Intelligent Robust Control for Uncertain Nonlinear Multivariable Systems using Recurrent Cerebellar Model Neural Networks

A Polyh Hug Vol. No. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No hu-hug h hg-hh hug F ho hh-m L 4* I J. Ru Elo Sy Rh Do hug-sh Iu of S hology o-yu E-l: hhh@u.u. Dp of Ell Egg Yu Z Uy hug-l o-yu E-l: 988@l.yzu.u. Dp of og S X Uy X h 4* opog Auho Shool of Ifoo S Egg X Uy X h; Dp of Ell Egg Ioo fo Bg D Dgl og Yu Z Uy hug-l o-yu E-l: l@u.yzu.u. Iu of Illg Egg Sy Joh o Nu Fuly of Ifo Óbu Uy 4 Bup Hugy -l: u@uobu.hu Ab: h pp lop llg obu ool lgoh fo l of u ol ulbl y by ug u-bll-olulo-ooll RMA lg o hology. h popo ool lgoh o of p RMA obu ooll. h p RMA g ooll ulz o l lg o ooll h p of h p RMA o-l u by h p l fo h Lypuo fuo. B o h H ool ppoh h obu ooll ploy o ffly upp h flu of ul ppoo o b h l lg o ooll h p RMA o h h obu g pfo of h y b gu. Flly opu ulo ul o hu ho u h-l obo pulo pfo o fy h ff fbly of h popo ool lgoh. h ulo ul of h h lop ool lgoh o oly gu h y bly bu lo h ll obu g pfo. 7

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No Kyo: u-bll-ol-ulo-ooll RMA; lg o ool; H ool; ol ulbl y Iouo I y ool of u ol y h b o of h op fo y ool gg. Vou ool ffo h b ulz o g lyz h u ol y. Slg o ool SM h b of poful obu h fo oollg h ol y h u ] ]. h o oug fu of SM o y p o f y po l ub jo ]. Ho pl pplo SM uff o g. O h u h y ol h y b fful o ob o. h oh h bu h gu of uy bou uo h lg uy bou of u o h obu h; ho h ll l h ool pu hg. Nul o NN po l g uh plll ful ol glzo poful ppoo pbl o h NN h b ppl fo y fo ool ]-]. So gf ul h h popy of NN p lg o h ufoly ppo by pu-oupu l o ol ppg o lo ub. B o h popy ub of h h popo h NN-b p lg o ooll hh ob h g of h lg o ool h obu h h NN h o-l p lg bly; o h h bly og obu of h y b po 7]-9]. Fo pl L Hu p NN-b hyb p lg o ool y 7]; h ppoh NN u opo ooll. I 8]. p uo-lg o ool h ulz o plll ul o o lz ul ool o ool; hu h y pfo b po h hg b l. I 9] D ou fo-b lg o ool h bou of u lo o u. Ho h bo ppoh uff h opuol oply. O h ul o uu p NN b lf ffo ul o FNN ] ] 8] 9] u ul o RNN 4] ] 7]. A o FNN ppg. Moo h gh up of FNN o o ulz h l o foo o h h fuo ppoo o h g. Fo RNN of pul h bly o l h yg pu o oupu hough h o ul pol opo 7]. hu RNN y ppg o goo ool pfo h p of uoll y. Ho o fo 8

A Polyh Hug Vol. No. FNN o RNN h lg lo ll h gh up ug h lg yl. hfo h ff of NN l pobl ug o-l lg. bll-ol-ulo-ooll MA lf o-fully o ppo-l o oy o h olppg p-fl ]; o ol h f z-gog pobl h lg fful uly lbl yp of ul o NN. opg o ul o MA po goo glzo pbly f lg bly pl opuo ] ]. h o h ly b ho o b bl o ppo ol fuo o o of o y uy ]-]. Fo h o MA h op ly fo h lo-loop ool of opl yl y lu 4]-7]. Ho h jo b of g MA h h pplo o l o pobl u o h h o uu. I o o ol h MA pobl p h g of SM h obu h h pp lop llg obu ool lgoh fo l of u ol ulbl y lg o hology. h popo ool y op of p u MA RMA obu ooll. h p RMA g ooll ulz o l lg o ooll h p of h p RMA o-l u by h p l. Moo b o h H ool ppoh h obu ooll ploy o ffly upp h flu of ul ppoo o b h l lg o ooll h p RMA o h h obu g pfo of h y b gu. Flly o pl p o uppo h ly of h popo ool lgoh. Sy Dpo o h h-o ulbl ol y p h follog fo: f u y h u u u u ] h ool pu o of h y y ] h y oupu o - ] h o of h y 9

