ࡱ> 2:< 789k~345678e ^ r{Iws|KLqxz|9 bjbj;JllllHHH8>H JI2jJd V"VVV/X Ib4}e!!!!!!$J jnElg+X/XggEcVV;cccgVlVcgccgjV,@lSV^J `s-h@?H JC2S0uW ScSVEU ILI`TE U ZAGREBU FAKULTET ELEKTROTEHNIKE I RA UNARSTVA DIPLOMSKI RAD br.1364 MODELIRANJE TRENJA PRIMJENOM RBF NEURONSKIH MRE}A Dinko Vranjea Zagreb, srpanj 2003. Zahvaljujem se prof.dr.sc. Nedjelju Periu, Doc. dr.sc. Ivanu Petroviu i dipl.ing. Jadranku Matuaku na pomoi pru~enoj tijekom studija i pri izradi diplomskog zadatka. Iznad svega, veliko hvala mojim roditeljima i sestri na podraci i strpljenju. SADR}AJ  TOC \o "1-3" \h \z  HYPERLINK \l "_Toc43847537" 1. UVOD  PAGEREF _Toc43847537 \h 5  HYPERLINK \l "_Toc43847538" 2. TRENJE  PAGEREF _Toc43847538 \h 7  HYPERLINK \l "_Toc43847539" 2.1. Definicija trenja  PAGEREF _Toc43847539 \h 7  HYPERLINK \l "_Toc43847540" 2.1.1. Vrste trenja  PAGEREF _Toc43847540 \h 7  HYPERLINK \l "_Toc43847541" 2.1.2. Modeli trenja  PAGEREF _Toc43847541 \h 9  HYPERLINK \l "_Toc43847542" 2.2. Modeli trenja za kontakt guma-asfalt  PAGEREF _Toc43847542 \h 15  HYPERLINK \l "_Toc43847543" 2.2.1. Fizikalna zbivanja u kontaktu automobilskog kota a i ceste  PAGEREF _Toc43847543 \h 15  HYPERLINK \l "_Toc43847544" 2.2.2. Stati ki modeli trenja za u kontakt automobilskog kota a i ceste  PAGEREF _Toc43847544 \h 18  HYPERLINK \l "_Toc43847545" 2.2.2. Dinami ki modeli trenja u kontaktu automobilskog kota a i ceste  PAGEREF _Toc43847545 \h 19  HYPERLINK \l "_Toc43847546" 3. NEURONSKE MRE}E  PAGEREF _Toc43847546 \h 21  HYPERLINK \l "_Toc43847547" 3.1. Bioloaka osnova razvoja neuronskih mre~a  PAGEREF _Toc43847547 \h 21  HYPERLINK \l "_Toc43847548" 3.2. Modeli umjetnih neurona  PAGEREF _Toc43847548 \h 22  HYPERLINK \l "_Toc43847549" 3.2.1 Stati ki modeli neurona  PAGEREF _Toc43847549 \h 22  HYPERLINK \l "_Toc43847550" 3.2.2. Dinami ki model neurona  PAGEREF _Toc43847550 \h 25  HYPERLINK \l "_Toc43847551" 3.3. Modeli i osnovna svojstva neuronskih mre~a  PAGEREF _Toc43847551 \h 26  HYPERLINK \l "_Toc43847552" 3.3.1. Stati ke neuronske mre~e  PAGEREF _Toc43847552 \h 26  HYPERLINK \l "_Toc43847553" 3.3.2. Dinami ke neuronske mre~e  PAGEREF _Toc43847553 \h 27  HYPERLINK \l "_Toc43847554" 3.3.3. RBF Neuronska mre~a  PAGEREF _Toc43847554 \h 27  HYPERLINK \l "_Toc43847555" 3.3.4. Svojstva neuronskih mre~a  PAGEREF _Toc43847555 \h 30  HYPERLINK \l "_Toc43847556" 4. HIBRIDNI ALGORITAM U ENJA PARAMETARA RBF NEURONSKE MRE}E  PAGEREF _Toc43847556 \h 32  HYPERLINK \l "_Toc43847557" 4.1. U enje neuronskih mre~a  PAGEREF _Toc43847557 \h 32  HYPERLINK \l "_Toc43847558" 4.2. Algoritam rasporeivanja srediata neurona ulaznog sloja mre~e  PAGEREF _Toc43847558 \h 34  HYPERLINK \l "_Toc43847559" 4.3. Algoritam podeaavanja varijanci neurona ulaznog sloja mre~e  PAGEREF _Toc43847559 \h 35  HYPERLINK \l "_Toc43847560" 4.4. Algoritam podeaavanja koeficijenata sinapti kih te~ina izlaznog sloja mre~e  PAGEREF _Toc43847560 \h 37  HYPERLINK \l "_Toc43847561" 4.5. Usklaivanje meusobnog rada algoritama u enja  PAGEREF _Toc43847561 \h 38  HYPERLINK \l "_Toc43847562" 5. PROJEKTIRANJE MODELA TRENJA RBF NEURONSKIM MRE}AMA  PAGEREF _Toc43847562 \h 40  HYPERLINK \l "_Toc43847563" 5.1. Rezultati simulacija  PAGEREF _Toc43847563 \h 44  HYPERLINK \l "_Toc43847564" 5.1.1. Modeliranje signala linearne brzine  PAGEREF _Toc43847564 \h 44  HYPERLINK \l "_Toc43847565" 5.1.2. Modeliranje signala kutne brzine  PAGEREF _Toc43847565 \h 50  HYPERLINK \l "_Toc43847566" 6. ZAKLJU AK  PAGEREF _Toc43847566 \h 56  HYPERLINK \l "_Toc43847567" 7. LITERATURA  PAGEREF _Toc43847567 \h 57  HYPERLINK \l "_Toc43847568" 8. SA}ETAK  PAGEREF _Toc43847568 \h 58  HYPERLINK \l "_Toc43847569" 9. ABSTRACT  PAGEREF _Toc43847569 \h 59  HYPERLINK \l "_Toc43847570" 10. }IVOTOPIS  PAGEREF _Toc43847570 \h 60  1. UVOD Zbog komercijalnih razloga i sve veih zahtjeva na sigurnost putnika, automobilska industrija ula~e velika sredstva u razvoj novih tehnologija koje e omoguiti bolje ponaaanje automobila na cesti. Sustav ko enja vozila klju an je dio za osiguranje takvog ponaaanja. Tokom zadnjih nekoliko godina razvijeni su razni sustavi koji poboljaavaju ponaaanje automobila na cesti, poglavito ko enja. Neki od njih postali su standard u danaanjoj automobilskoj industriji, poput sustava upravljanja ko enjem vozila (ABS,engl., Antilock Breaking System), sustav upravljanja vu nom silom (TCS engl., Traction Control System) i sustav upravljanja dinami kim vladanjem vozila (VDS engl., Vehicle Dynamics System). Nagli razvitak elektroni ke industrije rezultirao je u lako dostupnim i jeftinim elektroni kim komponentama s pomou kojih se gore navedeni sustavi lako realiziraju. Za razvitak navedenih sustava nu~no je imati dobar uvid u proces koji se dogaa u kontaktu automobilske gume i ceste. To je proces trenja koji je po svojoj prirodi izrazito nelinearan proces. Zbog toga se teorija linearnog upravljanja procesima ne mo~e primijeniti na njega. Mada ne postoji jedinstvena teorija upravljanja nelinearnim procesima, s vremenom su razvijene mnoge teorije koje se mogu vrlo dobro primijeniti u odreenim uvjetima. Jedna od njih je teorija neuronskih mre~a. Najbitnije svojstvo neuronskih mre~a je sposobnost aproksimacije proizvoljne nelinearne funkcije sa ~eljenom to noau. To ini neuronske mre~e pogodnima za identifikaciju i upravljanje nelinearnim procesima. Razvijeno je viae vrsta neuronskih mre~a, a u ovom radu koriste se neuronske mre~e s kru~nom osnovicom (engl., Radial Basis Functions). U enje mre~e odvija se rekurzivno (on-line), a parametri mre~e podeaavaju se hibridnim algoritmom. Poato RBF neuronske mre~e spadaju u stati ke neuronske mre~e, potrebno im je dodati vanjske dinami ke lanove. U drugom poglavlju ovog rada opisana su fizikalna zbivanja u kontaktu automobilskog kota a s podlogom, model trenja za kontakt automobilskog kota a i ceste, te opi stati ki i dinami ki modeli trenja. U treem poglavlju dan je pregled teorije neuronskih mre~a od bioloakog neurona do stati kih i dinami kih mre~a. Poseban naglasak stavljen je na neuronske mre~e s kru~nom osnovicom koje se koriste u ovom radu. U etvrtom poglavlju opisan je rekurzivni hibridni algoritam u enja RBF neuronskih mre~a. Svaki od triju dijelova algoritma matemati ki je opisan pseudokodom. U petom poglavlju dani su rezultati u enja modela trenja. Mre~a je u ila linijsku i kutnu brzinu proairenog LuGre modela trenja s razli itim brojem neurona. 2. TRENJE 2.1. Definicija trenja Prema standardu DIN 5281 trenje definiramo kao otpor koji se javlja izmeu povraina nalijeganja dvaju tijela i suprotstavlja se meusobnom gibanju bilo klizanjem, bilo kotrljanjem ili valjanjem (trenje gibanja - kineti ko trenje), ili onemoguuje gibanje (trenje mirovanja - stati ko trenje). 2.1.1. Vrste trenja Tri osnovne vrste trenja su trenje klizanja, trenje kotrljanja i trenje valjanja. Trenje klizanja javlja se u slu ajevima kada translacijske komponente brzina v1 i v2 za to ke dvaju tijela ( 1 i 2 ) u kontaktu nisu jednake, tj. postoji neka relativna brzina. Pojava trenja klizanja prikazana je na slici 2.1.  