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Sliding Mode Control with Nonlinear Sliding Surfaces
Copyright © ! 996 IF AC 13th Triennial World Congress . San FrJncisco . USA
2b-30 2
SLIDI NG MODE CONT ROL WITH NONL INEAR SLIDI NG SURFA CES Vi ctor K. Chu and Masay oshi Tomiz uka Departm ent of M echanical Enginee ring Univers ity of Califor nia at Berkele y Berkele y, Calif orn ia 94720, USA
Abstra ct. Traditi onal sliding mode conLrol1 ers use lin ear sliding surfaces, which m ight not. fit. t he glohal dynam ic proper ty of nonlioear plants . As a result, sliding m ay not take place due to actua tor sa tura tion and even insta bility m ay be induc.ed when t he error is large. Fur t hermore, linear sliding surfac€8 m ay c:ause large cha tteri ng in digital implem enta~·. ion. These problems may be solved by replacing lim'ar sliding surfaces with nonlinea r sliding surfaces. Two met hods for d esigni ng nonlim 'ar sliding su rfa ce~ a re suggest ed in th is paper. T he advanta ges of nonlin ear sliding sUl'faces a nd t h ~ ir approx ima.t ion by fu zzy logic are exam ined in det ails for a rotat ional inver ted pendulu m . K e yw ords . Slidi ng Mode Control , Fuzzy Modelin g, Nonline ar System s, Discrete Digit al Dynam i c Contro l
L INT RODlI CTION Slid ing Mode Co ntrol (S MC) is a popula r robuRt CODtrol m et hod for non linear system s satisfyi ng the matching r.oudit ion. The design of sliding mode control lers involves t.he followi ng two ste ps : 1) d~~ s lgn of a sliding surface , whi(:h represe nts the dr,si red stable system d ynamics, and 2) detiign of a co ntrol law whi ch m akes t he sliding surface attracti ve. When the slid ing t:i urface represenl.s lin(~ar dyna mics, th e magnitu tie of cOll tro l in put required to keep the system sl.a te on r.he ~ lidin g surface usua.lly i n t: rt ~illies as t he m agnitll rtp. of errors increases (J a bbar i d al., 1990 b). As a consequ euc.€, when the control force is bounde d, only a par t of the sliding sur face, called t hf' ..liding regime, is tlt.t radive. O ut~ id e t. he .sliding n~yl m f' , t. here is no gu a rantc(' for eit.h r.r sta bili t.y or perform a nce. Furt her more. sincr. tJw linear I;lid ing surface. m ay not be
na t ural to t.he pla nt, it may ca use la rge approac hing a ngles ) the an gles a t which t he system sta te ap proache s t he sliding surface. La rge approaching angles may ca.use a large scaJe chat teri ng along t he sliding ~ urfac e when the sliding mode controller is diseret ized (i.e. wi t h a fi ni te sampling ti me) . T his sit:u.atio n m ay be worsened at large velociti es. Therefo re) t he :$tate is exp ected to de viate from a linear sliding surface over a sampli ng period wh en t he error is large even if ir is d ose to th e surface at one samplin g inslance. J abba r i et al. (1 990.. ) a nalyzed th e hcha vior of C'ontinu ous time pla nts over one sampling period under di gital conl,rol and shown th a t the state must stay outside a conn sur roundin g the linear sliding surface at a IX\.mplin g instance in order that t he slid ing variable decreases its m agni t ude over one sampling p eri od . Th e area of t he non-att ractive cone will increase if Lhe ma.gl1ilude of t.h 4 ~ ~ l op e of t he linear sliding sur fac~ is increa.. ;;;ed. This an alysis shows tha t th e use of the linear slid ing ~ urfacf! m akes perform a nce Ilonuni -
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form .
where
The problem s mention ed above can be partly solved if we use SMC with nonlinear sliding surfaces. \\Then noolinear sliding surfaces are properl y designed , the sliding ffgune may be enlarge d and optima lity conditio ns, such as the mi'limu m en ergy consumpt.i on or the minimu m ti m.e usage, may bo embedd ed in [,he system. Dwyer a nJ Sira.· Ramirez (1988) have shown the idea of nonlinear sliding surfaces hut they have nOI. provided a design algorith m. Jabbari et al. (199010) US", paraholic sliding surfaces on the design of the ncar minillJUm time control for linear seco nd order system s with force constra int. Lee (1991) has propose d t,o use c1Ibic polynom ials as sliding sHrraces to improve transien t respolI:-oe. In this pa.p cr~ we propose to design sliding mode control based 0 11 non linear sliding surface s that aTe formulated in .... way t o guaran tee th~ (l.':Iympt.otic stability. In ilddition~, we suggest. two m ~t hods to design the uonlin ear sliding s urfaces that may solve the saturat iOH / perfonn anc.r. problc:ms associa ted with linear sliding surfaces. The first. method is based on analysi s and/or expr.rience. All existing techniq ues for designi ng linear sliding surface may be include d in t.his method . Tht' seco nd meth od uses desired con/ml pmJiles and the reVlTSf intC'. gration lIIE>thod to generat ion lIonJinc ar sliding s llrfac~. The input. constra int as well as certa.in pr.rform anre criteri on may be emb edded in this design method .
In either design meth od, thE' /uzz.tJ (ogle, whi ch is an uniye rsal approx imator (Wang, 1992) , G1H provide th e required mathem atical descript.iun for nonline ar functions used to define nonline ar sliding func t ions. Advant ages of non linear sliding surfaces a nd their implem entatio n by fuz zy logic are detailed for a rota tion<'J.1 inverted pendu-
lum .
The rest. of this paper it] organiz ed as follows . The sliding mod e control law based on non linear sliding surfaces is derived in Sed ion 2. The d(,Bign of nonlinc ar sliding surfaces is di sc ussed in Section 3. In Sect.ioIl4, advanta ge of nonline ar sliding s urfac.es is df>tailed for a rot,a tional inverted pf!ndulum. Condu sions a.re givf! n in Section 5_
{x,x , .. . ,x (1l-1 .I}T is the system state vecis the i-th time derivat ive of x, and u is the control input , The sign of b(x) is assume d t o be know n; fix) and b(x) are assume d to) be bounde d by the following inequal ities: X:=
tal, X(i)
If(x) - j (x)1 $ F(x)
(2)
_1_ < b(x) < (3(x) (3(x) - b(x) -
(3)
where j(x) and b(x) are th., known e.stimat es of fix ) and b(x), respect ively. F(x) alld (3(x) are a.o;umed to be know" functio ns (Slot.ine and Li , ]991).