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No f uo bu bou ooh ol fuo uo bu bou ool pu g ] l bou ub. Au h h ol ol of h ulbl ol y b p u f h f h ol fuo of f h ol o g of. By ppoply hoog h ool p ubly gg h ool pu h o b ho o b po f bl. If h l ub u lu h ulbl ol y b b ] Δ Δ u f f l u f h Δ f Δ o h y u l f o h lup uy f. Δ Δ u f l h b p oupu uo follo: ] l u f B A y 4 h I I I A I B I. h obj of ool y o g ubl ooll uh h h y o joy. ] - o bg h f h g o h g o o of h y f. ] h f joy y uo b p B A.

A Polyh Hug Vol. No. Subg 4 fo g A B f u l ]. Slg Mo ool Sy Slg o ool SM o of h ff ol obu ool h po y y h popy o u o h y y ooll h lg o ] ]. I gl SM g b o o ph h h hg ph h lg ph. h y joy h po of bfo hg h lg uf ll h hg ph. O h y joy hg h lg uf y o l log h lg uf o h og h lg ph. h h of h ooll y h lg o h y of h y by h ppf lg uf p of u. I o o pl SM h f p o l lg uf h ol h lo-loop pfo bl p. h g h ool l uh h h y jo fo o h lg uf y o. hu h lg hyppl b f : h uo: λ K 7 K λ I λ I I] f h ll oo of h I λ I λ I λ I 8 h op lf hlf-pl hh h Lpl opo. h po of SM b o o ph h h hg ph h h lg ph h. If h lg o o h lg uf h h oo of h y go by h l ffl uo p 7 ho bho by h lg uf g ] ]. hu h g o o y polly o zo o h pf g b ypolly h. hu h ool obj bo h g of ool l o fo. A uff oo fo h hbl of h lg hyppl h y p o hoo h ool l uh h h follog hg oo f: σ 9

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No h σ ll po o. g h of boh of 7 ug yl K K A f u l. hfo l lg o ooll u hh gu h hg ISM oo u fy h follog oo: K A f u l ] σ. If h y y h lup uy ly o l lg o ooll b g follo o fy uly u ISM f l K A σ g ] h g g fuo σ g σ... σ... σ. Ho pl pplo h yl fuo o ply o h lup uy ly uo. hfo h l lg o ooll uobbl. hu llg obu ool lgoh b o RMA lg o hology popo h follog o o h obu g pfo. 4 Illg Robu ool Algoh h ofguo of h llg obu ool lgoh hh o of p RMA obu ooll p Fg.. h ool y u o h follog fo: u u ARMA u R h u p RMA u obu ooll. h p ARMA R RMA u g ooll ulz o h l lg ARMA o ooll h p of h p RMA o-l u by h p l fo h Lypuo fuo. h obu ooll u ploy o ffly upp h flu of ul ppoo o b h l lg o ooll p RMA o h h obu g pfo of h y b gu. R

A Polyh Hug Vol. No. Fgu h ofguo of h llg obu ool y 4. Dpo of RMA A RMA popo ho Fg. hh o ly. h RMA opo of pu p oo oy p h u gh p-fl p gh oy p oupu p. Ru U p o o p o O p gh Moy Sp Ipu Sp I Oupu Sp O Rp-Fl - Sp R Aoo Moy Sp A Fgu Ahu of RMA

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No h gl popgo h b fuo h p b follo. Ipu p I : Fo g p p p p ] h h ub of pu bl h pu bl p u b uz o go ll l og o g ool p. h ub of l oluo. Aoo oy p A : Sl l b uul blo h ub of blo uully g h o ul o o. A o b oo oy p h opo. I h p h b blo pfo p-fl b fuo h u fuo op h h p-fl b fuo hh b p p p fo 4 b h p h oupu of h -h p-fl b fuo fo h - h pu h h. I o h pu of h blo b p p p h h u gh o h lu of hough ly. I l h h pu of h blo o h oy hh o h p foo of h o p y ppg. Fgu p h h g of o-ol RMA h f 4 h ub of l opl f blo; hh p o blo B B p o b blo B B. By hfg h bl l ff blo ll b b ob. Fo blo B B fo p blo B B fo p pobl hf l fo h o ly; B B f fo p B B f fo p fo h h ly; B g B h fo p B g B h fo p fo h fouh ly. h p-fl b fuo of h blo h p h h jubl p. 4

