatevadia, rTulia da rTulad aris damokidebuli

sawyis pirobebze. amis gamo Tu diferencialuri

gantolebebi maRali rigisaa, umjobesia

gamoviyenoTYlaplasis gardaqmnis meTodebi miTumetes, rom

es meTodi iZleva saSualebas gamovyoT wevri, romelic

iTvaliswinebs sawyis pirobebs.

64

laplasis gardaqmnebilaplasis gardaqmnebilaplasis gardaqmnebilaplasis gardaqmnebi

laplasis gardaqmnis meTodi damyarebulia

operatiuli aRricxvis Teoriaze. am gardaqmnebis

saSualebiT SegviZlia mravali saxis funqciebi,

rogoricaa sinusi, kosinusi, milevadi sinusoidaluri

funqciebi da eqsponencialuri funqciebi gardavqmnaT S kompleqsuri cvladis algebrul funqciebad.

diferencirebis da integrebis operaciebi SeiZleba

Seicvalos kompleqsur areSi algebruli moqmedebebiT.

amrigad, mudmivkoeficientebiani Cveulebrivi diferencialuri gantolebebi laplasis gardaqmnebis

saSualebiT gardaiqmnebian algebrul gantolebebad,

romelTa saSualebiTac SesaZlebloba gveZleva

ganvsazRvroT sistemis an elementis gadacemis funqciebi.

am ukanaskneliT martivad gardavqmniT da gavaanalizebT

marTvis sistemis modelis blok-diagramas. garda amisa,

gardaqmnis Sedegad miRebuli gantolebebidan SesaZlebelia

ganisazRvros damokidebuli cvladi, romlis laplasis

Sebrunebuli gardaqmniT aRniSnuli sidide gardaiqmneba

droiT areSi moqmed funqciad. gardaqmnis es procesi SegviZlia gamoviyenoT diferencialuri gantolebis

amonaxsnis misaRebad (miiReba rogorc gardamavali

maxasaiaTebeli, e. i. kerZo amonaxsni, aseve erTgvarovani

amonaxsni, e. i. damyarebuli reJimis mniSvneloba).

rom movaxdinoT f(t) funqciis laplasis gardaqmna, is

unda akmayofilebdes pirobebs:

1. f(t) gansazRvrulia da uban-uban diferencirebadia

dadebiTi ricxviTi RerZis gaswvriv [0;∞ ] 2. f(t)=0, Tu t>0.

65

3. arsebobs iseTi dadebiTi ricxvi A da a, rom

sruldeba piroba: atAetf ≤)( (roca ∞<≤ t0 ).

es piroba imas niSnavs, rom f(t) funqciam ar unda miaRwios usasrulobas eqsponencialur funqciaze

adre, roca ∞→t (aRniSnuli sami piroba

kmayofildeba marTvis sistemaSi gamoyenebuli

signalebis TiTqmis yvela SesaZlo saxeebisaTvis) f(t) drois funqciis laplasis calmxrivi gardaqmna

ganisazRvreba calsaxad furies integralidan da

gamoisaxeba ase:

∫∞

−=0

dtetfSF st)()(

an igive gamosaxuleba SeiZleba Caiweros

Semdegnairad :

);()( tfSF = [ ])()( tfLSF =

(miRebuli X(t), f(t), g(t), h(t) an sxva funqciebis

laplasis gardaqmna Caiweros Sesabamisad X(S), F(S), G(S), H(S)…….).

am gamosaxulebaSi cvladi S warmoadgens laplasis

operators da aris kompleqsuri ricxvi ωα jS += (S cvladiT miRebul ares uwodeben S ares anu laplasis

ares).

laplasis gardaqmna warmoadgens calmxrivs,

radganac integrireba xorcieldeba sazRvrebSi [0;∞ ]. es imas niSnavs, rom f(t) funqciis yvela mniSvneloba t=0

66

momentamde uaryofiTia da ganixileba, rogorc nulis

toli. es daSveba ar zRudavs laplasis gardaqmnis

gamoyenebas wrfivi sistemebis amocanebis amoxsnisas,

radgan praqtikuli sakiTxebis gadawyvetis dros

miRebulia, rom funqciis mniSvneloba iwyeba t=0 momentidan. es mosazreba gamodis realuri fizikuri

sistemis Tvisebidan: reaqcia (Sedegi) ver uswrebs win

SeSfoTebas (mizezs). aRvadginoT pirveladi funqcia (e. i. f(t)), saWiroa movaxdinoT laplasis Sebrunebuli gardaqmna:

( )∫∞+

∞−=

j

j

st dseSFj

tfσ

σπ2

1)(

[ ],)()( 1 SFLtf −=

sadac σ aris namdvili sidide da aRemateba F(S) funqciis gansakuTrebuli wertilebis namdvil nawils. laplasis gardaqmniT miRebul F(S) funqcia warmoadgens

kompleqsur funqcias, romelic Seicavs rogorc Re(S) namdvil nawils, aseve Im (S) warmosaxviT nawils, e.i.

)()()( SISRSF me +=

am gamosaxulebis moduli udris 22me IR + , xolo faza

anu kuTxe ϕ aris e

m

R

Iarctg . yvelaferi es gamosaxulia

nax. 12. a) da b)-ze.

67

a) b)

nax. 12

ganvixiloT avtomaturi marTvis sistemebSi

gamoyenebuli signalebis laplasis gardaqmnis magaliTebi.

magaliTi:

davuSvaT f(t)=

≥<

0

00

tA

t

Tu,

Tu,

sadac A aris mudmivi sidide. Cans, rom es aris Aeαt eqsponencialuri funqciis kerZo SemTxveva, roca α=0. gveqneba:

S

A

S

Ae

S

Ae

dteAdtAeAL

t

ttt

tttt =−=−===∞=

−

∞∞−

∞−

∞−

∫∫∫

δδ

δδ

0

0

00

)(

0

S-fesvTa are

ϕ1

S1=(α1+jω)

jIm(S)

68

im SemTxvevaSi, Tu S fesvis namdvili nawili α>0 gamosaxulebis pirveli wevris zRvari udris nuls,

gveqneba

S

AAL =)(

laplasis gardaqmna samarTliania yvelgan, garda

polusisa, sadac S=0. Tu maSin saqme gvaqvs erTeulovan-

safexurovan funqciasTan da Sesabamisad, laplasis

gardaqmnas eqneba saxe:

nax. 13.

sadac 1(t) ganisazRvreba ase:

1(t)=

><

0

00

tA

t

Tu,

Tu,

fizikurad, safexurovani funqcia t=0 momentSi

armoadgens mudmiv signals, romelic myisierad miewodeba

sistemis Sesavals (t=0 momentSi). laplasis gardaqmnis meTodi Zalian moxerxebulia,

radganac es meTodi iZleva saSualebas gamovyoT wevri, romelic iTvaliswinebs sawyis pirobebs

U(t) anu1(t)

1

L[1(t)]=1

0 t

69

)(.... 11

1

1 tLFydt

dya

dt

yda

dt

ydaL

n

n

nn

n

n =

++++ −

−

−

)(.... 11

1

1 tLFLydt

dyLa

dt

ydLa

dt

ydLa

n

n

nn

n

n =++++ −

−

−

laplasis gardaqmnis cxrilidan vipivoT, rom

∑=

−

−−−=

n

kk

kknn

n

n

dt

ydSLyS

dt

ydL

11

01

sadac y0-iT da misi warmoebuliT aRiniSnulia sawyisi

pirobebi t=0-sTvis. magaliTi:

vTqvaT n=3, maSin

)(20

20

023

3

3

dt

yd

dt

dySySLyS

dt

ydL ++−=

CavsvaT gamosaxuleba da davajgufoT L(y)-is Semcveli wevrebi. gamovyoT calke sawyisi pirobebisagan

wevrebi (B1) da SemoviRoT axali aRniSvnebi Ly=y(S), LF(t)=F(S), gardaqmnebis Semdeg miviRebT:

∑+=++++ −− 11

11 1 BSFSySaSaSa n

nn

n )()()..........(

70

[ ]∑+

++++= −

−1

11

1 1

1BSF

SaSaSaSy

nn

nn

)(..........

)(

amis Semdeg Tu movaxdenT laplasis ukugardaqmnebs

orive nawilisagan, davasrulebT amocanis amoxsnas, maT

Soris sawyisi pirobebic iqneba gaTvaliswinebuli, magram

sanam amas gavakeTebdiT, SeviswavloT gantoleba ufro

yuradRebiT. es mogvcems saSualebas gavigoT, ratom aris

es meTodi aseTi popularuli inJiner avtomatikebSi.

gantolebis Sesabamisad, gamosasvleli signalis

y(S)-is gardaqmna udris ori mamravlis namravls. pirvel maTgans inJinrebi uwodeben sistemis gadacemis funqcias,

meores ki Sesaslel funqcias.

gadacemis funqcia Seicavs srul informacias

sistemis zemoqmedebaze, maT Soris gare zemoqmedebis

gardaqmnaze da sawyisi pirobebis gavlenas funqciis

gardaqmnaze (sawyis SeSfoTebaze). Tu yvela sawyisi

piroba nulis tolia, gadacemis funqciis da Sesasvleli

signalis gardaqmna gansazRvraven sistemis reaqcias gareSe

F(S) zemoqmedebaze.

71

Ubwj!!JJJ!

dinamikuri sistemebi, modelirebis sqemebidinamikuri sistemebi, modelirebis sqemebidinamikuri sistemebi, modelirebis sqemebidinamikuri sistemebi, modelirebis sqemebi

biologiuri sistemebis gamosaxvis Tanamedrove

xerxebi grafikuli sqemebis saSualebiT dafuZnebulia

gamosaxvis tradiciul meTodebze, romelic gamoiyeneba

bioqimiaSi, fiziologiaSi da sxva biologiur

disciplinebSi. ase im struqturuli sqemebis saxiT,

romlebmac Tavis droze didi gavrceleba hpova analogiur

gamoTvliTi manqanebis programirebaSi.

amJamad gamoyenebuli sqemebi inarCuneben im

sizustes, romelic damaxasiaTebelia analogiuri

modelirebisaTvis, magram amave dros ar aris gamiznuli viwro interesebisaTvis da aqvs saerTo daniSnuleba

gamosakvlevi bio-samedicino modelebis struqturuli

Tvisebebis TvalsaCinod warmosadgenad.

romelime procesis an sistemis modelirebis sqemis

misaRebad saWiroa misi maTematikuri aRwera. obieqtebis

aseTi aRwera xSirad mocemulia R – uri rigis

diferencialuri gantolebebis formiT, romelic

gantolebis umaRlesi rigis mimarT aris amoxsnili, e. i.

),........,,,........,..........,,,( )1()1( Zxxxxxxxxfdz

xd kmmmm

kiiiiik

i

k−− ″′″′=

(3.1.1) mi .......,2,1=

sadac ix - sistemis cvladia (damokidebuli cvladi), Z –

damoukidebeli cvladi, xolo )1(..........,,, −″′ k

iiii xxxx - ix -

is warmoebulia Z – iT.

72

gantoleba SeiZleba warmovadginoT, rogorc

pirveli rigis gantolebaTa sistema, amisaTvis saWiroa

SemovitanoT axali cvladebi

12,1,

12

01

0

................

,

−−− ==

″==

′==

=

ki

kiki

ii

i

ii

i

ii

xdz

dyy

xdz

dyy

xdz

dyy

xy

maSin gantoleba (3.1.1) miiRebs Semdeg saxes

);......,,;,( 1,101,20,01,

Zyyyyyyfdz

dykmmmkiii

ki−−

− =

yovelTvis SeiZleba CaiTvalos, rom gamosakvlevi

sistema aRiwereba Semdegi saxis gantolebiT.

miZxxxfdz

dxmi

i ..........,2,1),........,,( 21 ==

Tu Z damoukidebel cvlads warmoadgens t dro,

maSin diferencialuri gantolebaTa sistema gadaiwereba

Semdegis saxiT da mas dinamikuri sistema ewodeba.

mitxxxfx mi ,.......,2,1,);.........,,( 21 ==′

73

dinamikuri sistemebi aRweren fizikuri, qimiuri da

biologiuri movlenebisa da procesebis mimdinareobis

droSi.

sistemis amonaxsni zogierTi SezRudvebis

gaTvaliswinebiT drois 0t momentSi iwodeba sistemis

sawyis pirobebad. sawyisi pirobebi Cveulebriv Caiwereba

Semdegi saxiT

1x , 2x , ………. mx cvladebs zogjer uwodeben sistemis

fazur koordinatebs. fazuri koordinatebis erTobliobis

mocemas gansazRvravs sistemis mdgomareoba drois

mimdinare momentSi.

grafikuli sqemebi, romlebic gamoyenebulia

sistemis struqturis aRsawerad, unda asaxavdnen

garkveuli sisruliT kavSirs fazur koordinatebs Soris.

ganasxvaveben sistemis grafikuli warmodgenis ramdenime tips. Tu sqemaSi aisaxeba mxolod ZiriTadi

struqturuli kavSirebicalkeul cvladebs Soris,

romelic iZleva warmodgenas sistemaSi kavSirebis

xasiaTze, maSin aseT sqemas ewodeba sistemis blok-sqema.

blok-sqemis simboloebis yvelaze misaReb formad SeiZleba

CaiTvalos sqemis calkeuli elementebis warmodgena

nebismieri zomis sworkuTxedebis saxiT operaciis

miTiTebiT, romelsac asrulebs mocemuli kvanZi. blok-

00

2002

1001

)(

............

)(

)(

mm xtx

xtx

xtx

=

==

74

sqemis detalizaciis xarisxi SeiZleba iyos sxvadasxva,

romelic SeiZleba mkvlevaris mosazrebebs eyrdnobodes.

modelirebis sqema warmoadgens gantolebaTa

sistemis mkacr grafikul asaxvas, simboloebi, romelic

gamoiyeneba calkeuli elementebisaTvis unda iyos

standartizebuli, xolo struqturis yvela elements

unda hqondes detalizaciis erTi da igive done.

Cveulebrivad iyeneben elementebis aRniSvnis standartul

nusxas, romelic warmodgenilia nax. 14-ze

75

nax. 14

modelirebis sqema aigeba Semdegnairad: yovel

fazur koordinats Seesabameba erTi maintegrebeli

elementi (integratori), xolo Semdgo standartuli

simboloebis saSualebiT grafikulad gamoisaxeba kavSirebi

integratorebs Soris, gantolebis merjvena nawilis

Sesabamisad.

magaliTi: gamovsaxoT blok-sqema da modelirebis sqema martivi dinamikuri sistemisaTvis. ganvixiloT

sisxlZarRvTa sistemis elementis aRwera elastiuri

rezervuaris saxiT. vTqvaT cvladi x - aRwers sisxlis

raodenobas, romelic imyofeba mocemuli sisxlZarRvis

monakveTSi – rezervuarSi. maSin

76

yx −=′ ω

sadac ω da y - aris mocemuli rezervuaridan sisxlis

Semodinebisa da gamodinebis tempebi. Sesabamisad

CavTvaloT sisxlis Semodineba damoukidebel gare

signalad. gamodinebis tempi upirveles yovlisa

damokidebulia rezervuarSi wnevaze, romelic ganisazRvreba Semdegi gantolebiT

)(1

axC

p −=

sadac P – rezervuarSi wnevaa, C – sisxlZarRvis elastiurobis koeficientia, a -konstantaa – rezervuarSi dauZabavi moculoba, anu sisxlis iseTi moculoba,

romelic SeiZleba iyos rezervuarSi P wnevis awevis

gareSe. maSin gadinebis tempi iqneba

)(1

bpR

y −=

sadac R – sisxlZarRvis winaaRmdegobaa, b – konstantaa, romelic Semdgom rezervuarSi wnevis tolia, sadac

sisxli Caedineba.