EMBED Visio.Drawing.4  Slika 2.1. Trenje klizanja. Trenje kotrljanja javlja se u slu ajevima kada postoji kontakt izmeu dvaju tijela u jednoj to ci ili u liniji. Relativno gibanje sadr~i rotacijsko gibanje s polom P u to ki (liniji) kontakta u kojoj le~i pol rotacije. Za to vrijeme su translacijske komponente brzine u to ki dodira jednog i drugog tijela jednake. Pojava trenja kotrljanja prikazana je na slici 2.2.  EMBED Visio.Drawing.5  Slika 2.2. Trenje kotrljanja. Trenje valjanja predstavlja kombinacija trenja klizanja i trenja kotrljanja. Kinematika gibanja dvaju tijela koja su u meusobnom kontaktu odreuje koli inu zastupljenosti pojedine vrste trenja. Trenja valjanja naj eae tipi no susreemo u valjnim le~ajevima, meu zubima zup anika i sli no. Sila koja nastaje kao posljedica trenja proporcionalna je normalnoj komponenti sile koja tijela dr~i u kontaktu. Koeficijent proporcionalnosti te dvije sile nazivamo koeficijentom trenja i ozna avamo ga s  EMBED Equation.3 . On povezuje normalnu komponentu sile  EMBED Equation.3  i silu trenja  EMBED Equation.3  preko slijedeeg izraza:  EMBED Equation.2 . (2-1) Istra~ivanja u posljednjih pedesetak godina su pokazala da koeficijent trenja mo~e biti promjenjiva veli ina. Usprkos tome, uobi ajeno je da se koeficijent trenja u prora unima uzima kao konstantna vrijednost. U istim promatranim uvjetima (ista relativna brzina i ista normalna komponenta sile meu tijelima), trenje valjanja je manje od trenja klizanja. Osim toga ato je prisutna manja sila koja se opire gibanju, dinamika trenja valjanja lakae se da opisati nego dinamika trenja klizanja. 2.1.1.1. Priroda trenja kotrljanja Kod pojave kotrljanja na mjestu dodira dvaju tijela teoretski nema relativnog gibanja kao ato je prikazano na slici 2.2.. U realnom kontaktu, ato mo~emo vidjeti na slici 2.3. zbog deformacije na mjestu dodira dolazi do relativnog gibanja pri ulasku u kontakt odnosno izlasku iz kontakta. Na mjestu ulaska u kontakt vlada tla no naprezanje, dok na mjestu izlaska iz kontakta vlada vla no naprezanje. Tla no naprezanje potpoma~e stvaranje kohezivnih veza koje se u podru ju vla nog naprezanja razbijaju. Silu trenja se mo~e promatrati kao posljedicu nesimetri nog deformiranja podloge uslijed stvaranja i razbijanja veza. Kod trenja kotrljanja nema izra~enog stati kog trenja ato se mo~e objasniti polaganim ula~enjem i izla~enjem u i iz kontakta.  EMBED Visio.Drawing.4  Slika 2.3. Realni kontakt 2.1.2. Modeli trenja Istra~ivanja sile trenja su pokazala da ona ne ovisi samo o iznosu brzine gibanja, ve da ovisi i o nekim drugim parametrima poput polo~aja, brzine prirasta pogonske sile, vremena provedenog u mirovanju, promjeni brzine, kao i o "povijesti" gibanja (smjeru, iznosima ubrzanja itd.). 2.1.2.1. Stati ki modeli trenja Kako su spoznaje o trenju napredovale, tako su se razvijali i modeli trenja. etiri uobi ajena stati ka modela trenja prikazana su na slici 2.4.  EMBED Visio.Drawing.5 a) EMBED Visio.Drawing.5 b) EMBED Visio.Drawing.5 c) EMBED Visio.Drawing.5  d)Slika 2.4. Stati ke karakteristike trenja Na slici 2.4. a) prikazan je takozvani Coulombov model trenja nazvan po Coulombu koji ga je otkrio 1785 godine. U ovom modelu sila trenja Ftr se opire gibanju konstantnim iznosom FC, neovisnim o iznosu brzine i veli ini kontaktne povraine, ali uz predznak ovisan o smjeru. Ovakvo trenje se osim Coulombovo trenje joa naziva suho trenje ili kineti ko trenje. Matemati ki ga je mogue opisati funkcijom sgn:  EMBED Equation.3 , (2-2) Coulombova sila FC je proporcionalna normalnoj komponenti sile FN, a ovisnost o materijalu i stanju dodirne povraine izra~ena je koeficijentom trenja (:  EMBED Equation.2  (2-3) 1866 godine, Reynolds je proairio Coulombov model trenja modelom viskoznog trenja prikazanim na slici 2.4 b). Viskozno trenje proporcionalno je brzini (, a ra una se prema izrazu:  EMBED Equation.2  (2- 4) gdje je kv - koeficijent viskoznog trenja. Na slici 2.4 c) prikazan je prethodni model proairen stati kim trenjem Fs ( Morin, 1833 godine). Daljnjim proairenjem zahvaljujui Stribecku koji je uo io negativnu viskoznu karakteristiku trenja razvio se model koji je prikazan na slici 2.4 d). Negativna viskozna karakteristika o ituje se kao opadanje sile trenja s porastom brzine neposredno nakon ato nastupi gibanje. Sila trenja opada sve do iznosa Coulombovog trenja da bi potom po ela rasti po zakonu viskoznog trenja. 2.1.2.2. Dinami ki modeli trenja Dinami ki modeli trenja naj eae polaze od pretpostavke vlaknaste strukture kontaktne povraine guma-cesta, prikazane na Slici 2.5..  EMBED Visio.Drawing.4  Slika 2.5. Asperitna struktura kontaktne povraine Sila trenja koja se suprotstavlja gibanju, odreena je vrstoom asperitnih veza. Nakon ato iznos vanjske sile prijee iznos vrstoe asperitnih veza dolazi do njihova pucanja te nastupa gibanje. U toku gibanja, zbog relativne brzine meu asperitima, nema dovoljno vremena za ponovnu uspostavu veza, te stoga ponovno formiranje asperitnih veza mo~e nastupiti tek kada gibanje prestane. Za izgradnju asperitnih veza potrebno je neko vrijeme, a paralelno s izgradnjom veza raste i njihova vrstoa, te se to o ituje kao porast iznosa stati kog trenja ovisno o vremenu provedenom u mirovanju. Razvijeni su razli iti dinami ki modeli. U nastavku su opisani Reset-integrator i LuGre modeli. Reset-integrator model nastao je kao pojednostavljena varijanta  vlaknastog modela (engl. bristell model). Vlaknastim modelom je uzeta u obzir geometrija vlaknaste strukture asperitnog sloja. Sila trenja je posljedica velikog broja veza meu vlaknima kontaktnih povraina, te je njezin iznos zbroj naprezanja svake veze posebno. Ukoliko je naprezanje manje od vrstoe veze, dolazi samo do elasti ne deformacije vlakana. Kad iznos naprezanja pojedine veze prijee njenu vrstou, dolazi do pucanja stare i uspostavljanja nove veze. Nova veza se javlja meu vlaknima smjeatenima na pravcima koji su otklonjeni od pravca gibanja te ne omoguuju vrstou prethodne veze. Broj veza ovisi o relativnoj brzini meu kontaktnim povrainama. Kako je sila trenja izravno ovisna o broju veza, njezin iznos je u funkciji relativne brzine. Ovaj model dobro opisuje slu ajni karakter trenja, ali je neefikasan s obzirom na veliko vrijeme ra unanja. Da bi se skratilo vrijeme ra unanja, a zadr~ala dobra svojstva u opisu fenomena trenja, razvijen je model zasnovan na cikli kom integriranju (engl. reset integrator model) prikazan na Slici 2.6.  EMBED Visio.Drawing.5  Slika. 2.6. Simulacijski reset-integrator model. LuGre model se kao i vlaknasti model temelji na spoznaji o fizikalnim svojstvima vlakana kontaktne povraine. Za razliku od prethodnog modela, gdje se promjena sile trenja vezuje uz promjenu broja veza koje trenuta no egzistiraju, u ovom se modelu promjena sile trenja vezuje uz pregib asperitnih vlakana. Promjena pregiba vlakana z opisana je jednad~bom:  EMBED Equation.2 , (2-5) gdje je: z - pregib vlakana, vr - relativna brzina izmeu kontaktnih povraina, (0 - konstanta krutosti vlakana, g(vr) - funkcija koja opisuje povezanost promjene pregiba vlakana i relativne brzine v prema izrazu:  EMBED Equation.2 , (2-6) gdje je: (C  Coulombovo trenje, (S  stati ko trenje, vs  Stribeckova brzina. Ukupna sila trenja ra una se prema izrazu:  EMBED Equation.2 . (2-7) gdje je: (1  koeficijent priguaenja, (2  koeficijent viskoznog trenja, Simulacijski LuGre model prikazan je na slici 2.7.:  Slika. 