Robust Control Design for Nonlinear Uncertain Systems with an Unknown Time-Varying Control Direction

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH [7] L. Hsu, Variable srucure model reference adapive conrol using only I/O measuremen: General case, IEEE Trans. Auoma. Conr., vol. 35, no. 11, pp , [8] L. Hsu, A. D. Araújo, and R. R. Cosa, Analysis and design of I/O based variable srucure adapive conrol, IEEE Trans. Auoma. Conr., vol. 39, no. 1, pp. 4 1, [9] L. Hsu, A. D. Araújo, and F. Lizarralde, New resuls on VS-MRAC: Design and sabiliy analysis, in Proc. Amer. Conr. Conf., San Francisco, 1993, pp [10] L. Hsu and R. R. Cosa, Variable srucure model reference adapive conrol using only inpu and oupu measuremen: Par I, In. J. Conr., vol. 49, no., pp , [11] L. Hsu and F. Lizarralde, Redesign and sabiliy analysis of VS-MRAC sysems, in Proc. Amer. Conr. Conf., Chicago, 199, pp [1], Experimenal resuls on variable srucure adapive robo conrol wihou velociy measuremen, in Proc. Amer. Conr. Conf., Seale, 1995, pp [13] P. Ioannou and K. Tsakalis, A robus direc adapive conroller, IEEE Trans. Auoma. Conr., vol. 31, no. 11, pp , [14] S. M. Naik, P. R. Kumar, and B. E. Ydsie, Robus coninuous-ime adapive conrol by parameer projecion, IEEE Trans. Auoma. Conr., vol. 37, no., pp , 199. [15] K. Narendra and A. Annaswamy, Sable Adapive Sysems. Englewood Cliffs, NJ: Prenice-Hall, [16] S. S. Sasry and M. Bodson, Adapive Conrol: Sabiliy, Convergence and Robusness. Englewood Cliffs, NJ: Prenice-Hall, [17] V. I. Ukin, Sliding Modes and Their Applicaion in Variable Srucure Sysems. MIR, [18] A. C. Wu, L. C. Fu, and C. F Hsu, Robus MRAC for plan wih arbirary relaive degree using variable srucure design, in Proc. Amer. Conr. Conf., Chicago, 199, pp Robus Conrol Design for Nonlinear Uncerain Sysems wih an Unknown Time-Varying Conrol Direcion Joseph Kalous and Z. Qu Absrac In his paper, a coninuous robus conrol design approach is proposed for firs-order nonlinear sysems whose dynamics conain boh nonlinear uncerainy and an unknown ime-varying conrol direcion. The so-called conrol direcion is he muliplier of he conrol erm in he dynamic equaion, and i effecively represens he direcion of moion under any given conrol. A nonlinear robus conrol is designed o on-line and coninuously idenify sign changes of he unknown conrol direcion and o guaranee sabiliy of uniform ulimae boundedness. The proposed robus conrol design requires only hree condiions: he nominal sysem is sable, he conrol direcion is smooh, and he uncerainy in he sysem is bounded by a known funcion. The necessiy of hese condiions is esablished in his paper. Coninuiy of he proposed robus conrol is achieved by using a so-called shifing law ha changes smoohly he sign of robus conrol and racks he change of he unknown conrol direcion. Analysis and design is shown by using he Lyapunov s direc mehod. Index Terms Lyapunov sabiliy, robus conrol, uncerain sysems. and sabilizing robus conrollers have been developed, for example, he resuls in [1], [3], [10], [16] [19], and [1]. The uncerainies admissible in hese resuls can be of eiher mached ype or unmached ype, and hey are generally bounded by known nonlinear funcions of he sae. However, inpu-relaed uncerainies analyzed in he exising resuls are assumed o be of fixed and known signs; ha is, he direcion of moion of he sysem is fixed and known under any choice of conrol. The assumpion of priori knowledge of he conrol direcion being known makes robus conrol design sraighforward. In [5], he problem of designing robus conrol was sudied for he class of sysems wih an unknown bu consan and boundedaway-from-zero conrol direcion. The objecive of his paper is o generalize he resuls in [5] so ha sysems wih an unknown and ime-varying conrol direcion can be reaed. This exension is nonrivial since he ime-varying conrol direcion is allowed o cross zero an infinie number of imes and since idenificaion mus now be done fas and coninuously o keep racking he changing conrol direcion. As before, he Lyapunov s direc mehod is uilized o proceed wih he design and o ensure he sabiliy of uniform ulimae boundedness. Similar o he resuls inroduced in [5], coninuiy of he proposed robus conrol is achieved by designing he so-called shifing law, which coninuously and smoohly seers he sign of our robus conrol. In