TAP CHl KHOA HQC A C6NG NCHg CAC TRVr6NG P

TAP CHl KHOA HQC A C6NG NCHg CAC TRVr6NG P»l HQC KV THliAT * S6 89 - 2012

DltV KHiftN TRU'(;)(T THfCH NGHI G I A N T I A P DtJNG Mi^NG RBF INDIRECT ADAPTIVE SLIDING MODE CONTROL USING RBF Ddng Sr Ihiin ChSu, Duimg HoAl NghTa, Nguyin Diix: Thdnh

Tnr&ng Dgi hgc Bdch khoa - DHQG Tp, Ho Chi Minh Nh$n ngdy 31 th&ng 10 nfiin 2011, ch^p nh$n ddng ng&y 1 thdng 6 nam 2012

T6IVI TAT

Bdi bdo ndy di xuit mdt phuxmg phdp diiu khiin thich nghi gidn tiip cho hi phi tuyin.Md hinh dii fuong dux?c nh$n d^ng tn/c tuyin dgs trin md hinh m^ng na-ron hdm co s& xuyin tdm (goi Id m$ng RBF). 0(/s trin md hinh nh$n ddng duryc, tdc gli di xuit phuong phdp diiu khiin hi phi tuyin theo phuong phdp truot. Lu$t diiu khiin di/a trin md hinh duoc cip nh$t liin tgc. Tinh 6n dinh vd h^i tg cda lu$t dmu khiin duoc chdng minh ddng /y thuyit cda Lyapunov. CAc md phdng trin con lie nguoc duoc 5t> dgng di minh hga cho phuong phdp da di xuit.

lir khda' Oi^u khidn thich nghi gidn tidp. m^ng ncy-ron hdm c o sd xuyfin tdm.

ABSTRACT

This paper describes an indirect adaptive sliding mode control The models of the system are identiTied on - line based on radldal basis function neural network (RBF neural network). Depending on the Identified model, a sMing mode controller is proposed for nonlinear systems. The controller is updated continuosly depending on the on - line identified model. The stability of the contrt^ system is analysed using Lyapunov's theory. Simulation results are given to illustrate the proposed control system.

Keywords: Indirect Adaptive Control, Radial Basis Fuction Neural Network, 1. Gl6lTHI$U

Trong thge tl, phdn Idn cdc he thing diu Id ede h? phi tuyen.Tinh phi tuyen eua h? thdng, dg khdn^ chinh xde trong do ludng vd dg khdng chdc chdn thdng sd md hinh Idm eho bdi todn thiet ke bg dieu khien cho h^ phi tuyin rdt khd gidi quylt [I]-[I01. Tuy nhien nhu edu thilt kl bg dieu khien eho he thong phi tuyen la vdn de cap thiet.Tir nhQng ndm 50 den nay, dieu khiin thich n ^ i bdt ddu dugc nghien cuu vd lien tgc phdt trien. Cd rat nhieu phuang phap dieu khiin thieh nghi khde nhau dugc dua ra nhu diiu khien thich nghi trgc tiep hay gidn tiep, diiu khiin thfch nghi ket hgp vdi dieu khiin trugt [l]-[8], diiu khiin cuon chilu [8], diiu khiin thieh nghi vdi bg gidm sat [8],[ 9],... Cdc phuang phdp nh^in dang doi tugng trong diiu khien thieh nghi cung linh hoat giua logic md [ I ]-[5],[7] hay m^ng noron [8]-[ 10].

Do h? thdng phi tuyen khd phdn tich vd chung minh nen viee dua ra nhthig phuang phap chung dieu khien he phi tuyin la khdng the Ci day tdc gia de xudt mdt phuong phdp thill

ke bd diiu khien trugt thich nghi gian tiip dung msmg RBF.

