High Performance Adaptive Robust Control for Nonlinear System with Unknown Input Backlash

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1 Jon 48h IEEE Confeene on Deon n Cono n 8h Chnee Cono Confeene Shngh, P.R. Chn, Deebe 6-8, 9 FB.4 Hgh Pefone Ave Rob Cono fo Nonne Sye wh Unknown In Bkh Jn Go, Bn Yo, Mebe, IEEE n ASME, Qngwe Chen n Xobe W Ab A hgh efone ve ob ono (ARC goh eveoe fo of nonne ye wh nknown n bkh, e nene n nen nonnee bne. De o he nheen yn nonne ne of bkh, exng ob ve ono eho ny fo on ng on-ne ee on fo n oxe nveon of he nknown bkh. A e, even n he bene of ohe nen nonnee n bne, yo o kng n neve be heve by onno onoe fo he non-ooh he of bkh. Exeen e o how h ne onoe one efoe bee hn onoe nng he eee bkh nvee wh oey ee o oveee bkh g. Unke ny exng ono hee, he bkh nvee no one n h e. A new ney eeze oe fo bkh eene n he ojeon goh e fo ee eon. he ooe ve ob ono w on of hee : oe oenon e, he ob ono e n ob feebk e. he ARC ene h oe-oo gn e bone n heve he kng whn he ee eon. Son e e he efone of he ARC. B I. INRODUCION ACKLASH he e oon n ono ye h evoehn, eeon ey n eeogne eve wh hyee. I one of he o on nonnee h he efone of ee n oon ono n n, obo, oove, oon n ohe on. he ono of ye wh bkh h been he bje of y ne 94. Lne onoe hve been nvege, nng PID onoe, hgh-oe ne onoe, e feebk onoe. Fo bkh nonney, nvee oenon of bkh e egne wey boh n non-ve n n ve eng[. An ve nvee of bkh w on o ne he effe of bkh nonney n Mn eeve Feby 5, 9. h wok w oe n by gn fo Non Lbooy Inegen Cono of Chn n n by Doo Pog Fonon of he Mny of Eon of Chn (788,he Non N Sene Fonon of Jng Povne of Chn(BK844 n Jng Senf n ehnoog So Pn - In (BE96. Jn Go, Qngwe Chen n Xobe W e wh he Shoo of Aoon, Nnjng Unvey of S. n eh., Nnjng, Chn. he e- e: goj@.nj.e.n, n wxb@.nj.e.n. Coeonng hone: Bn Yo wh he Shoo of Mehn Engneeng Pe Unvey, We Lfyee, IN 4797 USA(e-: Byo@e.e. [,[, b he n onon ho be y e. A ooh nvee of bkh w eveoe o oenon he effe of bkh wh bk-eng oh n [4, whee he evon of he ono n w e o ge he onoe, whh ybe nvbe. Bkh oenon ng ne newok[5,[6 o fzzy og [7,[8 h been e n feebk ono ye. Fo hoe negen oenon, ne newok o fzzy og wee ny e fo neon of he nveon eo fo he exeen nonney oxon by. he oon fee of he nvee hee h hey ey on he onon of n nvee bkh o ge he effe of he bkh nonney. Exeen by [9 how h ne onoe one efoe bee hn onoe nng he eee bkh nvee wh oey ee o oveee bkh g, he eon beng h eeen noe ne heng n he nvee. I w noe h he ne onoe one o vee he bkh g y ne ony he oo oen of ne (n no he o ven ne he bkh g. A one o n he een oehenve vey e wh 96 efeene on onong ehn ye wh bkh[ h wek on n he bkh g (.e., by y oweng he oe-oo ye bnwh when he g oen gh be vngeo. In h e, new ney eeze oe fo bkh eene n new oh fo ve ob ono (ARC of ne o nonne ye wh bkh noe who onng he bkh nvee. Be ony on he nve one n ee-we eon of bkh, he nknown bkh nonney ney eeze goby wh bone oe eo. he ARC onoe on of hee : one he oe oenon wh he hy ee ee. Fo he eon, e ooon feebk e o bze he non ye. he one ob feebk e e o ene he effe of vo oe nene. he ARC ono w ene h oe-oo gn e bone n heve he kng whn he ee eon. Coe on e e o o e he effevene of he ohe. h e ognze foow: Seon II e he obe of h noe, whee he ne oe of bkh noe. In eon III, he ooe ARC hee eene. In eon IV, on e e eene o /9/$5. 9 IEEE 7675

