Rejection of Unmatched Disturbances in Sliding Mode Control

Cupyrigh t © 191}6 IFAC 13th Triennial World Congress. San Frnncls\:o. USA

2d-ll 2

REJECTION OF UNMATCHED DISTURBANCES IN SLIDING MODE CONTROL Y. A. Jiang

O. .T. Cleme nts

T. H<'l,ket h

School of 81('ctrical Engineering, Univer sity of J\ew South lValf's, Sydney 2052, A lIst[ali a Abstr act: Sliding mode! c.ontrol of linea r systems with unmatc h ed input dl.':lLurbanres is conside red in this paper. f\!lat che d uJlcerta inties an d dh;turba nc.es , which lie in tJw range space of Lh e input s ignals, can be ea.c;;ily ha ndl ed with s liding nl(ldr. control associat.ed wi th a fix cd sliding surfa ce. T o deal wit.h unmat." hed dist.urb ances, a dynami ca l slidin g s urrace is necessa ry. The pos."ibiliLy of drivi ng t.he ~t at e of the co nt.rolled SystClIl to a dynami cal sliding surface is ex pl o it-ed whi ch m ay result in an asy mpt.otic ally sta ble (J ut put in the presenc.e of th e uumatc h ed input dist.u rbances .

K ey words . Variabl e structu re

cont.rol ; ad apt.ive co ntrol j d isturba nce n dection .

1. INTR ODU CTIO N Slid ing mode control lers are ro bust. to m a tched un cerLai ntie$; ami insens it ive to mat ched disturb ances which li e in the l'iLuge spaCf" o f lhe inpu t m atri x (Dorl ing and Zin ober, 1986 ; Slolin" a ut! Li, 19HI; lIt ki n, 1977) . A central fe at. uJ'(~ or sliding m ode (·ont.rol systems is that sLates a.rc forc/'d O il to a sJi di n~ su rface by a. discollt.inuous contro l si!4nal whi ch O VC I" (".O lllffl a ll matche d uncerta inties and di s turbanc es. The pl ~ rfor m all ce of the closed-l oop system is g iven by sp~·'r.ify ill g the sliding s urface (Durtin g and L;ino ber, 1980). TIH~ undesir ed dis<".on tinuo us " d. uat.ion bdlavio ul' ('a n ht' a ll(~ viatec.l by linear a.pproxi mation togd. her with intc~ra l co nt.rol act io ns (Slot-ille a nd 1.1. UHIl). Howeve r , ill the prcsell CI' of lInmatc hcd un c.crtaint y and dist. urb ance. f he ni c~ prop ert.ies of Rlitling m ode cont.rol <:aJ'lnot. be lllaint.a inf'd . A number of ap p roaches (Chen and (;. L cit.lll a.nll , 1987: I\.wan . 1995; Rya.n an d

Corle;s , 1984; Sp urgeoll and Davies, \ 993) have been propose d to handle unm at.ch! ~ d un cert ainty. In Ryau and Corle" ( 1984) and S~urgeon and Davies (1993) , unmat.c hed uncerta inty is deal t with by choosjn g a s uitab le s lidi ng surface . T hi:;; method all ows a. fre e structu re ror th e IIn("('ctaint.y , but. Iirnits Its m agni t ude. Moreover, t.h e effec ts o f input disturbanC!~s on the perforrn ance of the contro l syst.em have not bt!en conside red. In Kwan (1995) , a dynami cal s lidi ng snrface is ll sed which ca n ha ndl e large m o delling un cert ;\inty if ce rt.aill extende d m atching conditio lls are satisfi ed . An advanta ge of t his m et.hod is t.hat it all ow!:) for rejectio n of dist urb ances. The require d dynami cs irnpli(·d by the sliding surface are obt.ained by using ad a pt.ive method s : f,he idea UIlderlying this m et hod is similar to t.hat o f back-st. epping control (KencIJ akopou los , et al,. 199 1). Th is paper conside rs sliding m ode control of linear SYRkms with uum aLchf'd input, di::-turb ances. It. is assume d that a ll sta ff'S of the system ,It'e availab le and the desired perform ance of t he system is specifle d by a given outPll t perfor m ance measur eml :nt vector. The syst.em is

3473

It follows that. cont.rol law u :::: -M tt(x) robus tl y stabilise~ the system (.5) . Constan t. input/o utput sr.aled optima l H,~ control t.echniq ues can he used to design SlKh a. feedbac k gain /\' (Jia.ng an d ClemeH ts, 1991). Furt.her . since 'WI(J.} and w:!{t) a.re bounde d , the followin g result follows directly from high gain robust control t.h eory (Corlc~s and Leitmar m, IDR I; Dorling and ZiHober, 1986). Le lnlua 1 If Xr (t) fm//aw of thr form 11(1)

lS

hounded.