Xet h$ thong phi tuyen b^c n vdi tin hieu vdo u. tin hi?u ra y [2,3,5,6].

y'"' = f(y'""".y""".-.y)+g(y'"'" y)u (i) Trong dd y'"' = •-• . Mgt bilu diln trang thdi

dt eua h^ thdng nhu sau

(2)

= flx)-fg(x)u

vdix = I\|. X2. ... , XnJ^la vecta trang thai nxl, trong dd ham f, g Id cac ham tran, kha vi bac n.

So dd khdi he dieu khien thieh nghi gian tiep dugc cho nhu hinh I.

T^P CHl KHOA HQC & C6NG NGHf CAC TRirfifNG D^l HQC KV THll^T * S6 89 - 2012

Th6ng sA ciia md hinh dii tupng

•

'

Bg diiu khiin

u Nhftn dtmg dii

lucmg

DII lucmg di^u khi£n

Hinh L Sa dd khdi hi ihdng diiu khien thich nghi gidn liip

Phin tiip theo gidi thi^u md hinh RBF diing dl nh§n d^ng dii tugng phi tuyin. Phin 3 gidi thi^u dieu khiin trugt In djnh hda vd trugt bdm. Trfin co sd dd, phin 4 phdt triln bg diiu khien trugt thieh nghi gidn tiip diiu khiin h^

phi tuyen. Tinh h^i tg dugc phdn tich dga vdo ly thuyet ciia Lyapunov.Kit qud md phdng va ket lugn duge trinh bdy lln lugt trong phan 5 va phan 6.

2. M 6 HINH RBF

Gia thilt cd the xdp xi cdc hdm f vd g bdi ede hdm ca sd xuyen tam (radial basis function, RBF) nhu sau [5]

g{x) = Q'Jl+(o^

(3) (4) Trong dd 6/= [8^,8^,... O^J', 8,= [ 6^1 6*2 ... 0^ y Id cdc vecto thdng sd,^ = [ipn'Pfi—^ft.^V, <tg^ [<Pgt<Pg2 —^p,,fla cdc vecta dS lieu, ^r, =e^ °' •'^, (*g,=e^ "" '^ la cdc hdm ea sd xuyen tam(ir, vd \i^ la cac vecta tham ehieu Uf, a^ la eac hang sd, nf vd ng la sd thdng sd, ||.|| la chudn 2 cua vecta, (y^va^y^, la sai sd md hinh vd bi eh|n nhu sau

\^f\<^f (5)

K|<^R (6) Cdu true cCia md hinh mang ham ca sd

xuyen tdm nhu sau.

y'"'=f(x) + g(x)u (7)

\\) vdg(x) Idcdc xip xi eua f{x) vd g(x)

g(x)=i;M„=iV, (9)

8f vdOy Id cdc vecto thdng si cua md hlnh.D$t

, f#= 7 (10)

y"" = ef^+9g^Li = 9 t f " ) Ghi chu. Sd phan tir trong mang RBF (nf vd ng) cang Idn thi chat lugng xap xi cdng cao nhung khoi lugng tinh toan Idn. Vi vay trong thuc tl nen chgn nf va ng viia dii ddp irng yeu e^u sii dgng eu the.

3. DIEU KHIEN TRUgTT 3.1.On djnh hoa

Trong phan nay, muc tifiu diiu khien la xac dinh u sao cho x -->0 khi t ->oo.

Dinh nghTa ham trugt S S = x„-(i/(Xi,...,A„_,)

( ( ' ( X p ^ n - , ) =

(12)

(13) 8 = 0 xac djnh khdng gian eon trong R"

ggi la mat trugt. Thay (12) vd (13) vao (2), ta duge bieu dien trang thdi cua he thdng con bac n-1 tren mat trugt.

(14) x„_, --a|X, -...-a„_,\„.|

He thdng (14)ed da thii'c dac trung s""' H-a^^iS""^ +...-Ha2S + a| =0 (15)

T»P CHl KHOA HQC * CpNC NGH$ CAC TRtTftNC D»l HQC Kf THU;»T * S6W-2QII Bi cic qu; 4(10 pha cila h$ th6ng (14)

ti^n vi [0] khi t -»<x>, chpn cic h$ sdai, aj

&n.iSao choda thijrc (IS) Hurwitz. TCr (12) la cd

= fl:x) + g(!i)u-(-a,x,-...-a._,x. ,) (16)

= f(x) + g(x)u+(a|Xj i-...+a^,,x„) BSta, =tO,a,, ... ,a..,r.(l6)li*lh4nh:

S = Hx) + g(x)u+a^x (17) Di S -V 0, Iu|it diiu Idiiin u dupe xic djnh sao cho SviS ludn n^i d^u.