he area of adapive conrol, many resuls dealing wih sysems conaining an unknown consan conrol direcion, referred o herein as he unknown high-frequency gain, have been obained. For example, Nussbaum [15] proposed for linear ime-invarian sysems an adapive conrol algorihm whose sign changes an infinie number of imes as is argumen ends o infiniy. In [1], Mudge and his coworkers developed an adapive conrol design scheme for general linear sysems wihou requiring he knowledge of high-frequency gain by following he same principle as ha in [15]. In he effor of exending hese resuls o nonlinear sysems having unknown conrol direcions, wo adapive conrol schemes have been proposed for simple firs-order nonlinear sysems [], [6]. In [] and [6], a hyseresis mehod was used o consruc an adapive conrol ha jumps a finie number of imes over he infinie ime horizon. Along he line of nonlinear robus conrol, a coninuous robus conrol design scheme was developed in [5] for uncerain nonlinear sysems wih an unknown bu consan conrol direcion. Compared wih he exising resul, he proposed resul is he firs aemp in addressing he conrol design problem in he presence of an unknown bu smooh conrol direcion, whose sign change is of unlimied number and may occur anyime. The paper is broken ino he following secions. Sysem descripion, necessary assumpions, and he problem of designing robus conrol are described in Secion II. Robus conrol design is proceeded in Secion III by he Lyapunov s direc mehod. A simulaion example is presened in Secion IV. I. INTRODUCTION For more han one decade, robus conrol design of nonlinear uncerain sysems has been an acive area of research. Several imporan classes of sabilizable uncerain sysems have been idenified, Manuscrip received March 14, 1995; revised Sepember 8, 1995, April 0, 1996, and July 1, J. Kalous is wih Loral Vough Sysems, Dallas, TX USA. Z. Qu is wih he Deparmen of Elecrical and Compuer Engineering, Universiy of Cenral Florida, Orlando, FL 3816 USA. Publisher Iem Idenifier S (97) II. PROBLEM FORMULATION We shall consider he nonlinear uncerain sysem described by _x = f (x; ) +1f 0 (x; ) +a()u(x; ) (1) where x < is he sae, u < is he conrol inpu, f (x; ) and 1f 0 (x; ) are known and unknown dynamics of he sysem, respecively. Symbol a() characerizes he ime-varying direcion of moion wih respec o a given conrol. The sysem is chosen o be scalar in order o highligh he developmen of robus conrol design /97$ IEEE

2 394 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH 1997 in he presence of an unknown, ime-varying conrol direcion. I is obvious ha (1) can be made of a vecor and ha, based on he resuls in [5], i can also be exended o be of higher order. The conrol direcion, sign[a()] of funcion a(); is unknown, and i may swich over ime among 01, 0, and 1 arbirarily. For idenifiabiliy, an upper bound will be imposed on he rae of change of funcion a(); and he necessiy of such an assumpion can easily be shown by conradicion. Since a() is allowed o change is sign, he case ha a() may be zero all he ime mus be considered. Noe ha a() 0 implies ha here is no conrol acion, no maer wha u is. In his case, moion and sabiliy of he sysem are governed solely by known and unknown dynamics, bu hese dynamics, if unsable, mus be compensaed. For his reason, wo necessary assumpions mus be made. Firs, known dynamics f (x; ) have o be sable (for he case ha a() 0 and 1f (x; ) 0). Second, he uncerainy in he sysem has o have he propery ha for some well defined funcion 1f (x; ); 1f 0 (x; ) = a()1f(x; ): Wihou a() being he common facor beween he erms of conrol and uncerainy, sabilizabiliy canno be guaraneed if he bounding funcion of he uncerainy is arbirary. Tha is, if funcion a() is muliplied only a he fron of conrol u; he sysem is sable only if he uncerainy is small in he sense ha i is dominaed by he robus sabiliy margin. This laer case is no considered in he paper, since sabiliy is a hand simply by leing u =0; herefore, he conrol design is rivial. Wih hese facs in mind, we can define he following assumpions for sysems in he form of (1). Assumpion 1: Under a coninuous conrol u o be devised, he dynamics of (1) are Caraheodory. Tha is, (1) has a classical soluion under any coninuous conrol. Assumpion : Funcion f (x; ) characerizing he dynamics of he nominal sysem is exponenially