2.7. Simulacijski LuGre model trenja. 2.2. Modeli trenja za kontakt guma-asfalt Prvo emo opisati fizikalna zbivanja do kojih dolazi prilikom kontakta automobilskog kota a i ceste. 2.2.1. Fizikalna zbivanja u kontaktu automobilskog kota a i ceste U automobilskoj industriji sigurnost putnika u automobilu je najva~nija stvar, te se pored ostalih sigurnosnih elemenata (krutost karoserije, zaatitne pregrade, zra ni jastuci, i sl.), mora osigurati najvea mogua efikasnost ko enja, odnosno ubrzavanja vozila. Na ponaaanje vozila u vo~nji najviae utje u sile koje se javljaju prilikom kontakta automobilskog kota a i ceste. Pogonski kota i vozila su tada u klizanju, te se kao posljedica klizanja javlja sila. Pri tome se klizanje s definira kao:  EMBED Equation.3 , ako je  EMBED Equation.3  i  EMBED Equation.3  (ko enje) (2-8)  EMBED Equation.3 , ako je  EMBED Equation.3  i  EMBED Equation.3  (vo~nja) (2-9) Klizanje s mo~e poprimiti vrijednosti u intervalu [0,1].  EMBED PBrush  Slika 2.8. Ovisnost klizanja i koeficijenta trenja. Glavni je sigurnosni problem kako u kriti nim situacijama postii naju inkovitije ko enje, odnosno ubrzavanje. U automobilima to osiguravaju dva elektroni ka sustava: ABS (engl. Antilock Breaking System) zadu~en za ko enje, te TCS (engl. Traction Control System) za ubrzavanje. ABS i TCS sustavi nastoje odr~ati takav iznos klizanja da sila izmeu kota a i ceste bude najvea mogua za trenutno stanje na cesti ato zna i maksimalni koeficijent trenja. Senzori moraju biti u stanju prepoznati kakvo je stanje ceste (suha cesta, mokra cesta, led, snijeg). Na slici 2.8. uo ljivo je da za razli ita stanja podloge razli it optimalan iznos klizanja. Prilikom vo~nje na svim kota ima ne mora biti jednaka sila (npr. pola vozila na suhoj podlozi, a druga polovica na ledu) te se mora osigurati mjerenje potrebnih veli ina i rad sustava takav da vozilo ostane na ~eljenoj putanji. Razmatra se pojednostavljeni model s koncentriranim parametrima koji opisuje kontakt automobilskog pogonskog kota a i ceste, predstavljen na slici 2.9. Ovaj model omoguava mjerenje sile izmeu kota a i ceste za svaki pogonski kota  posebno.  EMBED PBrush  Slika 2.9. Kontakt automobilskog kota a i ceste. Sustav na slici 2.9. mo~e se opisati jednad~bama:  EMBED Equation.3  , (2-10)  EMBED Equation.3  , (2-11) gdje je: m  masa kota a, J  moment inercije kota a, r  polumjer kota a, v  linijska brzina kota a, (  kutna brzina okretanja kota a, ur  moment na osovini, Ftr  sila trenja izmeu kota a i podloge. Modeli trenja koji opisuju zbivanja u kontaktu guma-asfalt naj eae su izvedeni iz openitih stati kih karakteristika trenja. ali ima i takvih modela koji su izvedeni na osnovi eksperimentalnih mjerenja. To su esto krivulje najmanjih kvadrata provu ene kroz skup izmjerenih podataka. 2.2.2. Stati ki modeli trenja za kontakt automobilskog kota a i ceste Pacejka predstavlja model, u literaturi esto nazivan  magi na formula . Model ima svojstvo da se dobro poklapa s eksperimentalno dobivenim podacima uz dobar odabir konstanti ci.:  EMBED Equation.3 . (2-12) Kiencke i Daiss opisuju model koji pored klizanja s uvodi dodatni parametar Ks, nagib stati ke karakteristike trenja u okolici to ke  EMBED Equation.3 .  EMBED Equation.3 . (2-13) Pseudo-stati ki modeli se esto koriste pri modeliranju ponaaanja guma. Pseudo-stati ki se nazivaju stoga ato mnogi modeli prikazuju ovisnosti i o brzini vozila v. Burckhardt predstavlja model s konstantama ci koje se odreuju tako da se dobije ato bolje poklapanje sa eksperimentalno izmjerenim podacima:  EMBED Equation.3 . (2-14) Budui da su svi ovi modeli izrazito nelinearni, i poato se pokazalo da parametri modela jako ovise o karakteristikama guma (sastav gume, profil, temperatura, tlak u gumi, i sl.), oni se ne koriste u on-line identifikaciji. Kao jedno od rjeaenja mo~e se koristiti pojednostavljene modele oblika npr.:  EMBED Equation.3 , (2-15) u kombinaciji sa linearnim rekurzivnim algoritmima identifikacije. 2.2.2. Dinami ki modeli trenja u kontaktu automobilskog kota a i ceste Razni dinami ki modeli trenja imaju osnovu u pretpostavci da je kontaktna povraina guma-asfalt vlaknaste strukture, kao ato je prikazano na Slici 2.10..  Slika 2.10. Model kontaktne povraine guma-asfalt Pretpostavka da je kontakt realiziran preko niza siuanih, bezmasenih i elasti nih vlakana koji se nalaze na kru~nom pojasu. Kontaktna povraina je kvadratnog oblika duljine L. Ukoliko je relativna brzina:  EMBED Equation.DSMT4  (2-16) razli ita od nule (postoji klizanje), vlakna se deformiraju i na taj na in tvore longitudinalnu silu F. Canudas de Wit i Tsiotras predstavljaju modifikaciju po etnog LuGre modela na na in da uvode parametar ( koji predstavlja stanje ceste (suhi asfalt, mokri asfalt, snijeg, led).  EMBED Equation.3 . (2-17) Prilikom izvoenja simulacija ustanovljeno da po etni LuGre model daje pogreaku u stacionarnom stanju, te je stoga bilo potrebno na initi korekciju modela. Deur predstavlja modifikacija danu u (2-18).  EMBED Equation.3  (2-18) gdje je ( empirijski dobivena konstanta, te L duljina kontaktne povraine automobilske gume i ceste. Jedna od dobrih strana ovog modela je u tome da ima vrlo sli no ponaaanje kao i modeli sa distribuiranim parametrima. 3. NEURONSKE MRE}E 3.1. Bioloaka osnova razvoja neuronskih mre~a Neuronske mre~e nastale su iz te~nji razvijanja matemati kih struktura koje bi bile u mogunosti oponaaati rad ljudskog mozga, te koriatenju tih struktura u rjeaavanju prakti nih problema. Razvijeno je viae razli itih vrsta neuronskih mre~a. U osnovi ih mo~emo podijeliti na stati ke i dinami ke neuronske mre~e. U ovom radu koristi se model stati ke neuronske RBF mre~e. Da bi razumjeli osnovnu strukturu umjetnih neuronskih mre~a potrebno je razmotriti osnovnu strukturu ljudskog mozga. Ljudski mozak se sastoji od oko 1011 neurona koji su meusobno povezani u slojeve, koji su opet meusobno povezani u slo~enu mre~u. Struktura mozga je preslo~ena, a mre~a neurona pregusta, te se zbog toga joa uvijek nije doalo do detaljnijih spoznaja o radu neuronske mre~e mozga. Ipak, bioloaki neuron, prikazan na slici 3.1. poslu~io je kao model za umjetni neuron. On se mo~e pojednostavljeno prikazati kao stanica sastavljena od tijela (soma), mnoatva dendrita i aksona.  