S = -Ksign(S) (18) vdi K 16 hang s6 duong vi:

r 1, s> 1 sign(S)- J-1 . S < -1

[ 0, S = 0 Lu$t diiu khiin dugc xic djnh tir (17) vi (18)

u=-!-r-fi;x)-a^x-Ksign(S)]

g(x)'- J Trong th\rc te ta khdng biet ehinh xic f(x) va g(x). Cic him niy dupe thay thi boi eie him xip xi ehung lan lupt la fl^x) vi g(x).

u = T;^r-f(x)-a;x-Ksign(S)] (19) Neu su dyng mo hinh RBF o (8) vi (9),

a;x-Ksign(S) (20) Dieu kien in djnh bin vihig ciia lu^t dieu khien (20) he thong hoi tiepdoi vdi sai so md hinh nhu sau.

Djnh ly I

Ta cd cic gia thiet sau, V X G D C R "

g(x) bj chin: 0 <g„,< g(x) <g„„

|f(x)| bj Chan Uen: |l(x)| < f.„

Dieu kien on djnh ben vimg eua luat diiu khiin (20) li

Smln I " 1 J Chung minh:

Thay(l9)v»0(17),tacd:

S = f(, Neu S •0 S

X) + it!i)-:r-\-iU, - s! X - Ksign(S)l+a.'x S • 0:

ft2S) + g ( x ) T / - [ - « ' * , - a , ' . x - K l + a.'x<0

K > -^[o,'^,llx)-g(x)0?^^-(g(x>4|^,)a:x]

s=fi:x) + g(x)Tr—[-8?#, - 5 j x + Kl + a|x>0

'^ > ^[e;^/x>-g{x)e;^,-(g(x)-el^,)ajx]

^>^^f^w^^g^w, +»;ik,x,|j

Dieu ki§n (21) dd dugc ehiing minh.

Ghi chu: Dl udc lugng ch$n dudi g„,n, vd ch^

tren g„^, ta cd thi su d^ng k^ thu^t setmembership identification. Vdn de ndy sg duge dl c$p din rong bdi bdo sau.

3.2.Dtlu khiin bdm

Trong dieu khiin bdm, chung ta thiet ke lu$t dieu khien u sao cho ngd ra ciia h$ thing ybam theo ngd ra mong mudn yj.

D$t: e = y - y, (22) Djnh nghia hdm trugt S:

S-e'""+a„.,e'"-"+... + a,e (23) Taed:

S = e'"' + a„_|e'""" +... + a.e

= Ux) + g(^)u-y|,"'+a,^,e"'""+.„-Ha,e(24)

^fl!x) + g(x)u-y!i'"+aje

T»P CHt KHOA HQC & C6NG NGHg CAc TRU'ONG BAI HQC K^ THU^T • S6 89 • 2012 Tucmgtv nhu diiu khiin In dinh hda, u A « . , . I f ;;'f , j ,„, „ , „ 1 dupe xic djnh sao cho S v i S trSi diu, nghia S = f e ) + 6 ( x ) T T - - [ - e , ^ , - a . e + yi"-Ksgn(S)J

li: - • 2 . S = -Ksign(S) (25) >'' * - • - ^^^

Vol KlihSng si duong. N i u S > 0 : Luat diiu khiin u dupe xic djnh tir

phuong trinh(24)vi (25) nhu sau: S=fl;x) + g(x)-f!—[-oj^ - a ' c + y™-Kl

1

Kl

•i=-r^[-f(x)-i'e+y>"'-Ksign(S)] (26) ,., , ' g(x)L J -yd +a.s<o

';>-^[C«'.''2<)-I(x)§'«>,-<g(x)-S;*,XaJx-.vr')]

g(x)L •-" - ' ' - • J Trong dd:

a. =[0,a„ ...,<•..,]' (27) e = [e,e e ' - " ] ' (28) "='!'•

Voi md hinh RBF * (8) vi (9), ta cd: '!' > — /™.»*', + S™."'/ + "'.'Zl'',-,',I + f, \y',,"\