OU11

th en; exists a con-

- d(t) = -iI1 x(t) - lV(J·,1)1 1I'1'(.,,) (.,,)- d( t) 11

s·u.r.h that fI(.r)

C01l.Vl'rg e!l

(10)

to d(t.) in jlnite lime for (/ny

smooth vector' fund-iol l d{t) E R pos.~ible moc/elling Imrt:rtaiTlt1(' ~'

lIl HI

the presenc e of all and flAn·

~ ./ l 'lI (t)

measur ement output z(t) small even with an unmatc hed input dist.urb a nce 'Wt(t).

3. DIST URBA NCE REJE CTIO N In this section, the rejectio n of the unmatched disturbance wr(l) by Ilsing all Ofl"-Sf! t function d(t) is considered when z(l) Ilt(l). 'Two mdhod s are propose d One method is to adaptively drlv(' Il(t) to cancel th f~ unmatc hed dist.urb ance Ft wt{t) so that the output .;ft) tends to zero. Thi~ is possible if z(J:) only depend s on Xl(t) , and t,he modelling un certai nty LlA 12(t) arid the input disturba .nce Ft1Llt(t) lie ill thl~ range space of A 12. The idea behind this metho d is si milar to that of certain parame ter estimatio n method s used ill ad :~ptive control (Narcn dra ami Annasw amy , 1989), Anothe r method is t.o use a.n integral functio n of :-ft) in the sliding surfac.e.

In control law (10) . /1.4 E R mxll is a constan t linear control gain. and N (x , t) E R ill XTIl is a nonline ar cont.rol gain a nd may he a uon lincar m at.rix function of oX and t.. Not.e t. hat, with tIre rantrol law given hy Lemma 1, t.hf': va ri ahlf' ,,(x) i!i (ot il.lly insensil .i ... ;~ to tht' mat.ch ed mode lling uncerta int.ies ~A 2r (l) and 6A:?:!(t ) and t.he lnatche d dist.urb an('p I/ I :!(/ ) .

For further develop ment, it is necessa ry to conside r the st.ability of follo\...·ing syst~m

Remar k 2 To redllc~ t he cha.ttering of the cont,rol signal u(t), in the con~rol law (10) . t.he sign funrtion lHay be rep laced by a saturat ion runct.lo lJ and an in tegra l control t.erm can bp uspd.

where w(t) is a bounde d disturh ance, A E R UXFI is an asympt otically stable matrix, ~( Xlt) E R mxm IS a. kn own m;ttrix function , and P > 0 E R"xn .

u(t) = -M 1'(1) -

lV, (" . t)sat("

- .1(1r' IV,!!J , t) (I'(x j

- d.
(1I)

- d(t))dt

wiLh (, > 0,

sat(", <) =

1

( 12)

and !:Iuit ablc i'unctio ns NI (It, t) and N'A ~l , t). As-sumi ng that a conl,rol law uU) :::: f(r., t) has been given which rmct'S JL{X(t.)) t.o cOII\'t' rge to d(l.) in fiui te time, then 1.1w co ntro l system ( !) is uomina tcd by i,(! ) z('/)

=

AKX,(t ) + ( A" + j, ..ldd(l) [C , - (',I'l.r dt) + C,d(l).

+ B(x, I)O(t) + wIt)

x(l)

'"

4,,(1)

Oft)

=

-<\>T ( x ,nBT[' x(t)

(14)

=

Note that when 111(1) 0 and PA + AT P < 0, it. b well-kn own frolll adapt ive t.heory (Naren dra a nd Annaswam y, 1989) l.hat. th e system (14) is uniform ly stable and is uniform ly (\''5ymptotically sLable if cI>(l~, t) is a persist.ent.ly e x(·.it.in g function . Howe ver, in the presenc e of a bounde d disturha llce l/I(t) , thl' !:Itability resul t for syt;tern (14) is unknowll to our best knowle dge.