77

gantolebebis gaerTianebiT vRebulobT

R

bax

RCx +−−=′ )(

1ω

modelirebis sqema warmodgenilia nax. 15-ze.

nax. 15.

78

Ubwj!JW!

marTvis impulsuri sistemebimarTvis impulsuri sistemebimarTvis impulsuri sistemebimarTvis impulsuri sistemebi

signalis daqvantvis saxeebisignalis daqvantvis saxeebisignalis daqvantvis saxeebisignalis daqvantvis saxeebi

signali, romelic daqvantulia doniT, iRebs

mxolod garkveul mniSvnelobebs. signali, romelic

daqvantulia droSi icvleba naxtomiseburad drois

fiqsirebul momentebSi. signali Seizleba erTdroulad

iqnas daqvantuli droSic da doniTac. zemoT xsenebuli

daqvantvis saxis mixedviT diskrituli a. m..s. iyofian

sam tipad: releurebi, impulsurebi da cifrulebi. a.m.s., romelSic signali daqvantulia droSi,

ewodeba impulsuri ; doniT- realuri.

nax 16-ze naCvenebia realuri sistemis

struqturuli Tema da mudmivi denis generatoris Zabvis

regulirebis opoziciuri sistema. Tu raime datvirTvis

nax. 16

79

gamo Zabva davarda, R daSuntdeba, agznebis Zabva gaizrdeba da amas mohyveba Ruzis momWerebze Zabvis

gazrda. avtorxevebis da mdgradobis realuri a.m.s. SeiZleba harmoniuli gawrfivebiT

daqvantvas, romelsac axorcielebs impulsuri

sistema uwyveti signalis impulsebis mimdevrobad

gardaqmnisas, ewodeba impulsuri modulacia. igi

mdgomareobs gamosasvleli impulsebis erT-erTi

parametris SecvlaSi, Sesasvleli (mamodulirebeli)

signalis damokidebulebiT impulsuri elementis

gamosasvlelze. impulsebis mimdevrobis mamodulirebeli

parametri SeiZleba iyos impulsis amplitud,amisi sigane da impulsebis gameorebis periodi. Sesabamisad arsebobs

impulsuri modulaciis sami saxe:

amplitudur_impulsuri; ganed-impulsuri da

droul_impulsuri modulacia. ukanaskneli iyofa fazur_

impulsurze da sixSirul_impulsurze. yvela SemTxvevaSi

impulsis forma modulaciis dros miiReba ucvlelad.

modulaciis mixedviT impulsuri a.m.s iyofa

nax. 17.

80

amplitudur_impulsurze, ganed-impulsurze da a.S.

impulsuri elementi SeiZleba Sediodes marTvis nebismier

funqcionalur rgolSi an miekuTvnebodes marTvis

obieqts.

impulsuri a.m.s ZiriTadi upiratesoba ganpirobebulia

signalis gadacemis ,,wyvetili,, xasiaTiT sistemis

sxvadasxva nawilebs Soris da mdgomareobs

mravalwertilovan marTvis SesaZleblobaSi, kavSirebis

xazebis mravaljerad gamoyenebaSi da gaZlierebul dabrkolebad mdgradobaSi.

nax. 18.

81

a.m.s impulsuri sa.m.s impulsuri sa.m.s impulsuri sa.m.s impulsuri sistemebis maTematikuri istemebis maTematikuri istemebis maTematikuri istemebis maTematikuri

aRweraaRweraaRweraaRwera

aseTi sistemebis maTematikuri aRwera Sesdgeba

uwyveti moqmedebis da diskretuli moqmedebis airisagan.

ukanasknelis maTematikuri aRwerisaTvis aucilebelia

gamosasvleli impulsebis formis, elementis statiuri

maxasiaTeblis da misi drois dayovnebis codna. impulsuri elementis statiuri maxasiaTebeli

warmoadgens mamodulirebeli parametris damokidebulebas

Sesasvleli mamodulirebeli signalebisagan. impulsuri

elementis arweris dros misi statiuri maxasiaTebeli

TamaSobs igive rols, rogorc uwyveti moqmedebis

rgolebSi. wrfiul miaxloebaSi, e.i. Sesasvleli signalis

cvlilebis diapazonis SesazRudad. impulsuri elementis

statiuri maxasiTebelis magivrad SegviZlia davaxasiaTod

gadacemis koeficientiT, gansazRvruliT am maxasiaTeblis

daxriT damyarebuli reJimis wertilSi; ar niSnavs diferencialuri gantolebis wrfilobas. wrfeuli

statiuri maxasiaTeblebis SemTxvevaSi FUM–is gansxvavebiT impulsuri elementebi arawrfivebia.

ukanasknelebi SeiZleba gavawrfivoT impulsebis signalis

SemosazRvruli mniSvnelobebisas impulsebis gameorebis

period Tu- Tan SefardebiT. davuSvaT, rom impulsuri sistema Sedgeba uwyveti

moqmedebis nawilisagan, romelic gadacemis funqciaaris wun(p) da wrfivi impulsuri elementisagan AUM –iT. am SemTxvevaSi impulsuri elementi xasiaTdeba impulsebis

formiT, dayovnebis droiT da gadacemis koeficientiT:

82

sadac X-impulsuri elementis Sesasvlelze moqmedi

signalis sididea. morigi periodis ganmeorebis Tu-s dasawyisi, xolo a-gamosasvleli impulsis amplitudaa

amave periodSi.

imisaTvis, rom movaxdinoT aseTi sistemis Seswavla

Tavdapirvelad saWiroa misi struqturuli sqemis sami

Semdegi gardaqmna: realur impulsur elements vSliT

idealurze da maformirebelze; gare zemoqmedeba

gadagvaqvs impulsuri elementis Sesasvlelze da sistemaSi moqmed realur uwyvet signalebs vcvliT

faqtiur diskretul signalebad.

nax. 19.

83

naxazze moyvanilia gardaqmnis samive tipi:

gardaqmnis Sedegad miviRebT sistemas; romelSic moqmedebs

marto diskretuli signalebi. es mniSvnelovnad

amartivebs

impulsuri a.m.s. aRweras, radgan Cveulebrivi

diferencialuri gantolebis magivrad gveqneba sxvaobiTi

gantolebebi, xolo laplasis Cveulebrivi gardaqmnis

magivrad ki laplasis diskretuli gardaqmna.

aseTi Secvlis SesaZlebloba dafuZnebulia imaze, rom

impulsuri elementis gamosasvleli signali ganpirobebulia Sesasvleli impulsis x(t) mniSvnelobiT

drois diskretul momentSi. impulsebis gameorebis

periodis Tn –s yovel dasawyisSi. amitom impulsebis

elementebss muSaobaSi araferi ar Seicvleba, Tu mis

Sesasvlelze moqmed uwyvet funqcias x(t) SecvliT

diskretuli funqciiT, romlis mniSvnelobac yoveli

periodis dasawyisSi e.i. drois momentSi nTn ( sadac n= 1,2,3,…) emTxveva uwyveti funqciis mniSvnelobebs.

danarCen momentSi ki udris nuls. aseT diskretul

funqcias uwodeben meserisebuls.

avtomaturi marTvis impulsuri sistemebis kvlevis avtomaturi marTvis impulsuri sistemebis kvlevis avtomaturi marTvis impulsuri sistemebis kvlevis avtomaturi marTvis impulsuri sistemebis kvlevis

meTodebimeTodebimeTodebimeTodebi

aseTi sistemebis Seswavla gviCvenebs, rom

impulsuri SeiZleba CavTvaloT uwyvetis eqvivalenturad,

Tu sruldeba utoloba:

84

sadac wf - gare zemoqmedebis umaRlesi sixSirea.

rodesac adgili aqvs ganed-impulsur modulacias an droul-impulsurs, maSin

Tu piroba sruldeba, maSin gamokvleva SeiZleba

vaWarmooT adre Seswavlili meTodebiT. Tu ar sruldeba

impulsuri a.m.s. unda SeviswavloT droSi daqvantvis

gaTvaliswinebiT, e.i. diskretuli gadacemis funqciis

aRweris daxmarebiT an Sesabamisi sixSiruli

maxasiaTeblebiT. es mianiWebs gamokvlevebs garkveul specifikas, magram TviT sizustis, mdgradobis da

gardamavali procesis xarisxis kvlevis meTodi ucvleli

rCeba.

wrfiuli impulsuri a.m.s. mdgradobis sakiTxis

Sesaswavlad gamoiyeneba rausgurvacis, , naikvistis da

mixaelovis kriteriumebi, romlebic modificirebuli

iqneba impulsuri sistemebis aRweris TaviseburebebiT.

impulsur sistemebSi sizustisa da xarisxis

gamokvlevisas gamoiyeneba igive kriterumebi da meTodebi,

rogorc uwyveti moqmedebis sistemebSi. Sesabamisad impulsuri sistemebis sinTezis meTodebic analogiuria

kerZod, aqac vrclad gamoiyeneba sinTezi logariTmul

sixSiruli maxasiaTeblis bazaze. gardamavali procesebis

ageba sruldeba

85

X*вих (z) =w(z)FНР (z)-is amoxsniT.

im SemTxvevaSi, rodesac saWiroa gamosasvleli

sididis mniSvnelobis povna Sualedur momentSi, e.i. Tn-intervalSi gamoyeneba modificerebuli laplasis

diskretuli gardaqmna, romelic iZleva saSualebas vipovoT Sesabamisi sistemis diskretuli gadacemis

funqcia W* (Z) da misi daxmarebiT gardamavali procesis

mrudi X(9t). impulsuri sistemebis koreqcia sruldeba

Cveulebrivi uwyveti moqmedebis makoreqtirebuli

rgolebiT; ukanasknelebs xSirad uwodeben impulsur

filtrebs.

cifroli a.m.s. ganmarteba da Tvisebebanicifroli a.m.s. ganmarteba da Tvisebebanicifroli a.m.s. ganmarteba da Tvisebebanicifroli a.m.s. ganmarteba da Tvisebebani

ufro zustad cifruli sistemebi ewodeba

sistemebs romlebSic aqvT adgili diskretuli signalebis

gardaqmnas cifruli saxiT.

cifruli sistemsbesis yvelaze srulyofilia

diskretul sistemebSi. maTSi mmarTvel rgols TamaSoben

cifruli gamoTvliTi manqanebi. cifruli a.m.s.-is

umetosoba SeiZleba warmovidginoT Semdegi blok-sqemis sax

86

Q sistemis uwyveti nawilia (xSirad SemTxvevaSi

marTvis obieqti). g da y–Sesabamisi uwyveti sididis

cifrulSi gardaqmnilia da SebrunebiT. es gardamqmnelebi

akavSireben uwyvet nawils cifrul gamomTvlel

manqanasTan, romlis Sesasvlelia da gamosasvleli warmodgenilia cifruli formiT.

uwyvet sididis cifrulSi gardaqmna mdgomareobs

misi donit a. droiT daqvantaSi da miRebuli diskretuli

mniSvnelobis warmodgenaSi cifrebis saxiT; aseT

gardaqmnas ewodeba kodur-impulsuri gardaqmna. amitom

cifrul sistemebs xSirad uwodeben kodur-impulsebs.

nax. 20.

87

Sesabamisad A-D gardaqmnas uwodeben kodirebas,

xolo D-A dekodirebass. arsebobs ramodenime saxis

kodireba. umartivesia erTeulovani. am SemTxvevaSi

diskretuli sididis yoveli mniSvneloba yvelaze

gavrcelebulia orobiTi, sadac gamoiyeneba 0 da 1.

cifruli a.m.s. ZiriTadi upiratesoba

ganpirovnebulia im SesaZleblobebiT, romelic

warmoiqmneba cifruli teqnikis gamoyenebiT. upirveles

yovlisa esaa sizuste, dabrkoleba, mdgomareoba da Zalian

rTuli martvis algoriTmebis realizacia; amave dros

marTTvisac damaxasiaTebelia mravalwertilovani marTva.

cifruli a.m.s. dinamikis Taviseburebanicifruli a.m.s. dinamikis Taviseburebanicifruli a.m.s. dinamikis Taviseburebanicifruli a.m.s. dinamikis Taviseburebani

doniT daqvantva cifrul sistemebsi iwvevs, iseve

rogorc realur sistemebsi, arawrfivobas. amis Semdeg maTSi gardamavali procesi xSirad iwvevs avtrexevebs.

magram radgan daqvantvis biji sakmaod mcirea, rxevebis

amplitudac mcirea.

cifrul a.m.s. sizustec ganpirovnebulia

daqvantvis bijit; romlis minimaluri sididec

SezRudulia obieqtis gamosasvleli sididis gazomvis

sizustiT da maTi cifrul formaSi gardaqmnis sizustiT.

daqvantvis bijis q–s SemcirebiT igi uaxlovdeba

impulcur sistemebs. Tu q Zalian mcirea, doniT daqvantva SeiZleba ugulebelvyoT da cifruli sistema

miaxloebiT ganvixiloT, rogorc impulsuri. aseT

impulsur sistemas (rodesac q=0) ewodeba zRvruli

impulsuri sistema. impulsur sistemebTan SedarebiT

88

cifruli a.m.s. dinamikis Taviseburebaa aucilebeli

droiTi dayovneba, romelic ganpirobebulia informaciis

damuSavebis erTi ciklis xangrZlivobiT.

cifruli a.m.s. koreqcia upirveles Yyovlisa

sruldeba cifruli gamomTvleli manqanis algoriTmis

Sesabamisi SerCevis gziT, romelic Sedis mmarTvel

mowyobilobaSi. amitom principialurad aq SeiZleba

gamoviyenoT nebismieri koreqcia, nebismieri algoriTmi,

romlis realizaciac SeiZleba movaxdinoT Sesabamisi

c.g.m._Ti. algoriTmis garTuleba ar iwvevs sizustis

Semcirebas.

89

Aavtomaturi marTvis adapturi sistemebis cneba da Aavtomaturi marTvis adapturi sistemebis cneba da Aavtomaturi marTvis adapturi sistemebis cneba da Aavtomaturi marTvis adapturi sistemebis cneba da

maTi klasifikaciamaTi klasifikaciamaTi klasifikaciamaTi klasifikacia

A

adapteri anu Sewyobadi sistema es aris

sistema,romelic avtomaturad Seewyoba gare pirobebs da

marTvis obieqtis Tvisebebis cvlilebas da

uzrunvelyofs am dros sasurvel marTvis xarisxs

marTvis mowyobilobebis sqemis da parametrebis

cvlilebebiT.

marTvis xarisxi ricxobrivad ganisazRvreba

xarisxis kriteriumiT (j) romelic afasebs avtomaturi

marTvis sistemis windawin amorCeul erT_erT

mniSvnelovan parametrs. es SeiZleba iyos sistemis muSaobis sizustis maCvenebeli,gardamavali procesis

xarisxis, mdgradoba da a.S.

zogad SemTxvevaSi xarisxis kriteriumi (j) warmoadgens funqcionals romelic damokidebulia Semaval

da gamomaval parametrze, rogorc avtomaturi marTvis

sistemebis optimalobis kriteriumi.calkeul SemTxvevSi j

SeiZleba iyos ubralod funqcia.

gare pirobebis cvalebadobis magaliaTi, romelic

moiTxovs adapturi sistemebis gamoyenebas,aris sistemaze

arastacionaluri gare zemoqmedeba,romlis

maxasiaTeblebic magaliTad speqtraluri simkvrive

icvleba farTo sazRvrebSi.am pirobebSi avtomaturi

marTvis sistemis muSaobis sizuste icvleba gare

zemoqmedebis statistikuri maxasiaTeblebis cvlilebasTan erTad. sistemis sizustis gazrdisTvis saWiroa vcvaloT

makoregirebeli rgolebis parametrebi gare zemoqmedebis

90

maxasiaTeblebis cvlilebasTan erTad. am cvlilebas

zogierT SemTxvevaSi asrulebs adamiani. adaptur

sistemaSi igi icvleba avtomaturad.

cvlad parametrebiani sistemis magaliTebs,

romelnic moiTxoven adaptur marTvas, warmoadgenen

TviTmfrinvi da kosmosuri xomaldi. maTi dinamiuri

maxasiaTeblebi icvleba droSi sawvavis masis, obieqtis

formis an aerodinamiuri Zalebis cvlilebisagan frenis

simaRlis cvlilebiT.

