Direct Adaptive Backstepping Control for a Class of MIMO Non-affine Systems Using Recurrent Neural Networks

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Naomi Barker
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1 Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog Dc Aapv Backsppg Coo o a Cass o IO No-a Syss Usg Rc Na Noks Chg-Hg IAENG J-Chh Ch Hao-Ha Chag Ch-g Ko a Ha-Hsag Chag Asac hs pap poposs a c aapv acksppg coo sch o a cass o -p-op oa ca o-a syss sg op c av a oks s ca DABC. h popos cos h avaags o av-as a ok zzy a ok FNN a op ack ay. Fo h ackg o oa o-a syss h o-aga o s aso o a sc-ack-k o. Ssqy h a ok as acksppg coo s vop. h a va coos a aca coo a appoa y s. I ao a os coo s sg o copsa h appoa o o s. Bas o h yapov appoach h aapv as a say aayss o -oop sys s oa. Fay sao ss o o-a o-ps sys s sho o osa h poac o o appoachs. I s p-p-p-op oa o-a sys acksppg av a ok aapv coo yapov ho I. INRODUCION I hs pap h acksppg chq s s o sg aapv coo o a cass o IO oa ca o-a syss. h acksppg chq povs a sysac aok a csv sg hooogy o oa syss [. h sg poc as h sa vaas as va coo ps; h h va coos s sg sp y sp. Fay h aca coo p ca oa. I sas h say y yapov say ho. Hov h ajo a s ha h sys s acy ko. I h a cay a a sac s h hy ay s a sa sys. ho popos a op-c av a ok sys o appoa h ko o sov hs po. Pvos as vop zzy syss a a oks o appoa h ko [- 6. ay sachs hav sho ha sg av ass ca achvs spo poac ok sz a ag ay [6. ho av a co h h zzy a ok o c h av as a ok. I o o o q h op hs ok as sppo pa y h Naoa Scc Coc aa R.O.C. coacs NSC-5--E-55-6-Y. Chg-Hg s h Dpa o Ecca Egg Ya-Z Uvsy Chg- aoya aa. pho: 6--6 : 7; a: a: ch@sa.yz... ack sch s s o vop a op c av a ok [-. Rcy y a ok as acksppg coo schs popos o oa a ca syss [. Accog o h a o as [ h o-a o sys ca as o sc-ack-k o sys hch ach ssys ca v as a a-k sys. hs h ss sazg coo o hs yp o asog sys y pc co ho [. Accog o h ss o [ h IO oa o-a a o-aga ca as o a k IO sc-ack-k sys; ho ca spy h copy o coo sg.g. o-a o-p sys [. Bss hs as ca aso cop h IO a o syss. a [ popos a NN-as coo o a h h sa-ack aza sys a o NNs o s o appoa o ko.. ga a co a ca sys yac. h c aapv acksppg coo sg op-c av a ok DABC s popos o a h a cass o IO oa ca o-a syss. h s a s o a h a va coos a aca coo [ 5-6. A os coo s sg o aa c o a h o yac og os a a sacs o ackg o. Accog o h yapov say appoach h aapv as a say o -oop sys a gaa. hs pap s ogaz as oos. Sco II ocs h po oao a h popos op c av a ok sys. h DABC coo schs a oc Sco III. Sco IV shos h sao ss o h-o o-a sys o-aga o a sho o osa h poac o h popos DABC. Fay cocso s gv. II. PREIINARIES A. Po Foao Cos h oog IO oa ca sys sa-spac psao F o K F y h [ K R K a h vco o og h sas o sys F [ F F K ISBN: IECS

