NEURO ADAPTIVE COMMAND SYSTEMS FOR ROCKETS MOVE

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1 Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN NEURO ADAPIE COAND SYSES OR ROCKES OE Romls LUNGU ihi LUNGU Avionis Dprtmnt Univrsit o Criov lt o Eltrothnis Blv Dbl No7 Criov Dolj ROANIA romls_lng@hooom Lm@hooom Abstrt thmtil mols o th mov o th rokts with big ttk ngls ommn lws n tomt nro ptiv sstms r prsnt Using sh strtrs on stis longitinl mov o n gil rokt tlb/simlink mol is lso obtin A lot o grphil hrtristis r prsnt in this ppr Kwors: nrl ntwork ptiv rokt mov longitinl AHEAICAL ODELS h mol o th rokt s mov is th on rom [] os sin os sin sin os () os sin tg os os sin sin os sin os whr is th light vloit ttk ngl sislip ngl o ths on s th momnts qilibrim qtions whr () r th ronmi momnts whih oprt ron o o o s p p ; () or vr goo ontrol o th gil ir ir rokts inlintion in [] n [] on intros n ronmi roll ngl is s; it vriis qtion os sin sin os tg (4) os h nglr vribls r grop in vtors (5) h son n th thir qtion () n qtion (4) m b prss nr th vtoril orm (6) whr os tg sin tg sin os os / os sin / os sin os os os sin sin os sin os tg NEURO ADAPIE COAND SRUCURES (7) Eqtion (6) is qivlnt with th ollowing qtions sstm in whih omponnt o th pso-ommn is istingish [4] (8) Similrl qtion sstm () m b srib b qtions in whih nothr omponnt o th pso-ommn is istingish (9) B noting with th ling ommn (vtor is th impos on) pso ommn m hv th orm p 6

2 Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN K () whr K is th bk oiints mtri (positiv in) Conorm to qtion (8) on m llt nglr vloit o th rokt sing n nmi invrsion trior loop () Pso ommn m b hosn o orm K () whr K is th bk oiints mtri (positiv in) n ptiv ommn or invrsion rror s ompnstion h ommn vtor m b llt with orml [] ˆ G () or with th ollowing qtion G K ; (4) is llt with th ollowing qtion (5) Anothr ontrol strtr m b obtin sing stbilit thor with Lipnov ntions i th ontroll objt (A) m b srib b th non linr qtions sstm [5] h (6) In igr on prsnts th blok igrm or bov qtions moling igr : Blok igrm or qtions (6) Lt s onsir now th ollowing s: sb sstm srib b ntion is wll known whil th on srib b is proimtl known; tht s wh th ommn mst b tr vribl h impos vtor is q t (7) his lw mst ssr th stbilit o th vribl in rpport with vribl (igr 4); q t (8) h son sb-sstm (srib b th son qtion (6)) to th lk o th trnl istrbns p ( t) m b srib b qtion v v (9) whr inpt v is pso-ommn I th ntion is invrtibl thn th nmi invrsion o m b pproimtl on; v I is known thn n i it is pproimtl known thn th invrsion o ntion is m with rror n th irst qtion (9) boms v p () whr hs th orm ˆ () v igr : Aptiv ommn sstm bs on nmi invrsion (vrint ) 6

3 Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN with ˆ llt ntion h ommn lw m b hosn [5] v v K v () whr th ommn in s K positiv in mtri n ptiv ommn or th invrsion rror ompnstion obtin rom th Sigm nrl ntwork; W I () with th tivtion ntion o th hin lr () I th inpt vtor W b w v (4) i ij i ij b n i i bis w ij th wights o onntions btwn lvl n v th wights o onntions btwn lvl n h strtr o sh nrl ntwork is prsnt in igr igr : Nrl ntwork s strtr Lrning rl is obtin sing Lipnov stbilit thor s blow [] Consiring robnis norm o mtri A A tra A (5) introing th ompt mtri ij W Z (6) with Z Z hoosing th inpt vtor o th nronl ntwork Z (7) I n stnr Lipnov ntion l trw W tr W (8) rom stbilit nlsis on obtins th trm v rom () v K Z Z (9) whr K n / h ontrol sstm strtr (PA-A) is prsnt in igr 4 In prtilr sstm (6) rprsnts non linr mol o n irrt (A) whih m b or mpl n ir ir rokt hs ; () rom th qivln o qtions (6) n (6) it rslts h n o orms (7) () with h son qtion (6) is qivlnt with th qtions sstm (8) whr n () whih m b sprt in two omponnts Eqtion (9) is qivlnt with qtions sstm (9) whr n v Compring qtions () n () on rslts tht mtri K pls th rol o mtri K n th rol o hs pls th rol o Compring now qtions (7) with () on obtins q t () with pso ommn o orm () p igr 4: Aptiv ommn sstm bs on nmi invrsion (vrint ) 6