(29)

NSuS<0: (32) trd thanh:

u = T ; ^ r - f ( x ) - a j e + yf - Ksign(S)l

g(x)l- J

u = . ^ r - e : ; ^ , - a : e . y r - K s i g „ ( S ) l (30) S = f e ) . g ( x ) ^ [ - 6 ; ^ ^ - a j e + y f + K ]

— y'."* + a e > 0 Dieu kien dn dinh ben vOng cua luat dieu '

khien (30) nhu sau. K>—\SI^ f(x)-g(x)il^ -(g(x)-il«l )(£}<,-yj")\

Ghi chu: De cd the xac djnh tin hieu dieu khidn u tir bieu thuc (26), tin hieu mong mudn

!{x)L Hay:

ya phii khi vi den cap n. K>—\ / ^ W , +?„,»', +"',2;|u,.,:t,| + »;|y;"'|

Dieu kien (31) da duoc chung minh.

Oia thiet sau, V x e D c R ^ P l j ^ K H I E N TRUgT THICH NGHI g(x) bj chin: 0 < g„.< g(x) < g , „ CAN TIEP DUNG M*NG RBF

|f(x)| bj chjn tren: |f(x)| < f„„ Trong phin nay, ta xet trudng hpp vector Diiu kien dn djnh bin vDng cua luat diiu *°"S sd 9, va §, dupe cap nhat true tuydn.

khien (30) la 4.1. Dieu khien on dinh hda

g _!_\ f or +„ w +wT\a x\*W lv'"ll Thay luat diiu khiin u U'ong (20) vao g.,.L >Zjl -I .1 »!•" IJ (I6),tacd:

S=f(x) + g(x)-f—[-eJ^^-aJx-Ksign(S)l + a'x (31)

Chimg minh.

Thay u trong (30) vio (24), ta cd: S = f(x)-fl:x) + (g(x)-g(x))« - Ks,gni.S) (33) Dat 0, = 0 ^ - 1 , , i j =e]^-e, .Tacd:

S = e ' ^ +^^(p 11 +01, *0),,u-Ksign(S) (34) 23

T/Sr CHl KHOA UQt & C6NG NCHt CAC TRUttNC D/^l HQC Kf THlI,iT * S6 89 - 2012 Chpn him Lyapunov:

*' = T S ' + - ! - f l ^ g , + J - e , ' f l , (35) 2 2a, ' 2a, " "

li him xic djnh duong vdi a^.a^ > 0.

Dao him hai vi, ta cd:

'Sle'fl, +oll^u + a<, + a),u - Ksign(S)]

(36) Nhin vio phuong trinh trdn, niuO, vi 9^

dupe diiu chinh true tuyin theo phutmg trinh

S-to)

Of = af,5^ (37)

(38) vi K phii thda dieu kien:

¥.>W,+W^\u\ (39) Thay (37, 38) vio (36), tacd:

I'= S(eJ^^ + e^^^a + <u^ + ffl,u - K!;gn(S)j

= S(is^ +a.,u-Ksign(S))<(»; +»'Ju|-/:)|S|

Niu K >W, +W^\u\ thi V xac djnh im, nghTa ii V se tiin vi 0 khi khi t -•or. khi dd :

X->0 i, -»o=>e^ ->e' i , -» 0 =5 0, -> 6'.

Nhu vay he thing sd dn dinh ti?m can vi vector thdng sd sg hpi ty vdi luat cap nhat (37.

38).