2 Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog F R K a sooh vco co [ K R s h coo p y y y R s h sys op [ K y [ K K os h a o sac sasyg ρ hch s h Eca o a ρ >. H sys sa vaas a aso ass o asa a s a q po. hogho hs sy h oog asspos o a s h cooay o h sys. F j j j Asspo : h qas > j ho R K j K. F Asspo : h qay > hos o a Ω R h a cooay go Ω.. F F > > F > a F > F >. Asspo : h sg ajcoy vco y s sooh a o. h coo ojcv s o sg h coo p sch ha h op y oos a s ajcoy vco y. O popos coo sg ho s as o h cocp o [. H h SISO o-a oa sys o IO o-a oa sys aga o. I ca ps as oos: y F y y y h [ [ y y y R R a F s sooh vco co. h c sga y a s vas y y y a ass o o. D h ackg o as y y a h cospog o vco as [. H y F c { F c H h c s a sg a a H Δ F c. h ack azao coo p o ca as c c c h s a coo p o saz az yac c s a aapv c a ok RNN coo sga sg o cac H a c s a aoa os coo p o copsa h appoao o. Ssg o ys Δ c y c { H c. 5 W sg as c y k 6 c h k [ k k ssg a 6 o a ys k k 7 hch ps ha. hs ca o y > choosg pop k so ha a oos o h poyoa ha s k s k a oca h op -ha pa. hs H s pcy cac y.. H a c h -oop sys s sa. As h scsso aov a RNN coo sho poy o appoa H. h ps o h RNN a a. Ovosy h op o c c h RNN s cy o RNN o poc h coo p. Accog o h pc co ho [ h ss a s Ω R a q hch s a co o a α c sch ha c α sass o a Ω R. α B. Op Rc Wav Na Nok I hs pap o achv hghy appoa accacy a sp p h covgc h FNN s o as a ov av-as NN. H co h avaags o FNN h av o popos a o-ay op c av a ok. h schac aga s pc Fg. hch z - os a ay. hs s copos o a p ay a av ay a h ay a op ay a a c ay. Uk h Gassa shp s covoa FNNs av a spaay ocaz. ho h ag o s o c ha FNNs co appoao. H h Gassa shp a pac y av ass a h s-c ay s pac y op c ay. W ca h sga popagao a h co o vy o ach ay. ay : Ip ay ack ay h ps o hs ay a h c ok p a pas ok op y h ghg v h [ K K R s h p. h op o hs ay s y h [ K K R a [ K K o os h o ok op. I s ca ha coas h op y hch sos h pas oao o ok. ay : H ay Wav ay Each o hs ay pos a av co. H h Gassa av co zωz p -z / s aop as h acvao co h ω s h sc qcy. Hc ISBN: IECS

3 Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog k k k ω p[ k k k k k h a k k a h asaos a ao h kh o h h p o h o o h oh av ay spcvy. ay : H ay I hs ay ach o cacas h poc o a p sgas.. k... k k k k [ K K k k k [ k K k K R h R k. I ca pss a vco oao as [ R h R a [ k [ k R. ay : Op ay Each o cacas h a coao o p vaas. ho h ph op s y p p p kp k o k h os h cocg acva gh va o kp h ph op assoca h h kh ay. I vco psao y [ y yp y o Accog o h aov oco h has ajsa paas a. h achc o s hs pap s sg o hav h avaags o ok h yac chaacscs. III. DIREC ADAPIVE S CONRO VIA BACKSEPPING DESIGN ECHNIQUE os o as sg acksppg appoach a o h ack aza oa syss.. h ko oas s sasy achg coo [- 6. Fo oa o-a sys hy a o va. As aov scsso oa o-a sys as oos a-k o o spy o sg appoach H o H y h H a ca oa. Sc h sys yac ay ko o p y a sac pacca appcao h a va coos a h a aca coo a cao pcsy oa. ho h say o h coo sys cao gaa. h yac a ok- s a aop o sa h a va coos a aca coo a o s h say o h coo sys sp h sc o h ca sys yac. h c aapv acksppg coo sch poys s o appoa h va coos ach sp. Fay h aca coo s oa y sg acksppg appoach aga. hs h as aapv acksppg coo as a sg as h - a û a va a aca coos ga y s spcvy. hy a s o a h a coos. Ros coos a sg o copsa h appoa os o coos. Bas o h vsa appoao ho [5 h ss a opa appoao - a o sch ha ε ε 7 ε ε h ε o appoao o vco; a a opa paas o a spcvy. Fo h s op ca ps as -. h sa o a sasy a ε ε ε ε h a. h azao chq s poy o hav h oog ayo paso o O. R ca ps as O ISBN: IECS