sable. Tha is, by he Lyapunov converse heorem, here exiss a differeniable funcion V 1(x; ) such ha, for some posiive consans 1; ; 3; and 4 ; 1 jxj V 1 (x; ) jxj ; j =@j 3 jxj; and =@ +( =)f (x; ) 0 4 jxj : Assumpion 3: Lumped uncerainy 1f 0 (x; ) can be rewrien as 1f 0 (x; ) = a()1f(x; ), and 1f (x; ) is bounded as, for all (x; ); j1f (x; )j (x); where (1) is known, coninuous, and locally uniformly bounded wih respec o x: Assumpion 4: Funcion a() is bounded and normalized in he sense ha ja()j 1:In addiion, a() is smooh in he sense ha is firs-order derivaive is bounded from above as j _a()j for a known consan >0: Assumpion 5: For all possible choices of he uncerainy 1f 0 (x; ) bounded by ( + )(x) for some consan >0; (1) wih u 0 does no have a finie escape ime ha is infiniesimal. Mos sysems in he form of (1) saisfy Assumpion 5. To see his conclusion, consider he choice of bounding funcion (x) ha (x) =jxj p for some p>0:then, i follows from Assumpion and from u =0ha, along every rajecory of (1), he ime derivaive of Lyapunov funcion V 1 (x) saisfies _V jxj +(+)jxj(x) 0 4 V p+1 V p+1= 1 : I is obvious from he upper bound on he growh rae ( _ V 1 ) ha Lyapunov funcion V 1, and herefore he sae x, will no become infinie insananeously. This resul holds for any choice of posiive real number p: Noe ha by he Taylor series expansion, polynomial funcions form a local basis for all differeniable funcions. Neverheless, o avoid he complicaion of proving Assumpion 5 for all possible expressions of bounding funcion (1); we choose o make he assumpion. Once (x) is given, one can simply proceed wih he proof of Assumpion 5 using he above argumen. Assumpion 5 is needed for on-line idenificaion of he unknown and ime-varying direcion. If Assumpion 5 does no hold, no idenificaion algorihm can idenify insananeously and keep perfecly racking an unknown changing conrol direcion so ha a robus conrol of he correc sign can be implemened. The reason ha he bounding funcion used in he assumpion is ( + )(x) raher han (x) is ha robus conrol mus have is magniude no smaller han (x) for compensaing for uncerainy of size (x) and ha he wors case of robus conrol iniially having a wrong sign and consequenly being desabilizing mus be considered. In he case ha he finie escape ime is infiniesimal, here is no conrol of finie magniude ha can pull he sysem back, even if he conrol direcion can be idenified insananeously. Based on hese assumpions, we can proceed wih robus conrol design in he nex secion. III. ROBUST CONTROL DESIGN The main approach for designing robus conrol for nonlinear uncerain sysems has been he Lyapunov s direc mehod. In is applicaion, deerminisic funcions are used o bound uncerainies, and several classes of uncerain sysems have been shown o be globally sabilizable [1], [3], [10], [16], [19], [1]. More recen developmen along his line of work is he recursive-inerlacing design procedure [17], [18] which is generic enough o overcome he common srucural condiions imposed on uncerain sysems. There are wo common feaures among hese resuls. Firs, global sabilizing robus conrollers are devised if bounding funcions of uncerainies are available. The selecion of a robus conroller is done hrough a Lyapunov argumen such ha robus conrol dominaes uncerainies in boh magniude and sign. Second, in hese resuls he conrol direcions are known in he sense ha afer choosing a robus conroller, he sign of he robus conrol deermines he signs of he ime derivaives of boh he sae and he Lyapunov funcion. I was shown in [5] ha he dominaing effec of robus conrol can be used o idenify a consan unknown conrol direcion and ha a coninuous robus conrol can be devised whose sign is changed smoohly. In his paper, robus conrol design is sudied for sysems wih an unknown and ime-varying conrol direcion, which can be viewed as an exension of [5]. Unlike he problem of dealing wih a fixed unknown conrol direcion, he ime-varying conrol direcion considered here is no bounded away from zero, he number of is sign changes is unlimied over ime, and a successful idenificaion algorihm mus rack closely is sign changes and be capable of handling he poenial problem of singulariy a zero crossings. Thus, he proposed exension is no rivial. Like he exising resuls, he design of a sabilizing robus conrol in he presence of a ime-varying