4.2. Dieu khien bam

Thay luat dieu khien u trong (30) vio (24), ta cd:

+ti]k)^\-S,t, -aJe+yi" -Ksgn(S)l

S = fl;x)-f(x) + (g(x)-g(x))" - *i>«'>(S) (40) Phuang trinh tr^n viit lai thinh:

-KslgniS)

(41) Hay:

S = e'^^+e[^ u+(u,+i!),u-/£!/gn(S) (42) Chpn him Lyapunov:

v=-s'+—e',o,*—S',e, (43) 2 2a,-'-' la,-'-' ' li him xic djnh duong a, .a, > 0.

Dao him hai vi, ta cd:

V = SS-B'IB,-O\0^

= Sl9^^ +0j^^ u + oi,+m„\i-K5igni.S)\

(44) Nhin vio phuong trinh tren, niu 6, vi Ojdupc diiu chinh true Uiyin theo phuong trinh sau.

\,=a,St, (45) 0, = a.S^ u (46) va K phii thda dieu kien:

KMr,+n;.|i,| (47) Thay (31. 32. 33) vao (30). ta cd:

' =•''(0,^, +0,.^ t/ + ru^ + ct}„u - KsigniSn

= s(iOf+ (u,u-Ksign(S)) < ((F, + W^ \u\ - K)\^

TAP CH! KHOA HQC & C6NG NGHf: CAC TBUdNG 0^1 HQC KV THU^T * S6 89 - 2012 NIU chgn K>Wj+W^ \u\ thi V xdc djnh

am, nghTa Id V se tiin vl 0 khi khi t ->«, khi dd:

Nhu v§y h§ thing se dn dinh ti§m e$n vd vector thdng sd sE h§i ty vdi lu§t c§p nh§t (45, 46).

5. K £ T QUA M 6 P H 6 N G

H? phuong trinh biin trgng thdi h§ eon ldc ngugc dugc cho nhu sau [2,3J:

gsinx, -m/xjCOSXtSinjc, /jm^ +/») /(4/3-mcos^ jc, l{m^+m))

cosx, l(m^+m) 1(4/3-mcos^X-, /(m^ + m)) I ,

Trong dd X| = 5 Id gde quay con lac dng vdi vj tri hudng I6n va x^=0. Khdi lugng cua xe '"t - ^^S' 1*0' liJ^ng con ldc m - 0, Ikg , ehieu dai canh tay eon ldc I = 0,5m, gia tdc trgng trudng g = 9,8m / s^.

Quy dao mong mudn:

Chgn mdt trugt:

S = 4e + e

De nhan dang h& thdng tren, chgn ngd vao mang RBF Id vector bien trang thai x, he sd a, =0,5, (Tg =0,5, trgng tam mang RBF

//^, = [-2;-2];;/^, =[2;2];//^3 =[-0,5;3];/i/, - [0,5;6];/i^5 =[0,1;-I0];/^^, =[-0,l;-0,5];^^, - [0;0];//^, = [-2;0];/i„= [2;0];//,3= [-0,5;0];

M,, = [0,5;0];/i,, = [-5;0];/i^, = [5;0];//^, - [0;0];

Chgn h$ s6 hgc ofi = 0,002, ttj =0,00013. Diiu ki§nd^u x^ =[p//I0;O].

Kit qud nh$n dgng hdm f vd g dugc eho trong hinh I vd hinh 2.

Hinh 2. Ket qud nhdn dang g Nhin vdo ket qua ta thdy h? thdng ude lugng dugc md hinh ddi tugng f, g. Trong hinh 1, sau khodng thdi gian ban ddu (0,5s), mang no-ron bat dau nhan dang duge hdm f, ham f nhan dang dugc gan bdng ham f thge te, sai sd khodng 8,6%.

Tuang tu nhu hdm f, mang na-ron cung dd nhan dang duge hdm g. Do ddc thd hdm g thi chi nh?ui d ^ g gan diing, khdng nhdn dang chinh xde eac dinh eua ham g. Tu^ nhien trong diiu khiin, difiu nay hoan toan chap nhan dugc.

