DSCC CONTROL OF RECURRENT NEURAL NETWORKS USING DIFFERENTIAL MINIMAX GAME: THE STOCHASTIC CASE

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1 Poceedis of he ASME Dyamic Sysems ad Cool Cofeece DSCC Sepembe -5,, Cambide, Massachses, USA DSCC- CONRO OF RECURREN NEURA NEORKS USING DIFFERENIA MINIMAX GAME: HE SOCHASIC CASE Ziia i Depame of Eieei Sae Uivesiy of New Yok Maiime Collee hos Neck, NY 65 zli@symaiime.ed Niwa Asai Depame of Elecical & Compe Eieei New Jesey Isie of echoloy Newak, New Jesey, 7 Niwa.Asai@ji.ed ABSRAC As a coiaio of o sdy, his pape eeds o eseach esls of opimaliy-oieed sabilizaio fom deemiisic ece eal ewoks o sochasic ece eal ewoks, ad peses a ew appoach o achieve opimally sochasic ip-o-sae sabilizaio i pobabiliy fo sochasic ece eal ewoks dive by oise of kow covaiace. his appoach is developed by si sochasic diffeeial miima ame, Hamilo-Jacobi-Isaacs HJI eaio, ivese opimaliy, ad yapov echie. A meical eample is ive o demosae he effeciveess of he poposed appoach. INRODUCION he pas wo decades have wiessed eomos advaces i eieei ad i compe sciece o bild aificial compaioal sysems ], amo which ece eal ewoks have bee applied o may scieific ad eieei fields, sch as sysem ideificaio ad cool, pae ecoiio, imae pocessi, ad modeli bioloical seso-moo sysems. heefoe, heoeical sdies o boh sabiliy ad coollabiliy of ece eal ewoks have bee heavily ivesiaed i he las few yeas ] - ]. Howeve, hese sdies pimaily focsed o deemiisic ece eal ewoks. I he mahemaical models of hese afoemeioed ewoks, hey do o coside he oise pocess ha is fah wih sial asmissio paiclaly i bioloical sysems. O he ohe had, ebos ] poied o ha i ode o develop mahemaical eal ewok specificaios which have dal ses as models of ielliece i he bai, ad as hihly effecive aificial iellie sysems whe implemeed i compes ad chips, we ms coside he sochasic eviome. Ufoaely, wih ead o he aalysis of sochasic ece eal ewoks, hee has bee lile wok i he lieae il he vey ece yeas ]. Hece, i is impoa o aalyically eploe he chaaceisics of sabilizaio ad coollabiliy fo ece eal ewoks de he iflece of sochasic pebaio. As a coiaio of o sdy i ], we pese i his pape a heoeical aalysis fo sochasic ece eal ewoks o achieve sochasic ip-o-sae sabilizaio i pobabiliy de a opimal cool saey, ad o aeae icemeal covaiace of sochasic pebaio o a pedefied level wihi sabiliy mais. By applyi he heoy of diffeeial miima ame o he sochasic ewoks, he appoach is developed by cosidei he veco of eeal ips as a playe ad he veco of sochasic disbace as he opposi playe. heefoe, a miima eilibim ca be achieved by popely coolli sochasic ece eal ewoks. I shold be poied o ha his pape develops a sochasic coepa of he disbace aeaio esls of hose i ]. PROBEM FORMUAION Based o he sadad fomlaio of sochasic ece eal ewoks ], we coside he followi sochasic ece eal ewok, which is deived fom he model of deemiisic ece eal ewoks defied i ] pls a addiive whie oise. Mahemaically, i ca be descibed by he followi Io-ype compac fom d A S d dψ Copyih by ASME