Slika 3.1. Shematski prikaz bioloakog neurona Akson se mo~e predstaviti kao tanka cjev ica iji je jedan kraj povezan s tijelom neurona, a drugi razdijeljen na mnoatvo grana. Krajevi ovih grana zavraavaju malim zadebljanjima koja u pravilu dodiruju dendtrite, a rjee dodiruju tijelo drugog neurona. Sinapsom nazivamo mali razmak izmeu zavraetka aksona prethodnog neurona i dendrita sljedeeg neurona. Funkcija aksona je da formira sinapti ke veze s drugim neuronima. Tijelo neurona generira impulse koji putuju kroz akson do sinapsi. Ti signali e doi do dendrita ovisno o sinapti kom prijenosu koji ovisi o viae faktora, izmeu ostalih i o proalim sinapti kim prijenosima. Na odreen na in, sinapse predstavljaju memorijske lanove bioloake neuronske mre~e. Kada signali stignu do tijela drugog neurona, oni za njega mogu biti pobuujui ili smirujui. Ukoliko je zbroj signala pobuujui, tijelo neurona e generirati impulse prema drugima neuronima. Iz opisanog, mo~e se zaklju iti da se rad bioloakog neurona odvija kroz dvije operacije: sinapti ka operacija ( pridodavanje va~nosti ulaznim signalima u neuron ) somatska operacija ( zbrajanje ulaznih signala i generiranje impulsa ovisno o rezultatu ) Ovaj prikaz bioloakog neurona samo na elno prikazuje njegov rad i postavlja okvir u kojem su se razvili prvi modeli umjetnih neurona. 3.2. Modeli umjetnih neurona Umjetne modele neurona mogue je svrstati u dvije osnovne skupine: stati ke i dinami ke modele neurona. 3.2.1 Stati ki modeli neurona McCulloch i Pitts razvili su prvi model neurona. To je bio jednostavan model koji je obraivao signale upravo pomou prethodno navedenih dviju operacija, somatske i sinapti ke. Taj jednostavni model neurona zove se perceptron.. Mno~enje svakog ulaznog signala xi s te~inskim koeficijentom wi predstavlja sinapti ku operaciju. Zbrajanje tako ote~anih signala i usporedba zbroja s pragom osjetljivosti neurona wn+1 predstavlja somatsku operaciju. Kod perceptrona, u slu aju kada je zbroj vei od praga osjetljivosti aktivacijska funkcija ( generira izlazni signal y iznosa 1, a ako je zbroj manji, generira se izlazni signal iznosa 0. Rosenblat (1962) je dokazao zna ajan teorem o u enju perceptrona: perceptron mo~e nau iti sve ato mo~e predstaviti. Pri tome je predstavljanje sposobnost aproksimiranja odreene funkcije, a u enje postupak koji sustavnim podeaavanjem parametara mre~e posti~e to da ona postane zadovoljavajua aproksimacija odreene funkcije. Na slici 3.2. mo~emo vidjet shematski prikaz perceptrona:  Slika 3.2. Shematski prikaz perceptrona Perceptron se mo~e matemati ki opisati sljedeima izrazima:  EMBED Equation.DSMT4  (3-1)  EMBED Equation.DSMT4  (3-2) gdje je:  EMBED Equation.DSMT4  - vektor ulaznih signala neurona, pobudni vektor;  EMBED Equation.DSMT4  - vektor sinapti kih te~inskih koeficijenata;  EMBED Equation.DSMT4  - prag osjetljivosti neurona;  EMBED Equation.DSMT4  - izlaz operacije konfluencije  mjera sli nosti; ulaznih signala sa sinapti kim koeficijentima;  EMBED Equation.DSMT4  - aktivacijska funkcija;  EMBED Equation.DSMT4  - izlaz neurona. Ako se vektor ulaza proairi lanom xn+1=1, mo~emo izraz (3-1) napisati na sljedei na in:  EMBED Equation.DSMT4 , (3-3) gdje su:  EMBED Equation.DSMT4  - proaireni vektor ulaznih signala neurona;  EMBED Equation.DSMT4  - vektor sinapti kih te~inskih koeficijenata proairen pragom osjetljivosti neurona. Gledano s matemati kog stajaliata, umjetni neuron se dijeli na dvije operacije: operaciju konfluencije (3-3) i aktivacijsku funkciju (3-2). Gledano s bioloakog stajaliata, operacija konfluencije predstavlja pridjeljivanje te~ine ulaznim signalima x(t) ovisno o akumuliranom znanju u sinapsama w(t). S matemati kog stajaliata, operacija konfluencije predstavlja skalarni umno~ak vektora x(t) i w(t) pa se mo~e promatrati kao linearno, te~insko preslikavanje iz (n+1)-dimenzionalnog prostora neurona  EMBED Equation.DSMT4  u jednodimenzionalni prostor  EMBED Equation.DSMT4 . Izlaz operacije konfluencije predstavlja mjeru sli nosti izmeu proairenog ulaznog vektora x(t) i vektora te~inskih koeficijenata w(t). Veina neuronskih mre~a ima u sebi skalarni umno~ak kao operaciju konfluencije, ali ne i RBF neuronske mre~e kod kojih se umjesto skalarnog umnoaka primjenjuje Euklidska udaljenost izmeu vektora x(t) i w(t). Aktivacijska funkcija ((() preslikava izlaznu vrijednost operacije konfluencije v(t) u izlazni signal neurona y(t) ograni enunutar [0,1] za unipolarne i [-1,1] za bipolarne signale. Stati ki neuron ne sadr~i dinami ke lanove pa stoga njegov izlaz ovisi samo o trenuta nim vrijednostima ulaznih signala i te~inskim koeficijentima. Navedeno svojstvo stati kog neurona ini stati ke neuronske mre~e strukturno stabilnima. 3.2.2. Dinami ki model neurona Za razliku od izlaza stati kog modela neurona, izlaz dinami kog modela neurona ovisi, osim o trenuta nim vrijednostima ulaza i te~inskim koeficijentima sinapti kih veza, i o proalim stanjima sinapti kih veza. Kod ovog modela neurona prostiranje signala odvija se unaprijed, ali i unatrag preko unutarnjih povratnih veza. Poopeni dinami ki model neurona, prikazan je slici 3.3., sastoji se od operacije konfluencijske, diskretnog dinami kog lana drugog reda, nelinearne aktivacijske funkcije promjenjiva nagiba i povratnog signala s izlaza neurona.  Slika 3.3. Poopeni dinami ki model neurona Matemati ki opis poopenog modela dinami kog neurona dan je sljedeim izrazima:  EMBED Equation.DSMT4 , (3-4)  EMBED Equation.DSMT4  , (3-5)  EMBED Equation.DSMT4  , (3-6) gdje je k oznaka diskretnog vremena. Ako nekim parametrima poopenog dinami kog modela neurona pridru~imo nepromjenjive vrijednosti, dobijemo modele neurona esto koriatenih neuronskih mre~a: MLP mre~e: a0=1,a1=a2=b1=b2=c1=0,ga=1; povratne neuronske mre~a: a0=1,a1=a2=b1=b2=0,ga=1; neuronske mre~e s vremenskim kaanjenjem: b1=b2=c1=0,ga=1; dinami ke neuronske mre~e: c1=0. 3.3. Modeli i osnovna svojstva neuronskih mre~a Neuronske mre~e mogu se klasificirati na viae na ina. Gledano sa strukturnog stajaliata, ovisno o modelu od kojeg su graene, neuronske mre~e mo~emo podijeliti na stati ke i dinami ke. S obzirom na slojevit na in organizacije neurona u mre~i, neuronske mre~e se dijele na jednoslojne i viaeslojne. Svaki sloj u pravilu prima ulaze iz prethodnog sloja, a svoje izlaze aalje sljedeem sloju. Nulti sloj nazivamo ulaznim slojem, posljednji sloj izlaznim. Svi ostali slojevi zovu se skriveni slojevi. 3.3.1. Stati ke neuronske mre~e Stati ke neuronske mre~e naj eae koristimo u primjenama poput identifikacije procesa, upravljanja procesima, obradbe signala i prepoznavanja oblika. Osnovna karakteristika stati kih neuronskih mre~a je ta ato su neuroni organizirani na unaprijedan na in. To zna i da su neuroni povezani tako sa ne formiraju povratne veze. Stoga stati ke neuronske mre~e ne sadr~avaju dinami ke lanove ato rezultira jako bitnim svojstvom strukturne stabilnosti. Poato nema dinami kih lanova, odziv stati ke neuronske mre~e ovisi isklju ivo o trenuta nom stanju ulaza i vrijednostima parametara mre~e. Ako ~elimo postii dinami ko ponaaanje, stati kim neuronskim mre~ama mo~emo dodati vanjske dinami ke lanove. Dva naj eaa tipa stati kih neuronskim mre~a su MLP i RBF neuronske mre~e. U ovom radu koriste se RFB neuronske mre~e te e one biti pobli~e opisane u nastavku. 