Research on Probability Density Estimation Method for Random Dynamic Systems

Transcription

1 73 A publaton of VOL. 5, 6 CHEMICAL ENGINEERING TRANSACTIONS Guest Edtos: Thun Wang, Hongyang Zhang, Le Tan Copyght 6, AIDIC Sevz S..l., ISBN ; ISSN The Italan Assoaton of Chemal Engneeng Onlne at DOI:.333/CET653 Reseah on Pobablty Densty Estmaton Method fo Random Dynam Systems Yanong N*, Ru Mn Depatment of Eletal Engneeng, Henan Insttute of Tehnology nyanong666@6.om In ths pape, the autho eseahes on pobablty densty estmaton method fo andom dynam systems. It ndate that, the Pobablty Densty Funton (PDF) uves and falue pobabltes of stohast systems wth dsjont falue domans, multple desgn ponts and dsontnuous esponse ae alulated effetvely and auately. Moeove, the Pobablty Densty Estmaton Method (PDEM) has muh hghe effeny, and s a potental and geneal appoah to attak the elablty analyss of omplated poblems. Subsequently, the unetanty popagaton and dynam elablty analyss of nonlnea andom system unde non-statonay extatons ae addessed. The fst passage falue teon and stohast hamon funton of powe spetum densty model of non-statonay andom poess fo eathquake aton ae utlzed. The PDEM an aheve effently and auately the dynam esponse, tansent PDF and elablty of nonlnea systems.. Intoduton Stat and dynam elablty analyss and unetanty popagaton of nonlnea stohast stutue ae mpotant eseah tops fo domest and aboad sholas, whh have wde applaton pospet n engneeng pate, and povde a sold theoetal bass fo elablty desgn of the lage sale and omplex stutue. Hong (3) extends the use of pobablty densty evoluton method (PDEM) nto the elablty analyss of nonlnea stohast stutues wth omplex pefomane funtons, and dynam esponse and elablty analyss of nonlnea stutue unde non-statonay extatons. Meanwhle, the nfluene mehansm of stohast unetanty popagaton s nvestgated. In elablty analyss and optmal desgn of stutues, thee exst some omplex lmt state funtons wth falue doman sepaaton, multple desgn ponts and dsontnuous esponse et. Fo suh knd of elablty analyss of stutues wth omplex lmt state funtons, FORM and SORM whh ae wdely used and hghly effent beome fal. Despte the numeal smulaton methods (suh as Monte Calo smulaton, subset smulaton and lne samplng et.) an solve ths poblem, but the omputatonal effeny s qute low. In addton, thee s no auate and effent omputng appoah to addess the andom dynam esponse and elablty analyss of nonlnea stutues subjet to non-statonay extatons. In eent yeas, Lu (3) poposed the pobablty densty evoluton method whh povdes a unfed soluton famewok fo andom vbaton and elablty analyss of nonlnea stutues. In ths pape, the PDEM s appled to solve the dffult and mpotant eseah subjet of elablty analyss of stutues wth omplated lmt state funtons and unetanty popagaton. Atually, PDEM an obtan the pobablty densty funton of andom stutues unde stat and dynam load, and s ndependent of any fom of explt and mplt lmt state funtons. Ths s the so-alled Cuse of Dmensonalty. So, n ode to effetvely analyses hgh dmensonalty data, t s a pvotal step to edue the dmensonal membes. Yang s (3) pape s to exploe a new featue seleton way and popose a featue ankng method to edue featue s dmensonaltes. In hs pape, the pnple of edung featue s dmensonaltes s befly ntodued, and the pnpal ways of featue dmensonalty eduton s evewed. Moeove, the Pobablty Densty Estmaton Method (PDEM) has muh hghe effeny than the Monte Calo smulaton, and s a potental and geneal appoah to attak the elablty analyss of omplated poblems. a novel featue ankng appoah s poposed. A smplfed appoah s ntodued to deal wth unsupevsed data. At last, the algothm poposed n hs pape s ealzed by MATLAB, and many Please te ths atle as: N Y.R., Mn R., 6, Reseah on pobablty densty estmaton method fo andom dynam systems, Chemal Engneeng Tansatons, 5, DOI:.333/CET653