2,2 Nonline ar Slidirlg Surface s Let X d denote t,he desired o utput a nd i: :== (Xd - ;c) den ote th(' tracking error. A nonline ar sliding surface for Lhe sysLern (I) is written .., follows:
where, Ai 'S a re static nonIine ar functio ns satisfyi ng \Ix E R
with 0 < " i $ Pi < alwa.ys be writ.ten as .<;
=i
00.
The sliding surface (4) can
{TI-l )
-+-.ci == 0
(5)
For second order system s (u =: 2), for exa.mple, wc have s=i+A (r.)=O
(6)
If Ai 'S a re replaced by positive constan ts {Ad;=l ,.. Eq. (4) be.comes a linear slidlng surface.
,1'1 -1 ,
Stabilit y of t.he sliding dynami cs, which is the dynami cs e.mbedded in the sliding surffl(·e , is stated in the following theorem ,
Th eo rem 1. The system descrih ed in (4) is ~2-stahlc and the equilib rium point , {:.i:,i , ... ) i( n- 2 )} 0 , is globally asympLotically stable.
=
This theorem ili proved by first decomp osing (4) into n - t first orde.r differen tial equatio ns and th en a pplying Ut. circle criterio n on each of Ihem (Vidya.,agar, 1978).
2 SLIDIN G MODE CONTR OL
2.1 The Plant The plant dynami cs to be consid ered for applyin g SMC is assumed ill t.he following fOfIll. : X (H)
= fix ) + "(x) u
(I)
2.3 Control Law The sliding mode control law based on nonline ar sliding surface s call be designe d by usiJlg the Lyapun ov 's direct method . Let V = > 0 bn a. candida te Lyapuno\'
iS2
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fun ction . It can be shown that thE' foll owing control law makes ,i uegativ e definite a.nd , hence, is CL stabiliz ing control law: u=
(7)
with it := X~l )
k? (fJ -
- - f + .~ 1)1.,1+ p (F + (+ h(s))
where ( is a strictly positive number , and h(s) is defined
a,.., follows:
h( s) =
(i-)
2
Anothe r problem associa ted wi t h the linear sliding mode control law is related to sys!;em perform ance. The controllaw cancels the system nonline arity and then replace it with the linear dynami cs. The impose d linear dynamics may not be na tural for t.he plant. Further more, the nonlin ear dynami cs may bc better mainta ined for system st.a.bility and/or perform ance instead of being callceJed. Therefore ) when the system state move~ along a linear sliding surface , the pnform ance is nol. necessarily optima l. Nonline ar sliding surface s may be able t o have the nat.ural plant dynami C",s as well as the optima l dynami cs embedd ed at th e design stage.
(8)
This pooit ive functjo n, h(s), makes V decreas e rapidly when Isl is greater t,han \11 , whkb is a positive number .
3.2 De."yn Method s
In pra.cticf', the sign an functio n : sgn(-), in (7) is rf!placed by a sahJrat ion function ,
Two met.hods for designi ng nonline ar sliding surfaces are suggest ed. Th.ese meth.od s arc suited for h.andIing the problem s stated in Section ~1 . 1 . The first one ii:i based on analysis and /or experience. The second method is based. on th e 1'f~ tJerse integro hotl of the ~ystem dynami c equatio n.
sat(s /
{~1 s/
if s > ~, if s < -~ , otherwi se.
The saturat ion functio n helplj in rf'ducing chatter ing when the syst,clII statl' is cloSE' t.o t he sliding surface . Howeve r : thf! sliding mode does not. t.ake place exactly.
:1. DESIGN OF NON LINEA R SLIDIN (; SURFACES in this section, we: will state some problem s associa ted with the {iH C(I?' sliding mode cuut1'01 . i.e. sliding mode cont.rol basf'd on linear ~liding surfaces. These problem s motivat e th e use of non linear sliding surfaces. We will t.hen pra pmiC several possibl f\ ways tc. rle~ign nonline ar sliding surfaces.
;3. 1 Problem s with Lmear Slidillg Surface s
The first problem associa.t.ed with the- lmear sliding mode contml arises due to the saturat ion of control inputs. [n order to maintai n linear dYl1am in. by feedback control, the magnit uJe of the co nt.rol input. is normall y proporI,ional t.o tht> dist.allce bet.ween t.hp. sysl,em state and the target state. Therefore, if t.he ~ystem ~tate is far away from th e target stat.e and the input satura.tes) the controller may not be a ble t.o keep th e system state staying on (or close to) the sliding ~urface (Sira-R amirez ) 1994). The input saturat ion problem may call se even instabil it.y for soo1(' l:iysterns. For this reason , it is often more meani ngful t.o design nunlin ear sliding ~ ur[aces that may at.t.ract and hold the: syst.em stat.e no matter where it is in tbe !:I~ate space.
3.2.1. Analy,i s/Experience Method When the trackin g error is ,.;mall , a nonline ar sliding surface may be replace d hy a linear sliding surface with vc.ry little effect. on Lhe system performa.nc~. For this reason ~ one may first. design il. linear sliding surface by applyin g any existing techniq ue such as the pole plar.c. ment method . This linear sliding surface will be useJ t.o determ in e the slope of thf' nonlineaT sliding surface when t.racking error is zero. For exampl e, if constan ts {Adi=I ,.... Il-L are used t o b:pecify a linear sliding surface, then thf:> following constra int must be satisfied: A;(O)= J.,.
io=I, .. . ,,,-1
(9)
where Ai( ·)'S (i = I .... , n - 1) are used t o specify a non linear sliding surface .
Anothe r coustra int is necessa ry for handlin g the input sat uration problf.m: i.e.