,garda obieqtis cvladi TvisebebiT da cvladi

gareSe pirobebiT,adapturi marTva, gamoiyeneba obieqtebSi, romelTaTvis arasrulad aris mocemuli muSaobis

pirobebi an Tvisebebi. am SemTxvevaSi adaptaciis procesi

SeiZleba iyos erTjeradi.

nax.1-ze naCvenebia adapturi avtomaturi marTvis

sistemis zogadi blok_sqema. aq rogorc adre: O marTvis

obieqtia; yy-marTveli mowtobiloba, romelic Sedgeba

ZiriTadi mmarTveli mowyobilobisgan (yy0) da adaptaciis

mmarTveli mowyobilobisagan (yya). amgvarad, yya zomavs

F,X,U, da YY. am monacemebiT ganisazRvreba marTvis

xarisxis kriteriumis mniSvneloba J(U,Y,X,F), misi gadaxra da algoriTmis mixedviT (Yyya-Si

gamomuSavdeba iseTi y ,romelic mospobs arasasurvel

gadaxras.)

91

rogorc cans, adapturi avtomaturi marTvis

sistema Seicavs or marTvis konturs-ZiriTads da

adaptaciis. adaptaciis kontursaTvis marTvis

obieqtiamTliani avtomaturi marTvis sistema. igi qmnis

marTvis meore dones.

adaptaciis sistema SeiZleba muSaobdes mudmivad,

periodulad an erTjeradad, ukanasknel SemTxvevasi igi

gamoiyeneba sistemis avtomaturi regulirebisatvis.

adapturi marTvis sistemaSi ganasxvaveben marTvis

sistemebs stabilizaciiT da marTvis xarisxis optimizaciiT. marTvis xarisxis stabilizaciiT

avtomaturi marTvis sistemis amocanaa Seikavos marTvis

xarisxis kriteriumi j garkveul doneze an garkveul

nax. 20.

92

diapazonSi. adaptaciis konturi asrulebs am kriteriumis

stabilizacias. avtomaturi marTvis sistema xarisxis

optimizaciiT anxorcielebs xarisxis kriteriumis

optimaluri mnisvnelobis moZebnas da mis Sekavebas.

ZiriTad mmarTvel mowyobilobaSi (yy0) cvlilebaTa

xasiaTiT, romelTac anxorcielebs yya adaptaciis

procesSi, adaptur sistemebSi ganansxvaveben TviTmwyobad

sistemebs. TviTmwyobad avtomatur marTvis sistemebSi

adaptacia xorcieldeba yya –is parametrebis mniSvnelobiT,

xolo TviTorganizebadSi yy0-is struqturis cvlilebiT.

93

TviTmwyobadi marTvis sistemebi marTvis TviTmwyobadi marTvis sistemebi marTvis TviTmwyobadi marTvis sistemebi marTvis TviTmwyobadi marTvis sistemebi marTvis

xarisxis stabilizaciiTxarisxis stabilizaciiTxarisxis stabilizaciiTxarisxis stabilizaciiT

mocemul sistemaSi (nax. 21.) adaptacia mdgomareobs mocemuli xarisxis kriteriumis j mniSvnelobis

stabilizaciaSi Ziritadi mmarTveli mowyobilobis

(ganwyobis) daregulirebis cvlilebiT.

aseTma avtomaturi marTvis sistemebma ZiriTadi

gamoyeneba hpova dinamiuri parametrebis stabilizaciaSi.

nax.2-ze naCvenebia swored aseTi avtomaturi sistema da

adaptaciis marTveli mowyobiloba.

xSirad adaptaciis mowyobilobaSi agreTve

gamoiyeneba gamomTvleli mowyobiloba etalonuri

modeliT. modelis gamoyeneba amartivebs gamoTvliT mowyobilobas, radgan j gamoiangariSeba da misi gadaxdis gansazRvra icvleba obieqtis gamosasvlelze signalebis

sxvaobis gansazRvriT.

gardamavali procesis xarisxis arapirdapiri

maCveneblis magivrad SeiZleba sixSiruli an gardamavali

maxasiaTeblis stabilizacia.

nax.21-ze naCvenebia avtomaturi marTvis sistemis funqcionaluri sqema amplitudur-sixSiruli maxasiaTeblis

stabilizaciiT wnp wertilSi.

94

nax. 21.

95

Ubwj!W!

wrfivi avtomaturi marTvis sistemebis wrfivi avtomaturi marTvis sistemebis wrfivi avtomaturi marTvis sistemebis wrfivi avtomaturi marTvis sistemebis mdgramdgramdgramdgradobadobadobadoba

marTvis sistemis sinTezisa da analizis dros

yvelaze mniSvnelovani sakiTxia sistemis mdgradoba drois

nebismier momentSi. mdgradobis gansazRvris sxvadasxva meTodebi arsebobs, romelTa gamoyeneba damokidebulia

imaze sistema wrfivia Tu arawrfivi.

sistemis mdgradoba ganisazRvreba ase: sistema

mdgradia Tu is ubrundeba Tavis sawyis, an masTan axlos

mdebare damyarebul reJims, am sistemis sawyisi

mdgomareobidan gamoyvanis SemTxvevaSi, masze nebismieri

gare zemoqmedebisas. nax 22-ze moyvanilia gardamavali

procesis tipiuri maxasiaTeblebi aramdgrad (nax. 22 a)

da mdgrad (nax 22 b) sistemebSi.

96

nax. 22

Tu sistema aramdgradia, maSin sakmarisia mcire

zemoqmedebac, rom masSi daiwyos procesi, romelic

gamoiyvans sistemas wonasworobis mdgomareobidan da

gamosasvleli sidide ganuwyvetliv gaizrdeba. es procesi SeiZleba iyos perioduli (nax. 22 amrudi 1) an rxeviTi

(nax. 22 b mrudi 2).

rxeviTi ganSladi procesi, SeiZleba aRiZras

sistemis gadacemis koeficientis ganusazRvrelad

gazrdiT, ris gamoc mmarTveli mowyobiloba iwyebs

zedmetad energiulad zemoqmedebas obieqtze, raTa

moaxdinos gamosasvleli signalis gadaxris kompensireba.

am SemTxvevaSi x-is yoveli morigi dabruneba nulovan

mniSvnelobaze mmarTveli mowyobilobis zemoqmedebiT

iwvevs X mrudis mier abscisTa RerZis sul ufro mzardi siCqariT gadakveTas da procesi mTlianobaSi gaxdeba

droSi ganSladi.

97

mdgradi sistemis SemTxvevaSi (nax. 22 b) gardamavali

procesi, romelic gamowveulia obieqtze raime

zemoqmedebiT, droSi milevadia da sistema kvlav

ubrundeba Tavis damyarebul mniSvnelobas.

mdgradobis moyvanili cneba, gansazRvravs sistemis

damyarebuli reJimis mdgradobas. magram sistema SeiZleba

muSaobdes mudmivad cvladi zemoqmedebis pirobebSi, roca

damyarebuli reJimi saerTod ara gvaqvs. am pirobebis

gaTvaliswinebiT mdgradobas SeiZleba mivceT ufro

zogadi ganmarteba: sistema mdgradia, Tu misi gamomavali

sidide rCeba gansazRvruli, sistemaze sididiT gansazRvruli zemoqmedebis pirobebSi.

ar aris rTuli vaCvenoT, rom Tu gardamavali

procesi sistemaSi qrobadia, maSin sistema akmayofilebs

bolo pirobasac.

ganvixiloT razea damokidebuli sistemis

mdgradoba da rogor ganisazRvreba: amisaTvis mivmarToT

sistemis dinamikis gantolebas:

fPWx )(3=

sadac:

)(

)()(

PW

PWPW

fx

+=

13

;)(

)()(

PQ

PRPW =

)(

)(

PQ

PRW

fx

fxfx =

;)(

)()(

PD

PMPW =3

;)(

)(f

PD

PMx ⋅= fPMxPD )()( =

98

am erTgvarovani wrfivi gantolebis amoxsna zogad

saxeSi warmoadgens ori mdgenelis jams:

x(t)=xTav(t)+xiZ(t)

sadac xTav(t) aris D(P)x=0 erTgvarovani gantolebis

zogadi amonaxsni, xolo xiZ(t) – araerTgvarovani

gantolebis kerZo amonaxsni, romelic aRwers sistemis

iZulebiT reJims, romelic myardeba gardamavali procesis

damTavrebis Semdeg. sistema iqneba mdgradi, Tu gardamavali procesi

xTav(t) nebismier zemoqmedebaze iqneba qrobadi, anu drois garkveuli periodis Semdeg xTav(t) miiswrafis nulisaken.

xTav(t)-s diferencialuri gantolebis amonaxsns,

rogorc cnobilia aqvs Semdegi saxe:

∑=

=n

i

ti

iectx1

λ)(Tav

sadac Ci-integrirebis mudmivebia, romlebic

ganisazRvrebian sawyisi pirobebiTa da SeSfoTebebiT; iλ -

maxasiaTebeli gantolebis fesvebia

0=)(λD

mravalwevri )(λD aris 4.1 gantolebis marcxena

nawili, sadac diferencirebis operatori P Secvlilia λ

kompleqsuri cvladiT.

mravalwevri )(λD warmoadgens sistemis W3(P) gadacemis funqcias.

)()()( λλλ QRD +=

99

sadac R( iλ ) da Q( iλ ) – W(P) gadacemis funqciis

mricxveli da mniSvnelia gaxsnili sistemisaTvis, masSi

P-s iλ -iT Secvlisas.

gardamavali procesi Tavx (t) warmoadgens mdgenelTa

jams, romlis ricxvi ganisazRvreba maxasiaTebeli

gantolebis fesvTa raodenobiT, anu sistemis gantolebis

rigiT.

zogadad iλ warmoadgenen kompleqsur ricxvebs.

amasTan isini qmnian fesvTa SeuRlebul wyvilebs:

iiii jωαλ ±=+1,

sadac iα - SeiZleba iyos, rogorc dadebiTi aseve

uaryofiTi sidide. yoveli aseTi fesvebis wyvili iZleva

gardamavali procesis mdgenelebs:

)sin()( 1)(

1)(

iit

itj

itj

ittj

itj

i teCeCeCeeCeC iiiiiii ϕωαωωαωαωα +′=+=+ −+

−+

+

sadac iC′ da iϕ ganisazRvrebian Ci da Ci+1-is saSualebiT.

rogorc vxedavT, es gamosaxuleba warmoadgens

sinusoidas, romlis amplituda droSi icvleba

eqsponentis kanoniT.

Tu 0<iα , maSin procesi iqneba qrobadi, xolo Tu

0>iα maSin miviRebT araqrobad–ganSlad rxevebs. Tu

0=iα , romelic Seesabameba wyvil warmosaxviT fesvebs,

miviRebT sinusoidur rxevebs mudmivi amplitudiT.

100

amdenad procesis qrobis aucilebel pirobas

warmoadgens rom maxasiaTebeli gantolebis fesvebis

SeuRlebuli wyvilebis namdvili nawili iyos uaryofiTi.

kerZo SemTxvevaSi, roca 0=iϖ , gvaqvs namdvili

fesvi ii αλ = . gardamavali procesi t

iieC αwarmoadgens

eqspnentas, romelic iqneba an qrobadi, an ganSladi imis

da mixedviT 0<iα -ze Tu 0>iα -ze.

zogadad gardamavali procesi sistemaSi Sedgeba an

rxeviTi an perioduli mdgenelebisagan. yovel rxeviT

process Seesabameba maxasiaTebeli gantolebis

SeuRlebuli wyvili kompleqsuri fesvebi, xolo

aperiodul process – namdvili fesvebi.

moyvanilia sxvadasxva tipis wevrebi, romlebic

gvxvdeba erTgvarovani gantolebis amonaxsnSi da aseve

maxasiaTebeli gantolebis fesvebis mdebareoba S sibrtyeze, anu kompleqsur ricxvTa sibrtyeze. am

sibrtyeze kompleqsuri ricxvebi gamoisaxeba marTkuTxa koordinatTa sistemaSi, romelSic namdvili ricxvebi

gadaideba horizontalur RerZze, xolo warmosaxviTi

vertikalurze. Tu gamoviyenebT S sibrtyis cnebas,

SegviZlia vTqvaT, rom mdgradi sistemis maxasiaTebeli

gantolebis yvela fesvi mdebareobs warmosaxviTi ricxvTa

RerZidan marcxniv, e. i. marcxena naxevarsibrtyeSi.

fesvis mdebareoba warmosaxviT RerZze aRniSnavs,

rom sistema imyofeba mdgradobis zRvarze.

sistemis mdgradobis dasadgenad ar aris saWiro

maxasiaTebeli gantolebis fesvebis povna imis gamo, rom damuSavebulia iribi niSnebi, romliTac SeiZleba

gavarkvioT fesvebis namdvili nawilis niSani. am irib

niSnebs ewodebaT mdgradobis kriteriumebi.

101

imis gaTvaliswinebiT, rom 02 >∆ , bolo utoloba

daiyveneba 03 >a -ze. amdenad, mTlianobaSi mdgradobis

pirobebi mdgomareobs yvela koeficientisa da boloswina

minoris dadebiTobaSi. n=4

432

23

14

0)( aaaaaD ++++= λλλλλ

420

31

420

31

4

0

00

0

00

aaa

aa

aaa

aa

=∆

mdgradobis piroba:

;0;0 10 >> aa

;030212 >−=∆ aaaa

00

42

12340

11233 >−∆=−∆=∆ aaa

aa

aaa

5

0344

=

>∆=∆

n

a

552

42

33

24

15

0)( aaaaaaaD ++++++= λλλλλλλ

102

mdgradobis piroba, Tu vimoqmedebT analogiurad,

daiyvaneba imaze, rom yvela koeficienti da ori minori

iyos dadebiTi: 02 >∆ ; 04 >∆ .

rogorc vxedavT mdgradobis pirobebi raus-hurvicis

kriteriumiT, rTuldeba sistemis rigis zrdasTan erTad.

amasTan Zalian maRali rigis sistemebisaTvis,

garTulebulia gavarkvioT rgolebis calkeuli

parametrebis, romlebic Sedian gantolebis koeficientebis

SemadgenlobaSi, gavlena sistemis mdgradobaze. es

ganpirobebulia imiT, rom rogorc wesi erTi da igive

parametrebi erTdroulad gantolebebSi Sedian ramodenime

koeficientis saxiT, amitom raus-hurvicis kriteriums iyeneben SedarebiT dabali rigis sistemebisaTvis. sistemis

sinTezis amocanis amoxsnisaTvis, rodesac saWiroa

SevarCioT sistemis parametrebi, raus-hurvicis

kriteriumi meoTxe rigis zemoT zalian rTulia.

magaliTisaTvis ganvixiloTYmesame rigis sistema,

romelic Sedgeba sami rigis statikuri rgolebisagan.

aseTi sistemis gadacemis funqcias aqvs Semdegis saxe:

)1)(1)(1()(

321

321

+++=

PTPTPT

KKKPW

am SemTxvevaSi

)1)(1)(1()())( 321321 ++++=+= λλλλλλ TTTKKKQRD

e. i.