and unknown conrol direcion involves a Lyapunov argumen and he concep of having he conrol dominae he unknown dynamics. As in [5], an iniial guess is made on he sign of he unknown conrol direcion, idenificaion of he unknown conrol direcion is hen carried ou on-line coninuously, and a robus conrol is designed whose sign is modified according o he idenified conrol direcion. The key resuls in his sudy are ha despie he fac ha he unknown conrol direcion may change is sign an infinie number of imes, idenificaion and racking can be done using he concep of size dominaion, and since a coninuous conrol usually produces a beer ransien response han disconinuous conrols, a smooh ransiion of he sign of he robus conrol can be achieved. Inuiively, he basic idea emanaes as follows: begin wih some guess ^a() of a(); design a robus conrol

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH in erms of ^a(); idenify a() on line; formulae shifing law _^a() so ha he robus conrol is coninuous; and le all hree designs be coordinaed and inegraed in such a way ha robus sabiliy is ensured. Robus conrol design in he presence of a ime-varying conrol direcion consiss of hree inegraed pars: 1) selecion of robus conrol using he Lyapunov s direc mehod in a similar fashion as ha wih a known conrol direcion; ) a scheme ha on-line idenifies he unknown ime-varying conrol direcion; 3) formulae a coninuous shifing law ha changes he sign of robus conrol smoohly in racking he idenified sign of he imevarying conrol direcion. These hree pars are deailed as follows. Par I: In his par, a candidae of robus conrol is seleced. As shown in [1] and [19], robus conrol in he absence of an unknown conrol direcion can be seleced o be 1 j(x)j + u r = 0 j(x)j (x)(x) () + 3 where >0is a design parameer, and (x) =( =)(x): Noe ha ju r j = j(x)j + j(x)j j(x)j + 3 (x) (x): (3) The above inequaliy holds since ( )(j(x)j + 3 ) j(x)j j(x)j j(x)j + j(x)j (4) in which he Cauchy Scharwz inequaliy a + b ab is used. In he presence of he ime-varying, unknown conrol direcion, we propose he following robus conrol: u(x; ^a()) = ^a() () 1 u r (5) where u r is defined in (), ^a() represens on-line esimae of funcion a(); and () >>0 is a design ime funcion such ha for all 0 j^a()j (): (6) Funcion (); o be defined shorly, is o normalize robus conrol (5) so ha is magniude is compaible wih conrol () and herefore wih he class of uncerainies under consideraion. I follows from (3) and (6) ha ja()[1f (x; ) +u]j(x)+ju rj( )(x): (7) Therefore, by choosing small such ha 4; we can always ensure ha he condiion in Assumpion 5 holds, which is why (6) was imposed on conrol (5). Noe ha besides he requiremen of 4; in (5) or () can be chosen freely by he designer, and he smaller is, he smaller he ulimae bound on he sae. As inroduced in [5], he concep of size dominaion means ha in a Lyapunov argumen, he erm associaed wih he conrol dominaes in magniude, and hus in sign, he erm(s) associaed wih uncerainies. If size dominaion is possible, he sign of he ime derivaive of he Lyapunov funcion is dicaed by ha of he erm associaed wih he robus conrol, and herefore sabiliy of he conrolled sysem can be ensured. Since ^a() may become zero and since () in robus conrol (5) mus be bounded from zero, we choose funcion () o be j^a()j if j^a()j () = (8) if j^a()j <: The exac role of funcion ; a design consan, will become clear shorly. Obviously, choice (8) of () makes (6) hold. In he case ha he conrol direcion is ime invarian, seing () === j^a( 0 )jreduces conrol (5) o ha in [5] (in [5], ^a( 0 ) is se o be eiher 1 or 01 so ha () =1): As will be shown in he upcoming sabiliy analysis, size dominaion of robus conrol over uncerainy is achieved in a Lyapunov argumen if u > 1f(x; ) (9) or equivalenly j^a()j () 1 j(x)j + j(x)j j(x)j j(x)j + 3 j(x)j: Therefore, we know ha he size dominaion is accomplished under he following condiions: j(x)j (10) and j^a()j : (11) Noe ha esimae ^a() has o be able o cross zero in order o rack he idenified version of a() and ha whenever j^a()j <; funcion ^a() may soon have zero crossing. The size dominaion in erms of (9) is no longer possible, and he sign of a() may no be idenified correcly. Since insabiliy will occur if he robus conrol provides a wrong conrol sign for a period of ime, ^a() mus rack a() closely. However, i is