2 whee R is he sae of sochasic ece eal m ewok, R is he ip, sally m, A dia,, I R ad >, S s,, s ] R is a veco fcio ad is compoe s i is a simoidal fcio defied below, m R, R ae weih maices descibi he coecios of hidde ad op layes, ad Ψ is a - dimesioal idepede iee pocess wih icemeal covaiace d, i.e., E dψdψ } d whee is a kow boded fcio aki vales i he se of oeaive defiie maices. e shall fis iodce he followi wo defiiios. Defiiio : e s defie a oeaive boded fcio as follows: R he fcio will be sed as a playe o oppose he cool sial i ode o solve a sochasic diffeeial ame poblem addessed i his pape. Defiiio : he fcio of s i possesses he followi popeies: s i is boded o R; s i is piecewise aalyic ad sicly iceasi o R, ds i i.e., < < M i ad M i < fo all R d i ; i s i whe i. Remak: Based o Defiiio, i is impoa o poi o ha Model is siificaly diffee fom mos models epoed i he lieae. he acivaio fcio s i i his pape epeses a class of eeal oliea fcio ha does o have o be he widely sed simoid b fcio s a e i i / c. heefoe, Model ecompasses a mch lae class of sysems. DESIGN OF OPIMAY SOCHASIC INPU-O- SAE SABIIZAION e fis ewie he sysem of he sochasic ece eal ewok as d A S d dψ d Now le s coside a cadidae sochasic yapov fcio E, which is he same as he oe ive i ] E Fom ], he ifiiesimal eeao of he sochasic diffeeial eaio is ive as E E f E } E E E whee f E A S, E, I, hs E E, ad I. } E S Fo he secod em S, we ca similaly apply he eaio 7 i ] hee, ad hs esli i 5 M S 6 Sbsii 6 io 5, we have M E } 7 e shall e discss how o fid a opimaliy-oieed cool fo he sochasic ece eal ewok o achieve sochasic ip-o-sae sabilizaio i pobabiliy. Fom he cocep of diffeeial miima ame 5], ], he followi eeal sochasic oliea sysem affied i he oise Ψ ad cool is well kow d f d dψ d 8 If we pse a diffeeial ame poblem which ses defied i Defiiio as a playe o oppose he cool, ad sppose ha hee eiss a posiive opimal vale fcio, which saisfies he followi HJI eaio f 9 Copyih by ASME

3 he he followi cool is a opimal sabilizi cool which miimizes he cos fcioal τ d E J lim, whee > is a desi paamee, boh ad > fo all, ad he wos case is e ow ake he ifiiesimal eeao of he sochasic diffeeial eaio 8 wih he opimal vale fcio : f Fo he model of he sochasic ece eal ewok, if we coside he yapov fcio E as he opimal vale fcio, ha is he solio is ive by, we he have he followi eaios S f } 5 6 he he sbsiio of he eaio io he HJI eaio yields he e elaio S 7 Based o he above, he iealiy i 7 ca be wie as } M } M 8 Fom 8, we ca se p M 9 Simila o ], e s defie he followi scala fcio ] ] φ he a cool sial ca be deemied as φ wih he assmpio of. Usi he eaio, he cool sial is eivale o φ φ By compai wih ad si 6, 7 ad, we obai φ ad S Now assmi ha, ma, 5 Copyih by ASME

4 we ca obai he followi heoem. heoem: Give he sysem, hee eis a posiivedefiie fcio ad a sicly posiive fcio i which, sch ha he feedback cool law 6 achieves boh sochasic ip-o-sae sabilizaio ad ivese opimaliy wih espec o a meaifl cos fcioal τ τ d E J lim, 7 fo he wos case I 8 Poof: Sep : Cosidei a posiive-defiie sochasic yapov fcio ha is he same as, he ifiiesimal eeao of he sochasic diffeeial eaio is ive by } S 9 Sbsii 6 io 9, we have } M } M he sbsiio of he cool law 6 io yields M } ] ] By he defiiio of 9, we obai } ] ] ih he assmpio of 5, we kow ha hs } ] ] ] ] I I heefoe wheeve By he defiiio of sochasic ip-o-sae sabiliy 6], we kow ha he sysem descibed by achieves sochasic ip-o-sae sabilizaio wih he cool law 6. Sep : e s coside ad. By, we have S Fom 6, we have M S Usi he iealiy ad elaios, ad 5, he epessio ive above ca be wie as ] ] M ] ] Copyih by ASME