322

13

0)( aaaaD +++= λλλλ

103

sadac

321213322113210 ;; TTTaTTTTTTaTTTa ++=++==

3213213 11 KKKKKKKKa =+=+=

mdgradobis piroba, rogorc wesi n=3 sistemisaTvis, daiyvaneba Semdeg utolobamde:

;0321 >TTT 133221 TTTTTT ++ >0 ;

321 TTT ++ >0 K+1>0

0)1())(( 3213211332213021 >+−++++=− KTTTTTTTTTTTTaaaa

pirveli sami utoloba ar aris saintereso Tu Cven

vzRudavT ganxilvas drois mudmivebis dadebiTobiT, anu

viciT rom drois mudmivebi dadebiTia.

meoTxe utoloba SezRudvebs adebs K-s uaryofiT

mniSvnelobebs. uaryofiTi K-s absoluturi mniSvneloba

1-ze naklebi unda iyos.

realuri, mniSvnelovani SezRudvebi sistemis

parametrebis SerCevaze bolo utolobidan gamomdinareobs.

es utoloba ufro mexerxebulia gadavweroT Semdegnairad:

1)111

)((321

321 −++++<TTT

TTTK

es utoloba gviCvenebs, rom drois mudmivebis

nebismieri mniSvnelobis dros sistemis mdgradoba dairRveva K-s usasrulo gazrdiT. K-s zRvruli

104

mniSvneloba mdgradobis SesanarCuneblad ganisazRvreba

sistemis drois mudmivebis mniSvnelobebidan gamomdinare.

1)111

)((321

321 −++++=TTT

TTTKzR

rausrausrausraus----hhhhurvicis kriteriumiurvicis kriteriumiurvicis kriteriumiurvicis kriteriumi

es algebruli kriteriumia, romliTac mdgradobis

kriteriumi daiyvaneba rigi utolobebis Sesrulebaze,

romlebic akavSireben erTmaneTTan diferencialuri

gantolebis koeficientebs. es kriteriumi sxvadasxva

formaSi iyo SemoTavazebuli ingliseli maTematikosis rausis da Sveicarieli maTematikosis hurvicis mier me-19 saukunis bolos. moviyvanoT es kriteriumebi damtkicebis

gareSe.

aviRoT mravalwevri:

nnnn aaaaD ++++= −

− λλλλ 11

10 ...........)(

mravalwevris koeficientebisagan matrica:

=∆n

nn

n

aa

a

aaaa

aaa

aaaa

aaa

2

1

6420

531

6420

531

................................0

0.................................0

.................................................

.................................................

0............0

0.................0

0................

0........................

−

−

105

sadac vTliT rom 0a >0, rac yovelTvis SeiZleba

uzrunvelyoT mravalwevris (-1)-ze gamravlebiT. am

msazRvrels ewodeba hurvicis msazRvreli. mas aqvs n striqoni da n sveti. pirveli striqoni Seicavs yvela

kent koeficientebs bolomde, romlis Semdeg striqoni

Seivseba n saWiro ricxvamde nulebiT. meore striqoni

Seicavs yvela luw koeficientebs da aseve mTavrdeba

nulebiT. mesame striqoni miiReba pirvelisagan, xolo

meoTxe – meorisagan Tu maT wavwevT marjvniv erTi

elementiT.

ganTavisuflebeli adgilis marjvniv iwereba nuli. analogiurad unda moxdes yvela striqonis waweva

marjvniv, ris Sedegadac mTavar diagonalSi moxvdeba

TanmimdevrobiT yvela koeficienti 0a -s garda.

mdgradobis piroba mdgomareobs im moTxovnaSi, rom

hurvicis msazRvreli da misi yvela diagonaluli minori

dadebiTi unda iyos. es minorebi Semoxazulia

gamosaxuleba (4.4.2)-Si punqtiriT. ganvixiloT

mdgradobis hurvicis kriteriumi ramodenime n-isaTvis. vTqvaT n=1 maSin:

10)( aaD += λλ

mdgradobis piroba daiyvaneba ramodenime utolobis

Sesrulebaze:

0;0 10 >> aa

106

aqedan, vTqvaT pirveli rigis rgoli gadacemis

funqciiT 1+P

K

τ aris mdgradi, xolo rgoli gadacemis

funqciiT 1−P

K

τ - aris aramdgradi.

n=2 ;

212

0)( aaaD ++= λλλ

2120

12

0aa

aa

a==∆

mdgradobis piroba iqneba: 0;0 10 >> aa ; ;02 >a

2∆ >0. magaliTad, rgoli romlis gadacemis funqciaa

1222

1 ++ PTPT

K aris mdgradi, Tu mniSvnelis yvela

wevris win aris plusi.

n=3

322

13

0)( aaaaD +++= λλλλ

31

20

31

3

0

0

0

aa

aa

aa

=∆

107

mdgradobis piroba 0;0 10 >> aa

0302120

312 >−==∆ aaaa

aa

aa

0323 >∆=∆ a

mixailovis mdgradobis kriteriumebimixailovis mdgradobis kriteriumebimixailovis mdgradobis kriteriumebimixailovis mdgradobis kriteriumebi

es aris grafikuli kriteriumi da is efuZvneba

)(λD mravalwevris ganxilvas. )(λD mravalwevrSi λ -s

nacvlad CavsvaT warmosaxviTi cvladi ωj .

)()()( ωωω dd jVUjD +=

sadac )(ωdU - namdvili nawilia, romelic miiReba

)(λD -s im wevrebisagan romelic λ -is luw xarisxs

warmoadgenen, xolo )(ωdV - warmosaxviTi nawilia,

romelic miiReba )(λD mravalawevris λ -is kenti

wevrebisagan. warmovadginoT )( ωjD - hodografis saxiT

kompleqsur sibrtyeze (nax. 3.4.1).

108

ω - yovel mniSvnelobas Seesabameba )(ωdU -sa

da )(ωdV mniSvnelobebi da wertili kompleqsur

sibrtyeze. roca ω =0 funqcia )( ωjD = na , anu

hodografi iwyeba namdvil RerZze, roca

∞→ω funqcia )( ωjD -is ganusazRvrelad izrdeba.

mixailovis kriteriumi formulirdeba

Semdegnairad: sistema mdgradia, Tu hodografi

)( ωjD iwyeba namdvil dadebiT naxevarRerZze, Semowers

koordinatTa saTaves saaTi isris moZraobis

sawinaaRmdegod da gaivlis TanmimdevrobiT n kvadrants, sadac n aris sistemis rigi.

nax. 23-a-ze hodografi 1 ekuTvnis mdgrad

sistemas, xolo hodografi 3, 4, 5 aramdgrad

sistemebs.

sistema imyofeba mdgradobis zRvarze Tu

hodografi gadis koordinatTa saTaveze. marTlac am

SemTxvevaSi arsebobs ω -s iseTi mniSvneloba roca

)( ωjD =0. e. i. sistemis maxasiaTebel gantolebas aqvs

wyvili SeuRlebuli warmosaxviTi fesvebi ωλ j±= . es

ukanaskneli aris piroba sistemaSi mudmivi amplitudiT

rxevebis warmoqmnisa da misi yofna mdgradobis

nax. 23.

109

zRvarze. nebismierma mcire zemoqmedebam SeiZleba

sistema gadaiyvanos an aramdgrad (hodografma

gadainacvlos marjvniv), an mdgrad mdgomareobaSi,

hodografma gadainacvlos marcxniv.

)( ωjD hodografis agebisas, upirveles yovlisa

pouloben koordinatTa RerZebTan mis gadakveTi

wertilebs. amisaTvis )(ωdU =0 gantolebidan

pouloben sixSiris im mniSvnelobas, romelic

Seesabameba hodografis warmosaxviT RerZTan

gadakveTis wertils, da Semdgom am mniSvnelobas

Casvaven )(ωdV -s gamosaxulebaSi. Semdeg Rebuloben

Sesabamis koordinatebs. analogiurad pouloben

)( ωjD -s gadakveTis wertils namdvil RerZTan da

warmosaxviT )(ωdV nawils utoleben nuls.

rom ar gvqondes saqme ω -s maRal xarisxebTan, )( ωjD hodografis ageba SeiZleba vawarmooT sistemis

rgolebis mixedviT. warmovidginoT )( ωjD Semdegnairad:

)()()()()( ωωωωω jQjRjQjRjD ii

ii

Π+Π=+=

sadac )( ωjRi da )( ωjQi - erTkonturiani

sistemis i -uri rgolis amplitudur-fazur-sixSiruli

funqciis mricxveli da mniSvnelia.

gamosaxulebis Tanaxmad )( ωjD hodografis

agebas iwyeben calkeuli rgolebis hodografebebis da )( ωjRi da ωjQi ( ) hodografebis agebiT. Semdeg ageben

)( ωjR da )( ωjQ hodografebs )( ωjRi da ωjQi ( )

hodografebis gadamravlebiT. hodografebs amravleben

110

cnobili veqtorebis gadamravlebis wesebis Tanaxmad,

iseve rogorc sixSiruli maxasiaTeblebis dros

calkeuli rgolebis sixSiruli maxasiaTeblebis

saSualebiT. ω -s yoveli mniSvnelobisaTvis mudulebs

(veqtorebis sidideebi, romlebic gavlebulni arian

koordinatTa saTavidan hodografis Sesabamis

wertilamde) amravleben, xolo argumentebs (am

veqtorebis fazebi) Sekreben.

111

naikvistis mdgradobis kriteriuminaikvistis mdgradobis kriteriuminaikvistis mdgradobis kriteriuminaikvistis mdgradobis kriteriumi

es kriteriumi SemoTavazebuli iyo 1932 wels

amerikeli mecnieris naikvistis mier da iZleva

saSualebas vimsjeloT Sekruli sistemis mdgradobaze,

gaxsnili sistemis )( ωjW amplitudur-fazur

sixSiruli maxasiaTebliT.

Tavdapirvelad CamovayaliboT es kriteriumi im

SemTxvevisaTvis, roca cnobilia, rom sistema gaxsnili

mdgomareobaSi mdgradia. maSin Sekruli sistemis

mdgradobis piroba daiyvaneba im moTxovnamde, rom

gaxsnili sistemis amplitudur-fazur sixSiruli

maxasiaTebeli ar Semowvdes (-1, j0) wertils.

nax. 24.

112

nax. 24-ze 1 da 4 maxasiaTeblebi ekuTvnis mdgrad

sistemebs, xolo maxasiaTebeli 3 aramdgrad sistemas.

maxasiaTebeli 2 ekuTvnis mdgradobis zRvarze myof

sistemas. Tu magaliTad, aramdgrad sistemaSi

SevamcirebT gadacemis koeficients, maSin misi

amplitudur-fazur sixSiruli maxasiaTebeli

SeikumSeba koordinatTa saTavisagan, ris gamoc sistema

gaxdeba mdgradi. da piriqiT Tu gadacemis koeficients

gavzrdiT maSin mdgradi sistemis amplitudur-fazur

sixSiruli maxasiaTebeli Semowvdeba (-1, j0) wertils

da sistema gaxdeba aramdgradi. rogorc vxedavT moyvanili formulirebebi

ekuTvnis sistemebs, romlebic mdgradni arian sistemis

Ria mdgomareobaSi. erTkonturiani sistemis dros

mdgradoba yovelTvis SenarCunebulia Tu sistema

Sedgeba mxolod mdgradi rgolebisagan. Tu gvaqvs

adgilobrivi ukukavSirebi, maSin unda gaisinjos am

kavSirebiT Seqmnili konturebis mdgradoba.

im sistemebisaTvis, romlebic ar arian mdgradni

Ria mdgomareobaSi naikvistis kriteriumi

formulirdeba Semdegnairad: Sekrul mdgomareobaSi sistemis mdgradobisaTvis saWiroa Ria sistemis

amplitudur-fazur sixSirulma maxasiaTebelma moicvas

wertili (-1, j0). amasTan mis mier namdvili uaryofiTi

naxerRerZis (-1, j0) wertilis marcxniv gadakveTis

ricxvi zevidan qveviT unda iyos 2

K-iT meti

ukumimarTulebiT gadakveTis ricxvze, sadac K aris gaxsnili sistemis W(P) gadacemis funqciis polusTa

dadebiTi namdvili nawiliT.

nax 24-ze magaliTis saxiT moyvanilia gaxsnili

sistemis amplitudur-fazur-sixSiruli ma-

113

xasiaTeblebi, romelic aramdgradia Ria mdgomareobaSi,

radganac gvaqvs marjvena fesvebi, magram mdgradia

Caketil mdgomareobaSi.

maxasiaTebeli 1 Seesabameba K=1, xolo

maxasiaTebeli 2 K=2-s. amdenad zogad SemTxevevaSi

naikvistis kriteriumis gamoyenebis dros aucilebelia

winaswar ganvsazRvroT W(P)-s marjvena polusTa

ricxvi. erTkonturiani sistemisaTvis, rodesac W(P)-s mniSvneli warmoadgens calkeuli rgolebis gadacemis

funqciis mniSvnelTa namravls am ricxvis gansazRvra

Zneli ar aris, radgan W(P)-polusebs warmoadgens calkeuli rgolebis gadacemis funqciis polusebi.

maravalkonturian sistemebSi, gansakuTrebiT ki

jvaredianad gadakveTil kavSirebSi, K-s gansazRvra

Zalian garTulebulia da amitom aseT pirobebSi ar

aris rekomendirebuli gamoviyenoT naikvistis

kriteriumi.

114

Ubwj!WJ!

marTvimarTvimarTvimarTvis biologiuri sistemebis biologiuri sistemebis biologiuri sistemebis biologiuri sistemebi

mdgomareobis cneba biologiur mecnierebebsa da marTvis mdgomareobis cneba biologiur mecnierebebsa da marTvis mdgomareobis cneba biologiur mecnierebebsa da marTvis mdgomareobis cneba biologiur mecnierebebsa da marTvis

TeoriaSiTeoriaSiTeoriaSiTeoriaSi

biologiur mecnierebebSi intuiciurad Seiqmna

warmodgena sisitemis mdgomareobis Sesaxeb, rogorc

maCveneblebisa da maxasiaTeblebis farTo wreze. romlebic

gansazRvraven sistemis funqcionirebasa da reaqcias

gansxvavebul garegnul faqtorebze. mag. fiziologiaSi

saubroben organizmis normalur mdgomareobaze, Tu misi

yvela cvladi da maTi reaqciebi gare stimulebze imyofeba dasaSveb sazRvrebSi. ufro farTod termini

„mdgomareoba“ biologiaSi gamoiyeneba rogorc bio-

sisitemis zogierTi integraluri maxasiaTebelis analogi.

termini „mdgomareoba“ fiziologiaSi erT

SemTxvevaSi aRniSnavs ricxviTi maxasiaTeblebis

konkretul nakrebs, romlebic aRweren organizmis an misi

sistemis struqturasa da funqcionirebas, magaliTad,

sisxlis mimoqcevis sitemis mdgomareoba adamianis

organizmSi SeiZleba aRvweroT Semdgei saxiT: gulis

SekumSvis sixSire 68 dartyma wuTSi, sistoluri da diastoluri wneva - 150 da 110 mmHg. sisxlis

wuTmoculoba 7 l/wT.

sxva SemTxvevebSi „mdgomareobis“ cneba gamoiyeneba

rogorc maklasificirebeli termini, romelic gvaZlevs

organizmis an misi struqturisa da funqcionirebis raime

garkveul Sefasebas. termini „mdgomareoba“ am SemTxvevaaSi

exeba raodenobrivi maxasiaTeblebis nakrebis

erTmaneTisagan mcired gansxvavebul mTlian jgufebs.