4 h [ [ [ O s h hgh-o. Ssg 6 o a h hav h sa o o a a D ε - 5 D ε 6 h D a h ca s a ass o o y ; D a ko posv a. Usay cao oa pacca appcaos. H hov a aapv sch s s o sa. ho h oog ho ca oa. ho : Cos h IO oa ca o-a sys sasyg Asspos a. h aapv as o h acksppg coos a sg as 7. h a a posv aapv paas a h os coos h a aapv o sao a sg as oos [ h [ os h pso vss a s a o- sa va o h ca o. Hc h asypocay covgc o ackg o a h sys say ca gaa. Poo: As h aov scsso. h coo sch sg ho s ca DABC. Fg shos h cogao o DABC. IV. SIUAION RESUS I hs sco h sao ss o a o-a o-ps sys as sho Fg. a ps. hs sao ap shos h poacs o o popos appoachs. Cos h ackg coo o o g-o-o o ps [. As sho Fg. h o os h vca pa a o cocg jos a v y oq coo. A coa ocs a go h. h oog qaos o oo ca v y s s s s g g φ h a a h gaz cooas a a h oqs acg o h cocg jos o h o a o. h qaos ca h spc o a h. [5 6 s [5 s [5 s [5 s [5 s 5 [5 s [5 s [5 s [ [ 6 [ [ [ g g g g g h o ps sys ca sa-spac psao h [ s h vco o asa sas [ s h vco coo ps a 5 Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog ISBN: IECS

5 Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog h j j a o a R. Usg h ss o [ h yac qaos o oo a y 6 h [ [. Ovosy sys 6 s oa o-a cao hav h sazg coo y ack azao appoach. H h coo ojcv s o s o aapv acksppg coo sch ha h sas oo a sg o c ajcoy π π asypocay. h a coos s [ 5 a h sg paa s k h aapv 5 paa as o DABC - a 5... h ok sc s sc as [ a h a va as s ha j s j s [ j s.a s. h a j sac s D [s. h sao ss o DABC - a sho Fgs. a 5. Sa ajcos a sho Fg. so : aca ops; ash : c ajcos. Fg 5 shos h cospog coo ocs a ackg os. I ca o ha DABC - appoach pos a has as sazg ao. sco. V. CONCUSION hs pap has sccssy ps h DABC coo sch o a cass o IO oa ca o-a syss o-aga o. I h DABC coo syss s s o a h a va coos a aca coo. I ao h os coos a sg o copsa h appoa os o s. Accog o h yapov say appoach h aapv as o o g paas a oa a h say o h coo sys s gaa. o vy h cvss o h popos coo sch ca sao o o-a o-ps sys o-aga o hav ps o sa h cvss a poacs o o appoach. REFERENCES [ B. Ch X. a S. og Aapv zzy op ackg coo o IO oa ca syss IEEE as. o Fzzy Sys Vo. 5 No. pp [ C. S. Ch Ros aapv coo o ca IO oa syss- oa akag-sgo zzy appoao as appoach IEE Poc. Coo hoy App. Vo. 5 No. pp [ B. R. Chg Fzzy Na Nok Bas Aapv Backsppg Coo Dsg o a Cass o Noa Syss as hss o Ya Z Uvsy aa 7. [ E. V. Gs a C. P. Boa Aapv a-as acksppg coo o ca IO oa syss IEEE Iaoa Jo Coc o Na Noks pp [5 H. Ha C. Y. S a Y. Spako Aapv coo o a cass o oa syss h oay paaz zzy appoaos IEEE. as. o Fzzy Syss Vo. No. pp. 5-. [6 C. F. Hs C.. a.. Wav aapv acksppg coo o a cass o oa syss IEEE as. o Na Noks Vo. 7 No. 5 pp [7 N. Hovakya E. avsky a A. Sasa Dyac vso o oa--coo syss va -sca spaao pa Joa o Dyaca a Coo Syss Vo. No. pp [ Y. Q. J H. W a W. J. G Aapv coo o a cass o oa syss as o zzy-a appoach IACS coc o Copaoa Egg Syss Appcaos Vo. pp [. Ksc I. Kaakopos a P. Kokoovc Noa a Aapv Coo Dsg N Yok: Wy 5. [ Y. S. Qag X. Zhag a O. Kayak Ros a aapv acksppg coo o oa syss sg RBF a oks IEEE as. o Na Noks Vo. 5 No. pp [ C. H. a C. C. g Icao a coo o yac syss sg c zzy a oks IEEE as. o Fzzy Syss Vo. No. pp [ C. H. a C. C. g F g o shp o zzy a syss Asa Joa o Coo Vo. No. pp [ C. J. a C. C. Ch A av-as a-zzy sys a s appcaos Poc. IEEE I. Jo Co. Na Noks pp. -6. [ C... Y. Ch a C. H. Ch RCAC hy coo o IO ca oa syss sg sg-o choogy IEEE as. o Na Noks Vo. No. pp [5 C.. a Y. J. o Dcopg coo y hachca zzy sg-o coo IEEE as. o Fzzy Syss Vo. No. pp [6 C.. a C. S. G. Na Fzzy Syss: A No-Fzzy Sygs o Ig Syss Pc-Ha NJ 6. [7 F. J. a C. H. O- ga-g IP coo sg RFNN IEEE as. o Aospac a Ecoc Syss Vo. 7 No. pp [ Y. J. a W. Wag Aapv zzy coo o a cass o ca oa oa sys Ioao Sccs Vo. 77 Iss pp [ S. ao a..ga Aapv zzy coo o a cass o SISO oa oa syss Fzzy Ss a Syss Vo. 5 No. pp [ K. S. Naa a K. Pahasaahy Icao a coo o yaca syss sg a oks IEEE as. o Na Noks Vo. No. pp. -7. [ J. K. Pak S. H. Hh a S. H. K Dc aapv coo o oa oa syss sg s-scg a oks IEEE as. o Na Noks Vo. 6 No. pp [ Y. F. Pg a C.. Ig hy coo o ca oa syss sg a c ca o acao coo IEE Poc. Coo hoy App. Vo. 5 pp [ Y. F. Pg. H. a C.. Chog Dsg o op c CAC acksppg coo sys o ackg poc ajcos IEEE Iaoa Jo Coc o Na Noks pp [ C. D. Sosa J. E.. Hy a R. H. Cavao Aapv coo o o oo sg av oks IEEE as. o Syss a a Cycs Pa B: Cycs Vo. No. pp. -5. [5 S. og a Y. Dc aapv zzy acksppg coo o oa syss IEEE o Pocg o h Fs Iaoa Coc o Iovav Copg Ioao a Coo Vo. pp [6 R. J. Wa a K. Y. Hsh ackg coo sg o oo apao va zzy a ok IEEE as. o Fzzy Syss Vo. pp. -7. [7. X. Wag Aapv Fzzy Syss a Coo: Dsg a Say Aayss Egoo Cs Pc-Ha NJ. [ A. Yog C. Cao N. Hovakya a E. avsky Coo o a oa o-p sys va yac vso a ISBN: IECS

Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog -sca spaao Aca Coo Coc pp. -5 6. [ A. Ysk a F.. s Fack azao sg a oks Aoaca Vo. No. pp. 65-66 5. [ W. Zhag a S. S.G A goa pc co ho ho a po a s appcaos o coo o o-a syss o hgh sos Joa o ahaca Aayss a Appcaos Vo.

5-5 6 7 sc R c sga Sy s op R c sga Sy s op - 5 6 7 sc Fg : Sao ss - Sa ajcos ash : ; so : DABC -. ackg o 5-5 5 6 7 5-5 5 6 7.5 - -.

6 Pocgs o h Iaoa Coc o Egs a Cop Scss Vo I IECS ach - Hog Kog -sca spaao Aca Coo Coc pp [ A. Ysk a F.. s Fack azao sg a oks Aoaca Vo. No. pp [ W. Zhag a S. S.G A goa pc co ho ho a po a s appcaos o coo o o-a syss o hgh sos Joa o ahaca Aayss a Appcaos Vo. No. pp z y z o y o o Wav co ay ay ay ay Fg : Schac aga o op c av a ok sc R c sga Sy s op R c sga Sy s op sc Fg : Sao ss - Sa ajcos ash : ; so : DABC -. ackg o ackg o sc Fg 5: Sao ss- Coo oc a ackg os. Fg : Do ps sys [. - Aapv as - coo Ros coo Aapv as - coo - Ros coo Aapv o Aapv o sao sao Fg : Dc aapv s coo va acksppg coo sch DABC s. û Noa Pa ISBN: IECS

11/8/2002 CS 258 HW 2

11/8/2002 CS 258 HW 2 /8/ CS 58 HW. G o a a qc of aa h < fo a I o goa o co a C cc c F ch ha F fo a I A If cc - c a co h aoa coo o ho o choo h o qc? I o g o -coa o o-coa? W ca choo h o qc o h a a h aa a. Tha f o o a h o h a:.