impossible for ^a() o correcly rack a() a all ime. To eliminae he possibiliy of insabiliy, we shall devise a sign idenificaion scheme which successfully idenifies he sign of a() unless eiher a() is abou o change is sign or x is very small. I is by limiing he occurence of idenificaion failure o be only wihin he described regions in he spaces of x and a() ha he sabiliy of he overall sysem can be guaraneed. Par II: In his par, a scheme o on-line idenify he conrol direcion is synhesized using he concep of size dominaion. Since he conrol direcion is unknown, any conrol will fail o ensure sabiliy wihou a correc idenificaion of he conrol direcion. If a() is consan, idenificaion of he conrol direcion needs o be performed only for a very small inerval of ime as was proposed in [5], and robus conrol can hen be operaed forever based on he one-ime idenificaion. Since he conrol direcion considered in his paper is a smooh ime funcion, is sign changes, possibly an infinie number of imes, mus be accouned for coninuously. In order o idenify sign[a()]; muliplying = on boh sides of (1) and hen inegraing he boh sides yield, for all [ 1; ] [ 0 ;1) dx 0 f(x();) d = a( ) 1f(x();)+ u(x()) d: (1) Noe ha he inegrand funcions are all coninuous and ha he sign of erm (=)u is he negaive of he sign of ^a(): Recall ha under condiions defined in (10) and (11), u(x) will dominae uncerainy 1f (x; ) in magniude as specified by (9). Therefore, i follows from he mean value heorem ha under condiions (10) and (11), he sign of a() can be idenified by he following equaion if boh a() and ^a() do no change heir signs in he inerval [ 1 ; ]: sign[a()] = 0sign[^a()]sign dx 0 f(x();) d : (13) Wih size dominaion of robus conrol over uncerainy, (13) can be used o idenify he correc conrol direcion, so long as a() and

4 396 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH 1997 ^a() do no change heir signs in he inerval [ 1 ; ]: If here is a sign change in eiher a() or ^a(); he resul from (13) mus be considered o be wrong. Since ^a() is known, is sign change will be known as well so ha proper precauion can be buil ino he idenificaion and robus conrol schemes o overcome any possible wrong idenificaion. In fac, as will be shown in he upcoming sabiliy analysis, he soluion is simply o make he convergence of ^a() o an esimae of sign[a()] fas enough, regardless of wheher he esimae of sign[a()] is righ or wrong. On he oher hand, since a() is unknown, is sign change canno be prediced. Therefore, (13) may poenially produce a wrong sign a any ime, which is a major sabiliy problem. Our soluion o his sabiliy problem is o selec judiciously he inegraion inerval [ 1 ; ] and o use he fac ha he nominal sysem is exponenially sable. Specifically, we choose he inegraion inerval as follows: firs le = and 1 = 0 T so ha he inegraion inerval is a sliding window of he curren ime insan. Second, selec he inerval such ha is lengh T saisfies he following inequaliy: T< (14) where is defined in Assumpion 4, and 0 < 1 is he design parameer inroduced in (8). The role of consan allows us o analyze he effec of zero crossing of a() and o ake proper precauions in our robus conrol scheme. Specifically, if ja()j (namely ja( )j) is greaer han or equal o ; here will be no sign change in he inerval s [ 1; ]because of he choice of T in (14). Tha is, as far as a() is concerned, he conrol direcion will be idenified correcly if ja()j >: (15) I is when ja( )j < ha here may be a sign change occurring in he inerval. Thus, assuming ha a correc idenificaion of conrol direcion implies sabiliy (which will be guaraneed by designs of he robus conrol and he shifing law), robus sabiliy can be esablished by sudying wheher he conrolled sysem is sable if ja()j <: Recalling ha he nominal sysem obained by seing a() = 0is exponenial, we know ha robus sabiliy can be esablished for he sysem (wih he robus conrol of a wrong sign) by simply leing be sufficienly small. Deails on selecing will be given in he sabiliy proof. In fac, such a reamen of zoning is inuiive since, if ja()j < 1; sign idenificaion is no possible anyway as a() is ime varying. Summarizing he above discussions, we know ha he conrol direcion found from (13) is correc if (10), (11), and (15) hold. Par III: As soon as an esimae of sign[a()] becomes available, a shifing law can be consruced so