115

swored, am azriT unda gavigoT klinikuri termini,

„damakmayofilebeli mdgomareoba“ an „saSualo sirTulis

mdgomareoba“, romelic exeba mTlian organizms.

termini “biosistemis mdgomareoba”, intuiciuria da

naTelia ar aqvs mkacri gansazRvreba. marTvis TeoriaSi

gvaqvs sistemis mdgomareobis cnebis mkacri gansazRvreba.

dinamikuri sistemis mdgomareoba - es aris

ricvxebis umciresi nakrebi, romelic aucilebelia

mocemul iqnes drois mocemul momentSi, raTa iyos

saSualeba sistemis maTematikuri aRweris CarCoebSi.

viwinaswarmetyveloT misi qceva sxva momaval momentSi, T>T0.

arsebobs ricxvebis am nakrebis gansazRvris martivi

xerxi (es aris e.w. sistemis mdgomareobis veqtori),

romelic dafuZnebulia modelirebis sqemebis gamoyenebaze.

modelirebis sqemebma biosistemebis kvlevaSi didi

gamoyeneba hpoves. amitom mdgomareobis veqtoris arsi

asaxsnelad Cven gamoviyenebT am xerxs.

mdgomareobis bunebrivi veqtori sistemaSi

warmoadgens veqtors, romelic Sedgenilia modelirebis

sqemaSi yvela integratoris gamomavali cvladebisgan. mdgomareoba aris veqtori, romelic Sedgeba misi yvela

integratoris gamomavali cvladebisa an fazuri

koordinatebisagan. biologiuri sistemebisaTvis es

gansazRvreba sakmaod moxerxebulia. yvela cocxali

sistema aris Ria sistema, romelSic mimdinareobs

nivTierebaTa da energiis cvlis procesebi.

arseboben biologiuri sistemebis modelirebis

farTo klasi, romelSic ZiriTadad yuradReba mimarTulia

nivTierebaTa cvlis energiis maragisa da xarjvis

procesis aRwerisaken. aseT modelebs kompartmentuls uwodeben. Ria sistemebis kompartmentul modelebSi

116

integratorebis gamomavali signalebi yovelTvis aRweren

energiis an nivTierebaTa marags sistemaSi, gansazRvraven

maT dones, raodenonobas, an koncentracias. biosistemis

mdgomareobis veqtori warmoadgens nivTirebaTa

koncentraciis veqtors am sistemaSi.

mdgomareobis sivrcis meTodi marTvis TeoriaSi

mivida klasikuri midgomis Secvlamde, romelic

dafuZnebulia „Savi yuTis“ ideaze, roca sistemis

gamokvlva xdeba mxolod Semaval-gamomaval sidideebs

Soris damokidebulebis analizis mixedviT. „Savi yuTisa“

meTodi rTul sistemebSi aRmoCnda aradekvaturi, ramdenadac gamomavalis reaqcia erTsa da imave Semaval

sisideze aseTi aRweris dros araerTmniSvnelovani

aRmoCnda.

sistemis aRwera gadacemis funqciiT Siga

cvaldebis da Semaval da gamomaval sidideebs Soris

kavSiris damyarebis saSualebas ar iZleva; roca es

cvaldebi icvlbian, sistemis Tvisebac icvleba. reaqcia

erTsa da imave Semaval signalze damokidebulia mis

mniSvnelobaze. es nakli ispoba sistemis mdgomareobis

cnebis ganxilvis SemTxvevaSi. sistemis mdgomareobis codna saSualebas iZleva yovelTvis erTmniSvnelovnad

ganvsazRvroT misi nebismieri maxasiaTebeli da is

warmoadgens sistemis amomwurav aRweras.

117

mdgomareobis gantolebamdgomareobis gantolebamdgomareobis gantolebamdgomareobis gantoleba

ramdenadac sistemis aRsawerad sakmarisia drois

nebismier momentSi misi mdgomareobis veqtorisa da

Semavali veqtoris mocema, amdenad marTvis Tanamedrove

TeoriaSi sistemebis dinamikuri Tvisebebi aRiwereba

gantolebebis daxmarebiT, romlebic Seexeba sistemis

gamomaval signalebs, isini gamoisaxebian mdgomareobis

veqtoriT calke gantolebis saSualebiT. e.i aseT

formaSi sistemaSi aRiwereba wyvili gantolebiT,

romelTa Soris pirveli - diferencialuri gansazRvravs

sistemis mdgomareobas da amiT iZleva mis dinamikur Tvisebebsac. meore - algebruli gansazRvravs sistemis

gamomaval sidideebs misi mindinare mdgomareobiT.

aRvniSnoT ra sistemis mdgomareobis veqtori x-iT,

Semavali veqtori v-Ti, gamomovali ki y-iT, SeiZleba

CavweroT uwyveti determinirebuli sistemis gantoleba Semdegi saxiT.

x(t)= Φ [x(t), v(t)]

y(t)= Φ [x(t), v(t)] (1)

118

(1)-pirvel gantolebas ewodeba mdgomareobis

gantoleba, meores gamomavali gantoleba. mdgomareobis

gantoleba Cveulebriv diferencialuria. gamomavali

gantoleba ki algebrulia.

orive gantoleba dawerilia veqtorebisaTvis.

rodesac msgavsi gantolebebi gadaiwereba koordinatebis

mixedviT, e.i amoiwereba cal-calke TiToeuli

koordinatebisaTvis mdgomareobis veqtori, maSin sawyisi dimamikuri sistema ganisazRvreba ara erTi, aramed

ramdenime mdgomareobis ricxviT (tolia mdgomareobis

veqtorisa da gamomaval veqtorSi komponentebis

ricxvisa).

sistemisaTvis, romlic aRiwereba wrfivi

diferencialuri gantolebebiT, mdgomareobis gantoleba

Caiwreba Semdegi saxiT:

…………………………………………………………………………

Gamomavali signalebis gantoleba moicma algebrul

gantolebaTa sistemiT.

119

sadac ,…….. - mdgomareobis cvladebia,

,……… Semavali veqtorebia, …… gamomavali

veqtorebia, a, d, b, c,- koeficientebia.

sadac, A(m×m), B(m×l), C(r×m), D(r×l)-matricebia. Tu

koeficientebi a, d, b, c- icvlebian droSi, maSin matricebs

eqnebaT cvladi elementebi.

x= v= y=

x, y, v - mdgomereobis, Semavali da gamomavali

veqtorebia Sesabamisad, SevniSnoT rom A matrica yvelgan kvadratulia, danarCeni matricebis kvadratuloba ki araa

aucilebeli. veqtoruli gantolebis wyvils zogjer

uwodeben Semaval-gamomaval-mdgomareobis gantolebebs. am

SemTxvevaSi amboben, rom TviTon sistema aRiwereba,

rogorc „Semaval-gamomaval-mdgomareoba“. sistema,

120

romelic aRiwereba zemoTmoyvanili gantolebebiT,

SeiZleba grafikulad gamovsaxoT (nax. 24)

nax. 24

es sqema savsebiT analogiuria erTganzomilebiani sqemisa,

romelic miiReba im SemTxvevaSi, rodesac x, y, v

Cveulebriv skalaruli sidideebia. xolo A, B, C, D ricxviTi koeficientebia, im gansxvavebiT, rom veqtorul

kavSirebs Cven gamovsaxavT ormagi farTo isriT.

magaliTi: grodinzis sunTqviTi hemostazis gantolebebs aqvs sami Sesavali da ori gamomavali sidideebi da

aRiwereba ori diferencialuri gantolebiT:

(9.3.5)

121

sadac -Semavali signalia (CO2-is koncentracia SesunTqul haerSi).

-Semavali signalia (CO2-is gamoyofis

tempi metabolizmis procesSi).

- Semavali signalia (damxmare konstanta).

-gamomavali signalia (CO2-is koncentracia arterialur sisxlSi).

-gamomavali signalia (CO2-is koncentracia qsovilebSi).

, b, c, d, g–parametrebia (koeficientebi, romlebic ganisazRvrebian sasunTqi sistemis

struqturiT).

modelirebis sqema moyvanilia nax. 9.3.2-ze

nax. 25.

122

sqemaSi gvaqvs ori integratori gamomavali

cvladebiT da . es cvladebi qmnian sistemis

mdgomareobis veqtors. Tu aRniSnavT = , =

miviRebT mdgomareobis veqtors

grodinzis modelia ar miekuTvneba kompartmentul

models, amitom misi mdgomareobis veqtori ar aRwers

sistemaSi nivTierebis dones. nivTierebis dones

warmoadgens modelis gamosasvleli signalebi.

wamovadginoT grodinzis sunTqvis modeli

matricul formaSi da CavweroT misi mdomareobis

gantoleba. SemovitanoT Semdegi aRniSvnebi:

, , - Semavali sidideebi.

= , = - mgdomareobis cladebi.

, - gamomavali sidideebi.

sistemis mdgomareobis gantolebas aqvs Semdegi saxe:

gamomavali sidideebis gantolebas aqvs Semdegi saxe:

123

matricebis saSualebiT es gantolebebi SeiZleba ase

CavweroT:

sadac matricebs aqvT Semdegi saxe:

, , ,

124

konveqciuri transportikonveqciuri transportikonveqciuri transportikonveqciuri transporti

biosistemebSi xSirad es Tu is komponenti

SeiZleba gadadiodes erTi kompartmentidan meoreSi ara

uSualod, aramed romelime gadamtanebis saSualebiT.

magaliTad, organizmSi Jangbadi atmosferodan sisxlSi

xvdeba SesunTquli haeris meSveobiT, xolo sisxlidan organomde gadamtans sisxlis nakadi warmoadgens. imave

sisxlis nakadi warmoadgens siTburi energiis gadamtans

kompartmentidan kompartmentSi, xolo amosunTqul haers miaqvs siTbo garemoSi. ekologiur doneze aseT procesebs

aqvs adgili, rodesac mcenareuli Teslebis gadatana

xorcieldeba qaris an cxovelebis saSualebiT.

epidemiologiur procesebSi epidemiis gavrceleba xdeba

gadamtanebis saSualebiT, romlis rolsac zogjer

satransporto nakadebi warmoadgenen.

konveqciur gadatanas fizikur movlenebSi maSin aqvT adgili, rodesac gadasatani nivTiereba imyofeba

125

moZrav siTxeSi an gazSi. nivTierebis rezultiur

moZraobas ewodeba konveqciuri transporti.

Tu, romelime zedapiris mimarT gadis siTxis an

gazis normaluri nakadi U moculobiTi siCqariT, maSin

CUy =kon

sadac, kon

y - nivTierebis gadaadgilebis tempia konveqciiT,

xolo C – nivTierebis koncentracia nivTierebis matarebelSi. samganzomilebiani SemTxveveisaTvis, imis analogiurad, rogorc amas adgili hqonda difuziis

ganxilvis dros SeiZleba CavweroT konveqciuri difuziis

gantoleba

,)()()(

CDz

CW

y

CV

x

CU

t

C ∆=∂

∂+∂

∂+∂

∂+∂∂

sadac, D – konstantaa, [ ]TUVU ω= - siCqaris

samganzomilebiani veqtoria, x, y, z RerZebis gaswvriv. Tu U veqtori konstantaa, maSin miviRebT

CDCUt

C ∆=∇⋅+∂∂

sadac ∇⋅U operatori ganisazRvreba Semdegnairad:

,z

Wy

Vx

UU∂∂+

∂∂+

∂∂=∇⋅

126

e. i. konveqciuri transportireba aRiwereba difuziis

Cveulebrivi gantolebiT, romelSiac damatebiT gvaqvs

wevri, romelic asaxavs koncveqciis movlenas.

kompartmentul modelebSi Cveulebriv sakmarisia

gamoviyenoT nivTierebis matarebliT gadatanis gantoleba,

romelic Semdegnairad Caiwereba:

QXy =

sadac y – komponentis gadatanis tempia. x – done an igive gansaxilveli komponentis

koncentracia matarebelSi.

Q – matareblis nakadis tempi.

ganvixiloT, magaliTad siTbogadacemis procesi amosunTqul haerSi. vTqvaT haeris xvedriTi

siTbotevadoba aris C0, filtvebis qsovilebis

temperatura Q1, garemos temperatura Q0, sunTqvis

intensioba V. maSin siTbos kargva y gamoisaxeba

TanafardobiT

VQQCy )( 010 −=

127

urTierTqmedeba da kompopentTa urTierTqmedeba da kompopentTa urTierTqmedeba da kompopentTa urTierTqmedeba da kompopentTa

urTierTgardaqmnaurTierTgardaqmnaurTierTgardaqmnaurTierTgardaqmna

biosistemebis funqcionirebisas maTSi myofi

komponentebi ucvlelni ar rCebian. sacicocxlo

procesebis arsi, anu sicocxlis arsi mdgomareobs

SemdegSi: komponentebis, romlebic biosistemas miewodeba

garedan da ikeTen marags masSi, urTierTqmedeben

erTmaneTTan, Sedian sxvadasxva reaqciaSi, gardaiqmnebian

da erTvebian nivTierebaTa da energiis mimocvlaSi, utilizdebian, xolo biosistemisaTvis arasaWiro

nivTierebebi kvlav garemos ubrundeba.

amdenad, nivTierebebi, romlebic organizms

miewodeba, ganicdian rig qimiur gardaqmnebs, riTac

uzrunvelyofen energiis miwodebas da organoebisa da

qsovilebis struqturis Seqmnas. meores mxriv mudmivad

mimdinareobs qsovilTa daSlisa da martiv nivTierebaTa

warmoqmnis procesi. es ori tipi procesebisa (anabolizmi

da katabolizmi) erTad warmoqmnis imas, rasac

metabolizmis procesi ewodeba. rTuli qimiuri procesebi erTiandebian kargad organizebul marTvis procesad da

xorcieldebian nervuli da humoraluri sistemebiT.

am sakiTxebis ganxilvis dros unda viqonioT

mxedvelobaSi maTi metismeti sirTule da Cveni codnis

arasakmarisi done organizmisa da calkeuli organoebis

funqcionirebis Sesaxeb. amitom, rogorc wesi, qimiuri

gardaqmnebis da marTvis procesebis detaluri suraTis

warmodgena SeuZlebelia. miuxedavad amisa, mainc arsebobs

zogierTi tipiuri xerxebi komponentTa

urTierTgardaqmnisa da urTierTqmedebebis (moqmedi masebi

128

kanoni, fermentaciuli reaqciebis gantolebebi)

Sesaswavli martivi modelebisaTvis.

129

moqmed masaTa kanonimoqmed masaTa kanonimoqmed masaTa kanonimoqmed masaTa kanoni

biosistemis komponentebad organizaciis yvela

doneze upirveles yovlisa SeiZleba ganvixiloT qimiuri

nivTierebebi, romlebic miewodeba organizms da

utilizdeba masSi.

Tu sistemis kompartmentSi warmoebs qimiuri

reaqcia A, B,.…, nivTierebebTan, romlebic gardaiqmneba M, N, …... nivTierebebad, maSin moqmed masaTa kanoni misTvis Caiwereba Semdegi saxiT

.............. ++⇔++ NMBA νµβα

sadac ricxvebi α , β , ……µ ,ν ….. cnobilia, rogorc

stexiometriuli koeficientebi.

reaqciis umniSvnelovanes maxasiaTebels warmoadgens

reaqciis siCqris K koonstanta. Tu A, B, ……, M, N, …. nivTierebebis koncentraciebs – avRniSnavT XA, XB, ……, XM, XN, … maSin droSi am koncentraciebis cvlileba

ganisazRvreba gantolebiT

...).....(...........νµβαNMBA

NMBA XXXXKdt

dX

dt

dX

dt

dX

dt

dX−==−=−===

xSirad bioqimiuri reaqciebisaTvis aRmoCndeba, rom

zogierT komponenti, romelic Rebulobs monawileobas reaqciaSi, gvaqvs garkveuli siWarbiT. masSi maTi

cvlilebis ugulvebelyofa xdeba gantolebis marjvena

130

mxareSi Sesabamisi koncentracia icvleba konstantiT.

magaliTad Tu yvela nivTiereba, A N-is garda gvaqvs

siWarbiT, maSin gantolebaSi marjvena mxare Seicvleba ναNA XXK1− -iTi. am SemTxvevaSi moqmed masaTa kanoni

miiRebs saxes

ναNA

NA XXKdt

dX

dt

dX1−=−=

sadac K1 – axali konstantaa.

biosistemebSi mimdinareoben rTuli specifiuri

qimiuri reaqciebis kompleqsebi. maTi modeliereba

warmoadgens damoukidebel amocanas, Tumca cocxal

organizmebSi nivTierebaTa mimocvlis modelirebisas SesaZloa sasargeblo aRmoCndes qimiur reaqciaTa

maTematikuri modelireba, romelic damuSavebulia qimiur

teqnologiaSi.