3.3.2. Dinami ke neuronske mre~e Razlog zbog kojeg se dinami ke neuronske mre~e koriste znatno rjee od stati kih neuronskih mre~a je to ato njihova stabilnost nije zagarantirana. U enje dinami kih neuronskih mre~a znatno je zahtjevnije od u enje stati kih. Izlaz dinami ke neuronske mre~e ovisi o njezinim po etnim stanjima ato ote~ava u enje jer loae odabrani po etni uvjeti mogu dati loae rezultate. Kada primjena stati kih neuronskih mre~a ne daje dobre rezultate opravdano je koristiti dinami ke neuronske mre~e kao npr. kod modeliranja procesa s velikim mrtvim vremenom. U tom slu aju potrebno je da neuronska mre~a sa uva informacije na neodreeni vremenski period, a stati ka mre~a s vanjskim dinami kim lanovima to nemo~e je ima kona ni impulsni odziv (engl. Finite Impulse response, FIR). Naj eae koriatene dinami ke neuronske mre~e su Hopfieldove, Elmanove i NARX dinami ke neuronske mre~e. 3.3.3. RBF Neuronska mre~a RBF neuronska mre~a, prikazana slici 3.4., je dvoslojna stati ka neuronska mre~a. Funkcija ulaznog (nultog) sloja je da prosljeuje ulaze u mre~u na ulaz prvog sloja koji je sa injen od neurona s aktivacijskim funkcijama s kru~nom osnovicom. Po tome su RBF neuronske mre~e i dobile ime (Radial Basis Functions  funkcije s kru~nom osnovicom).  Slika 3.4. RBF Neuronska mre~a Drugi (izlazni) sloj mre~e, koji je ujedno i njezin izlazni sloj, sastoji se od perceptrona s linearnom aktivacijskom funkcijom jedini nog poja anja. Naj eae koriatene aktivacijske funkcije u prvom sloju su: EMBED Equation.3, (thin-plate-spline funkcija) EMBED Equation.3, (Gaussova funkcija) EMBED Equation.3, (viaekvadrati na funkcija) EMBED Equation.3, (inverznaviaekvadrati na funkcija) U ovom radu koriste se aktivacijske funkcije s Gaussovom funkcijom, prikazanom na slici 3.5.. Ovakve RBF neuronske mre~e joa se nazivaju Gaussove RBF neuronske mre~e.  Tablica 3.1. Gaussova funkcija. Matemati ki, RBF mre~a se mo~e opisati na slijedei na in:  EMBED Equation.DSMT4  ; (3-7)  EMBED Equation.DSMT4  (3-8)  EMBED Equation.DSMT4 , (3-9)  EMBED Equation.DSMT4 , (3-10)  EMBED Equation.DSMT4 , (3-11)  EMBED Equation.DSMT4 , (3-12) gdje je:  EMBED Equation.DSMT4  - vektor ulaza u mre~u, dimenzije n(x);  EMBED Equation.DSMT4  - vektor ulaza u prvi sloj mre~e;  EMBED Equation.DSMT4  - vektor ulaza u drugi sloj mre~e;  EMBED Equation.DSMT4  - vektor izlaza operacije konfluencije l-tog sloja mre~e;  EMBED Equation.DSMT4  - vektor izlaza l-tog sloja mre~e, l=0,1,2;  EMBED Equation.DSMT4  - matrica koordinata srediata RBF neurona, dimenzije n(1)n(0) ;  EMBED Equation.DSMT4  - matrica te~inskih koeficijenata sinapti kih veza drugog sloja mre~e, dimenzije n(2)n(n(1)+1);  EMBED Equation.DSMT4 - vektor aktivacijskih funkcija l-tog sloja mre~e. RBF neuronska mre~a ima sposobnost aproksimirati proizvoljnu nelinearnu kontinuiranu funkciju. Tri parametra RBF neuronskih mre~a odreuju njenu aproksimacijsku sposobnost. To su: polo~aj srediata neurona, varijance aktivacijskih funkcija neurona, te~inski koeficijenti izlaznog sloja mre~e. Pomou razli itih algoritama u enja ovi parametri se podeaavaju da bi se dobilo odgovarajue ponaaanje mre~e. RBF mre~e iznimno su u inkovite u slu ajevima kada je mogue unaprijed rasporediti srediata neurona i odrediti iznose varijanci RBF neurona. Time se u enje mre~e svodi na podeaavanje te~inskih koeficijenata izlaznog sloja. Tada vladanje RBF neuronske mre~e postaje linearno ovisno o parametrima, ato je velika prednost. Meutim, za dobiti kvalitetne rezultate na ovaj na in potreban je jako veliki broj neurona. Da bi se to izbjeglo, postupak u enja mre~e proairuje se i na podeaavanje srediata i varijanci neurona RBF mre~e. To proairenje nam omoguava znatno smanjenje broja RBF neurona. U ovom slu aju ponaaanje RBF mre~e postaje nelinearno ovisno o parametrima. 3.3.4. Svojstva neuronskih mre~a Bez obzira na tip neuronske mre~e, sve imaju neka zajedni ka svojstva koja ih razlikuju od konvencionalnih ra unskih tehnika: Paralelna raspodijeljena obradba informacija Sposobnost istodobnog prihvaanja viae ulaznih informacija paralelno i distribuirana spremanje tih informacija. Takav na in spremanja omoguava viae prednosti od kojih je najva~nije svojstvo redundantnosti. U enje i adaptacija Ovo svojstvo omoguava mre~i da obrauje neprecizne i loae o uvane podatke. Mre~a ima odreenu sposobnost poopavanja kada joj se na ulazu pojave podaci koji nisu bili u uzorku na temelju kojih je mre~a nau ena. Univerzalni aproksimator Ovo je najva~nije svojstvo neuronskih mre~a. Omoguava aproksimaciju proizvoljne kontinuirane funkcije do ~eljene to nosti. Viaevarijabilni sustavi Viaevarijabilnost je strukturno svojstvo neuronskih mre~a. Sklopovska implementacija Razvijena su razli ita rjeaenja koja omoguuju sklopovsku implementaciju neuronskih mre~a. 4. HIBRIDNI ALGORITAM U ENJA PARAMETARA RBF NEURONSKE MRE}E 4.1. U enje neuronskih mre~a Razvijen je veliki broj raznih algoritama u enja neuronskih mre~a pomou kojih se podeaavaju iznosi sinapti kih te~inskih koeficijenata. Po na inu u enja mogue ih je podijeliti na algoritme u enja temeljene na pogreaci (engl. error based algorithms), algoritme u enja temeljene na izlazu mre~e (engl. output-based learning algorithms) i algoritme u enja s oja anjem (engl. reinforcment learning algorithm). algoritmi u enja temeljeni na pogreaci; Drugo ime im je algoritmi s u iteljem jer im je potreban vanjski referentni signal (u itelj) s kojim usporeuju dobiveni odziv neuronske mre~e generirajui signal pogreake. Na temelju signala pogreake algoritam podeaava sinapti ke koeficijente. Ovi sealgoritmi mogu primijeniti samo ako je unaprijed poznato ~eljeno vladanje neuronske mre~e, odnosno podaci na temelju kojih se mre~a u i moraju sadr~avati parove vrijednosti ulazno-izlaznih signala. algoritme u enja temeljene na izlazu mre~e; Nazivaju se joa i algoritmi bez u itelja, jer ne zahtijevaju vanjski referentni signal. Podaci na temelju kojih se mre~a u i sadr~e samo vrijednosti ulaznih signala u mre~u. algoritme u enja s oja anjem; Ovaj tipa algoritma se primjenjuje kada nije dostupan signal koji definira ~eljeno vladanje neuronske mre~e, ali nam je dostupan signal koji predstavlja kriti ku ocjenu njezinog vladanja u smislu dobro/loae. U identifikaciji, modeliranju i upravljanju nelinearnim dinami kim procesima naj eae je poznato ~eljeno vladanje neuronske mre~e pa se za njezino u enje primjenjuju algoritmi temeljeni na pogreaci. Kao mjera pogreake koristi se neka kriterijska funkcija ili kriterij kakvoe (((). Osnovni problem koji treba rijeaiti kod primjene postupka nelinearnog optimiranja za u enje neuronskih mre~a jest izra unavanje gradijenta kriterijske funkcije po parametrima mre~e. Problem je uspjeano rijeaen primjenom algoritma povratnog prostiranja izlazne pogreake kroz mre~u. Dakle, cilj algoritama u enja je pronai minimum kriterijske funkcije. Postoje dva osnovna na ina minimiziranja funkcije ((() na kojima se temelje algoritmi u enja