2 74 datasets s used to expement. A lot of oss-valdaton and othes expemental esults demonstate the valdty, feasblty and advantage ove othes of ou appoah.. The Chaatest of andom dynam systems The stohast dffeental equaton and dynamal systems an geneate a andom dynam system. Ths s a hot ssue of eseah n eent yeas. The most bas and mpotant of the theoy of the dynam system s to eveal the dynams of the long-tem development of the system whh popetes and the bas dynam haatests. Jn (4) gves a speal knd of osllato, the osllato s omposed of n han, and the han dynams system an be used to desbe the Hamlton funton. The nteaton between heat bath and han though dffeental equatons (often efeed to as the system) s to desbe. Afte a sees of devaton, the man use of Ito s fomula, get the multplatve egod theoem of the oespondng stohast dynam system. Qu (4) gves a multplatve egod model needs to meet the ondtons. The seond seton aodng to the ondtons of the fst seton lsts the sx speal models and llustates them espetvely wth multplatve egod theoem. As n pate the systems have unetantes, the ontol poblem an not use a smple detemnst model to desbe, and t need to ombne ontollng a system and leanng a system unetanty to an ssue. The natue of ontol s that: on the one hand, the ontol sgnal an make the system output towads the desed goal (alled ontol aton); Howeve, two hands ae ontadtoy, and the fome eques the ontol sgnal to hange smoothly, whle the latte wants to mantan etan ampltude of motvaton, so ontol need tade-off. In L s (4) pape, lnea, nonlnea stohast system and mult-model nonlnea stohast systems ae studed. Fo the unknown paametes wth Gaussan whte nose stohast lnea systems, usng "utlty funton" t s pesented a tade-off of leanng and ontollng of the ontol stategy. On the one hand, t an ontol the system towad the desed output: on the othe hand, t an lean the unknown lnea paametes. Smulaton esults show the valdaton of suh appoah. Wth unknown paametes of the non-lnea stohast systems ontol poblem, t s poposed a leanng and ontollng optmzaton algothm. Contolle an not only ontol the output to tak the desed output, but also an usng RBF netwoks onlne lean unknown paametes of nonlnea systems. Yang s (4) smulaton esults show supeoty of the algothm. Fo a lass of unknown paametes of nonlnea mult-model swthng stohast systems, t s poposed an algothm, whh use RBF netwok to lean nonlnea funtons onlne, and Bayes posteo pobablty to estmate the model, aodng to the ost funton obtan the ontol sgnal. By smulaton, the algothm an obtan a swthng tme of system exatly and an auately tak hanges n the system. 3. The pobablty densty estmaton method fo andom dynam systems The equaton of bas funton s as equaton () as follows: ( C u e ) u () j jkl k l kj k Unde the lnea elatonshp, bas equaton s shown n equaton (): ( e u ) () j jkl k l kj k The lnea dffeental equaton an be expessed nto the followng smplfed foms: L(, ) f ( x, ), L(, ) T( ) J (3) In whh, T ( ) C, t ( ) e, ( ) k k (4) k j jkl l j jk k Consde an nfnte stuaton, we have the equaton (5) n the followng: (x) (x) (5) Consde the popagaton, nstead the equaton (3) wth the followng fom: C(x) C C (x), e(x) e e Then we have equaton (7) to (): (x), (x) (x), (6)

3 C C C, e e e,, (7) The ontanng nlusons an be smplfed nto the followng ntegal equaton set: f ( x, ) f ( x, ) S ( x x)(l F(y ) + V R)T f (y )] S(y )dy In vew of the followng elatonshp k3x3 e dx 3 k3 g( (8) ( ) (9) Equaton (8) an be onveted nto the followng fom: J () f ( y, ) f ( y, ) S( y y, )L F(y, )d y+ g( y y, ) f (y, )dy s s In whh, S s ylnde oss seton, y ( x, x), and g( y y, ) kdk g( k, )exp( k (y y )) d k ( ) k k (, ) () Suppose k 3=, g(k,) an be obtaned fom Equaton (8) Fo suh knd of mateal, geneal fom of equaton () s expessed as followng equaton (-4): Gk ( k, ) [ ] k kkk m mk k k k k k e 5 k k g (3) ( k, ) e 5 ( k, ) m k,, C C 66 C 44, 5 44 C44 The fst one s the funton fo andom dynam systems n the fom of: ( ) (, ) ( ) () g t t t 75 () (4) ( e ) C (5) (6) ( ) ( ) (, ) ( ) () h t t t t (7) (8) h(, t) f ( t ) g(, ) d In whh, f(t) s the oespondng funton defned as the followng (9): t [ ] f ( t) ( t) (9) Afte g (,t) s obtaned, h (,t) an be easly obtaned fom Equaton (8). So, we have:

4 76 t e d sn( kt) g( k, t) ( t) e k ( ) k t () () ( ) k g(, t) e g(k, t) d k Va Equaton (), () an be onveted nto: g(, t) ( t) d, sn( k[ t k os ]) dk ( ) () In the equaton, the followng popety s adopted: sn( k os ) d (3) Fo defnng, we nomalze t k sn kdk lm e sn kdk lm Re (4) Thus, () an be epesented nto: () t d g(, t) Re ( ) os t Thus, Equaton (5) an be epesented as: (5) ( t) e d ( t) ds ( ) e oshe ( ) s s osh s (6) g(, t) Re Re Wthn the unt yle to solve (6), the followng an be obtaned: ( t) g(, t) Re (7) snh g(, t) ( t ) t ( ) (8) h expesson an be obtaned: ( t ) (, ) t t h t t ln t Thus, g (, ) must be detemned, that s: (9) t g(, ) e g(, t) dt (3) Put (8) nto (3) to obtan: g (, ) / e t t dt (3)

5 77 osh g(, ) e d (3) z osh H() z e d (33) As pe (33), g (,) of Equaton (3) an be obtaned: (, ) ( ) (34) g H 4 The funton omponent defned fo Foue tansfom an be obtaned n the followng equaton (35)-(39). G (, ) { H ( ) [ H ( q)] m m H ( )} k k k 4 yyk e (, ) ( ) H( ) m 4 5 (35) [ f ( q)] f ( ) f ( ) y (36) e (, t) mg3 (, t) 5 C 44 (37) In whh, g epesents output of Equaton (38) g (, t) ( t ) t ( ) (38) Meanwhle, t also epesents output fo equaton (9) ( t ) t t h (, t) t ln t (39) The expement esult of pobablty densty evoluton method (PDEM) and the Gaussan Kenel Pazen Estmaton (GKPE) algothm s shown n the fgue. Fom the expement, we an get that the PDEM an aheve effently and auately the dynam esponse, tansent PDF and elablty of nonlnea systems. Fgue : The expement esult of pobablty densty evoluton method (PDEM) and the Gaussan Kenel Pazen Estmaton (GKPE) algothm

6 78 4. Conluson In ths pape, the autho eseahes on pobablty densty estmaton method fo andom dynam systems. As n pate the systems have unetantes, the ontol poblem annot use a smple detemnst model to desbe, and t need to ombne ontollng a system and leanng a system unetanty to an ssue. The natue of ontol s that: on the one hand, the ontol sgnal an make the system output towads the desed goal (alled ontol aton); the othe hand, the ontol sgnal an edue the unetanty of system paametes (alled leanng the paamete unetanty). The PDEM s appled to solve the dffult and mpotant eseah subjet of elablty analyss of stutues wth omplated lmt state funtons and unetanty popagaton. Atually, PDEM an obtan the pobablty densty funton of andom stutues unde stat and dynam load. Refeenes Hong X., Chen S., Qatawneh A., Daqouq K., Shekh M., Mofeq A., 3, Spase pobablty densty funton estmaton usng the mnmum ntegated squae eo, Neuoomputng, 5-8. Jn X.L., Wang Y., Huang Z.L., Paola M.D., 4, Constutng tansent esponse pobablty densty of non-lnea system though omplex fatonal moments, Intenatonal Jounal of Non-Lnea Mehans, L B., Pang F.W., 4, Impoved adnalzed pobablty hypothess densty flteng algothm, Appled Soft Computng Jounal, Lu Y., Chen L.Y., Y H., 3, Submane pessue hull butt weld fatgue lfe elablty pedton method, Mane Systems, 9-9. Pe Y.J., Evatt R., 3, Hawkes, Sanghoon Kook. Tanspoted pobablty densty funton modellng of the vapou phase of an n -heptane jet at desel engne ondtons, Poeedngs of the Combuston Insttute, Qu X., Huang Y.X., Zhou Q., Sun C., 4, Salng of maxmum pobablty densty funton of veloty nements n tubulent Raylegh-Bénad onveton, Jounal of Hydodynams, Se. B, Yang Q.S., Tan Y.J., 4, A model of pobablty densty funton of non-gaussan wnd pessue wth multple samples, Jounal of Wnd Engneeng & Industal Aeodynams, Yang Y., Wang H.F., Pope S.B., Chen J.H., 3, Lage-eddy smulaton/pobablty densty funton modelng of a non-pemxed CO/H tempoally evolvng jet flame, Poeedngs of the Combuston Insttute, Zhu J.Y., Gu W.H., Yang C.H., Xu H.L., Wang X.L., 4, Pobablty densty funton of bubble sze based eagent dosage pedtve ontol fo oppe oughng flotaton, Contol Engneeng Pate, 8-4.