Lemln a 2 Conside r system. (U) satisfyi ng the following assumpt .lrl'flS

(J) <1>(:1:, t) is bounded and uniform ly contin1WUS for all x and t. and

(2) th ere (yists a f' > 0 E R II >OI, and a Q > 0 E R 71 x n sfllis/yin g lhe 1-:r;apunotJ equatio n

+ F,Wl(l)

( 13) Not.e that d(l.) i~ a. fl'(,c design paral1le kr . for a given H , it is possihle to ch uose a d(t) so that .:(1,) is ~ma.ll. 1n th~ rest. of this paper , t.he design of tb C' off-set d(t) is considereo wit.h I.h~ int.e nt.ion of kee plJ'l~ t he pcrform
Then sysl.em. (14) is uniform ly ,~t.ahl(' in the sen.. e that bolh x(t) and (x, nB(t) arc un ifo rmly bounded .

Proof: Conside r the LyaplIll ov fu nd io]) candida te

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with P > 0 giVCH by a,·"sul11ption (2). It follows froIll ( H) and (8) tha I.

HI) ::; - x T(l )Qx(t.) + ,, 1 (t)f'w(1)

=

_(QL'_~ Q-~i'u . )T(Q!x_~Q-tPwj

+ :tIl,T eq ::"'l pw.

Sillce .1I.!(t) is bo uuded , .1:(0 is uniform ly bounde d aDd HO is 0(1), To show t.ht~ b o ul1d(~dness of q,(.r.. t) 8(t ), conside r the solutioll of ('q uat.ion (l4)

~(t) = ,,-A ',,(O)

-1'

,-A" -T )(T) O( TjrlT

n

aod a Q, > 0 such thai, 'he Lyapull ov equatio n (7) hold s, Con,ide r

=

d(1) 4'(1)0(1) (15) where
11(1)

= -rq, T (I)Ai"P ,J:,(t), r > 0,

(1 6)

Tf I' (x ) converg e. to th e off-,"( d(t) , then the state ;r,(t) converg es t.o zero.

Propo sition 1 Conside r' th~ system (1) and an off·sel d(l) g;"fI! by (l ,IJ. If Ihen exists a f eedback gain K so t.hat Lyapun mJ equatio n ( i) holds, lh en lhere exists a co nt.rolla w of thf form of (111) su ch that lhe closed-l oop

,r;ysie. m is t'obusfly sta ble with bounded output and control

signals. . 14oran1ff" , if Assump tion I is s atit;fi ed, t.hen I'(t) tend, 10 d(t) an d ",(I.) len ds to =C1'O with bounded control "iigna/,,,.

Not.e (·hat

M", Let.

11,,(t)11

,= sup(liw , (t)IIl IIAI I,;;,' ,

:S M r , Thell, it follows t.hat

Ill"~ p- ,"'- Tl (T)9( T)rfTI I :S 2M.T + !VI",.

"

Proof': First. conside r d(t) as a n input signal. From the assump tions a.nd Lc:mrna 1 t.1l ere exists a control law of the form of (10) so Ibat t,he dosed-l oop system is robustly stable wi~h bounde n ,>utpllt a nd con,trot signa.l if x,(I) is bounde d, I<. furth'" follows Ihal Olt) is also bounde d ancl thus d(t) given by (15 ) is smooth . Tbis

implies from Lemma 1 th a.t the (".ontrol law drives the

No~e front ( 14) t.ll at 8(1) is llllifoflHly continu ous since 0(1) is I>o uoeled, Thus, ( x (t) , 1)0(1) is also IIoiformly continu ous since bot.1t (J' . t) au d 0(1) a re uniform ly con-

tinuolls. Thus, it. follows from Propo.... it.io n I given in .Ji angei al.. ( I \l\](i) t.1",(. <1>( xl'), I)O(t) is bound ed for all I.