2 FB.4 e he effevene of he ooe oh. In eon V, onon e wn. II. PROBLEM SAEMEN A. Sye Moe Fo y, he foowng of eon-oe nonne ye onee: x& = x x& = w + φ( + ( ( y = x w = B( whee x = [ x he e veo, w, y e he n n o eevey. φ( x, = [ φ(, L, φ ( he known he fnon, = [,, L he veo of nknown onn ee, n ( eeen he e nen nonney nng exen bne. he o nonney w = B( ebe bkh he. he ono objeve o egn ono w fo ( o ene h oe-oo gn e bone n he kng eo y( y ( jbe whn he ee eon. Aon : e n known bone e Ω :, } Ω { : j n j j x, j =, L Aon : he nen nonney n be bone by ( B whee B ove onn. B. Bkh Che A bkh nonney hown Fg. w( B Fg. Bkh Moe ony, n be ebe by: B w( = B( ( B f & > n w( = ( B ( = ( B f & < n w( = ( B ohewe whee he onn oe of he ne, B, B e onn ee. w( en no hnge o n w (. Fo he eveoen of ono w, he foowng on e e: Aon : he bkh ee, B, B e nknown, b he gn e known. In gene, e >, B >, B <. Aon 4: he bkh ee e whn known bone: < n x < ( B n B ( B x ( B n B ( B x < Aon 5: he bkh o w( no vbe fo eeen. Fo he bove, we n ewe he bkh oe ( w( = B( = + b ( ( Whee b ( oe eo fo ney eeze of bkh n n be e fo ( n ( B f & > n w( = ( B b ( = B f & < n w( = ( B (4 ohewe Le : wh (4 n on, b ( bone, n fe b ( ρ (5 whee ρ he e-bon, whh n be hoen ρ = x{( B, ( B } x Cobng ( n (, one obn x& = x x& = + φ( x + b ( + ( y = x Defne onn n e-vyng ( h h + ( = b ( + ( (7 Coney, (7 ve he bkh oeng eo b n he ogn ye nen nonney ( no he oonen(o ow feqeny oonen n ey n he hgh feqeny oonen (. By n (6 7676

3 FB.4 Aon n Aon 4 n (9, one obn Ω { : } whee ( Ω { n e known. : } III. DESIGN OF ADAPIVE ROBUS CONROLLER In h eon, he ve ob ono (ARC egy [[ w be egne. A n [[, he f e o e ojeon ye on w e o heve onoe enng o on oe ee n he foowng. Le ˆ enoe he eon of, n be he = ˆ. he foowng ojeon-ye eon eo ee on w[4 e: & ˆ = P oj ( Γτ, ˆ( Ω ˆ f ˆ = x n > P oj = f ˆ ˆ ( = n n < ohewe whee Γ gon x of on e n τ n on fnon o be ynheze fhe on. Sh ee on w h he foowng ebe oee. A ny e nn,.e., : (P ˆ ( Ω (P ( τ Γ P oj ˆ ( Γτ Defne = e& + ke = y& y& + ke (9 = y& ( y& ke x xeq whee : xeq y& ke, e = y y he o kng eo n k > ove gn. If o onvege o zeo, he o kng eo e w be o onvege o zeo ne G = E( / P( = /( + k (8 ( be nfe fnon. So, he e of he egn o ke obe. Dffeenng (9 n nong (6 (7, one obn & = x& x& eq ( = + φ x& eq + + ( h: & = + ( φ x& eq + + ( ( = + ϕ + ( whee = [ ϕ = [ φ x& eq ( = ( he foowng ARC ooe o egn foow: = + = + = ˆ ϕ ˆ& = P oj( Γ ϕ ( ( In (, eeen he oe oenon wh he hy ee eeˆ,whh e by ng n on-ne on ojeon goh (. eeen he ob ono e n whh e ooon feebk o bze he non ye n ob feebk e e o ene he effe of vo oe nene fo gnee ob ono efone n gene. Nong Aon, Aon 4, n (P, hee ex h h he foowng wo onon e fe: ( (4 ( ( + ϕ + ( ε (5 whee ε n e egn ee whh n be ε by. Een,(5 how h ynheze o one he oe nene ong fo boh e nene n nen nonnee. An (4 o ke e h ve n ne o h oen nefee wh he fnony of he ve ono. Rek : One exe of fyng (4 n (5 n be hoen : = [ x ϕ + x (6 4ε whee =, = x x n x n 7677