131

fermentaciuli reaqciebifermentaciuli reaqciebifermentaciuli reaqciebifermentaciuli reaqciebi

Tu gantolebebi, romlebic aRweren moqmed masaTa

kanons, erTnairad axasiaTeben cocxal da aracocxal

bunebas, maSin fermentuli reaqciebi, sadac fermentebi

muSaoben, rogorc katalizatorebi – reaqciis

damaCqarebeli, ZiriTadad specifiuria biosistemebisaTvis

(Tumca fermentaciuli reaqciebi mimdinareoben ara marto

cocxal organizmebSi – in vivo, aramed qimiuri

eqsperimentis pirobebSi, sinjaraSi – in vitro). fermentebi – es sxvadasxva cilebia, romlebic

organizmSi katalizatoris funqciebs asruleben.

nivTierebebi, romlebic fermentebis saSualebiT

Rebuloben monawileobas reaqciaSi, substratebi ewodebaT.

zogjer fermentisa da substratis garda reaqciis

gansaxorcieleblad saWiroa damatebiTi nivTierebebi –

kofermentebi.

fermentebis zemoqmedeba vlindeba reaqciis siCqaris

cvlilebaze, kofermentebi moqmedeben zogjer rogorc

zogierTi qimiuri jgufebis gadamtanebi, zogjer ki rogorc fermentebis moqmedebis gamaZliereblebi

(aqcivatorebi).

E fermentis S substratTan urTierTqmedebiT

miiReba ES kompleqsi, sadac substratis reaqciuli unary

aweulia – substrati garadaiqmneba reaqciis produqtad,

xolo fermenti ubrundeba sawyis mdgomareobas:

PEESSEKK

K+→⇔+

−

2

1

132

Cveulebriv substratis koncentracia fermentis

koncentracias aWarbebs. stacionarul pirobebSi

Tavisufali fermentis koncentracia X1 da kompleqsSi

Sekruli fermentis koncentracia X2 mudmivebia. kinetikur gantolebas aqvT Semdegi saxe:

2211121 )( XKKSXKXX ++−=′−=′−

stacionarul reJimSi 021 =′=′ XX

fermentis saerTo koncentracia X3 aris E

21 XXE +=

stacionarul reJimSi procesis mimdinareobis tempi

22XKPSy ==−= &&

maSin Tu CavsvavT X2-is stacionarul mniSvnelobas,

romelic advilad SeiZleba vipovoT fermentaciuli

reaqciis siCqaris gantoleba:

MKS

ESKy

+= 2

sadac .1

21

K

KKKM

+= − gantolebas ewodeba mixaelis-

mentenis gantoleba.

133

S substratis koncentraciis gazrdiT fermentaciuli

reaqciis siCqare izrdeba da uaxlovdeba maqsimalurs

EKy 2max =

mixaelis-mentenis gantoleba warmoadgens bioqimiuri procesebis aRmwer models da erTgvarad gaxda

sabazo modeli sxvadasxva saxis kvlevebisaTvis.

134

ekosistemebSi pekosistemebSi pekosistemebSi pekosistemebSi populaciaTa SorisopulaciaTa SorisopulaciaTa SorisopulaciaTa Soris

urTierTqmedebaurTierTqmedebaurTierTqmedebaurTierTqmedeba

populaciaTa Soris urTierTqmedebis umartives

models warmoadgens volteras modeli ~mtacebeli-

msxverpli~, romelic ganixileba Semdeg pirobebSi. sakvebi

msxverplisaTvis sakmarisia, xolo mtacebeli ki ikvebeba

mxolod msxverpliT. msxverplis namati drois mcire

monakveTisaTvis proporciulia misi raodenobis,

mtacebelis namati ki proporciulia msxverplisa da

mtaceblis raodenobaTa namravlisa, rac Seesabameba mtaceblisa da msxverplis SesaZlo Sexvedris sixSires.

mtaceblis bunebrivi sikvdilianoba proporciulia misi

raodenobisa.

models aqvs Semdegi saxe:

)(

)(

121222

212111

XXdt

dX

XXdt

dX

γε

γε

+−=

−=

sadac X1 da X2 msxverplisa da mtaceblis raodenobebia

Sesabamisad.

1ε -msxverplis bunebrivi namatis koeficientia.

2ε -mtaceblis bunebrivi Semcirebis koeficienti.

12γ da 21γ - koeficientebia, romlebic gansazRvraven

mtceblisa da msverplis ~konkurentunarianobas~.

135

volteras gantolebam farTo gavrceleba Hhpova

sxvadasxva biologiuri procesebis modelirebisas.

mtaceblisa da msxverplis ~konkurentunarianobis~

gaTvaliswinebas mivyavarT kosticinis gantolebamde

)(

)(

222121222

111212111

XXXdt

dX

XXXdt

dX

γγε

γγε

−+−=

−−=

sadac 11γ , 22γ - koeficientebia, romlebic iTvaliswineben

Siga konkurencias (populaciis matebis siCqaris

Semcireba, mis raodenobis zrdasTan erTad).

koeficientebis mniSvnelobebis Zalian farTo

diapazonSi sistemas aqvs ukiduresad mdgradi wertili

(kvanZi). es ki niSnavs, rom ekosistemaSi garkveuli saxeobis obieqtebis raodenobis ryevebs adgili ara aqvs

da gvaqvs stacionaruli gauwonasworebeli reJimi.

amJamad ekologiuri sistemebis maTematikuri

aRwerisaTvis umTavresad iyeneben ferxiulsti-pirdis

models, romelic Cawerilia sistemisaTvis, sadac aris n saxeoba:

niXXxxn

jjiij

iii ........,,3,2,1,

1

1

=+=′ ∑=

αβ

ε

136

sadac iε - namatis Siga koeficientebia.

iβ da ijα - populaciebis erTmaneTze gavlenis

koeficientebia.

komponentTa utilizaciakomponentTa utilizaciakomponentTa utilizaciakomponentTa utilizacia

biosistemis sicocxlisaTvis saWiroa sistema

mudmivad xarjavdes energias, risTvisac saWiroa man

miiRos aucilebeli nivTierebebi. Tavisufali energiis

mudmivi modineba – es aris mxolod erTi piroba

stacionaruli mdgomareobis SesanarCuneblad. meore

piroba ki aris – iseTi organizaciis arseboba romelic

uzrunvelyofs am energiis STanTqmas da gamoyenebas. ganvixiloT, magaliTad, bioqimiuri (ujredis

doneze) da fiziologiuri (organizmis doneze) sistemebi.

yoveli ujredi damoukideblad sxva ujredebisagan unda

uzrunvelyofdes Tavis energetikul moTxovnebs im

energiiT, romelsac is Rebulobs nivTierebaTa cvlis

procesSi.

ujredis moTxovnebi SeiZleba gamovsaxoT

metaboluri energiis katalizaciis tempiT – erTeul

droSi daSlili atf-is raodenoba. dabejiTebiT SeiZleba

iTqvas, rom ujredSi energoxarjvis tempi gansazRvravs masSi metaboluri procesebis mimdinareobis temps, Tumca

ujredSi metaboluri procesebis cvlilebis tempi da

ujredis doneze nivTierebaTa cvlis regulaciis

procesebi bolomde dResac ar aris Seswavlili. rogor

xdeba qimiuri energiis miwodeba misi utilizaciis

137

adgilze, rogori saSualebebiT xdeba nakadis moZraobis

mimarTulebis regulireba_jerac ucnobia. magram rom

unda arsebobdes Zlier ujredSiga sistema es cxadia.

egreTwodebuli zedapiris energetikuli kanonis

Tanaxmad (rubneris kanoni) organizmSi mimocvla,

ganisazRvreba genetikuri faqtorebiT da sxeulis

zedapiris proporciulia. nivTierebaTa cvlis tempi miT

ufro intensiuria, rac ufro naklebia sxeulis woniTi

da xazovani parametrebi. rubneris kanoni ricxobrivad

gansazRvravs metabolizmis mocemuli tempis sidides,

romelic Semdgom SenarCunebulia fiziologiuri meqanizmebiT, miuxedavad garemo pirobebis cvlilebisa (ase

magaliTad, garemoSi Jangbadis koncentraciis Semcireba an

momateba).

organizmSi yoveli ujredis energoxarjvis tempi,

saboloo jamSi damokidebulia genetikurad gansazRvruli

umaRles makoordinebel centrebze. yoveli ujredi unda

asrulebdes Tavis sfecifiur funqcias tempis

uzrunvelyofaSi. yvela ujredis tempi ikribeba da

vRebulobT imas, rasac vuwodebT mTliani organizmis

nivTierebaTa cvlis temps. metabolizmis intensivoba xSirad gamoisaxeba Tboproduqciis tempiT (magaliTad

kkal/sT).

vlaparakobT ra biosistemebSi nivTierebebisa da

energiis utilizaciis tempebze, Cven gvinda avsaxoT erTi

damaxasiaTebeli Tavisebureba, romelsac

biokibernetikisadmi miZRvnil literaturaSi saTanado

yuradReba ar eqceva. laparakia imaze, rom nivTierebebisa

da energiis nakadebi romlebic miemarTebian biosistemis

kompartmentebSi da toveben maT ar arian Tanabarni.

138

erTi nawili am nakadebisa ganpirobebulia TviT

procesebis mimdinareobis cxovelmoqmedebiT. es nakadebi

sistemaSi da kompartmentebSi arRveven wonasworobas.

meore nawili nakadebisa ki mimarTulia darRveuli

wonasworobis aRsadgenad kompartmentebSi da atarebs

aRmdgenisa da maregulirebris xasiaTs da mimarTulia

balansis SesanarCuneblad. aseTi nakadebis tempi mTlianad

damokidebulia sistemis regulatorul meqanizmebze, maTi

sidideebi regulirdeba imis da mixedviT Tu rogoria

sistemis sacicocxlo moTxovnebi.

xazi rom gausvaT principul gansxvavebas am ori nakadis tempebis tipebs Soris, SemovitanoT maTTvis

specialuri aRniSvnebi.

procesebi tempebi, romlebic uSualod arian

damokidebulni biologiuri funqciebis SesrulebasTan

iwodebian pirvelad tempebad, xolo tempebi im

procesebisaTvis, romlebic mimarTulni arian darRveuli

balansis aRsadgenad, iwodebian meorad tempebad.

cocxali ujredebis funqionirebis dros pirvelad

da meorad temepebs gansazRvruli adgili eTmobaT.

pirvelad tempebs am procesebSi ganekuTvneba energoxarjvis tempebi, anu nivTierebaTa cvlis tempebi.

energo xarjva yovel ujredSi unda Seivsos daxarjuli

energiis tolfasi energiis SevsebiT.

amdenad meoradi tempebi swored daxarjuli

energiis aRdgenis tempebs ganekuTvneba. saSualod

nivTierebaTa da energiis tempebi, romlebic mimarTulni

arian darRveuli balansis aRsadgenad, unda iyos

pirveladi nakadebis temepebis toli yovel kompartmentSi.

organizmis energetikul sistemaSi Jangbadis

utilizaciis tempi SeiZleba CavTvaloT pirvelad tempad. metabolur sistemaSi Jangbadis miwodebis tempi

139

ukavSirdeba utilizaciis temps da amis gamo Jangbadis

miwodebis tempi SeiZleba CavTvaloT meorad tempad.

naxSirorJangis warmoqmnis tempi myarad aris

dakavSirebuli Jangbadis moxmarebis tempTan da ar aris

damokidebuli fiziologiuri regulaciis meqanizmebze,

ris gamoc es tempi meoradia. mocemuli nivTierebaTa

cvlis pirobebSi siTboproduqciis tempebi pirveladia,

xolo siTbogadacemis tempebi, romlebic fiziologiur

meqanizmebzea damokidebuli, meoradi.

nivTierebaTa da energiis utilizaciis tempebi,

romlebic biosistemebis struqturis SenarCunebas xmardeba, ganisazRvrebian maTi masiTa da moculobiT, maTi

rogorc biologiuri masalebis specifiurobiT da

sivrciTi agebulebiT. umravles SemTxvevaSi SeiZleba

CavTvaloT, rom energiis nakadis tempi, romelic ixarjeba

struqturis SesanarCuneblad, an mudmivia, an

proporciulia biosistemis mdgomareobis cvladisa iX .

j – kompartmentisaTvis energiis xarji struqturis

SesanarCuneblad avRniSnoT )1(

jω - iT.

140

Ubwj!WJJ!