neuronskih mre~a: rekurzivni i nerekurzivni. Prema nerekurzivnom na inu, funkcija ((() se minimizira na na ina da se promjene parametara mre~e akumuliraju preko svih N vektora mjernih podataka i tek nakon toga se parametri mre~e stvarno promijene. Algoritmi u enja utemeljeni na ovom pristupu zovu se nerekurzivni algoritmi u enja (eng. Batch, off-line, block data learning algorithms). Prema rekurzivnom na inu , funkcija ((() se minimizira na temelju lokalne kriterijske funkcije. To zna i da se parametri mijenjaju nakon svakog vektora mjernih podataka. Algoritmi temeljeni na ovom pristupu nazivaju se rekurzivnim algoritmima u enja (engl. recursive, on-line, adaptive or data learning algorithms). Prednost rekurzivnih algoritama u enja dolazi do izra~aja kada svi mjerni podaci nisu poznati prije po etka u enja, ve se prikupljaju tijekom u enja; u tom se slu aju nerekurzivni algoritmi ne mogu primijeniti. U ovom radu koristiti se rekurzivni hibridni algoritam u enja RBF neuronskih mre~a. Parametri RBF neuronske mre~e koji se mogu podeaavati su: a) polo~aj srediata neurona ulaznog sloja mre~e, b) varijance neurona ulaznog sloja mre~e i c) te~inski koeficijenti sinapti kih veza izlaznog sloja mre~e. Svi se parametri RBF mre~e mogu podeaavati jedinstvenim algoritmom u enja. Meutim, zbog razli itog utjecaja na vladanje mre~e svake od navedene tri grupe parametara, bolji se rezultati mogu o ekivati ako se za svaku grupu parametara primijeni zaseban algoritam u enja, prilagoen na inu djelovanja tih parametara na vladanje mre~e. Sva tri algoritma skupa ine hibridni algoritam. U sljedeim potpoglavljima detaljno se opisuju pojedini algoritmi od kojih se sastoji hibridni postupak u enja neuronske mre~e. 4.2. Algoritam rasporeivanja srediata neurona ulaznog sloja mre~e Srediata neurona ulaznog sloja mre~e potrebno je rasporediti po o ekivanom prostoru u kojem se kreu ulazi u mre~u. Postoji viae na ina rasporeivanja srediata neurona. Najjednostavniji na ini su linearno rasporeivanje i slu ajno rasporeivanje. I uz ovakve jednostavne na ine rasporeivanja mogue je aproksimirati bilo koju nelinearnu funkciju, ali je pri tome esto potreban jako veliki broj neurona. Stoga se koriste algoritmi koji optimalno rasporeuju srediata neurona tako da su guae postavljena u podru ju vrijednosti koje ulaz u mre~u poprima eae. Na taj na in mogue je postii jednako dobru aproksimaciju nelinearne funkcije s mnogo manjim brojem neurona. U ovom radu koristi se algoritam K-srednjih vrijednosti koji dijeli ulazni prostor mre~e u K cjelina, te svaku cjelinu aproksimira jednim srediatem neurona tako da ukupno kvadratno odstupanje bude minimalno. Pseudo-kod rekurzivne verzije algoritma k-srednjih-vrijednosti glasi: Izra unaju se udaljenosti srediata svih RBF neurona od vektora ulaza i odreuje se njemu najbli~i neuron: dj(k)=dist(x(k)-sj(k-1)), j=1:n1 k=arg(min(dj(k))), j=1:n1 gdje su: x - vektor ulaza u mre~u, dimenzije nx, s - vektor polo~aja srediata ulaznih neurona, dimenzije n1, k - oznaka koraka iteracije (diskretno vrijeme), n1 - broj neurona u prvom sloju mre~e, nx  broj ulaza u neuronsku mre~u. Srediate neurona najbli~eg ulaznom vektoru dodatno mu se pribli~i za: sk(k)=sk(k-1)+ (c (k) " (x(k)-sk(k-1)). (4-1) Koeficijent u enja (c lagano pada prema nuli prema izrazu:  EMBED Equation.DSMT4  , (4-2) gdje je int(.) funkcija "cijeli broj" (integer). Po etni raspored srediata neurona odabire se prema nekom jednostavnom pravilu, npr. ravnomjerni ili slu ajni raspored. U ovom radu odabran je slu ajni raspored u o ekivanom prostiru vrijednosti. Algoritam rasporeivanja srediata neurona ulaznog sloja mre~e nazivamo joa grupiranje. 4.3. Algoritam podeaavanja varijance neurona ulaznog sloja mre~e Zadaa ovog algoritma je odreivanje optimalnih vrijednosti varijanci ve rasporeenih neurona u cilju postizanja bolje pokrivenosti ulaznog prostora mre~e. Varijanca neurona je parametar koji najmanje utje e na ponaaanje mre~e, ali se ipak njihovim podeaavanjem mo~e postii br~a konvergencija procesa u enja mre~e. U ovom radu koristi se metoda P  najbli~ih susjeda uz odabir P=2. To je ujedno i najuobi ajenija metoda. Varijanca j-tog neurona izra unava se prema izrazu:  EMBED Equation.2 , (4-3) gdje su: Pj - broj najbli~ih susjednih neurona koji se promatraju, obi no isti za sve neurone u sloju, jp - indeksi prvih P susjeda j-tog neurona Varijance neurona odreene na ovaj na in su uglavnom daleko od optimalnih. Stoga se primjenjuje odreena modifikacija ovog algoritma koja donosi poboljaanje mijenjanjem broja promatranih susjednih neurona na sljedei na in. Definira se srednji kut neurona kojem se podeaava varijanca j-tog neurona:  EMBED Equation.2 , (4-4) gdje su: P( - broj promatranih susjeda (obi no P( =2), j1 - indeks prvih P( susjeda j-tog neurona, sj - koordinata srediata j-tog neurona. Zatim se varijanca neurona (faktor oblika) odreuje prema standardnom pravilu P-najbli~ih susjeda, uz linearnu vezu izmeu srednjeg kuta neurona ((j) i broja razmatranih susjeda Pj. Broj razmatranih susjeda odreuje se prema pravilu:  EMBED Equation.DSMT4 , (4-5) gdje je: Pmax - najvei broj susjeda koji se mo~e razmatrati, npr. Pmax=5, Pmin - najmanji broj susjeda koji se mo~e razmatrati, npr. Pmin=1, Nint - funkcija "najbli~i cijeli broj" Na kraju se prema izrazu (4-3) odreuju varijance svakog neurona. Najjednostavniji na ini odabira po etnih vrijednosti varijanci su ili da sve varijance imaju istu vrijednost ili da su odabrane kao slu ajni brojevi iz odreenog opsega. Neato bolji rezultati se mogu postii ako se po etne varijance odaberu off-line interpolacijom, na osnovi polo~aja srediata neurona. Algoritam podeaavanja varijanci neurona ulaznog sloja mre~e nazivamo joa interpolacijom. 4.4. Algoritam podeaavanja koeficijenata sinapti kih te~ina izlaznog sloja mre~e U ovom radu je primijenjen standardni RLS algoritam. On podeaavan koeficijente sinapti kih te~ina izlaznog sloja mre~e tako da ukupno kvadratno odstupanje izmeu izlaza procesa i mre~e bude minimalno. Prednost RLS algoritma je u njegovoj brzoj konvergenciji. Vektor izlaza iz prvog sloja mre~e definiramo na sljedei na in:  EMBED Equation.DSMT4 . (4-6) Zatim definiramo matricu te~inskih vrijednosti sinapti kih veza:  EMBED Equation.DSMT4  (4-7) Matrica kovarijanci P(t) definira se kao pravokutna matrica dimenzija n1n1. Vrijednosti te~inske matrice ra unaju se sljedeim rekurzivnim postupkom:  EMBED Equation.DSMT4  (4-8)  EMBED Equation.DSMT4  (4-9)  EMBED Equation.DSMT4  (4-10)  EMBED Equation.DSMT4  (4-11) U prethodnim izrazima ( je faktor zaboravljanja, a (0 njegova kona na vrijednost. Poato ovakva izvedba RLS algoritma ima tendenciju divergiranja uslijed numeri kih pogreaaka, u prakti noj se izvedbi simetri na matrica P rastavlja na diagonalnu i donju trokutastu matricu te se algoritam piae za takvu konfiguraciju matrica. Taj postupak se zove UDUT dekompozicija. Algoritam podeaavanja koeficijenata sinapti kih te~ina izlaznog sloja mre~e nazivamo joa adaptacijom. 