A Polyh Hug Vol. No. Ly 4 Ly Ly Ly Vbl p B B. f h 8 B B f B f.9 4 7 S.8.8 B. b B B B g -. B B B b b -.9 B B Vbl p B g Bg -. -. -.9 -...9. B B b Ly B B B 7 B f Ly Ly B 4 B h g 8 Ly 4 Fgu A o-ol RMA h 4 Rp-fl p R : A fo by blo f o B B B ll p-fl. h ub of p-fl ul o B b b h uy. h -h ul-ol p-fl fuo f b p p p fo h ] ] ] b p o fo ] f. h ul-ol p-fl fuo p 7 h ] ] ]. 4 gh oy p : Eh loo of R o pul jubl lu h gh oy p b p o p ] p 8 o o p p o o

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No h p p p p ] p lu of h p-h oupu o h h -h p-fl. o h og gh Oupu p O : h oupu of RMA h lgb u of h gh h gh oy p o p p p fo p. 9 o h oupu of RMA b p o oo o o o o ]. p O I h o-ol ho Fg. h oupu of RMA h u of h lu p-fl B B B B B b b f B f BgBg h h pu.8.8. h hu of RMA g o h h g of pl uu h y h. h ol of h u loop o o h p lu of h p-fl b fuo h oo oy p. hu h RMA h y h. 4. Robu ooll Dg Subg fo yl u u] σ g ]. ISM Au h opl RMA o ooll u uh h ISM * u o h l lg ARMA * * * * * * * u u p ε ε ISM ARMA h ε... u ou o o;... ] * * * * * h opl p o of ply. Ho h opl RMA o b ob; hu g RMA u o h opl RMA. Fo h ool l b follo: u u p u u ARMA R R h h o of * * * * * ply. hu h y uo b p u * ARMA ε u ARMA u R ] σ g ]

A Polyh Hug Vol. No. 7 ] ] * * g R σ u ε ] ] * g R σ u ε 4 h * *. Moo h lzo hu ploy o fo h ul-ol p-fl b fuo o plly l fo. h po of ylo b ob β * * * β h ; ; * ; * ; * β o of hgh-o. Moo f ] ] 7 ]. 8 Rg b ob h β *. 9

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No Subug 9 o 4 yl β β ε ur] σ g ] * β ε ur] σ g ] ω u ] σ g ] * h h ppoo o ω β ε. I of h of ω o pf R H g pfo 8] ] h gol po o lg- pb uo o. If h y h l oo h h g pfo b up L ] H h. h ho h uo ll b h ppoo o y oupu fuo. If h h of u o g ool hou ppoo uo 8]. hfo h follog ho b po. ho : o h h-o ulbl ol y p by. h llg obu ool y f hh h p l of RMA g - h obu ooll g 7. h h obu g pfo b h fo h pb uo ll... h R=g ] gol. 4 8

A Polyh Hug Vol. No. 9 R R R u I. 7 Poof: h Lypuo fuo g by V. 8 g h of h Lypuo fuo ug yl V ] ] g R σ u ω 9 I b o h ] g σ o 9 b ] V ] R u ω. 4 Fo - ug 7 4 b V ] ] ] ]. 4 Aug ] L gg h bo uo fo o yl V V ]. 4

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No S V h bo uly pl h follog uly V. 4 Ug 8 h bo uly ul o h follog ] hu h poof opl.. 44 Sulo Rul o llu h ff of h popo ool y ppl o ool hu ho u h-l obo pulo. Moo p fuzzy ul o ooll AFNN 9] h popo RMA ppl o h o y fo opo. Epl : hu ho u A ypl hu ho u o of o l o R o po o uo L o ol o g ho Fg. 4. L R L u u u g Fgu 4 hu ho u h y uo of h hu u ] g u 4 R u L 4 R