Now, c on5id(~ r the syst em (5) wit.ll t.h e intent.ion of developin g a n off-set f1l11(·(.ion d(l) which cancels the unm atched dist.urJ)(UlCe F 1 wdl) . Tlw following assump ti o ns are n'quireci.

ASSUl nption 1 Th ere exist. all unknow n constan t Q' i: R , CI.n unknow n C(Jnst,'\Ilt. vector 0 E R\ a.nd a known , boulHJed and smoo t.h mat.rix fun ction 4l (t) of appropr iat,e dimensions sueh t.hat

slate onlo the , urface JI(x ) = d(t) in fioil" tim e if d(l) is bo unded . To show t,hat. ",(t) and d(l) are uniform ly bounded, consiu(>.r the exists (t PI > 0 satisCying the Lyapun ov equat.ion (7) . Applyin g Lemma 21.0 system (13) associa ted with PI , it is oonrluderl t.bat X l (t) and d(t) an! uniform ly bounde d. Now, suppose that Assump tie,n 1 is satisfied and consiner a Lyapull o\! rUllct.ion candida te

oE (1)

/ 11:!

ha.<; full

COIUlIllt

rank,

V

wh ere ii = 0 - ,,0(1), It follow s from (13), (15) and (7) t.hat the derivati ve of V(l) along t.he traj ectories of (13) is

V Not.0. that tli e un certa inry a.nd di~1.1Jrbancf· must sat.isfy an exr ~ ud ed mal.chi ng con dir.io Jl. Suppose that. t.he re px ist.!) a r\· ~o t.hat. . . . lK de fin ed by (6) is a robll stJ.y sT.able mat.rix . Th(,1I thet·e. exist. a Pi > 0

I T 1 -T , = :ix, P,x, + 2,,8 r- 8

Hence

= -x ,T QJx. , - x,T f',il, ·,(od(l) - Oj = - j;""[ Q1Xl :s; O. XI

E L:,: nL('o,) and

ii E

. -1 or 8

Loo. Note frolll (13) that

Xl E L oo · It follows from Barb alat.'s Lemma (Slot.ine and Li, B)O!) th at. ;Cl(t) c.onverg es to ze ro a nd hounoed .

3475

BCt)

is

Renlar k 3 If th e. ll1 o d ~ llill g unct'-rta in Ly 6.All al so lies in the r<'1llge space o r .4 12 , then for any hounde d L\..All t here: exists a sufficien tly large J":' so t.h at AJ;," is robustly stnblc ( .Jirt.Il~. et aL , 1996 ). Mo reove r , in t.his case, it is possibk t o ha ndl e th e UIH'Nt<.: co nst r nd,ive ly c.ompar cd to t.he well- kllowll bark.st epping apPl'(lach (K an cl lak o p o u~ 10'. " "/" I ~n l ) .

For the s ,y-st ~ m ( 1) , when 1.lw o ut.pu t perfor mance z(/) can br: const ru d,e d wit.h ('2 = 0, t.Jlt~n as a consequ ence of Lemm a 2 , a sliding· lIlo\le ('o ut roller witb an adapt.ivdy adj llst.ed off- s,,{ d(t) giv<·n by (I!») drives z(l. ) Lo zero ill tlw prf:'s~~n ('e o r tile UlIlIl at"hc d dist.urb ance 1/,'1(1.) whi r h !;a tisfi es A:";SI1 JlI. ptioll 1. Ho\vevcr , ."inrc the off-sel d(t.) g iv@ 1l by ( L,) can only drive :.I~I(t) t o zP.ro. (l llly n bo unded .:( t ) call be obtaine d when C"1 f:- O. NOIe t,11al. ott)

= [I', -

C,f"]J, ,(t)

+ C , d(t) .

( 17)

\Vhen th e fo rm o f I,he inpu t. di sturban ce W l is known, it. is possib le t.o (' l1oo:-w It.n off-st't IIU) ~o that ::(t) - 0 wit,h a n OB-zero a:~ l(t). A l11 e th o d ; 0 ha nrll e this fo r the general ca.<;e willlw distllsscti ill rtn ()lher paper , and only a simple integral co nt rol is cons idc l'('d Iwre.