4.5. Usklaivanje meusobnog rada algoritama u enja Sva tri algoritma rade odvojeno od drugih ali koriste parametre koje drugi algoritmi izra unavaju u istoj iteraciji. Promatrajui rad algoritama s matemati kog i logi kog stajaliata dolazimo do zaklju ka da se mora poativati pravilan slijed izvoenja algoritama. Postoje tri ispravna na ina: a) Obavlja se samo algoritam adaptacije. Algoritmi rasporeivanja i interpolacije ne izvraavaju se u promatranoj iteraciji hibridnog algoritma. b) Obavlja se algoritam interpolacije, a nakon njega algoritam adaptacije. Algoritam rasporeivanja ne izvraava se u promatranoj iteraciji hibridnog algoritma. Obavlja se algoritam rasporeivanja, nakon njega algoritam interpolacije, a zatim algoritam adaptacije. Nepoativanje ovakvog slijeda izvoenja algoritma dovodi do loaeg procesa u enja. Na slici 4.1. prikazana je ispravna hijerarhija unutar hibridnog algoritma u enja:  EMBED MSDraw  Sl. 4.1. Vremenska hijerarhija algoritama u enja mre~e Najbolji rezultati se posti~u za slu aj kada se samo obavlja algoritam adaptacije, zatim slu aj kada se obavlja algoritam interpolacija pa adaptacije, i najmanje dobri rezultati se posti~u kada se izvode sva tri algoritma. Algoritam grupacije je najgrublji pa njegovo esto izvoenje ima loae posljedice na proces u enja. Stoga se algoritam grupacije izvodi ili s razli itim vremenom uzorkovanja ili se primjenjuju neki drugu uvjeti kojima se posti~u razli ite frekvencije izvoenja algoritama. 5. PROJEKTIRANJE MODELA TRENJA RBF NEURONSKIM MRE}AMA U ovom poglavlju opisuje se na in na koji je projektiran model trenja u kontaktnoj povraini automobilskog kota a i ceste RBF neuronskom mre~om. Model trenja koji se modelira je modificirani LuGre model. On se razlikuje od standardnog modela trenja po tome ato uvodi parametar ( - koeficijent stanja ceste. Matemati ki opis ovog modela dan je jednad~bama (2-16), (2-17) i (2-18). Simulacija je provedena u programskom paketu MATLAB 6.0. Ulazni signal u model trenja je moment na osovini ur. Predstavljen je Band Limited White Noise blokom u Matlabu s s parametrima: Snaga auma : 0.1 Vrijeme uzorkovanja 0.2s Uzorak: [4771] Signal je na izlazu iz bloka poja an za 5. Signal momenta na osovini ur koji je koriaten u svim simulacijama prikazan je na slici 5.1.:  EMBED Word.Picture.8  Slika 5.1. Pobudni signal ur. Signal ur poprima vrijednosti iz intervala [-10, 10]. Negativan predznak predstavlja usporavanje, a pozitivan ubrzavanje vozila. Izlazni signali modela trenja su kutna brzina ( i brzina vozila v. Parametar ( se mo~e dr~ati konstantnom ili promjenjivom vrijednoau kako bi predstavljala konstantne ili promjenjive vremenske uvjete Simulacijski model pomou kojeg se obavio proces u enja prikazan je na slici 5.2.. Model trenja (engl., One wheel contact model) se sastoji od nekoliko podsustava, dok se RBF mre~a nalazi u podsustavu zbog preglednije prikazivanja vrijednosti parametara.  Slika 5.2. Simulacijski model Po etne vrijednosti parametara RBF neuronske mre~e odabrane su na slijedei na in: srediata neurona (vektor S) kao slu ajni brojevi u opsegu [-50,50], sinapti ke te~ine (vektor W) kao slu ajni brojevi u opsegu [-20,20], varijance (vektor () kao slu ajni brojevi u opsegu [0,20], broj ulaza u mre~u i broj neurona odreuje se eksperimentalno, faktor zaboravljanja (=0.95, (0=0.995, faktor u enja (c=0.95 (u enje vrai tako da je u po etnom trenutku (c relativno velik a zatim lagano pada, vektor diag ima sve elemente postavljene na 1000, vektor offdiag ima sve elemente postavljene na 0. RBF neuronska mre~a je stati ka neuronska mre~a. Da bi se mogla primijeniti za modeliranje nelinearnih dinami kih procesa potrebno joj je dodati vanjske dinami ke lanove. Na slici 5.3. prikazan je postupak u enja na taj na in proairene neuronske mre~e:  EMBED Word.Document.8  Sl.5.3. Shematski prikaz postupka identifikacije nepoznatog dinami koga procesa pomou RBF neuronske mre~e Veli ina regresijskog vektora odreena je brojem proalih stanja ulaza i izlaza odgovarajueg NARX modela. Odabran je broj ulaza u neuronsku mre~u nx=4. Regresijski vektor sa injavaju lanovi: u(k-1), u(k-2), yp(k-1), yp(k-2). Broj neurona odreuje se eksperimentalno. Simulacijom su usporeivane karakteristike srednjeg kvadratnog odstupanja neuronskih mre~a s 5, 10, 15, 20 i 40 neurona u ulaznom sloju. Poato se prije po etka svake simulacije neuroni rasporeuju na slu ajan na in, a brzine konvergencije u enja mre~e u prvim koracima ovisi o po etnom rasporedu neurona, srednje kvadratno odstupanje nije uvijek najbolji pokazatelj. Ipak ono dovoljno dobro ukazuje na to da mre~a s 5 neurona najloaije u i proces, te da poveanje na broj od 40 neurona ne poboljaava ponaaanje mre~e, a znatno usporava proces u enja. Na slijedeim slikama prikazana su srednja kvadratna odstupanja za n1=10,15,20 neurona u ulaznom sloju. Rezultati pokazuju da je odstupanje najmanje za sli aj u enja mre~om s 20 neurona u ulaznom sloju. U enje za slu aj n1=10 neurona u ulaznom sloju odvija se najbr~e, ali i najloaije. U nastavku e biti prikazani rezultati u enja gdje e se dodatno moi usporediti ponaaanje mre~e s razli itim brojem neurona.  EMBED Word.Picture.8  Slika 5.4. Srednje kvadratno odstupanje za razli it broj neurona  EMBED Word.Picture.8  Slika 5.5. Srednje kvadratno odstupanje za razli it broj neurona (uveano) 5.1. Rezultati simulacija Simulacije su se vraile u Simulinku (dio programskog paketa Matlab). Hibridni rekurzivni algoritam u enja izveden je u jednom funkcijskom bloku. Takoer, dodano je i kaanjenje na izlazu iz RBF neuronske mre~e da bi se poniatio Matlabov slijed izvoenja blokova koji je uzrokovao pogreano ra unanje pogreake (mre~a je kasnila jedan korak u praenju procesa). Proces je u en sa n1=10 i n1= 20 neurona u ulaznom sloju. U oba slu aja broj ulaza u RBF neuronsku mre~u bio je nx=4. Oblik ulaznog signala momenta na osovini ur prikazan je na slici 5.1., a ulazni signal koeficijenta ceste ( postavljen je na konstantnu vrijednost 2. 5.1.1. Modeliranje signala linearne brzine Prvo su vraene simulacije koristei mre~u s 10 neurona u ulaznom sloju. Na slici 5.6. prikazan je izlazni signal linearne brzine vozila, a nadalje na slikama 5.7. i 5.8., prikazani su zajedno izlazni signali iz mre~e i procesa.  