A Polyh Hug Vol. No. L u L 47 h u u u u ] o h ool pu ] o h l ub. h h u h bl. hu h olg L o of ho y f ] ]. h y uo 4- L 47 b f u 48 h g R f g. L R L L.. p h l ub g... o p p.. h phyl p of ho u u R R ΔR g g Δg L L ΔL R L Δ Δ h g h ol lu ΔR Δg ΔL Δ Δ o h uo ol -yg pubo 9]. h ol lu g R g. L.. h -yg pubo R / Δg. Δ ΔL. Δ..o / Δ.. h jo o fo h f ol oupu h ho 4 4 h h pu gl o h f ol. h l oo of h hu ho u h f ol g. h f pu u po gul gl. Fo h popo ool h h lg hyppl g. h popo RMA hz :

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No ub of pu bl: ub of l fo h bl: l glzo: 4 l/ blo f ub of blo fo h bl: blo/ly 4 ly b 8 blo ub of p-fl: p-fl/ly 4 8 p-fl ly p-fl b fuo: p p / ] fo 8. h pu of RMA ; hl h pu p of pu gl olz h { ] ] ]}. h l of h u fuo ully ].8..4.4..8] h 4 7 8 l. fo 8. h lg of RMA l I8 8. I 44 h pf uo o gol R.I. h ulo ul of AFNN fo h hu ho u ho Fg.. h jo of h y plo Fg. - fo L ply. h o ool ffo u u u p Fg. -f. Moo h lg hyppl ho Fg. g-. h ulo ul of RMA fo h hu ho u ho Fg.. h jo of h y plo Fg. - fo ply. h o ool L ffo u u u p Fg. -f. Moo h lg hyppl ho Fg. g-. Fo h ulo ul b h h popo RMA po b ool pfo h ll g o h h AFNN.

A Polyh Hug Vol. No. Fgu h ul ulo of AFNN fo h hu ho u - h jo of h y -f h o ool ffo g- h lg hyppl

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No po olg ol ool pu ol u f po olg ol Slg hyppl b g Iuo u p L Slg hyppl h ool pu p u Slg hyppl ool pu p u Fgu h ul ulo of RMA fo h hu ho u - h jo of h y -f h o ool ffo g- h lg hyppl 4

A Polyh Hug Vol. No. Epl : A h-l obo pulo Fgu 7 A h-l obo pulo A h-l obo pulo p Fg. 7. h y uo g follo ]: τ τ g M 49 h 4 4 4 M 4 4 4 4 g g g g..o. τ.

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No I 9 ] h gul poo o h jo loy lo o ply τ h pu ou o g h gy o M h h ool/pl τ h l ub. h lo of gy g 9.8 /. h l ; h l lgh; h ho h oo f o ; f bl. I bl o h o of g. h l of y p g bl... ]. ] 4 bl h y p of obo pulo... j ] j j.. ]...4.. g. g. g 4. g.8 g.7 g h y uo b p f u h Δ ] ] f M g ] M M τ u ]. h f jo b f ol oupu uo fuo ff. h. h f ol b... fo. h l oo of h obo pulo g.... h l oo of h f ol g. h f pu u po gul gl. h. uo fuo o u. Fo h popo ool h h lg hyppl g. h popo RMA hz : ub of pu bl: ub of l fo h bl: l j

A Polyh Hug Vol. No. glzo: 4 l/ blo f ub of blo fo h bl: blo/ly 4 ly 8 blo b ub of p-fl: p-fl/ly 4 8 p-fl ly p-fl b fuo: p p / ] fo 8. h pu of RMA ; hl h pu p of pu gl olz h {..]..]..]}. h l of h u fuo ully ]...9...9..] h 4 7 8 l. fo 8. h lg of RMA ho I8 8. I 44 h pf uo o gol R.I. h ulo ul of AFNN fo h h-l obo pulo ho Fg. 8. h jo of h y plo Fg. 8-f fo ply. h o ool u u u ffo hyppl p Fg. 8g-. Moo h lg ho Fg. 8j-l. 7

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No 8

A Polyh Hug Vol. No. Fgu 8 h ul ulo of AFNN fo h h-l obo pulo -f h jo of h y g- h o ool ffo j-l h lg hyppl h ulo ul of RMA fo h h-l obo pulo ho Fg. 9. h jo of h y plo Fg. 9-f fo ply. h o ool ffo u u u p Fg. 9g-. h lg hyppl ho Fg. 9j-l. Fo h ulo ul opo h popo RMA lo h b ool pfo h ll g o h h AFNN. Moo h hg phoo AFNN h b uh u by pplyg RMA. 9