ha he sign of robus conrol is changed smoohly from a possibly wrongly guessed conrol direcion o he idenified conrol direcion and is hen updaed all he ime. The shifing law given in (16), shown a he boom of he page, is chosen such ha he overall sysem is robusly sable during he ime period in which smooh ransiions of he sign of robus conrol are accomplished, where he iniial guess of he conrol direcion a(); denoed by ^a( 0 ); is chosen o be eiher 1 or 0 or 01. k s is a consan gain saisfying k s p 4 (17) and funcion w() is he idenificaion resul of sign[a()] defined by w() =0sign[^a()] sign 0T dx 0 f(x();) d : 0T Implemenaion of shifing law (16) requires w(): No maer wheher he idenificaion resul w() is he correc value of sign[a()] or no, w() has only hree discree values: 01, 1, and 0. As soon as w() picks up is new value, w() will remain o be a consan unil one of inequaliies (10), (11), and (15) fails. During he period ha w() is consan, we have according o (16) ha d[^a() 0 w()] =[^a()0w()] d^a d d 0 k s p j^a()0w()j: Solving he above differenial inequaliy, we know ha he esimae ^a() will converge o w() in a coninuous fashion, in a finie ime, and wihou any overshoo, ha is j^a()j 1: (18) If w() = sign[a()]; he esimae ^a() will converge quickly o he correc conrol direcion. In fac, he gain k s defined in (17) for shifing law (16) is chosen such ha he ime for ^a() o converge o sign[a()] is less han T defined in (14). This choice ensures ha ^a() is capable of racking sign[a()] excep for he case ha a() is in he zone of ja()j < and is on he verge of changing is sign. In he case ha he idenificaion resul is wrong, he esimae may converge o a wrong direcion. Sabiliy analysis mus be done o show ha any wrong idenificaion is emporary and ha he sysem sae will no escape from he sabiliy region anyime. Since (10) has been incorporaed ino he shifing law, we have w() = sign[a()] by implemening he shifing law if (11) and (15) hold. Tha is, sabiliy mus be concluded for he cases ha hese wo inequaliies fail. Global sabiliy of (1) under robus conrol (5) and shifing law (16) is analyzed by he Lyapunov s direc mehod using he Lyapunov funcion candidae V (x; ^a()) = V 1(x; ) +V (^a();) (19) where V 1(1) is ha given in Assumpion, and 1 V (^a();)= [a()^a() 0ja()j] ja()j = 1 [ja()j3=^a ()+ja()j 3= 0 sign[a()]ja()j 3=^a()]: Noe ha for any differenial funcion q(); jq()j p and sign[q ()]jq()j p wih p>1 are also differeniable. The following heorem shows he sabiliy of uniform and ulimae boundedness for (1). Theorem: Assume ha (1) saisfies Assumpions 1 5. Then, under robus conrol (5) and shifing law (16), he sysem sae can be made uniformly and ulimaely bounded by leing be small enough. if j(x)j () 0 if j(x)j < _^a() = 0sign[^a() 0 w()] 1 1 p k s j(x)j (16)

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH Proof: Differeniaing (19) along he rajecory of (1) yields f(x; ) + a() 1f(x; ) +a() u [1 + ^a ()] ja()j sign[a()] 0 3 ^a() ja()j _a() + a() [a()^a() 0ja()j]_^a ja()j 0 4 jxj +ja()j1j(x)j 0a()^a() j(x)j + j(x)j () j(x)j + 3 j(x)j+ 3 4 [1 + ^a ()] ja()j sign[a()] 0 3 ^a() ja()j _a() + a() ja()j [a()^a() 0ja()j]_^a: (0) Noe ha ^a() is bounded by one as shown by (18). Thus, sabiliy may also be sudied by Lyapunov funcion candidae V 1 (x; ): I follows ha _V 1 + f(x; ) +a() 1f(x; ) + u 0 4 jxj +ja()j1j(x)j 0a()^a() j(x)j + j(x)j () j(x)j + 3 j(x)j: (1) We shall sudy he sabiliy resul by invesigaing he following four cases. Case I j(x)j <: In his case, (10) does no hold, and herefore robus conrol does no dominae he uncerainy in size. In his case, w() may no be sign[a()]: I follows from Assumpion 4, shifing law (16), (6), and (18) ha he ime derivaive (0) of Lyapunov funcion V is bounded from above as _V 0 4 jxj +ja()j1j(x)j+ja()j 1 j^a()j () j(x)j + j(x)j j(x)j + 3 j(x)j [1 + ^a ()] ja()j + 3 j^a()j ja()j j _a()j + a() [a()^a() 0ja()j]_^a ja()j 0 4 jxj ++3: Noing ha ^a() is frozen because of _^a() = 0;here is no need o have a negaive definie erm associaed wih he variables in V (^a();):in fac, i follows from he same inequaliies ha he ime derivaive (1) of Lyapunov funcion V 1 is bounded from above as _V jxj +ja()j1j(x)j+ja()j 1 j^a()j () 0 4 jxj +: j(x)j + j(x)j j(x)j + 3 j(x)j From eiher of he above resuls, one can conclude using he sabiliy heorem in [1] ha he sysem is uniformly and ulimaely