'Ix E R (10) where Li E ~ is a bound for .\d ·) . For int.uitive understandin g of (10), note th.at s indud" " [A n _ ,(·)·· ·A,(·)] i as a cOlnpone.nf.. The proper I:hoice of Lt may require analysi s , comput er simulat ions , or experie nce on th e system . 3.2.2. Reverse lrltegra tion Method
For second order systems , an alterna tive way to design nonJinear sliding surfaces is provide d by the reverse integmtio n method. To use this meth od , one must det.er-
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Umu
F=====::::::.?. 7--;';""~ ~;;;me:-"""~ i .
e,
Neer Mill TinIt "'in Energy
o ••-- .. -_______ "____. -- ---- 0--.-- ----------
·Um..
tL___~======:I o
u
ErrOt
Fig. 1. Sample Control Profile); mine a desired cont rol pmfile (DCP). This profile outlines the desi red (antral effor t that. SMC will generat.e wh en the system state is m ovill g al ong the sliding surface. A properl y desig ned DC P will make SMC achieve certain des irco perfofm anc(> ,
f or exampl e , one may use the' following functio n
aB
the
When the analysl s/exper ie fl l:e design meth od is used , one can transfo rm constraint.s into fuzzy rtJ/C.s . If the reverse integratioH met hod is used, onc can usc some method s t.o automa tically generat.e fuzzy rules making the fuzzy system fi t those in put/ou tput data to any desi red accnrac y (Ch u and Totllizu ka, 1995).
(11 )
4 . EXAM PL E
de,-;ir"cd con trol pro/ill" :
udr,,) = _ 0 .511" U
max
tan-'
(_x 1'9 11
ta]] (0 .45,,-))
Fig. 2. Rotatio nal Inverted Pendulu m
where ti lll 3.)( is a bound for the ...:o ntro l a nd ±r90 are the locations where u d r~a.ch es HO% of its maxim um / minimum values . By varying rg\). o ue ca ll generat e va rious DC' P )s . For exa.m ple, when r 9 0 is dOSf' to zero, u d represen t.s a minimu m lim.~ eQ(ltrol {o r a ba.ng-ba ng c.ontrol ) . When T'g a is a positivf> small value , uti represe nts a near minimu m time control. When 1'90 approa ches infini ty, u d becomes a. zero functio n and represt' nts a minimu m ene rgy cont.ro l, whic..: h on ly m akes sense to syst.em s ha ving il. central ma.nifold passing t.hrough tht". origiu. Sample DCP ', defi"",j by Eq . (11) are shown in Fig. 1.
Once the desired control profile il":i determ ined , it can be Ilsed to make th e slidi ng tiurfa (':(' lJy doing th e reverse integrat ion twice with ini t ia l state> (,s, -t:) and (--S , t") res pectively. where J and ( ar f' t.iuy pos it.ive number s. The.se reverse integra tions will ~~enerat e two trajecto ri es o n th e seco nd a.nd fort.h quadra nts of the x - .i: phase plane. The uni o n of the:se two t. raject.ories is th e 110nl1near sliding surface.
3.3 Fuzzy Modelin g
Nonli near sliding surfaces t hat. ~~rc designe d by the propused method s m ay be h ighly non linear and ma.y not be easily describ ed by polynom ials and ex ponenti al functio ns . NevcrLh f!less, the fuzzy logic, which is a.n universal approxi m ator (Wang ) 1992), (~an provide the requir~d ma themat.ical descrip tion for no nlinf'.ar fuuctio ns defining nonline ar slidi ng functio nti.
The Rot(ltional In vert ed Pendul um (RIP) mod el prepared by Misawa et al. (199f,) is used as the test bed . As shown in Fig _ 2, both links of the RIP rotates on a verti ca.l plane . Jts dy namic equatio ns can be writ ten as follows:
9, = /, -1 ' b, u 9, = h + b,u
( 12)
( 13) Details of the plan parame ters and the dynami cs can be fo "nd in Misawa et ai , (1 995). Most plant parame ters are assu med t.o have a 5% unferta inty. The RIP h:.s two degrees of freedom but has only one controL As a result , the standar d SMC design meth odology can not be directly applied t o this system. For t his reason, t.h e' control ler del>igu is carried o ut in t wo 8teps. First, we des ign a SMC fo r th e outer link using Eq. (13). This SM C can stab ilize 0" but 0, may drift fro m t he origin anrl eventua lly th e system will become unstabl e . Therefo re, a tim p.-varyi ng 8~ is used as an in· direct co nt.rol to st.abilized th e base link.
4.1 Controt of the Outer Link Three non linear sliding surface s ar~ designe d for the otJter link of the RIP by using t.he reverse -ir,t egration metbod . However , only one will be used in the control of the whol e ill P. Eq_ 11 is lH:,ed to rl esign t hree desire-d control profiles . Setting r 90 t.o 0.1, 1, and 00 (ra d), we have three DCP's
2880
,.
where k,. and kd are positive cout rol gains. Both kp and kd are chosen tu b e 0.06 ill the simulat ion. This ap-
Win Tim. NellrMin Ti mc;l .~--. Mlrl E"ltI'gj •••. •.
"
proach is an improv ement of the m et.hod propose d by Ka waji and Maed a (199 1), where denoted as the virtual f:quilibr ium point th ere , only varies between a posit.ive t·onst.aut, a negativ e const ant I and zero .
0; ,
. ->0
Intuitively we can see t.hat if 8 1 is posit.ive (a.5Buming 9 1 is zero) I t.he above equatio n will make O~ a small negat ive number . In ord p.r to balance t he outer link at a p01)itive (J~I the SMC will eventu a lly mainta in a negative torque , which may drive 01 back to the origin .
-"
-,. Fig.
-6
-.
~
0 Error Irad)
2
6
:1. Nonline ar Sliding Surfaces
4.3 Simulat ion Results Two 8irnulation~ were made on the RIP. For the first one, O2 Wa>; initially set. at 0.3 (rad). All ot her initial states were sd to zero. The maxim um control was assume d t.o be 1 (N-m)_ As , hown in Fig . 5, the propose d strategy can effectiv ely driv ~ both 9, a nd (J2 back to th e origin.
tMUll -
~
... -..
fi g. 4. Control Torque s on Va.rious Sliding Surface s for t.he minzmu m time. control , t.he n f or minimu m time r:o nLrol , an d the minimu m f. tWTY.II control , respecti vely. The value of U maK is a.ljsumed l () be 0 03 (N-m).