EMBED Word.Picture.8  Slika 5.6. Signal linearne brzine v  EMBED Word.Picture.8  Slika 5.7. Signal linearne brzine v i izlaza mre~e  EMBED Word.Picture.8  Slika 5.8. Signal linearne brzine v i izlaza mre~e (uveano)  EMBED Word.Picture.8  Slika 5.9. Razlika signala linearne brzine v i izlaza mre~e  EMBED Word.Picture.8  Slika 5.10. Srednje kvadratno odstupanje  EMBED Word.Picture.8  Slika 5.11. Varijance neurona  EMBED Word.Picture.8  Slika 5.12. Sinapti ke te~ine neurona Ulazni prostor je dobro pokriven neuronima (40 srediata), od kojih se samo neki mijenjaju algoritmom. Podeaavanje varijanci se obavlja u prvih nekoliko koraka. Algoritam je aktivan i kasnije ali poato konvergira jako brzo i u enje traje mali broj koraka (slika 5.11.). Na slici 5.12. prikazane su sinapti ke te~ine svih 10 neurona ulaznog sloja. Njihovo podeaavanje algoritmom adaptacije se obavlaj tokom itavog u enja. I u slu aju kada one poprime konstantne vrijednosti, izllaz procesa mo~e poprimiti vrijednosti izvan do tada nau enog opsega te se sinapti ke vrijednosti moraju opet podeaavati. Kriterij koji je odreivao da li e se pojedini algoritam izvraavati je apsolutna vrijednost pogreake. Algoritam grupacije izvraavao se u slu aju da je apsolutna vrijednost pogreake vea od 0.1, interpolacija ako je apsolutna vrijednost pogreake vea od 0.01, i najfiniji, algoritam podeaavanja sinapti kih te~ina se izvodio ako je apsolutna vrijednost pogreake vea od 0.001. Algoritam adaptacije izvodio se preko 1500 puta tokom simulacije. Simulacija je ponovljena s RBF neuronskom mre~om od 20 neurona u ulaznom sloju. Rezultati su prikazani na slikama 2.13.-2.17., uz napomenu da je srednje kvadratno odstupanje prikazano bez prvih 5% podataka kada je odstupanje najvee kako bi se mogla dobiti kvalitetna slika.  EMBED Word.Picture.8  Slika 5.13. Signal linearne brzine v i izlaza mre~e(20 neurona)  EMBED Word.Picture.8  Slika 5.14. Signal linearne brzine v i izlaza mre~e (uveano 20 neurona)  EMBED Word.Picture.8  Slika 5.15. Srednje kvadratno odstupanje(20 neurona)  EMBED Word.Picture.8  Slika 5.16. Razlika signala linearne brzine v i izlaza mre~e s 20 neurona  EMBED Word.Picture.8  Slika 5.17. Sinapti ke te~ine neurona (20) 5.1.2. Modeliranje signala kutne brzine Isti postupak ponovljen je za izlazni signal kutne brzine (. Zbog aireg raspona koji poprimaju vrijednosti regresijskog vektora po etni raspored parametara neuronske mre~e postavljen je tako da obuhvaa airi opseg vrijednosti.  EMBED Word.Picture.8  Slika 5.18. Signal kutne brzine ( i izlaza mre~e(10 neurona)  EMBED Word.Picture.8  Slika 5.19. Razlika signala kutne brzine ( i izlaza mre~e (10 neurona)  EMBED Word.Picture.8  Slika 5.20. Srednje kvadratno odstupanje(10 neurona)  EMBED Word.Picture.8  Slika 5.21. Varijance neurona  EMBED Word.Picture.8  Slika 5.22. Sinapti ke te~ine neurona  EMBED Word.Picture.8  Slika 5.23. Signal kutne brzine ( i izlaza mre~e (20 neurona)  EMBED Word.Picture.8 Slika 5.24. Signal kutne brzine ( i izlaza mre~e (uveano, 20 neurona)  EMBED Word.Picture.8  Slika 5.25. Razlika signala kutne brzine ( i izlaza mre~e s 20 neurona  EMBED Word.Picture.8  Slika 5.26. Srednje kvadratno odstupanje(20 neurona) Kada usporedimo rezultate dobivene modeliranjem procesa mre~om s 10 neurona u ulaznom sloju s rezultatima dobivenim modeliranjem procesa mre~om s 20 neurona u ulaznom sloju mo~emo uo iti sljedee razlike: Mre~a s 20 neurona u ulaznom sloju bolje u i proces ato je i razumljivo jer se s veim brojem neurona posti~e bolja pokrivenost ulaznog prostora. Vrijeme potrebno za prora un mre~e s 20 neurona u ulaznom sloju je znatno du~e u odnosu na mre~u s 10 neurona u ulaznom sloju. Vrijeme uzorkovanja postavljeno je na 0.002s, 100 puta manje od vremena promjene pobude kako bi algoritam imao vremena podesiti parametre s obzirom na promjenu izlaznog signala procesa. Kada se mre~a dovoljno dobro nau i i algoritam prekine u enje, izlazni signal mre~e zadovoljavajue prati izlazni signal procesa. Meutim potrebno je jako dugo u iti proces jer zbog svoje nelinearne prirode i prirode pobude, izlaz procesa vrlo lako mo~e otii u podru ja koja nisu nau ena. 6. ZAKLJU AK U ovom radu modeliran je proaireni LuGre model trenja. Za modeliranje smo upotrijebili RBF neuronsku mre~u, a kao algoritam u enja primijenjen je rekurzivni hibridni algoritam u enja. Kao ulazni signal u proces koristili smo BLWN unaprijed definiranih parametara. Signali koje smo u ili su linearna brzina modela v i kutna brzina (. Poato te dvije brzine poprimaju razli ite vrijednosti a time i vrijednosti regresijskog vektora poprimaju razli ite opsege, po etni rasporedi neurona varijanci i te~inskih koeficijenata odabrani su posebno za svaki signal. Rezultati u enja su dobri s obzirom da se mre~a u i samo na temelju tekuih podataka, a ne na temelju bloka podataka kao u nerekurzivnim algoritmima. Loaa strana ovog na ina u enja je ta ato kod u enja dinami kih procesa vrijeme uzorkovanja mora biti dosta manje od vremena uzorkovanja pobudnog signala da bi algoritam stigao podesiti parametre. Algoritam grupiranja srediata neurona ima malo utjecaja na ponaaanje cjelokupnog algoritma u enja. Veliki broj neurona omoguava dobro pokrivanje ulaznog prostora tako da algoritam u malom broju koraka koji radi podeaava tek nekoliko neurona. Osim toga faktor u enja algoritma grupacije opada s brojem koraka tako da njegovo djelovanje postaje s vremenom zanemarivo. To je dobro poato je to dosta grub algoritam. Interpolacije obavlja podeaavanje varijanci neurona u prvih nekoliko koraka. Najbitniji algoritam koji se i naj eae izvraava je algoritam adaptacije. Ipak, on sam nije sposoban nau iti proces. Podeaavanja srediata neurona i varijanci je neophodno. Poato brzina konvergencije u enja procesa ovisi o po etnom rasporedu neurona loa raspored mo~e dosta usporiti konvergenciju. U ovom radu po etni parametri mre~e su se postavljali slu ajnim odabirom. 7. LITERATURA [1] I. Petrovi, N. Peri: Inteligentno upravljanje sustavima, predavanja, skripta, Zavod za automatiku i procesno ra unarstvo, Fakultet elektrotehnike i ra unarstva, Zagreb, 2001. [2] N. Peri, I. Petrovi: Identifikacija procesa, predavanja, skripta, Zavod za automatiku i procesno ra unarstvo, Fakultet elektrotehnike i ra unarstva, Zagreb, 2000/01. [3] H. Olson, K.J.Astrom, C. Canudas de Wit, M. Gaefvert, P.Lischinsky: Friction Models and Friction Compensation, European Jurnal on Control, 2000. [4] SIMULINK, Dynamic System Simulation for MATLAB, 1998, The Math Works Inc. [5] D. Novosel: Hibridni algoritmi u enja RBF neuronskih mre~a s primjenom za identifikaciju nelinearnih dinami kih procesa, Diplomski rad, 1996. 8. SA}ETAK U ovom radu analizirane su RBF neuronske mre~e i razli iti modeli trenja za kontaktnu povrainu automobilskog kota a i ceste. Za modeliranje koristili smo neuronsku mre~u s kru~nom osnovicom i rekurzivni hibridni algoritam za u enje. Model trenja koji smo modelirali je proaireni LuGre model trenja. 9. ABSTRACT This paper analyzes the RBF neural networks and different tire-road friction models. For modeling we used the radial basis functions with a recursive hybrid algorithm for learning. The friction model used in the experiment is the expanded LuGre model. 10. }IVOTOPIS Dinko Vranjea roen je u Splitu 3.travnja 1977. godine u Splitu. 1995. godine zavraava 3. gimnaziju u Splitu i iste godine upisuje prvu godinu na Fakultetu elektrotehnike i ra unarstva na Sveu iliatu u Zagrebu kao redovni student.. Na drugoj godini studija opredjeljuje se za smjer automatika. PAGE  PAGE 37  $~3456789:WXYbcd}~tjq5CJUaJj5CJUaJj{5CJUaJj5CJUaJ5CJaJj5CJUaJ 5CJaJj5CJUaJ5CJ\aJ 5CJ\5CJ\aJ 6CJ]CJ 5CJ\CJ OJQJ^J,,xz|~ "$@BDFHJLNPdh$dha$PRT~$dha$$a$9.  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