5 his meas ha is posiive defiie, fo all, ad is adially boded. Also, fom, i ca be see ha > whe φ. ha is ] ] > By si ad i ad, ca be wie io he followi fom I I Accodi o Dyki s fomla ], we have J, lim E τ dτ lim E τ dτ E ] lime I I dτ Fom he above eaio, we kow ha he opimal cool is a opimal solio o J 7 fo he wos disbace ad mi ma J, E ] heefoe, by cosidei he cool as a playe ad he oise covaiace as he opposi playe, a miima eilibim, d is achieved. his complees he poof. NUMERICA EXAMPE I ode o effecively descibe o esls, we pese he followi secod ode sochasic ece eal ewok d d 7 ah dψ d d d 8 ah Ψ whee,,,, ah S, ah, ad Ψ, Ψ ae whie oises ifomly adom wih he maide of Ψ i i,. Fi. shows he esl of ime esposes of wo saes ad fo his sochasic ece eal ewok wiho ay cool ips. Fi. shows he esl of ime esposes of wo saes wih he implemeaio of he opimally sochasic ip-o-sae sabilizi cool 6 a s. I ca be see ha he sysem achieves he epeced pefomace which cofoms o he heoeical aalysis i Secio III. CONCUSIONS his pape has peseed a ew desi o achieve opimally sochasic ip-o-sae sabilizaio i pobabiliy fo sochasic ece eal ewoks dive by oise of kow covaiace. he poposed appoach is developed by si sochasic diffeeial miima ame, Hamilo-Jacobi- Isaacs HJI eaio, ivese opimaliy, ad yapov echie. ih Defiiio, we have eeded o pevios eseach 7] o a mch lae class of oliea sochasic sysems. De o he difficly o solve he Hamilo-Jacobi- Isaacs eaio, fo sochasic oliea sysems, opimal sochasic sabilizaio seems o be a achievable oal i feedback desi. Howeve, a aleaive way has bee poposed i his pape o solve he poblem ad obai a opimal feedback coolle wih espec o a meaifl cos fcioal by si he kowlede of ivese opimaliy. I is believed ha he ew desi peseed i his pape wold iesify he applicaios of sochasic ece eal ewoks. REFERENCES ] ebos, P., 9. Ielliece i he bai: A heoy of how i woks ad how o bild i. Neal Newoks,, pp.. ] Esai,., ad Aik, S., 5. Global Sabiliy Aalysis of Neal Newoks wih Mliple ime ayi Delays. IEEE as. Aoma. Co., 5, pp ] Che, B., ad a, J.,. Global epoeial peiodiciy ad lobal epoeial sabiliy of a class of ece eal ewoks. Physics ees A, 9, pp Copyih by ASME

6 ] Maco, M., Foi, M., ad esi, A.,. O absole sabiliy of coveece fo oliea eal ewok models.. Ka e al. Eds. New eds i Noliea Dyamics ad Cool ad hei Applicaios, ece Noes i Cool ad Ifomaio Scieces, Spie, Beli, 95, pp. 9. 5] Sachez, E., ad Peez, J.,. Ip-o-Sae Sabilizaio of Dyamic Neal Newoks. IEEE as. Sysems, Ma, ad Cybeeics A,, pp ] H, S., ad a, J,. Global Asympoic Sabiliy ad Global Epoeial Sabiliy of Coios-ime Rece Neal Newoks. IEEE as. Aoma. Co., 7, pp ] Klawski, G., ad Bdys, M.,. Sable Adapive Cool wih Rece Newoks. Aomaica, 6, pp ] Aik, S.,. Global Asympoic Sabiliy of a Class of Dyamic Neal Newoks. IEEE as. Cicis Sys. I, 7, pp ] Soa, E., ad Qiao, Y., 999. Fhe Resls o Coollabiliy of Rece Neal Newoks. Sysems & Cool ees, 6, pp. -9. ] Soa, E., ad Sssma, H., 997. Complee Coollabiliy of Coios-ime Rece Neal Newoks. Sysems & Cool ees,, pp ] Ha, C., He, Y., Ha,., ad Zh,., 8. ph mome sabiliy aalysis of sochasic ece eal ewoks wih ime-vayi delays. Ifomaio Scieces, 78, pp. 9-. ] i, Z., ad Asai, N.,. Cool of Rece Neal Newoks Usi Diffeeial Miima Game: he Deemiisic Case, Sbmied o ASME Dyamic Sysems ad Cool Cofeece DSCC-. ] Das, S., ad Oloimi, O., 998. Noisy Rece Neal Newoks: he Coios-ime Case. IEEE asacios o Neal Newoks, 95, pp ] Ksic, M., ad De, H., 998. Sabilizaio of Noliea Uceai Sysems, Spie-ela, New Yok. 5] Basa,., ad Behad, P H -Opimal Cool ad Relaed Miima Desi Poblems: A Dyamic Game Appoach, d Ed., Bikhase, Boso, MA. 6] siias, J., 998. Sochasic ip-o-sae sabiliy ad applicaios o lobal feedback sabilizaio. Ieaioal Joal of Cool, 7, pp ] i, Z., a, Q., ad Schz, H., 9. Ivese Opimal Noise-o-Sae Sabilizaio of Sochasic Rece Neal Newoks Dive by Noise of Ukow Covaiace, Opimal Cool Applicaios ad Mehods,, pp Saes, ime FIGURE. SYSEM RESPONSE 6 Copyih by ASME

7 Saes, ime FIGURE. SYSEM RESPONSE 6 a s 7 Copyih by ASME