•••
._,
After a ppl ying the reven.e integration with the a bove DC P 's, we obta in thre:e non linear :sliding surfaces, as shown in Fig. 3. Co nt. rol torq ues co rresponding to system st at.~ o n th~8e nonline ar sliding s urfaces a.re s hown ill Fig. 4, togethe r wit.h the control t orques for the three i UH;UT s lidillg s urraet·.... t hat a rc tangent. to th ose three ll orliirH;'a r sliding surfac('2j a t t.h e ori gin . Not.e that. conLrol torques for linear sliding surfact's exc.eeds th e limit ra pidly as the error increases.
-0.2
O~--:.~.5-~-'~.5-'~--:'': . 5-'~--:'~.5-' Tim" (9IIcj
(a) Sta les
0.8
control - -
0.' 0.'
Sl\1C' with a ny of t.he above sliding surface s c.an achieve ~ aL isfactory perform a nce on th('! o tJt{~r link alone. Howcvt;r. only the SMC with t he lIonlille ar minimu m energy sl id ing surface will Iw used in t.he RIP co ntrol. By consllmiug less energy, the base hnk will be more llkcly not to deviate too fa r away from t.h(' origi n a.nd hence it will b (~ easier to driv(, it back.
•••o
.... .---- .- - --:;-~ - ._ - --
-0 ' \
~
- - ---1
-G.4 - \
·0.6 -0.8
-, .~--:0~.5-~--:'~.,-,~--:'~'-3~--:3~.5-" rir!. iMC)
(b) Contloi
4 _2 Control of the Bas e Link The following time-va rying 9~ is used Lo indirect ly stabilized the b(Js~; link :
(14)
Fig _5 . Nonline ar SMC with a Small Initial Error For the second simulat ion, initial values for 9 and 9 2 2 were chosen to he 2,,- (rad) and -3 (rad/s) , respect ively _ Since this initial staLe is far from the equilib rium point ,
2881
the nOllliut'ar sliding surface plays il critical rol e) as shown in Fig. 6. SMC)s with linear sliding s urfaces can hardly achiev e t his kind of pcrformallce.
(j- -
IS
-'- . -- <"'" , ".~
10
i"g W N
j "
.,
.r,
\
·10
\
"
-os
~
·6
.,
-2 0 2 TI'Ieta 2 En"O( (r.d)
6
guarantee the a."'Iyrnptotic stability of th e system. Two design me thods are suggested to generate non linear sliding surfaces - one is based on analysis and/or experience and the other is based on t he reverse integration method . The first method produces constraints ) while the second method produces data points representing nonlinear sliding surfaces. In dther ca....e, the fuzz y logic can be used to genera te desired nonlinear fundions that are used by nonlin ear sliding surfaCf~S. Simulation results on a. rotational in verted pendulum confirm the effectiveness of the proposed design methods and Msert t he SlIperiority of nOlllinear sliding surfaces over linear sliding surfac.es.
• 6. REFERENC ES
(a.) P ba!;e-Pla.nc Pl ol
tha1al Ihota2 ... •.
:!O.o _ _ _ ._._
.,
L-~~_~-L_~-L_~~
o
0.5
1.5 2 :2.5 TlmA (sac)
3
3.5
-4
(b) States
,.
control -
'-' '-' 0.2
~: ~---
--------.:;:;0 ---- -.= ----- - - - 1
·0.4 ·0.6
·0.8
., ,~-':o':,lL~-:,':.'-2~-:'':.'-3~-:3':.'-..J'· T"", ($(oc)
(c.) Control
Fig. 6. Nonlinear SMC with a Large lniLial Error
5. CO NC LUSIONS
\Ve ha ve proposed to design sliding mode control law hased on nonlin ear sliding su rfaces. The nonlinear sliding surfaces as well as t.he control law are fo rmulated to
Chu , V.K . and -'1 _Tomizuka ( 1995). Rule generation for fuzzy systems based on B-splioes. In : Proc . World Congress on Ncurnl Networks. pp. 1I 608- I I. Dwyer, lll, T.A.W. a nd H. Slra-Ramir.z (1988) . Variable st.ruct ure control of spacecraft attitude maneuverso A IAA J. of GuidmHt: . Control and Dynamics 11(3) , 262-70. Jabbari, A ., M. Tomizuka and T. Sakaguchi (1990a). Robust discrer.e-t.ime control of eontinuoUl:i time plant. In : U.S.A_ Sympos ium on Flexible Atclomation. pp . 573 - 79 . Jabbari , A., M. 1'o mizuka and T. Sakaguchi (1990b). Robu st. nonlinear con trol of positioning syste.ms with stiction . In: Proc. A merican Control Conference. pp. 1097- 1102. Kawaji) S. and T. Maeda (1991). Fuzzy servo control SYSt.CBl for an inverted pendulum. In: Pm<: . Interna tional Fuzzy Engineering Symposium. pp. 8 12- 82:" Lee, J.-J. (1991). Adaptive tracking controller of DC ser· vomo to rs. IEEE Tmu sadion on Circu its and Sy:;terns 37(4) ,905 12. Mi9awa, E. A., M. S. Arrill~ton and T. D. Ledgerwood ( 1995) . Rot ational inverted pendulum: A new control exp eriment. Amc1'ica n Co,llro! Conference pp. 29 --:13. Sira-Ramirez, H. and M . Rio.,-Bolivar (1994) . Sliding mode control of dc-to-de power converters via extended linearization. IEEE Transaction on CircU1Ls and Systems 41(10) , 652-liJ. Slotinc, J.E. and W . Li (I 991). Applied Non[m ear ControL Prcntice Ha.IL Vidyasagar , M. ( 1978). No n/inea,' System Analysis. Prentice Hall . Wang, L.X . (1992) . Fuzzy basis function , uni versal approximation, and orthogonal least squares learning . IEEE Tra ns. Neutra[ Netuarks 3(5) , 807-14 .