bounded. Case II ja()j : Noe ha alhough he esimae ^a() of he conrol direcion may converge o a wrong value, ^a() is bounded by one as indicaed by (18). I follows from Assumpion, (6), (4), (8), and (18) ha he ime derivaive (1) of Lyapunov funcion is bounded from above as _V 1 0 4jxj +1j(x)j j(x)j 0 4 jxj jxj(x): I follows from he resuls in [0] ha he sysem is semiglobal sable in he sense ha he sysem is uniformly ulimaely bounded and ha he sabiliy region can be made arbirarily large by leing be small enough. This is rue wheher he on-line idenificaion is done correcly or no. Case III j^a()j : In his case, he esimae ^a() is confined o be in a smaller se han is ulimae bound defined by (18). Since ^a() is small, he conribuion of he robus conrol may no be large enough o compensae for he uncerainy. Thus, his is he case where insabiliy can poenially occur. However, by he choice of gain k s in (17), his case will las a very small period of ime during which Assumpion 5 is saisfied [as ensured by (7)]. Therefore, by choosing small enough, no only can he ime inerval during which his case occurs be small enough, bu also he sae of he sysem will say in a finie region, specifically, he semiglobal sabiliy region defined in Case II. Case IV: j(x)j ; j^a()j >; and ja()j >: This case is he complemen of he union of he previous hree cases. Since condiions (10), (15), and (11) hold, idenificaion is done correcly. Tha is _^a() =0sign[^a() 0 sign[a()]] 1 1 p k s j(x)j : () () I follows from Assumpion 4, shifing law (), (4), and (18) ha he ime derivaive (0) of Lyapunov funcion V is bounded from above as _V 0 4 jxj +ja()j1j(x)j 0a()^a() j(x)j + j(x)j j(x)j j(x)j + 3 () +3+ Noe ha <() 1 and ha since j(x)j >: Hence a() [a()^a() 0ja()j]_^a: ja()j j(x)j + j(x)j j(x)j _V 0 4 jxj +[ja()j0a()^a()] 1 j(x)j + j(x)j j(x)j j(x)j + 3 () +3 + a() [a()^a() 0ja()j]_^a ja()j 0 4jxj 0k sja()^a() 0ja()jj +3 from which sabiliy of uniform and ulimae boundedness can be concluded. In summary, one can conclude he sabiliy of uniform ulimae boundedness for (1) by simply combining he above four cases. In he nex secion, a simulaion example is presened o illusrae our proposed robus conrol design.

6 398 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH 1997 Fig. 1. Sysem performance under a() = sin() and x(0) = : Fig.. Shifing law ^a (solid sign[a()]; dashed ^a(); doed u): Fig. 3. a() =0sin() and x(0) = (doed x; dashed ^a; solid sign[a()]): IV. SIMULATION To illusrae he proposed robus conrol scheme under an unknown ime-varying conrol direcion, simulaions were performed for (1) wih he following choices: known dynamics f(x; ) =0kx wih k =0:5and V 1 = 1 x ; uncerainy 1f =10sin()+[5+5cos()]x +10x 3 which is chosen o be unsable; bounding funcion (x) is chosen o be (x) = 10(1+x +jxj 3 ); unknown conrol direcion is given by ime-varying funcion a() = 6sin() (boh cases were simulaed). The rae of change is bounded by =; design parameers in shifing law (16) and in robus conrol (5) are =0:01;k s = 10; and =0:1; iniial guess of he conrol direcion is se o be ^a(0) = 0 (for all he cases simulaed); iniial condiions of he sae were chosen o be x(0) = 61 (boh cases were simulaed). The simulaions were carried ou using SIMNON. Sysem performance under all possible combinaions of iniial condiions of sae x and unknown conrol direcion a() are shown in Figs. 1, 3, 4, and 5. In Fig. 1, rajecories of he sae, he esimae of he conrol direcion, and he robus conrol are shown separaely. In Figs. 3 5, rajecories of x; ^a; and sign[a()] are combined ino one plo for each case. Conrol u is omied from hese figures for briefness since conrol rajecories are similar o ha in Fig. 1 and since he magniude of he conrol, if u is incorporaed ino he figures, will make he oher rajecories obscure. I should be noed from Fig. 4 ha he speed of ^a racking sign[a()] becomes slower as he sae x converges o he origin. This phenomenon is expeced from (10). To beer illusrae he behavior of he shifing law and is corresponding sabiliy propery, a porion of Fig. 1 (around =)is enlarged o be Fig. in which sign[a()] firs swiches is sign, ^a() soon follows, and in beween, he conrol emporarily has a wrong sign and drives he sysem in a wrong direcion (see he firs plo