4 Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN igr 5: Aptiv ommn sstm bs on nmi invrsion (vrint ) Consiring th inpts o th nrl ntwork rom igr 4 on obtin blok igrm rom igr 5 whr th rol o v rom (9) is pl b ; (9) boms K Z Z (4) whr / NEURO ADAPIE COAND OR HE ROCKE S OE Lt s onsir now th s o rokt s longitinl movmnt srib b qtions [5] s 4 (5) 57 whr s 4 Blok igrm o th los loop sstm (PA - A) is prsnt in igr 6; it hs bn obtin sing igrm bloks rom igr 4 n igr 4 B intiition o sstm (5) with sstm (6) n o sstm (9) with (6) on obtins v v h 4 (6) h vls o th othr prmtrs rom ig5 r: Z 5 K 6 or th lls o oiints k n k on kps onl th linr prt o th sstm rom ig5 ( ) Clos loop trnsr ntion 4 s H s is llt n it s prss s s ollows H s H s s k s k k k k s k k s s k k (7) B stting 77 n 5 th two oiints hv prssions k k k k k On hooss -orwr nrl ntwork with 8 (8) igr 6: Nro ptiv ommn sstm or th rokt s mov in vrtil pln sing mol (5) 64

Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN 84-485 inpt nrons nron on th hin lr n n otpt nron Ativtion

p n h nrl ntwork s otpt hs th orm W tnsig I (4) whr is th wights vtor o th inpt nrons W is th wights vtor o th hin lr

igr 7: im vrition o th rokt s ttk ngl (withot nrl ntwork) Using tlb/simlink mol rom igr 8 on obtins t (ig9) n rror t t t

5 Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN inpt nrons nron on th hin lr n n otpt nron Ativtion ntion or th hin lr nron is linr on whil tivtion ntion o th nrons rom th inpt lr hs tnsig orm (tngnt hprboli) tn sign (9) p n h nrl ntwork s otpt hs th orm W tnsig I (4) whr is th wights vtor o th inpt nrons W is th wights vtor o th hin lr nrons I inpts vtor I U Z (4) W [] Nglting trms n th iniil rspons (ig6) provs stbilition o ngl to its impos vl gr igr 7: im vrition o th rokt s ttk ngl (withot nrl ntwork) Using tlb/simlink mol rom igr 8 on obtins t (ig9) n rror t t t (ig9b) igr 8: tlb/simlink mol o th sstm rom igr 6 b igr 9: Dnmis o ttk ngl n rror or th sstm rom ig5 (with nrl ntwork) 65

6 Annls o th Univrsit o Criov Eltril Enginring sris No 8; ISSN CONCLUSIONS his ppr prsnts nonlinr mthmtil mols o th rokts mov with big ttk ngls lws n tomt ptiv ommn strtrs sing nrl ntworks On stis nro ptiv ommn sstm or th gil rokt s mov Rrns [] R Lng Lng N l C Cpis Stbilition ols n Strtrs or ov o vr nvrbl ling Objts h 6th WSEAS Intrntionl Conrn on Sstm Sin n Simltion in Enginring (ICOSSSE '7) ni Itl Novmbr - 7 pp 5 8 [] A Isiori Nonlinr Control Sstms Springhr Brlin 995 [] B rln A Clis Nrl Ntworks or Stbl Aptiv Control o Air to Air issils Proings o th AIAA Gin Nvigtion n Control Conrn Bltimor rln 995 pp [4] B rln A Clis Nrl Aptiv Nonlinr Atopilot Dsign or Agil Anti Air issil Proings o th Gin Nvigtion n Control Conrn Sn Digo Cliorni l 996 pp 8 [5] B rln A Clis Robst Aptiv Control o Unrtin Nonlinr Sstms Using Nrl Ntworks Proings o th Amrin Control Conrn Albqrq Nw io n

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören ME 522 PINCIPLES OF OBOTICS FIST MIDTEM EXAMINATION April 9, 202 Nm Lst Nm M. Kml Özgörn 2 4 60 40 40 0 80 250 USEFUL FOMULAS cos( ) cos cos sin sin sin( ) sin cos cos sin sin y/ r, cos x/ r, r 0 tn 2(