2882
2b-30 2
SLIDI NG MODE CONT ROL WITH NONL INEAR SLIDI NG SURFA CES Vi ctor K. Chu and Masay oshi Tomiz uka Departm ent of M echanical Enginee ring Univers ity of Califor nia at Berkele y Berkele y, Calif orn ia 94720, USA
Abstra ct. Traditi onal sliding mode conLrol1 ers use lin ear sliding surfaces, which m ight not. fit. t he glohal dynam ic proper ty of nonlioear plants . As a result, sliding m ay not take place due to actua tor sa tura tion and even insta bility m ay be induc.ed when t he error is large. Fur t hermore, linear sliding surfac€8 m ay c:ause large cha tteri ng in digital implem enta~·. ion. These problems may be solved by replacing lim'ar sliding surfaces with nonlinea r sliding surfaces. Two met hods for d esigni ng nonlim 'ar sliding su rfa ce~ a re suggest ed in th is paper. T he advanta ges of nonlin ear sliding sUl'faces a nd t h ~ ir approx ima.t ion by fu zzy logic are exam ined in det ails for a rotat ional inver ted pendulu m . K e yw ords . Slidi ng Mode Control , Fuzzy Modelin g, Nonline ar System s, Discrete Digit al Dynam i c Contro l
L INT RODlI CTION Slid ing Mode Co ntrol (S MC) is a popula r robuRt CODtrol m et hod for non linear system s satisfyi ng the matching r.oudit ion. The design of sliding mode control lers involves t.he followi ng two ste ps : 1) d~~ s lgn of a sliding surface , whi(:h represe nts the dr,si red stable system d ynamics, and 2) detiign of a co ntrol law whi ch m akes t he sliding surface attracti ve. When the slid ing t:i urface represenl.s lin(~ar dyna mics, th e magnitu tie of cOll tro l in put required to keep the system sl.a te on r.he ~ lidin g surface usua.lly i n t: rt ~illies as t he m agnitll rtp. of errors increases (J a bbar i d al., 1990 b). As a consequ euc.€, when the control force is bounde d, only a par t of the sliding sur face, called t hf' ..liding regime, is tlt.t radive. O ut~ id e t. he .sliding n~yl m f' , t. here is no gu a rantc(' for eit.h r.r sta bili t.y or perform a nce. Furt her more. sincr. tJw linear I;lid ing surface. m ay not be
na t ural to t.he pla nt, it may ca use la rge approac hing a ngles ) the an gles a t which t he system sta te ap proache s t he sliding surface. La rge approaching angles may ca.use a large scaJe chat teri ng along t he sliding ~ urfac e when the sliding mode controller is diseret ized (i.e. wi t h a fi ni te sampling ti me) . T his sit:u.atio n m ay be worsened at large velociti es. Therefo re) t he :$tate is exp ected to de viate from a linear sliding surface over a sampli ng period wh en t he error is large even if ir is d ose to th e surface at one samplin g inslance. J abba r i et al. (1 990.. ) a nalyzed th e hcha vior of C'ontinu ous time pla nts over one sampling period under di gital conl,rol and shown th a t the state must stay outside a conn sur roundin g the linear sliding surface at a IX\.mplin g instance in order that t he slid ing variable decreases its m agni t ude over one sampling p eri od . Th e area of t he non-att ractive cone will increase if Lhe ma.gl1ilude of t.h 4 ~ ~ l op e of t he linear sliding sur fac~ is increa.. ;;;ed. This an alysis shows tha t th e use of the linear slid ing ~ urfacf! m akes perform a nce Ilonuni -
2877
form .
where
The problem s mention ed above can be partly solved if we use SMC with nonlinear sliding surfaces. \\Then noolinear sliding surfaces are properl y designed , the sliding ffgune may be enlarge d and optima lity conditio ns, such as the mi'limu m en ergy consumpt.i on or the minimu m ti m.e usage, may bo embedd ed in [,he system. Dwyer a nJ Sira.· Ramirez (1988) have shown the idea of nonlinear sliding surfaces hut they have nOI. provided a design algorith m. Jabbari et al. (199010) US", paraholic sliding surfaces on the design of the ncar minillJUm time control for linear seco nd order system s with force constra int. Lee (1991) has propose d t,o use c1Ibic polynom ials as sliding sHrraces to improve transien t respolI:-oe. In this pa.p cr~ we propose to design sliding mode control based 0 11 non linear sliding surface s that aTe formulated in .... way t o guaran tee th~ (l.':Iympt.otic stability. In ilddition~, we suggest. two m ~t hods to design the uonlin ear sliding s urfaces that may solve the saturat iOH / perfonn anc.r. problc:ms associa ted with linear sliding surfaces. The first. method is based on analysi s and/or expr.rience. All existing techniq ues for designi ng linear sliding surface may be include d in t.his method . Tht' seco nd meth od uses desired con/ml pmJiles and the reVlTSf intC'. gration lIIE>thod to generat ion lIonJinc ar sliding s llrfac~. The input. constra int as well as certa.in pr.rform anre criteri on may be emb edded in this design method .
In either design meth od, thE' /uzz.tJ (ogle, whi ch is an uniye rsal approx imator (Wang, 1992) , G1H provide th e required mathem atical descript.iun for nonline ar functions used to define nonline ar sliding func t ions. Advant ages of non linear sliding surfaces a nd their implem entatio n by fuz zy logic are detailed for a rota tion<'J.1 inverted pendu-
lum .
The rest. of this paper it] organiz ed as follows . The sliding mod e control law based on non linear sliding surfaces is derived in Sed ion 2. The d(,Bign of nonlinc ar sliding surfaces is di sc ussed in Section 3. In Sect.ioIl4, advanta ge of nonline ar sliding s urfac.es is df>tailed for a rot,a tional inverted pf!ndulum. Condu sions a.re givf! n in Section 5_
{x,x , .. . ,x (1l-1 .I}T is the system state vecis the i-th time derivat ive of x, and u is the control input , The sign of b(x) is assume d t o be know n; fix) and b(x) are assume d to) be bounde d by the following inequal ities: X:=
tal, X(i)
If(x) - j (x)1 $ F(x)
(2)
_1_ < b(x) < (3(x) (3(x) - b(x) -
(3)
where j(x) and b(x) are th., known e.stimat es of fix ) and b(x), respect ively. F(x) alld (3(x) are a.o;umed to be know" functio ns (Slot.ine and Li , ]991).