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No

A Polyh Hug Vol. No. Fgu 9 h ul ulo of RMA fo h h-l obo pulo -f h jo of oluo h y g- h o ool ffo j-l h lg hyppl h pp popo llg obu ool y fo l of u ol ulbl y lg o hology. h popo ool y o of p RMA obu ooll. h p RMA g ooll ulz o h l lg

-H. h l. Illg Robu ool fo U Nol Mulbl Sy ug Ru bll Mol Nul No o ooll h p of h p RMA o-l u by h p l fo Lypuo fuo. B o h H ool ppoh h obu ooll ploy o ffly upp h flu of ul ppoo o b h l lg o ooll p RMA o h h obu g pfo of h y b gu. Flly h ulo ul of o ulbl ol y h o h ff of h popo ool h. Rf ] J. J. E. Slo. P. L Appl Nol ool. Egloo lff NJ: P-Hll 99 ]. M. L Y. J. Mo Douplg ool by Hhl Fuzzy Slg-Mo ooll IEEE. ool Sy hology Vol. No. 4 pp. 9-98 ]. M. L. F. Hu Supoy Ru Fuzzy Nul No ool of g Ro fo Sl Dl g IEEE. Fuzzy Sy Vol. No. pp. 7-74 4 4] J. H. P S. H. Huh S. H. K S. J. So.. P D Ap ooll fo Noff Nol Sy ug Slf-Suug Nul No IEEE. Nul No Vol. No. pp. 44-4 ]. F. Hu. M. L R.. Yh Supoy Ap Dy RBFb Nul-Fuzzy ool Sy Dg fo Uo Nol Sy Appl Sof opug Vol. No. 4 pp. - ]. M. L A. B. g. F. Hu. M. hug Ap ool fo MIMO U Nol Sy ug Ru l Nul No Iol Joul of Nul Sy Vol. No. pp.7-7]. M. L. F. Hu Nul-No Hyb ool fo Alo Bg Sy IEEE. Nul No Vol. 4 No. pp. - 9 8]. H. H. Y. hug F. M. Yu Nuo-Slg Mo ool h Applo o S Sy IEEE. Nul No Vol. No. pp. 4-4 4 9] F. D Dlz Slg Mo Ap ooll Dg B o Fuzzy Nul No fo Io U Nol Sy IEEE. Nul No Vol. No. pp. 47-48 ] J. S. Albu A N Appoh o Mpulo ool: h bll Mol Aulo ooll MA J. Dy. Sy. M. o. Vol. 97 No. pp. -7 97

A Polyh Hug Vol. No. ] F. J. ozlz-so A. R. Fgu-Vl A. A-Roguz lzg MA Ahu g IEEE. Nul No Vol. 9 No. pp. 9-4 998 ] J.. J S. L. Hug Hgh-O MS_MA Nul No IEEE. Nul No Vol. No. pp. 98- ]. M. L Y. F. Pg Ap MA-b Supoy ool fo U Nol Sy IEEE. Sy M yb P B Vol. 4 No. pp. 48-4 4]. M. L H. Y. L A Nol Ap l Fuzzy bll Mol Aulo ool Sy Dg fo Vo ol Moo IEEE. Iul Elo Vol. 9 No. 4 pp. 4- ]. F. u P. S. F. R. hg L. S. g Ap Fuzzy MA ool fo l of Nol Sy h Sooh opo IEE Po.-ool hoy Applo Vol. No. pp. 47-7 ]. H. L F. Y. hg. M. L A Eff Il yp- Fuzzy MA fo ho -S Po Syhozo IEEE. o yb Vol. 44 No. pp. 9-4 4 7]. M. L H. Y. L Dy P Fuzzy bll Mol Aulo ool Sy Dg fo Mg Lo Sy IEEE. ool Sy hology Vol. No. pp. 9-99 8] B. S. h. H. L Y.. hg g Dg of U Nol SISO Sy: Ap Fuzzy Appoh IEEE. Fuzzy Sy Vol. 4 No. pp. -4 99 9] F. J. L P. H. Sh Ap Fuzzy-Nul-No ool fo DSP-b P Mg L Syhoou Moo So D IEEE o o Fuzzy Sy Vol. 4 No. 4 pp. 48-49 ] Y.. hg Robu H ool fo l of U Nol -Vyg Sy I Applo IEE Po.-ool hoy Applo Vol. No. pp. -9 4 ] E. K Oupu Fb g ool of Robo Mpulo h Mol Uy Ap Fuzzy Log IEEE. Fuzzy Sy Vol. No. pp. 8-78 4 H