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 4, NO. 3, MARCH Fig. 4. a() =0sin() and x(0) = 0 (doed x; dashed ^a; solid sign[a()]): Fig. 5. a() = sin() and x(0) = 0 (doed x; dashed ^a; solid sign[a()]): in Fig. 1). Obviously, he simulaion shows ha his increase in jxj does no yield any insabiliy. In fac, he emporary increase of he magniude of he sae makes i possible o idenify he curren sign of a(), idenificaion in urn will provide he correc conrol direcion, and he sae is hen forced o converge back o he origin. Similar resuls can be seen around all poins a which he sign of a() is swiched. V. CONCLUSION Robus conrol of a class of nonlinear uncerain sysems is sudied. A sysem under consideraion conains no only nonlinear uncerainy bounded by a known nonlinear funcion of he sae bu also an unknown ime-varying conrol direcion. Wihou any priori knowledge of he conrol direcion excep for smoohness, a hreepar design is proposed for consrucing robus conrol via he Lyapunov s direc mehod. A resuling robus conrol is coninuous and consiss of an on-line algorihm o idenify coninuously he conrol direcion, a shifing law ha changes he sign of robus conrol smoohly, and a robus conrol modified from he sandard robus conroller. Essenially, he design only requires he bounding funcion of unknown dynamics and he maximum rae of change of he conrol direcion. Sabiliy of uniform ulimae boundedness is guaraneed. REFERENCES [1] M. J. Corless and G. Leimann, Coninuous sae feedback guaraneeing uniform ulimae boundedness for uncerain dynamical sysems, IEEE Trans. Auoma. Conr., vol. 6, pp , [] F. Giri, P. A. Ioannou, and F. Ahmed-Zaid, A sable adapive conrol scheme for firs order plans wih no priori knowledge on he parameers, IEEE Trans. Auoma. Conr., vol. 38, no. 5, pp , [3] S. Guman, Uncerain dynamical sysems A Lyapunov min max approach, IEEE Trans. Auoma. Conr., vol. 4, no. 3, [4] U. Helmke and D. Prazel-Wolers, Sabiliy and robusness properies of universal adapive conrollers for firs order linear sysems, Inern. J. Conr., vol. 48, pp , [5] J. Kalous and Z. Qu, Coninuous robus conrol design for nonlinear uncerain sysems wihou a priori knowledge of he conrol direcion, IEEE Trans. Auoma. Conr., vol. 40, no., pp. 76 8, [6] R. Lozano and B. Brogliao, Adapive conrol of a simple nonlinear sysem wihou a priori informaion on he plan parameer, IEEE Trans. Auoma. Conr., vol. 37, no. 1, pp , 199. [7] R. Lozano, J. Collado, and S. Mondie, Model reference adapive conrol wihou a priori knowledge of he high frequency gain, IEEE Trans. Auoma. Conr., vol. 35, no. 1, pp , [8] R. T. M Closkey and R. M. Murray, Exponenial convergence of nonholonomic sysems: Some analysis ools, in Proc Conf. Decision Conr., 1993, pp [9], Experimens in exponenial sabilizaion of a mobile robo owing a railer, in Proc Amer. Conr. Conf., 1994, pp [10] R. Marino and P. Tomei, Robus sabilizaion of feedback linearizable ime-varying uncerain nonlinear sysems, Auomaica, vol. 9, pp , [11] B. Marensson, Remarks on adapive sabilizaion of firs order nonlinear sysems, Sys. Conr. Le., vol. 14, pp. 1 7, [1] D. R. Mudge and A. S. Morse, Adapive sabilizaion of linear sysems wih unknown high frequency gain, IEEE Trans. Auoma. Conr., vol. 30, pp , [13] R. M. Murray, Z. Li, and S. S. Sasry, A Mahemaical Inroducion o Roboic Manipulaion. New York: CRC, [14] R. M. Murray and S. S. Sasry, Nonholonomic moion planning: Seering using sinusoids, IEEE Trans. Auoma. Conr., vol. 38, no. 5, pp , [15] R. D. Nussbaum, Some remarks on he conjecure in parameer adapive conrol, Sys. Conr. Le., vol. 3, no. 5, pp , [16] Z. Qu, Global sabilizaion of nonlinear sysems wih a class of unmached uncerainies, Sys. Conr. Le., vol. 18, no. 3, pp , 199. [17], Robus conrol of nonlinear uncerain sysems wihou generalized maching condiions, IEEE Trans. Auoma. Conr., vol. 40, no. 8, pp , [18], A new generic procedure of designing robus conrol for nonlinear uncerain sysems: Beyond he back-sepping design, in Proc Wkshp. Robus Conrol via Variable Srucure Lyapunov Techniques, Beneveno, Ialy, Sep. 1994, pp [19], Robus conrol of nonlinear uncerain sysems under generalized maching condiions, Auomaica, vol. 9, pp , [0] Z. Qu and J. Dorsey, Robus conrol of generalized dynamic sysems wihou maching condiions, Trans. ASME, J. Dyn. Sys., Meas., Conr., vol. 113, pp , [1] J. J. E. Sloine and K. Hedrick, Robus inpu oupu feedback linearizaion, Inern. J. Conr., vol. 57, pp , 1993.