2,2 Nonline ar Slidirlg Surface s Let X d denote t,he desired o utput a nd i: :== (Xd - ;c) den ote th(' tracking error. A nonline ar sliding surface for Lhe sysLern (I) is written .., follows:
where, Ai 'S a re static nonIine ar functio ns satisfyi ng \Ix E R
with 0 < " i $ Pi < alwa.ys be writ.ten as .<;
=i
00.
The sliding surface (4) can
{TI-l )
-+-.ci == 0
(5)
For second order system s (u =: 2), for exa.mple, wc have s=i+A (r.)=O
(6)
If Ai 'S a re replaced by positive constan ts {Ad;=l ,.. Eq. (4) be.comes a linear slidlng surface.
,1'1 -1 ,
Stabilit y of t.he sliding dynami cs, which is the dynami cs e.mbedded in the sliding surffl(·e , is stated in the following theorem ,
Th eo rem 1. The system descrih ed in (4) is ~2-stahlc and the equilib rium point , {:.i:,i , ... ) i( n- 2 )} 0 , is globally asympLotically stable.
=
This theorem ili proved by first decomp osing (4) into n - t first orde.r differen tial equatio ns and th en a pplying Ut. circle criterio n on each of Ihem (Vidya.,agar, 1978).
2 SLIDIN G MODE CONTR OL
2.1 The Plant The plant dynami cs to be consid ered for applyin g SMC is assumed ill t.he following fOfIll. : X (H)
= fix ) + "(x) u
(I)
2.3 Control Law The sliding mode control law based on nonline ar sliding surface s call be designe d by usiJlg the Lyapun ov 's direct method . Let V = > 0 bn a. candida te Lyapuno\'
iS2
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fun ction . It can be shown that thE' foll owing control law makes ,i uegativ e definite a.nd , hence, is CL stabiliz ing control law: u=
(7)
with it := X~l )
k? (fJ -
- - f + .~ 1)1.,1+ p (F + (+ h(s))
where ( is a strictly positive number , and h(s) is defined
a,.., follows:
h( s) =
(i-)
2
Anothe r problem associa ted wi t h the linear sliding mode control law is related to sys!;em perform ance. The controllaw cancels the system nonline arity and then replace it with the linear dynami cs. The impose d linear dynamics may not be na tural for t.he plant. Further more, the nonlin ear dynami cs may bc better mainta ined for system st.a.bility and/or perform ance instead of being callceJed. Therefore ) when the system state move~ along a linear sliding surface , the pnform ance is nol. necessarily optima l. Nonline ar sliding surface s may be able t o have the nat.ural plant dynami C",s as well as the optima l dynami cs embedd ed at th e design stage.
(8)
This pooit ive functjo n, h(s), makes V decreas e rapidly when Isl is greater t,han \11 , whkb is a positive number .
3.2 De."yn Method s
In pra.cticf', the sign an functio n : sgn(-), in (7) is rf!placed by a sahJrat ion function ,
Two met.hods for designi ng nonline ar sliding surfaces are suggest ed. Th.ese meth.od s arc suited for h.andIing the problem s stated in Section ~1 . 1 . The first one ii:i based on analysis and /or experience. The second method is based. on th e 1'f~ tJerse integro hotl of the ~ystem dynami c equatio n.
sat(s /
{~1 s/
if s > ~, if s < -~ , otherwi se.
The saturat ion functio n helplj in rf'ducing chatter ing when the syst,clII statl' is cloSE' t.o t he sliding surface . Howeve r : thf! sliding mode does not. t.ake place exactly.
:1. DESIGN OF NON LINEA R SLIDIN (; SURFACES in this section, we: will state some problem s associa ted with the {iH C(I?' sliding mode cuut1'01 . i.e. sliding mode cont.rol basf'd on linear ~liding surfaces. These problem s motivat e th e use of non linear sliding surfaces. We will t.hen pra pmiC several possibl f\ ways tc. rle~ign nonline ar sliding surfaces.
;3. 1 Problem s with Lmear Slidillg Surface s
The first problem associa.t.ed with the- lmear sliding mode contml arises due to the saturat ion of control inputs. [n order to maintai n linear dYl1am in. by feedback control, the magnit uJe of the co nt.rol input. is normall y proporI,ional t.o tht> dist.allce bet.ween t.hp. sysl,em state and the target state. Therefore, if t.he ~ystem ~tate is far away from th e target stat.e and the input satura.tes) the controller may not be a ble t.o keep th e system state staying on (or close to) the sliding ~urface (Sira-R amirez ) 1994). The input saturat ion problem may call se even instabil it.y for soo1(' l:iysterns. For this reason , it is often more meani ngful t.o design nunlin ear sliding ~ ur[aces that may at.t.ract and hold the: syst.em stat.e no matter where it is in tbe !:I~ate space.
3.2.1. Analy,i s/Experience Method When the trackin g error is ,.;mall , a nonline ar sliding surface may be replace d hy a linear sliding surface with vc.ry little effect. on Lhe system performa.nc~. For this reason ~ one may first. design il. linear sliding surface by applyin g any existing techniq ue such as the pole plar.c. ment method . This linear sliding surface will be useJ t.o determ in e the slope of thf' nonlineaT sliding surface when t.racking error is zero. For exampl e, if constan ts {Adi=I ,.... Il-L are used t o b:pecify a linear sliding surface, then thf:> following constra int must be satisfied: A;(O)= J.,.
io=I, .. . ,,,-1
(9)
where Ai( ·)'S (i = I .... , n - 1) are used t o specify a non linear sliding surface .
Anothe r coustra int is necessa ry for handlin g the input sat uration problf.m: i.e.