Sai sd nhdn dang ham g khoang 0,9%.

Ddp ling cua hB thdng dugc cho nhu sau.

Hinh 3- Ddp ung ciia con ldc ngugc.

Nhin vao hinh 3, ta thdy tin'hifu ra cua he thdng tien ve tin hieu mong mudn sau

TAPCElf KHOA HQr&C6NGNGH(:cACTRtrdNC DAI HQCKtTHUAT * 8689-2012 khodng thdi gian O.Ss.Sai s6 xde l(lp khodng

4%.

Tfn hi§u diiu khiin dugc cho trong hinh 4.

6. KtTLU^N

Trong bdi bdo ndy, tdc gid dS dl xuit phuong phdp diiu khiin thfch nghi gidn tiip diing m^ng RBF.Md hinh md td ddi tugng dugc nh0n d^ng diing md hinh RBF, trong dd thdng s i mgng RBF dugc c^p nh^t online.Lu$t diiu khiin ddm bdo hi In djnh dugc chiing minh diing I'jf thuyit Lyapunov.Lu$t diiu khi£n t r ^ cQng dugc kiim chdng qua md phdng trSn ddi tugng con lie ngirgc.

Hinh 4 Tin hi^u dl&u khiin.

TAI L l f U T H A M K H A O

1 W. S. Lin, C. S Chen, "Sliding - Mode - Based Direct Adaptive Fu22y Controller Design for a Class of Uncertain Multivariable Nonlinear Systems", Proceedings of the American Control Reference, Anchorage, AK 8 -10, 2002.

2. Wu Wang, "Adaptive Fuzzy Sliding Mode Control for Inverted Pendulum", Proceedings of the Second Symposium International Computer Science and Computational Technology, 26-28 Dec.

2009, pp. 231-234.

3. H. F. Ho, Y. K. Wong, A. B. Rad, "Adaptive Fuzzy Sliding Mode Control Design: Lyapunov Approach", The proceedings of the 5* Asian Control Conference, ASCC 2004.

4. S J. Huang, W. C. Lin, "Adaptive Fuzzy Controller with Sliding Surface for Vehicle Suspension Control", IEEE Transactions on Fuzzy Systems, Vol. 11, No. 4, Aug 2003, pp. 550 - 559.

5. S. Yu, X. Yu, M.O. Efe, "Modeling - Error Based Adaptive Fuz^ Sliding Mode Control for Trajectory - Tracking of Nonlinear Systems", The 29* Annual Conference of the ISEE, 2003.

6. V. Utkin, J. Guldner, J. Shi, "Sliding Mode Control in Elcciromeehanical Systems", Taylor &

Francis, 1999.

7. Z. M. Chen, J. G. Zhang, et ai, "Adaptive Fuzzy Sliding Mode Control for Uncertain Nonlinear Systems". Proceedings of the 2"'' International Conference on Machine Learning and Cybernetics, 2-5 November, 2003.

8. J. Y. Choi, J. A. Farrell, "Adaptive Observer Backstepping Control using Neural Network", IEEE Transactions on Neural Networks. Vol. 12 No. 5, September 2001, pp 1I03-III2.

9. F. Sun, Z. Sun. P. Y. Woo, "Neural Network - Based Adaptive Controller Design of Robotic Manipulators with an Observer", IEEE Transactions on Neural Networks, Vol. 12. No. 1. January 2001, pp. 54-67.

10. Ddng ST Thien Chdu, "Nh§n dgng h§ thdng dgng diing m^ng noron", Chuyen dinghien ciiu sinh, Dai hgc Bdch Khoa, D?i hge Qudc Gia tp. HCM. 2005.

Dia chi lien h?: Ddng ST Thien Chau - Tel: 0915.900.962. Email: chaudsl@yahoo.com Khoa Dien - Dien tir. Dgi hgc Tdn Dire Thdng, Tp. Hd Chi Minh Dudng Nguyin Hihi Thg, Phudng Tdn Phong, Quan 7, Tp. Hd Chi Minh.