rjvgfhnvtynekb vjltkbht,f

,bj-cfvtlbwbyj cbcntvt,ib

cbhsekbc bv ljytpt^ hjvtkbw f[kf ufvjb.tyt,f ,bj-

cfvtlbwbyj cbcntvt,bc vfhsdbc ghjwtct,bc fqothbcf lf

fyfkbpbcfsdbc hjujhw otcb fewbkt,tkbf ituhjdltc

cbcntvbc wdkflt,pt lf vfs ijhbc rfdibht,pt byajhvfwbbc

lblb hfjltyj,f^ hfw fhsekt,c ,bjkjubeh j,btmnib

vbvlbyfht ghjwtct,bc itcofdkfc& fvbnjv w[flbf cfzbhjf

vfhsdbc stjhbbc bctsb vtsjlt,bc ufvj.tyt,f^ hjvtkbw

141

hsekb ,bjkjubehb j,btmnt,bc ufvjrdktdbc lhjc tae’yt,f

vfsb aeymwbjybht,bc ‘bhbsflb rfyjyt,bc wjlyfc^ wjw[fkb

cbcntvt,bc jhufybpfwbekb ghbywbgt,bc itcofdkfc^

hjvtkybw epheydtk.jaty aeymwbjybht,bc cfbvtlj,bc

vfqfk vfxdtyt,kt,c& fcts vtsjlc ufytresdyt,f vlujvfhtj,bc

cbdhwbc vtsjlb& vbe[tlfdfl bvbcf^ hjv ,bjkjubeh

vtwybtht,fib vlujvfhtj,fc fmdc bynebwbehb lf fhf vrfwhb

ufycfpqdht,f^ vfhsdbc stjhbfib tc wyt,f vrfabjl fhbc

ufycfpqdhekb&

lbyfvbrehb cbcntvbc vlujvfhtj,f - tc fhbc hbw[dt,bc

evwbhtcb yfrht,b^ hjvtkbw fewbkt,tkbf vjwtvek bmyfc

lhjbc vjwtvek vjvtynib^ hfsf b.jc cfiefkt,f cbcntvbc

vfstvfnbrehb fqothbc xfhxjt,ib dbobyfcofhvtn.dtkjs vbcb

mwtdf lhjbc yt,bcvbth 0tt > vjvtynib&

hbw[dt,bc fv yfrht,bc ufycfpqdhbc [th[b ~t& o&

cbcntvbc vlujvfhtj,bc dtmnjhb` lfae’yt,ekbf

vjltkbht,bc cmtvt,bc ufvj.tyt,fpt^ hjvkt,vfw ,bj-

cfvtlbwbyj cbcntvt,bc rdktdfib ufycfresht,ekb flubkb

lfbrfdtc&

142

hjujhw wyj,bkbf .dtkf wjw[fkb cbcntvf fhbc qbf

cbcntvf^ hjvtkibw vbvlbyfhtj,c ybdsbtht,fsf lf tythubbc

wdkbc ghjwtct,b& ,bj-cfvtlbwbyj cbcntvt,bc vjltkt,bc

afhsj rkfcb^ hjvkt,bw fv ghjwtct,c fqothty

rjvgfhnvtynekb vjltkt,bc cf[bsff wyj,bkb& xdtekt,hbd

rjvgfhnvtync eojlt,ty ybdsbtht,bc hfqfw hfjltyj,fc^

hjvtkbw nhfycgjhnbcf lf ybdsbtht,fsf wdkbc ghjwtcib

qt,ekj,c vjyfobktj,fc^ hjujhw lfvjerblt,tkb thstekb&

fcts lfvjerblt,tk thstekc-rjvgfhnvtync ittcf,fvt,f

hfqfw fht^ fy vjwekj,f& fct vfufkbsfl^ jhufybpvbc

abpbjkjubeh cbcntvfib^ itb’kt,f ufvjd.js e,yt,b^

hjvkt,bw ufyc[dfdlt,bfy thsvfytsbcufy pjubthsb

ybdsbtht,bc rjywtynhfwbbsf lf tythubbs& ceysmdbc vjltkib

fctsb e,fyb itb’kt,f b.jc jhb - abkndt,bcf lf mcjdbkt,bc

htpthdefht,b^ hjvkt,bw thsvfytsbcufy ufyc[dfdlt,bfy

;fyu,flbc rjywtynhfwbbs& afhvfrjrbytnbreh

rjvgfhnvtynek vjltkt,ib ~afhvfrjrbytnbrehb vjltkt,b

ufvjb.tyt,f jhufybpvbc wfkrtek jhufyjt,pt ofvkt,bc

ptvjmvtlt,bc ufvjcfrdktdfl` e,yt,b ufybcfpqdht,f bvbs^

143

hfvltyfl sfyf,hfl rjywtynhbhlt,f afhvfrjkjubehb

ybdsbtht,t,b mcjdbkt,ib lf bcbyb itb’kt,f b.jc cfrvfjl

,tdhb& fldbkb lfcflutybf^ hjv rjvgfhnvtynekb vjltkbc

fqvothb lbathtywbfkeh ufynjkt,fsf hfjltyj,bc fy

cbcntvbc ufynjkt,bc hbub rjvgfhnvtynsf hfjltyj,bc

njkbf&

se hjvtkbvt ybdsbtht,f rjvgfhnvtynek cbcntvfib

uflfflubkt,f thsb rjvgfhnvtynblfy vtjhtib^ vfiby fv

ybdsbtht,bc hfjltyj,bc wdkbkt,bc ntvgb

iX i′ -ehb rjvgfhnvtynbcfsdbc ufybcfpqdht,f itvltub

ufynjkt,bs%

∑ ∑≠=

≠=

−+−+−=′n

ijj

n

ikk

iiiooiikjii dgyyyyX1 1

~1`

cflfw iX -ybdsbtht,bc hfjltyj,ff i -eh

rjvgfhnvtynib^ jiy -ybdsbtht,bc yfrflbc ntvgbf j -ehb

rjvgfhnvtynblfy i -ehib^ oiy lf ioy -ybdsbtht,bc yfrflbc

ntvgt,bf ufhtvjlfy i -eh rjvgfhnvtynib lf rjvgfhnvtynlfy

144

ufhtvjib itcf,fvbcfl ig lf id -eh rjvgfhnvtynib

ybdsbtht,bc ofhvjmvybcf lf enbkbpfwbbc ntvgt,bf&

rjvgfhnvtynekb fyfkbpbc lhjc ybdsbtht,fsf wdkbc

ghjwtcbc ‘bhbsfl vtmfybpvc lbaepbf ofhvjflutyc^ sevwf

ybdsbtht,fsf lf tythubbc wdkfib fhf yfrkt, hjkc

fchekt,ty mbvbehb lf abpbrehb ghjwtct,b& fct vfufkbsfl^

se ,bj-cbcntvf itbwfdc n - rjvgfhnvtync lf iX -rb fhbc

hjvtkbvt ybdsbtht,bc ljyt^ fy rjywtynhfwbf i -eh

rjvgfhnvtynib^ vfiby ybdsbtht,fsf uflfnfybc ghjwtct,b^

hjvkt,bw ufvjodtekbf rjywtynhfwbfsf c[dfj,bs

c[dflfc[df rjvgfhnvtynt,ib lf ufhtvjib fqbotht,bfy

ohabdb lbathtywbfkehb ufynjkt,bs

ni

XV

KXXKX

n

ijj

ii

oiijji

ii

,...,2,1

)V()(V

1

1

=

−+−=′ ∑≠= ~2`

cflfw )( ij XX − -ybdsbtht,bc ljytsf fy

rjywtynhfwbfsf c[dfj,ff^ jiK -velvbdb fhfefh.jabsb

145

rjtabwbtynt,bf^ V-ufhtvjib ufycf[bkdtkb ybdsbtht,bc

ljytf^ iV i -ehb rjvgfhnvtynbc vjwekj,ff& ufynjkt,f ~2`

itb’kt,f uflfdothjs dtmnjhbc cf[bs^ vfiby bc vlujvfhtj,bc

cbdhwbc ufynjkt,bc cf[tc vbbqt,c [1]&

V1 BXAX +=′

cflfw [ ]TnXXXX ..., 21= -vlujvfhtj,bc

dtmnjhbf^ [jkj V=[V] ufhtvjc ptvjmvtlt,bc dtmnjhbf

)(1 nnA × lf )(1 nnB × vfnhbwt,bf^ hjvtksfw fmds

itvltub tktvtynt,b

i

oii

n

ijj

oijii

ii

i

jiji

Kb

KKa

jiK

a

V

)(V

1

V

1

=

−−=

≠=

∑≠=

~4`

146

se i -eh rjvgfhnvtynib ufycf[bkdtkb ybdsbtht,bc

itcdkbc ntvgc iy -fqdybiyfds lf itvjdbnfys dtmnjhc

[ ]Tnyyyy ..., 21= vbdbqt,s ohabdb rjvgfhnvtynekb

vjltkbc ufynjkt,fc [2]

yBXAX 111 RV ++=′ ~5`

cflfw )(1 nnR × -lbfujyfkehb vfnhbwff tktvtynt,bs

iiir

V

1=

vfnhbwf R -bc cfiefkt,bs itcfZkt,tkbf rjvgjytynsf hbw[dbc vb[tldbs hjvkt,bw ufhtlfy itvjlbfy ufydcfpqdhjs vfsb rjywtynhfwbf i -eh rjvgfhnvtynib&

rjvgfhnvtynib ybdsbtht,fsf rjywtynhfwbbc

wdkbkt,f tae’yt,f ybdsbtht,fsf ityf[dbc rfyjyc^ t& b&

ybdsbtht,fsf vj[vfht,bc ghjwtct,b ufojyfcojht,ekbf^

ufhtlfy itvjcekb rjvgjytynt,bc ufhlfmvybs bv

ybdsbtht,t,fl^ hjvtkbw fewbkt,tkbf ,bjkjubehb cbcntvbc

w[jdtkmvtlt,bcsdbc& ybdsbtht,fsf vbojlt,bc ntvgt,b bv

147

rjvgfhnvtynt,ib cflfw bcbyb b[fh]t,bfy^ itcf’kt,tkbf

dfhtuekbhjs& vfsb cblblt lfvjrblt,ekbf [fh]dbc

ntvgt,pt lf ,eyt,hbdb ,bjkjubehb cbcntvbc

htuekfnjhekb vtmfybpvt,bc itcf’kt,kj,t,pt^ hjvkt,bw

wlbkj,ty epheydtk.jy ybdsbtht,bcf lf tythubbc ,fkfycb fv

cbcntvfib&

148

yf[&7 nax. 26.

149

‘bhbsflb bltf ofhvjlutybkb vjltkbht,bcf

vlujvfhtj,c cbcntvbc vfhsdbc [fhbc[bcf lf htcehct,bc

vfmcbvfkeh ufvj.tyt,fib& vfhsdbc htcehcbc mdti

buekbc[vt,f aeymwbjybht,bc ghjwtct,pt vfmcbvfkehb

hfjltyj,bc ptvjmvtlt,bc htfkbpfwbf^ hfw ybdsbtht,bc lf

tythubbc ,fkfycbcsdbc fhbc fewbkt,tkb& fvhbufl:

jhufybpvbc vfhsdbc htcehct,b yjhvfkeh gbhj,t,ib

ufyfobkt,ekbf itcf’kj lfhqdtdt,bc gjntywbfkehb

cbvhfdkbs lf .jdtk rjyrhtnek itvs[dtdfib vj,bkbpfwbfc

ertst,c cfhtuekbht,tk cblbltt,c&

vfhsdbc htcehct,bc bthfhmbekb byntuhfwbbcfsdbc

aeymwbjybht,bc .jdtkb ht;bvbcsdbc bmvyt,f fkujhbsvt,bc

cbcntvf^ hjvkt,bw vbvhfsekyb fhbfy vfhsdbc vbpyt,bc

epheydtkcf.jafl vfqfk bthfhmbek ljytt,pt&

tc vbpyt,b vbvfhsekbf ,bjkjubehb j,btmnt,bc

sdbsityfhxeyt,bcfsdbc lf ajhvekbhlt,f itvltuyfbhfl%

1& vfhsdbc evfqktcb vbpfyb - cbcntvbc aeymwbjybht,bc

epheydtk.jaf*

150

2& vtjht hbubc vfhsdbc vbpfyb - ,bjkjubehb j,btmnbc

/jvtjcnfpbc epheydtk.jaf^ ibyfufyb gbhj,t,bc velvbdj,bc

fye ,fkfycbc epheydtk.jaf*

3& vtcfvt hbubc vfhsdbc vbpfyb- ht;bvt,bc jgnbvbpfwbf

lf vfhsdbc ghjwtct,bc [fhbc[bc ufev]j,tct,f&

.jdtkb vfsufybcsdbc fhct,j,c cfvfhsfdb cblbltt,bc

ufhrdtekb cbvhfdkt lf vfhsdbc htcehct,bc ufvj.tyt,bc

sfdbct,eht,fyb^ hfw ,bjkjubehb j,btmnbc [fcbfsptf

lfvjrblt,ekb&

151

cocxali organizmis Tboregulaciis marTvis cocxali organizmis Tboregulaciis marTvis cocxali organizmis Tboregulaciis marTvis cocxali organizmis Tboregulaciis marTvis

sistemasistemasistemasistema

warmovidginoT sxeulis qsovilebi samSriani

struqturis saxiT (kani, kunTebi da birTvi).

temperaturiT x1, x2 da x3 Sesabamisad. siTbos wyaro

moTavsebulia birTvSi. siTbos modinebis tempi w3

ganisazRvreba nivTierebaTa cvlis intensivobiT. kunTebSi

siTbos modineba w2 dakavSirebulia kunTebis muSaobasTan.

siTbos nakadi birTvidan kunTebamde, kunTebidan

kanamde da kanidan garemoSi ganisazRvreba temperaturaTa

sxvaobiT (x3-x2), (x2-x1) da (x1-v1). sadac v1 aris garemos

temperatura. siTbos cvlis mprocesebi organizmSi

damokidebulia temperaturaTa sxvaobaze sxvadasxva ubans

Soris da Tbogamtarobis koeficientebze k3, k2 da

Tbogadacemis koeficientze k1.

nax. 27.

152

Tboregulacia dakavSirebulia ZiriTadad w3, w4 da

w1 (w1-aorTqleba) koeficientebze da aseve k2 da k1-ze

(k1-sisxlZarRvovani reaqciebi).

yovelive zemoTqmulis gaTvaliswinebiT

SesaZlebelia CavweroT Semdegi diferencialuri

gantolebebi, romlebic aRweren Tboregulaciis marTvis

sistemas:

sadac:

Ti=cipidi, i=1,2,3-drois mudmivebi TiToeuli SrisaTvis, ci-kuTri siTbotevadoba, pi-kuTri simkvrive, Sreebis sisqis maxasiaTebeli.

nax. 28.

153

simartivisaTvis davuSvaT, rom w=0. sistemaSi aris sami

mdgomareobis cvladi x1, x2 da x3. gare signali, romelic

moqmedebs sistemaze aseve samia w3, w2 da v1.

mdgomareobis gantoleba gadavweroT matricebis da

veqtorebis saSualebiT. gantolebis orive nawili gavyoT

drois mudmivaze Ti,

maSin mdgomareobis gantoleba Semdegnairad Caiwereba:

154

axla ganvixiloT sakiTxi imis Sesaxeb Tu romeli

signalebi unda aviRoT gamomavali signalis saxiT.

sadac d1, d2 da d3-fenebis sisqis maxasiaTebelia. y1 aviRoT, rogorc sistemis erTi gamosasvleli signali.

xSirad mkvlevarebs aseve ainteresebT birTvisa da

kanis temperatura. amitom y2 da y3 aviRoT Sesabamisad x1

da x2-is toli.

aseve saintereso iqneba siTbos nakadis tempebis

gansazRvra, amitom aviRoT y4 da y5 gamomavali signalebis saxiT.

155

aseve SesaZloa saintereso iyos organizmis saerTo

Tbogadacema, amitom SemovitanoT kidev erTi gamomavali

signali y6.

yovelive zemoTqmulis gaTvaliswinebiT davweroT

gamomavali signalis gantoleba:

sadac:

vpoulobT C da D matricebs:

156

axla ganvixiloT marTvis sistemis xarisxobrivi

maxasiaTeblebi. marTvadoba, mdgradoba, mgrZnobeloba da

dakvirvebadoba. aseve sistemis stacionaruli reJimi.

rac Seexeba mdgradobis dadgenas, roca saubaria

biologiur sistemebze, raTqmaunda aseTi sistemebi

mdgradia cvladebis sakmaod farTo diapazonSi. aseTi

marTvis sistemis mdgradobis Seswavla gvexmareba imaSi,

rom moxdes Sefaseba, Tu ramdenad kargad asaxavs sistema

mis mier aRmwer biologiur process.

ganvixiloT Cveni sistemis mdgradoba roca

cvladebi k1=k2=k3=T1=T2=T3=1. aseT SemTxvevaSi maxasiaTebel gantolebas aqvs

Semdegi saxe:

aqedan miviRebT gantolebas, romlis fesvebis

namdvili nawili aris uaryofiTi, anu ganlagebulia

marcxena naxevarsibrtyeSi. aqedan gamomdinare sistema

mdgradia sawyisi monacemebis mixedviT. sistema aseve

mdgradia Semavali signalis mixedviT anu asimptoturad

mdgradia.

stacionaruli reJimistacionaruli reJimistacionaruli reJimistacionaruli reJimi

SeviswavloT sistemis stacianuri reJimi.

x=-A-1Bv

157

sabolood miviRebT:

mgrZnobelobamgrZnobelobamgrZnobelobamgrZnobeloba

mgrZnobiaroba gviCvenebs Tu ramdenad icvleba

sistemis stacionaruli reJimi erTi garemodan meoreSi

gadasvlisas.

SevadginoT mgrZnobelobis matrica:

158

martivad SeiZleba SevniSnoT rom garemos

temperaturis cvlilebisas organizmis yvela fenis

temperatura Zlier icvleba.

marTvadoba da dakvirvebadomarTvadoba da dakvirvebadomarTvadoba da dakvirvebadomarTvadoba da dakvirvebadobabababa

marTvadobis dasadgenad unda SevadginoT

marTvadobis matrica.