'Ix E R (10) where Li E ~ is a bound for .\d ·) . For int.uitive understandin g of (10), note th.at s indud" " [A n _ ,(·)·· ·A,(·)] i as a cOlnpone.nf.. The proper I:hoice of Lt may require analysi s , comput er simulat ions , or experie nce on th e system . 3.2.2. Reverse lrltegra tion Method
For second order systems , an alterna tive way to design nonJinear sliding surfaces is provide d by the reverse integmtio n method. To use this meth od , one must det.er-
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Umu
F=====::::::.?. 7--;';""~ ~;;;me:-"""~ i .
e,
Neer Mill TinIt "'in Energy
o ••-- .. -_______ "____. -- ---- 0--.-- ----------
·Um..
tL___~======:I o
u
ErrOt
Fig. 1. Sample Control Profile); mine a desired cont rol pmfile (DCP). This profile outlines the desi red (antral effor t that. SMC will generat.e wh en the system state is m ovill g al ong the sliding surface. A properl y desig ned DC P will make SMC achieve certain des irco perfofm anc(> ,
f or exampl e , one may use the' following functio n
aB
the
When the analysl s/exper ie fl l:e design meth od is used , one can transfo rm constraint.s into fuzzy rtJ/C.s . If the reverse integratioH met hod is used, onc can usc some method s t.o automa tically generat.e fuzzy rules making the fuzzy system fi t those in put/ou tput data to any desi red accnrac y (Ch u and Totllizu ka, 1995).
(11 )
4 . EXAM PL E
de,-;ir"cd con trol pro/ill" :
udr,,) = _ 0 .511" U
max
tan-'
(_x 1'9 11
ta]] (0 .45,,-))
Fig. 2. Rotatio nal Inverted Pendulu m
where ti lll 3.)( is a bound for the ...:o ntro l a nd ±r90 are the locations where u d r~a.ch es HO% of its maxim um / minimum values . By varying rg\). o ue ca ll generat e va rious DC' P )s . For exa.m ple, when r 9 0 is dOSf' to zero, u d represen t.s a minimu m lim.~ eQ(ltrol {o r a ba.ng-ba ng c.ontrol ) . When T'g a is a positivf> small value , uti represe nts a near minimu m time control. When 1'90 approa ches infini ty, u d becomes a. zero functio n and represt' nts a minimu m ene rgy cont.ro l, whic..: h on ly m akes sense to syst.em s ha ving il. central ma.nifold passing t.hrough tht". origiu. Sample DCP ', defi"",j by Eq . (11) are shown in Fig. 1.
Once the desired control profile il":i determ ined , it can be Ilsed to make th e slidi ng tiurfa (':(' lJy doing th e reverse integrat ion twice with ini t ia l state> (,s, -t:) and (--S , t") res pectively. where J and ( ar f' t.iuy pos it.ive number s. The.se reverse integra tions will ~~enerat e two trajecto ri es o n th e seco nd a.nd fort.h quadra nts of the x - .i: phase plane. The uni o n of the:se two t. raject.ories is th e 110nl1near sliding surface.
3.3 Fuzzy Modelin g
Nonli near sliding surfaces t hat. ~~rc designe d by the propused method s m ay be h ighly non linear and ma.y not be easily describ ed by polynom ials and ex ponenti al functio ns . NevcrLh f!less, the fuzzy logic, which is a.n universal approxi m ator (Wang ) 1992), (~an provide the requir~d ma themat.ical descrip tion for no nlinf'.ar fuuctio ns defining nonline ar slidi ng functio nti.
The Rot(ltional In vert ed Pendul um (RIP) mod el prepared by Misawa et al. (199f,) is used as the test bed . As shown in Fig _ 2, both links of the RIP rotates on a verti ca.l plane . Jts dy namic equatio ns can be writ ten as follows:
9, = /, -1 ' b, u 9, = h + b,u
( 12)
( 13) Details of the plan parame ters and the dynami cs can be fo "nd in Misawa et ai , (1 995). Most plant parame ters are assu med t.o have a 5% unferta inty. The RIP h:.s two degrees of freedom but has only one controL As a result , the standar d SMC design meth odology can not be directly applied t o this system. For t his reason, t.h e' control ler del>igu is carried o ut in t wo 8teps. First, we des ign a SMC fo r th e outer link using Eq. (13). This SM C can stab ilize 0" but 0, may drift fro m t he origin anrl eventua lly th e system will become unstabl e . Therefo re, a tim p.-varyi ng 8~ is used as an in· direct co nt.rol to st.abilized th e base link.
4.1 Controt of the Outer Link Three non linear sliding surface s ar~ designe d for the otJter link of the RIP by using t.he reverse -ir,t egration metbod . However , only one will be used in the control of the whol e ill P. Eq_ 11 is lH:,ed to rl esign t hree desire-d control profiles . Setting r 90 t.o 0.1, 1, and 00 (ra d), we have three DCP's
2880
,.
where k,. and kd are positive cout rol gains. Both kp and kd are chosen tu b e 0.06 ill the simulat ion. This ap-
Win Tim. NellrMin Ti mc;l .~--. Mlrl E"ltI'gj •••. •.
"
proach is an improv ement of the m et.hod propose d by Ka waji and Maed a (199 1), where denoted as the virtual f:quilibr ium point th ere , only varies between a posit.ive t·onst.aut, a negativ e const ant I and zero .
0; ,
. ->0
Intuitively we can see t.hat if 8 1 is posit.ive (a.5Buming 9 1 is zero) I t.he above equatio n will make O~ a small negat ive number . In ord p.r to balance t he outer link at a p01)itive (J~I the SMC will eventu a lly mainta in a negative torque , which may drive 01 back to the origin .
-"
-,. Fig.
-6
-.
~
0 Error Irad)
2
6
:1. Nonline ar Sliding Surfaces
4.3 Simulat ion Results Two 8irnulation~ were made on the RIP. For the first one, O2 Wa>; initially set. at 0.3 (rad). All ot her initial states were sd to zero. The maxim um control was assume d t.o be 1 (N-m)_ As , hown in Fig . 5, the propose d strategy can effectiv ely driv ~ both 9, a nd (J2 back to th e origin.
tMUll -
~
... -..
fi g. 4. Control Torque s on Va.rious Sliding Surface s for t.he minzmu m time. control , t.he n f or minimu m time r:o nLrol , an d the minimu m f. tWTY.II control , respecti vely. The value of U maK is a.ljsumed l () be 0 03 (N-m).