Conside r a.n rwgenw ntcd

=

'HI)

=

idl) C hoose

Cl

where q is a bo unded unknf)\vn dist urbance t.o t he p~n­ dulurn , a > 0 is a const.ant unknow n pa.rame ter, and Wl,:l,3(l ) a.re bo unded unknown disturb ances. Note t.hat. this ~yst,em is in the equiva.l ent form of (3) and the unmat.ched m o tlelling un certain ty a sin(xt) and the unmak hed input dist,lub ances q COS(;Z: l ) and W2 (t) are ill LIl!~ ran ge spa.ce of X:I (.4 12 ). Dut, input disLur· bance tr l(t) iti unmatched ;,nd no t in the range space of X3(t) . III K\I. 'an ( 1995) , t.i lC input. dist.urb a nces WI(I.) and w:df) are ~f:t. to zero.

=

A robust. st.a.bjli~iltg feedbac k gain f{ (k t k2 ] can be chosen as k l > lal and 1.:2 > O. Let t.he upper bo und fo r " be kn own a.' lal S 11. The sliding mode 1'(/) Iw chosen mi

which r C ~Hll t s ill a transien t respo nse generated by a second-o rd er syst.em wit.h two poles lo cated at - .) , The m od ifi erl control law (11 ) is used

" = - '10.,,,(( 1'.4) -

(IS) law

' (1)

so (.hat system ( 18) is robll.U y s(able. Note that il(/) = z(l) , An off- s(~ t. d(t) ca.n he g iven a...;

= -L

1,'

=

x, (I) + x,(t) .

M ethod L Th e off·se.t. d(t ) is chosen as

(1 9)

d(1)

(2 2)

First consirlcr the case of C1, :::: 0 and ass llm e that t.he out.p ut. perform a nce is s pecifkd a.<;

SY.C; t.Clll

+ (.'"-"., (1) + :>it I 11",(I) + ( .4 " + LlA,,) x,( t)

I:)tate ~fcf:' dback

20
whMC the off-set del) will be designe d with (15 ) and (20) resp ectively .

C ,~ , (t )

(A'l

1,'

d(t) = q(t)fos (xd

with

q(t) ' (I )dl

(20 )

(23)

=-licos(x, )80t(" 1 + 0 29 x"

0.1),

q(U)

=O.

Metho d 2. T he off-sel. d( t ) is dlOsen as

4. EXAM PLE

d(t)

= - [' (i.
i"

Coltside r ttif' ex ,-UlIpJr. of a d ri vel! pe nd ulum propose d iu

Kwan (1990) .

Not~

+ ", , (I)

~,(t)

;'3(t)

- x,, (1)

+ x,, (I ) - qws(x !l + "'2(1) + H(I ) + "'"ll)

-lIsin(" , 1I

(21 )

= o.

(24)

that the same gain is use.l in both method s.

=

id/) ;; ,( 1.)

d(O)

=

Let init ial values "110) 1. " , (0) - 0.5 and " ,(0) = U a nd disl.urb a nces q = 4 an d /I 3( t) :::: 5 sin(t). SimulaLion re~ lIlt s a re s how n in Figur .. l(a-c.) (or the rollowing d isturban Cf'S (a)

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W1

== 0 and

U' 2

-= O.

:("""-l

I ..-'U!} "'1",1)_!w2,:~,

':~t~~~1

t __~__J ()

~

,

1

: ---- ---~~ 11

I II

~

III

Tlmc (1Ie(1.

'I ,,'. ""'''' '~~j

! ;r~-;J

REFE RENC ES

III

(l

1) " "'1", , )

Figu n' I : Solid : Method 2. Dashed: Method I.

(b) tvl = (c) tv l

I)

and 11,,(1)

= 2sin(1I"I) .

It is clea.r t.hat bot.h met hods l'es1Itt in good perform ance

when input, disturh an ces Iif: ill the range spa.ce of X3(t) . Wh ~n t he IIllma.tc:\H'd disturb ance U' j is applied , both me t.hods s t ill gi\,t~ a. ra irl y p;ood perform ance.

'#

0 and a..'>fSume t.hat the

output perform a.nrc i,<; specified as

, (I) = x dt) - .1'3(1) ·

Note thal , (I ) ha, unsl able "ros . The off-set d(t) is given by (23) fo r Method 1. and d(!) -v(t) with v( l) defin ed by (24) fo r Method 2 respect ive/y.