159

sabolood miviRebT:

matricis rangia sami. aqidan gamomdinare sistema

marTvadia.

ddddakvirvebadobaakvirvebadobaakvirvebadobaakvirvebadoba

dakvirvebadoba am SemTxvevaSi damokidebulia Cvenze.

SevarCevT ra sakmarisi raodenobis gamosasvlel signals. dakvirvebadobis dasadgenad SeviswavloT Sesabamisi

dakvirvebadobis matrica:

miviRebT:

160

misi rangi 3-is tolia. e. i. sistema dakvirvebadia.

161

rekomendirebuli rekomendirebuli rekomendirebuli rekomendirebuli literatuliteratuliteratuliteraturrrraaaa

1. ავტომატიზებული მართვის მოდელები: პეტრის ქსელები გ. გოგიჩაიშვილი, გ. სურგულაძე, დ. გულუა, მ. კაშიბაძე საქართველოს ტექნიკური უნივერსიტეტი.

2. მართვის ავტომატიზებული სისტემების ვიზუალური მოდელირების მეთოდები და ინსტრუმენტები (UML, MsVisio) გ. სურგულაძე, თ. სუხიაშვილი, გ. ნარეშელაშვილი. საქართველოს ტექნიკური

უნივერსიტეტი.

3. ავტომატიზებული მართვის მოდელები: მონაცემთა რელაციური მოდელი გ. გოგიჩაიშვილი, გ. სურგულაძე.

საქართველოს ტექნიკური უნივერსიტეტი.

4. იმიტაციური მოდელირება MATLAB- ში Simulink ნ. მჭედლიშვილი, ვ. სესაძე, ვ. კეკენაძე, გ. ჭიკაძე.

საქართველოს ტექნიკური უნივერსიტეტი.

5. კომპიუტერული მოდელირების სისტემა Matlab ნ. მჭედლიშვილი, ვ. სესაძე, ვ. კეკენაძე, გ. ჭიკაძე.

საქართველოს ტექნიკური უნივერსიტეტი.

6. მათემატიკური მოდელირების კურსი, ტომი 2 თ.ობგაძე, ლ. ობგაძე, ნ. მჭედლიშვილი, ი. დავითაშვილი, ნ.თუშიშვილი. საქართველოს ტექნიკური

უნივერსიტეტი.

7. Barab, A. -L.; Oltvai, Z. (2004). "Network biology: understanding the cell's functional organization". Nature

162

reviews. Genetics 5 (2): 101–113. doi:10.1038/nrg1272. PMID 14735121. edit

8. Covert; Schilling, C.; Palsson, B. (2001). "Regulation of

gene expression in flux balance models of metabolism". Journal of Theoretical Biology 213 (1): 73–88. doi:10.1006/jtbi.2001.2405. PMID 11708855. edit

9. Covert, M. W.; Palsson, B. . (2002). "Transcriptional regulation in constraints-based metabolic models of Escherichia coli". The Journal of Biological Chemistry 277 (31): 28058–28064. doi:10.1074/jbc.M201691200. PMID 12006566. edit

10. Edwards; Palsson, B. (2000). "The Escherichia coli MG1655 in silico metabolic genotype: its definition, characteristics, and capabilities". Proceedings of the National Academy of Sciences of the United States of America 97 (10): 5528–5533. Bibcode 2000PNAS...97.5528E. doi:10.1073/pnas.97.10.5528. PMC 25862. PMID 10805808. edit

11. Bonneau, R. (2008). "Learning biological networks: from modules to dynamics". Nature chemical biology 4 (11): 658–664. doi:10.1038/nchembio.122. PMID 18936750. edit

12. Edwards, J. S.; Ibarra, R. U.; Palsson, B. O. (2001). "In silico predictions of Escherichia coli metabolic capabilities are consistent with experimental data". Nature Biotechnology 19 (2): 125–130. doi:10.1038/84379. PMID 11175725. edit

13. Fell, D. A. (1998). "Increasing the flux in metabolic pathways: A metabolic control analysis perspective".

163

Biotechnology and Bioengineering 58 (2–3): 121–124. doi:

14. Hartwell, L. H.; Hopfield, J. J.; Leibler, S.; Murray, A. W. (1999). "From molecular to modular cell biology". Nature 402 (6761 Suppl): C47–C52. doi:10.1038/35011540. PMID 10591225.

15. Ideker; Galitski, T.; Hood, L. (2001). "A new approach to decoding life: systems biology". Annual review of genomics and human genetics 2 (1): 343–372. doi:10.1146/annurev.genom.2.1.343.

16. Kitano, H. (2002). "Computational systems biology". Nature 420 (6912): 206–210. Bibcode 2002Natur.420..206K.doi:10.1038/nature01254. PMID 12432404. edit

17. Kitano, H. (2002). "Systems biology: a brief overview". Science 295 (5560): 1662–1664. Bibcode 2002Sci...295.1662K. doi:10.1126/science.1069492. PMID 11872829. edit

18. Kitano (2002). "Looking beyond the details: a rise in system-oriented approaches in genetics and molecular biology". Current genetics 41 (1): 1–10. doi:10.1007/s00294-002-0285-z. PMID 12073094. edit

19. Gilman, A.; Simon, M. I.; Bourne, H. R.; Harris, B. A.; Long, R.; Ross, E. M.; Stull, J. T.; Taussig, R. et al. (2002). "Overview of the Alliance for Cellular Signaling". Nature 420 (6916): 703–706. doi:10.1038/nature01304. PMID 12478301.

164

20. Palsson, Bernhard (2006). Systems biology: properties of reconstructed networks. Cambridge: Cambridge University Press. ISBN 978-0-521-85903-5.

21. Kauffman; Prakash, P.; Edwards, J. S. (2003). "Advances in flux balance analysis". Current opinion in biotechnology 14 (5): 491–496. doi:10.1016/j.copbio.2003.08.001. PMID 14580578. edit

22. Segrè, D.; Vitkup, D.; Church, G. M. (2002). "Analysis of optimality in natural and perturbed metabolic networks". Proceedings of the National Academy of Sciences of the United States of America 99 (23): 15112–15117. Bibcode 2002PNAS...9915112S.

23. Wildermuth, MC (2000). "Metabolic control analysis: biological applications and insights.". Genome Biology 1 (6): REVIEWS1031. PMC 138895. PMID 11178271. edit

24. Maxwell, J.C. (1868). "On Governors". Proceedings of the Royal Society of London 16: 270–283. doi:10.1098/rspl.1867.0055. JSTOR 112510.

25. Routh, E.J.; Fuller, A.T. (1975). Stability of motion. Taylor & Francis.

26. Routh, E.J. (1877). A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Particularly Steady Motion. Macmillan and co.

27. Hurwitz, A. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts". Selected Papers on Mathematical Trends in Control Theory.

165

28. Donald M Wiberg. State space & linear systems. Schaum's outline series. McGraw Hill. ISBN 0-07-070096-6.

29. Liu, Jie; Wang, Golnaraghi, Kubica (2010). "A novel fuzzy framework for nonlinear system control". Fuzzy Sets and Systems 161 (21): 2746–2759.

30. Levine, William S., ed. (1996). The Control Handbook. New York: CRC Press. ISBN 978-0-8493-8570-4.

31. Karl J. Åström and Richard M. Murray (2008). Feedback Systems: An Introduction for Scientists and Engineers.. Princeton University Press. ISBN 0-691-13576-2.

32. Christopher Kilian (2005). Modern Control Technology. Thompson Delmar Learning. ISBN 1-4018-5806-6.

33. Vannevar Bush (1929). Operational Circuit Analysis. John Wiley and Sons, Inc.

34. Robert F. Stengel (1994). Optimal Control and Estimation. Dover Publications. ISBN 0-486-68200-5.

35. Franklin et al. (2002). Feedback Control of Dynamic Systems (4 ed.). New Jersey: Prentice Hall. ISBN 0-13-032393-4.

36. Joseph L. Hellerstein, Dawn M. Tilbury, and Sujay Parekh (2004). Feedback Control of Computing Systems. John Wiley and Sons. ISBN 0-471-26637-X.

37. Diederich Hinrichsen and Anthony J. Pritchard (2005). Mathematical Systems Theory I - Modelling, State Space

166

Analysis, Stability and Robustness. Springer. ISBN 3-540-44125-5.

38. Andrei, Neculai (2005). Modern Control Theory - A historical Perspective. Retrieved 2007-10-10.

39. Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. ISBN 0-387-98489-5.

40. Goodwin, Graham (2001). Control System Design. Prentice Hall. ISBN 0-13-958653-9.

41. Феликс Л. Элементарная математика в современном изложении. М.: Просвещение, 1967.

42. Малинецкий Г.Г. Хаос. Структуры. Вычислительный эксперимент: Введение в нелинейную динамику. М.: Наука, 1997.

43. Лайтфут Э. Явления переноса в живых системах. М.:Мир, 1977.

44. Балантер Б.И., Ханин М.А., Чернавский Д.С. Введение в математическое моделирование патологических процессов. – М., Медицина, 1980.

45. Рубин А.Б. Биофизика: В 2 т. Т.1: Теоретическая биофизика.-М.: Книжный дом «Университет», 1999.

46. Щукин С.И. Основы биофизики.Ч.1: Учебное пособие. М.: МГТУ, 2001. 36 с.

167

47. Амосов А.А., Дубинский Ю.А., Копченова Н.В. Вычислительные методы для инженеров. М., Высшая школа, 1994.

48. Котин В.В. Методическая разработка к курсовой работе по дисциплине «Математические модели макроуровня». МГТА, М.,1998.

49. Н.Бейли. Математика в биологии и медицине. М., Мир, 1970.

50. Беллман Р. Математические методы в медицине. - М.: Мир,1987.

51. Вольтерра В. Математическая теория борьбы за существование. М., Наука, 1976.

52. Свирежев Ю.М. Нелинейные волны, диссипативные структуры и катастрофы в экологии. М.:Наука, 1987.

53. Романовский Ю.М., Степанова Н.В., Чернавский Д.С. Математическое моделирование в биофизике.- М.: Наука,1975.

54. Рубин А.Б. Биофизика: В 2 т. Т.2: Биофизика клеточных процессов.-М.: Книжный дом «Университет», 2000.

55. Моисеев Н.Н. Математические задачи системного анализа. М. Наука, 1981.

56. Хомяков Д.М., Хомяков П.М. Основы системного анализа. Изд. механико-математического факультета МГУ им. М.В. Ломоносова, М., 1996.

168

57. Новосельцев В.Н. Теория управления и биосистемы. Анализ сохранительных свойств. М.: Наука, 1978.

58. Марри Дж. Нелинейные дифференциальные уравнения в биологии. Лекции о моделях. -М.: Мир, 1983.-397 с.

59. Копченова Н.В., Марон И.А. Вычислительная математика в примерах и задачах. М.,Наука, 1972.

60. Волков Е.А. Численные методы , М., Наука, 1987.

61. Косарев В.И. 12 лекций по вычислительной математике (вводный курс), М., МФТИ, 1995.

62. Турчак Л.И. Основы численных методов. М.,Наука,1987.

63. Вентцель Е.С. Овчаров Л.А. Теория вероятностей и её инженерные приложения. М.: Наука, 1988.

64. Гмурман В.Е. Руководство к решению задач по теории вероятностей и математической статистике. М., Высшая школа, 1997.

65. Ханин М.А., Дорфман Н.Л., Бухаров И.Б., Левадный В. Г. Экстремальные принципы в биологии и физиологии. М.,Наука, 1977.

66. Гродинз Ф. Теория регулирования и биологические системы. – М. Мир, 1966.

67. Мирошник И.В. Теория автоматического управления.

Линейные системы: Учебное пособие для вузов. - СПб.: Питер, 2005. - 336 с.

169

68. Повзнер Л.Д. Теория систем управления: Учебное

пособие для вузов. - М.: Изд. МГГУ, 2002. - 472 с.

69. Семенов Е.М. и др. Автоматика и автоматизация производственных процессов: Методические указания по выполнению лабораторных и практических работ. – СПб ГЛТА, 2004. - 43 с.

70. Орлов А.И. Менеджмент: Учебник. – М.: "Изумруд", 2003. URL: http://www.aup.ru/books/m151/

71. Кориков А.М. Основы теории управления: Мультимедийный учебник. – Томск: ТУСУР. URL: http://www.tcde.ru/docs_pub/demo/otu/otu.exe

72. Новиков Д.А., Петраков С.Н. Курс теории активных систем. М.: СИНТЕГ, 1999. – 104 с. URL: http://www.aup.ru/books/m110/file_46.pdf

73. Туманов М.П. Теория автоматического управления: Лекции.URL: http://elib.ispu.ru/library/lessons/Tihonov_2/index.htm.

74. Туманов М.П. Теория управления. Теория линейных систем автоматического управления: Учебное пособие. – МГИЭМ. М., 2005, 82 с. URL: http://window.edu.ru/window_catalog/files/r24738/5.pdf.

75. Бесекерский В.А., Попов Е.П. Теория систем автоматического регулирования. – М.: Наука, 1975.

76. Туманов М.П. Технические средства автоматизации и управления: Учебное пособие. – М.: МГИЭМ, 2005, 71 с.URL:

170

http://rs16tl.rapidshare.com/files/21651582/2889232/ Tehnicheskie_sredstva_avtomatizatsii_i_upravleniya.rar

77. Михайлов В.С. Теория управления. – К.: Выща школа, 1988.

78. Зайцев Г.Ф. Теория автоматического управления и регулирования. – К.: Выща школа, 1989.

79. Теория управления: Учебник / Под общ ред. А. Л. Гапоненко, А. П. Панкрухина. - М., 2004.

80. Виханский О. С, Наумов А. И. Менеджмент: Учебник 4-е изд, перераб. и доп. - М., 2006.

81. Райченко А. В. Общий менеджмент: Учебник. - М., 2006.

82. Аникин Б. А. Высший менеджмент для руководителя: Учебное пособие. - М., 2001.

83. Ансофф И. Стратегическое управление: Пер. с англ. - М., 1998.

84. Арин О. А. Россия в стратегическом капкане. - М., 1997.

85. Армстронг М. Стратегическое управление человеческими ресурсами: Пер с англ. - М., 2002.

86. Богданов А. А. Тектология: всеобщая организационная наука. - М., 1989.

171

87. Василенко И. А. Административно-государственное управление в странах Запада. – М., 2001.

88. Вудкок М., Френсис Д. Раскрепощенный менеджер. - М., 1999.

89. Глобальные тенденции развития человечества до 2015 года. Материалы Национального разведывательного совета США: Пер с англ. - Екатеринбург. 2002.

90. Друкер П. Ф. Энциклопедия менеджмента: Пер. с англ. - М., 2004.

91. Игнатьева А. В., Максимцов М. М. Исследование систем управления: Учеб. пособие для вузов. - М., 2000.

92. Кнорринг В. И. Теория, практика и искусство управления. Учебник для вузов. - М.,1999.

93. Корицкий Э., Нинциева Г., Шетов. В. Научный менеджмент: российская история. - СПб., 1999.

94. Кунц Г., О Донелл С. Управление: системный и ситуационный анализ управленческих функций: Пер. с англ. Т. - 1,2 . М., 1981.

95. Мескон М., Альберт М., Хедоури Ф. Основы менеджмента. - М., 2001.

96. Паркинсон С. Законы Паркинсона: Пер. с англ. - М., 1989.

172

97. Пирсон Б., Томас Н. Магистр делового администрирования: Пер с англ. - М., 2002.

98. Пригожин И., Стенгерс И. Порядок из хаоса. - М., 1986.