•••
._,
After a ppl ying the reven.e integration with the a bove DC P 's, we obta in thre:e non linear :sliding surfaces, as shown in Fig. 3. Co nt. rol torq ues co rresponding to system st at.~ o n th~8e nonline ar sliding s urfaces a.re s hown ill Fig. 4, togethe r wit.h the control t orques for the three i UH;UT s lidillg s urraet·.... t hat a rc tangent. to th ose three ll orliirH;'a r sliding surfac('2j a t t.h e ori gin . Not.e that. conLrol torques for linear sliding surfact's exc.eeds th e limit ra pidly as the error increases.
-0.2
O~--:.~.5-~-'~.5-'~--:'': . 5-'~--:'~.5-' Tim" (9IIcj
(a) Sta les
0.8
control - -
0.' 0.'
Sl\1C' with a ny of t.he above sliding surface s c.an achieve ~ aL isfactory perform a nce on th('! o tJt{~r link alone. Howcvt;r. only the SMC with t he lIonlille ar minimu m energy sl id ing surface will Iw used in t.he RIP co ntrol. By consllmiug less energy, the base hnk will be more llkcly not to deviate too fa r away from t.h(' origi n a.nd hence it will b (~ easier to driv(, it back.
•••o
.... .---- .- - --:;-~ - ._ - --
-0 ' \
~
- - ---1
-G.4 - \
·0.6 -0.8
-, .~--:0~.5-~--:'~.,-,~--:'~'-3~--:3~.5-" rir!. iMC)
(b) Contloi
4 _2 Control of the Bas e Link The following time-va rying 9~ is used Lo indirect ly stabilized the b(Js~; link :
(14)
Fig _5 . Nonline ar SMC with a Small Initial Error For the second simulat ion, initial values for 9 and 9 2 2 were chosen to he 2,,- (rad) and -3 (rad/s) , respect ively _ Since this initial staLe is far from the equilib rium point ,
2881
the nOllliut'ar sliding surface plays il critical rol e) as shown in Fig. 6. SMC)s with linear sliding s urfaces can hardly achiev e t his kind of pcrformallce.
(j- -
IS
-'- . -- <"'" , ".~
10
i"g W N
j "
.,
.r,
\
·10
\
"
-os
~
·6
.,
-2 0 2 TI'Ieta 2 En"O( (r.d)
6
guarantee the a."'Iyrnptotic stability of th e system. Two design me thods are suggested to generate non linear sliding surfaces - one is based on analysis and/or experience and the other is based on t he reverse integration method . The first method produces constraints ) while the second method produces data points representing nonlinear sliding surfaces. In dther ca....e, the fuzz y logic can be used to genera te desired nonlinear fundions that are used by nonlin ear sliding surfaCf~S. Simulation results on a. rotational in verted pendulum confirm the effectiveness of the proposed design methods and Msert t he SlIperiority of nOlllinear sliding surfaces over linear sliding surfac.es.
• 6. REFERENC ES
(a.) P ba!;e-Pla.nc Pl ol
tha1al Ihota2 ... •.
:!O.o _ _ _ ._._
.,
L-~~_~-L_~-L_~~
o
0.5
1.5 2 :2.5 TlmA (sac)
3
3.5
-4
(b) States
,.
control -
'-' '-' 0.2
~: ~---
--------.:;:;0 ---- -.= ----- - - - 1
·0.4 ·0.6
·0.8
., ,~-':o':,lL~-:,':.'-2~-:'':.'-3~-:3':.'-..J'· T"", ($(oc)
(c.) Control
Fig. 6. Nonlinear SMC with a Large lniLial Error
5. CO NC LUSIONS
\Ve ha ve proposed to design sliding mode control law hased on nonlin ear sliding su rfaces. The nonlinear sliding surfaces as well as t.he control law are fo rmulated to
Chu , V.K . and -'1 _Tomizuka ( 1995). Rule generation for fuzzy systems based on B-splioes. In : Proc . World Congress on Ncurnl Networks. pp. 1I 608- I I. Dwyer, lll, T.A.W. a nd H. Slra-Ramir.z (1988) . Variable st.ruct ure control of spacecraft attitude maneuverso A IAA J. of GuidmHt: . Control and Dynamics 11(3) , 262-70. Jabbari, A ., M. Tomizuka and T. Sakaguchi (1990a). Robust discrer.e-t.ime control of eontinuoUl:i time plant. In : U.S.A_ Sympos ium on Flexible Atclomation. pp . 573 - 79 . Jabbari , A., M. 1'o mizuka and T. Sakaguchi (1990b). Robu st. nonlinear con trol of positioning syste.ms with stiction . In: Proc. A merican Control Conference. pp. 1097- 1102. Kawaji) S. and T. Maeda (1991). Fuzzy servo control SYSt.CBl for an inverted pendulum. In: Pm<: . Interna tional Fuzzy Engineering Symposium. pp. 8 12- 82:" Lee, J.-J. (1991). Adaptive tracking controller of DC ser· vomo to rs. IEEE Tmu sadion on Circu its and Sy:;terns 37(4) ,905 12. Mi9awa, E. A., M. S. Arrill~ton and T. D. Ledgerwood ( 1995) . Rot ational inverted pendulum: A new control exp eriment. Amc1'ica n Co,llro! Conference pp. 29 --:13. Sira-Ramirez, H. and M . Rio.,-Bolivar (1994) . Sliding mode control of dc-to-de power converters via extended linearization. IEEE Transaction on CircU1Ls and Systems 41(10) , 652-liJ. Slotinc, J.E. and W . Li (I 991). Applied Non[m ear ControL Prcntice Ha.IL Vidyasagar , M. ( 1978). No n/inea,' System Analysis. Prentice Hall . Wang, L.X . (1992) . Fuzzy basis function , uni versal approximation, and orthogonal least squares learning . IEEE Tra ns. Neutra[ Netuarks 3(5) , 807-14 .
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