=

The initial value ~ ar C'! t.h e same. and the disturb ances a rc ~pyen as q

= 4,

Wj

= 1,

U!2(t) ==:

O.

Chen, Y. H. and G . Leitma nn (1987). Robust "c•• of ullcerta in spitems in I he absence of m a l.ching asslllnpti oll,.111t. 1. Control, 45 1527- 1542 . Corl ess. M..1. and G. Lellma nn (19S1). Continu O ilS sl.a t,e k-edback guarant eeing uniform uJtimat.e bounde dness fo r un c.ertain dyn a mic sy&t~m s, f E RE' TraIlS . A utomat . ConlT"OI. 26 ) 11 391144 .

= I ;"Id ",,( I ) = 2si n("I).

Now , co nsider tlw case o f C,:-

The rormer met hod requires the unmatc hed uncerta inty to sa tisfy certain ext.ende d matchi ng conditionl3, while t he Ia.tter does not. limit t.he st.ructu re of th e unmatc hed input. dist.urbance . The roLllstn ess of the proposed a pproaches are shown. Tt. should be noted t.hat m odelling uncerta inty and non-l inearit.y can also be viewed as input disturha.nces provided c, ~ rtain matchi ng conditio ns are sn.tisfied .

'w::!{ t)

= 5 s in( t).

The simul a.t.ion f('sul ts a,re shown in Figure l(d ). Not.e t.ha.t Met hod '2 produ ces a good perform ance , All simulat ions were done with SI Mll Llf'lK I'C / l. 2c.

5. CONC LUSI ON This paper suggest.s t hat the sliding m ode may be driven t.o a n off-set functio n for t,he rt~je c t io ll of input disturbances , Two approac hes are proposed 1,0 obtain t,he desi red off-set. functi o ll . One m et,hod adapt.ively adjusts the off-se t. functio n so that t.h t' input dist.urban ces can be cancelled. Anot ber llwl.hod ljSC'S t hI' off-set function as an input, compen s
Dorling, C. M. and A. S. I. Zinobcr (1986) . Two appro ac h~ t o h y pe rpJaJIt'! design in multiva ria.ble va ri a.ble struciu re conI rol syst.ems , Inl. .I. Control, 44 , 0:j-82. Jiallg, Y. A. , D. ,I. Clemen t, and T . Hesketh (1996) Sliding surface desigll with aD-sc aled HOCl metho d , 1996 [FAC Coy",s .•. Jian g, Y. A., D. J . C le men. s a nd T . Hesketh (1996) Rohust IcarninK control of a dllSS of Nonline ar syst ems, 15196 rFA C C ogre ss. Kanella kopo ul os , L, P. V. Kokoto vic and A. S. Morse ([991). Sys~ematic design of adaptiv e cont rollers for feedba.c.k linea.rlzablc systems , lEBE Tran.s. kalom. Contm/ , 36 . 1~41-12,13 . Kwan, C'. M. (UJ95). Sliding mode control of linear s),sl,ems wi t. h mismatc hed uncerta inties, A utam.hea, 31 :103-307. Narcnd ra, K. a nd A. Ann aswamy (1989) Stable Adapt i'!:e System.'). F:nglew(.od C liffs, NJ: PrenticeHall . Ryan, E. r. and M. Corless '1984). Ultima te boundcdness and asy mp totic :'l t,a bility of a class of uncerta in dynamica.l ~ys tj : ms via. continu ous and di~c o nt.inu ous feedba ck control , lMA 1 . .M"ath . Corilrol [n]. , 1 , 223-242 Slotine , J . .1. and W. Li (1991). Applied Non/inear Con /.rol. Englew ood C liffs, NJ: Prent.ice- Ha ll. Spmgeo n, S. K a nd R. Davif'.s (1993) , A nonlin ear control gtra t,egy for l'obw;;t. :;liding mode perfo[mall ce in t.he presence of unmatched un certaint y. Int. J. C ontl'Dl. 57, 1107-1123. Utkin , V. (1977) . Variable st.ru ct ure systems with s liding modes, IEEE T'r,ms . Automat. Co ntrol, 22, 212-222 .

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