4 FB.4 x n = [ = [ x n n x n x n I n be ove ey o how h he bove hoe of oe fy (4 n (5. Poof: he onon ( ey o fe by he ono w (6. Fo onon (, we n ge ε = [ ( x ϕ + x + ( ε ε ( x ϕ + x ε ε = x ϕ + x ( ϕ + ( h: ( + ϕ + ( ε So he onon ( o fe. heoe: Cone he ye ( onng of he ve ob onoe gven by ( n he ojeon ye ee on w (, gn n he eng oe oo ye e bone, n he o kng gnee o hve ebe nen efone n fn kng y n he ene h he kng eo nex bone by: k ε k ( e ( + [ e (7 k Poof: Cone he foowng Lynov fnon: V = hen he evon of V V& = & = ( + ϕ + ( + ( + ϕ + ( h: + ε V V + ε x & (8 ε V ( ex( k V ( [ ex( k k + (9 hen (7 e. he heoe how h he o kng eon gnee by eng ε n k eevey. heoe: Cone he ye ( onng of he foowng ee n he ve ob onoe gven ((, In he bene of nen nonnee(.e., ng ( =, yo oon kng o heve. Poof: Cone he foowng Lynov fnon: V = + Γ Nong (8 n (, hen he evon of V V& & = & + Γ & = ( + ϕ Γ ˆ + ( + ϕ Γ P oj( Γ ϕ + + [ ϕ Γ P oj( Γ ϕ By he Bb e, foow h ( =, whh e h e( =. heoe ne h he yo oon kng n be heve f he ye wok ony on he nge oe e of he bkh (.e., b ( onn b ( = b o ( =. b b IV. SIMULAION In h eon, he ooe ARC goh e o he foowng nonne ye: x& = x x& = w( + + ( y = x whee w( e + e x x + x n( x& + x he o of bkh ebe by: n(.[.5 & ( > n w( =.[.5 w( =.[ (.8 & ( < n w( =.[ +.8 ohewe he ve of hy ee e e, = =, =. 5 n he bon of he e hoen o be [.5,.5, [,, [.5,..he bkh ee of =., B =. 6 n B =.96 e e no known b whn he known nge of [.5,.8, B [.,., n B [.5,.5 eevey. he bne ( =.5n.he ono objeve o e he 7678

5 FB.4 ye foow he ee jeoy: y ( =.5[n( + n( + n( he n ve e hoen x ( = [.,, ( =, ˆ ( =, Γ = I5, k = 8, ε =.. Ang ng eo =.e, he on e e hown n fge -4. Fg. how he oon kng efone of ARC. Fg. how he oeonng kng eo n Fg.4 how he n ono gn of bkh. y,y kng eo Fg. kng efone of ARC Fg. O kng eo of ARC y y V. CONCLUSION In ono ye, bkh wh nknown ee n hy oonen y eveey he efone of ono. In h e, n ve ob ono ooe fo of onno-e nen nonne yn ye eee by n nknown n bkh. By ng gob ne oe of bkh, he ARC onoe eveoe who onng bkh nvee. he new ono w ene oe-oo gn e bone n heve he kng eon whn he ee eon. REFERENCES [ M Non, Pe-Oof Gn, Conong ehn ye wh bkh vey, Ao, vo.8, no., ,. [ o, G., Kokoov, P, Ave ono of ye wh bkh. Ao, vo.9, no.,. 5, 99. [ o, G., Kokoov, P, Conno-e ve ono of ye wh nknown bkh. IEEE n. Ao Cono., vo.4, no.6,.8 87,995.. [4 Jng Zho, Chengjn Zhng, n Chngyn Wen, Rob ve o ono of nen nonne n wh nknown bkh Nnonney, IEEE n. Ao Cono., vo.5, no.,.5-59, Mh 7. [5 Rko R. Se, n Fnk L. Lew. Bkh Coenon n Nonne Sye Ung Dyn Inveon by Ne Newok. In Po. of he EEE Inenon Confeene on Cono Aon, 999, Hw,. -7. [6 J. O. Jng, M. K. Son, n H.. Chng, Fon n o bkh oenon of ye ng ne newok n fzzy og, n Po. Ae. Cono Confeene., Boon, 4, [7 Chn-Y S, Mho Oy, n Heny Hong, Sbe Ave Fzzy Cono of Nonne Sye Peee by Unknown Bkh-Lke Hyee, IEEE n. Fzzy Sye., vo., no.,.-8, Feby [8 B. Yo n M. ozk, Ave ob ono of SISO nonne ye n e- feebk fo, Ao, vo., no. 5,. 89 9, 997. [9 Den, S. R. H., Sgeno, B. W., n Iono, H. N. Exeen evon of bkh nvee e o evooo wh ge n. n Po. 4h IEEE onfeene ono on,995, [ B. Yo, M. Aje, n M. ozk, Hgh efone ob oon ono of hne oo: n ve ob ono oh n ove exeen. IEEE/ASME non on Mehon, Vo., No.,.6-76, 997. [ B. Yo n M. ozk, Sooh ob ve ng oe ono of no wh gnee nen efone, n. of ASME, Jon of Dyn Sye, Meeen n Cono, vo. 8, no. 4, , 996. [ B. Yo, Hgh efone ve ob ono of nonne ye: gene fewok n new hee, n Po. of IEEE Confeene on Deon n Cono, , 997. [ L. X n B. Yo, Ave ob eon oon ono of ne oo wh neggbe ee yn: heoy n exeen. IEEE/ASME n. Mehon, vo.6, no.4, ,. [4 Goown, G. C., D. Q. Myne, A ee eon eeve of onno e oe efeene ve ono. Ao, vo., no.,.57-7, Fg.4 Cono gn he n of bkh of ARC 7679

flbc in Russia. PIWiREE COHORTS ARE NOT PULL- ING TOGETHER. SIGHTS AND SCENES IN ST. PETERSBURG.

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