SLIDING MODE CONTROL SYNTHESIS OF UNCERTAIN TIME-DELAY SYSTEMS

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1 568 Asian Jounal of Conol, Vol. 5, No. 4, pp Decembe 3 SLIDING MODE CONROL SYNHESIS OF UNCERAIN IME-DELAY SYSEMS Y. Olov, W. Peuquei, and J.P. Richad ABSRAC Sliding mode conol synhesis is developed fo a class of unceain ime-delay sysems wih nonlinea disubances and unknown delay values whose unpeubed dynamics is linea. he synhesis is based on a new delay-dependen sabiliy cieion. he conolle consuced poves o be obus agains sufficienly small delay vaiaions and exenal disubances. An admissible uppebound such ha he coesponding closedloop sysem emains globally asympoically sable fo each delay value less han his uppebound is deived. Pefomance issues of he conolle ae illusaed in a simulaion sudy. KeyWods: Sliding mode, ime-delay sysem, obusness. I. INRODUCION he pimay concen of he pape is obus conol of unceain ime-delay sysems wihin he famewok of sliding mode conol mehods. he sliding mode conol paadigm implies he delibeae inoducion of so-called sliding moions (i.e., he moions along a manifold whee he conol signal undegoes disconinuiies) ino he conol sysem and i consiss of wo seps [3]. Fis, a manifold, such ha if confined o his manifold he sysem has desied dynamic popeies, is designed. Second, a disconinuous conol law, which dives he sysem o his manifold in finie ime, is synhesized. he sliding mode conol saegy is o consuc a feedback ha guaanees a Lyapunov funcion, seleced fo a nominal sysem, o emain negaive on he ajecoies of he peubed sysem. he conolle, hus consuced, asympoically sabilizes he sysem and since he moion along he manifold poves o be uncouped by mached disubances, he closed-loop sysem is addiionally guaaneed o have song obusness popeies agains mached disubances. Due o hese advanages and simpliciy of implemenaion, sliding mode conolles have widely been used in vaious applicaions [4]. Manuscip eceived Januay 5, 3; acceped May, 3 Y. Olov is wih CICESE Reseach Cene, Eleconics and elecommunicaion Depamen, P.O.Box San Diego, CA , USA. W. Peuquei and J.P. Richad ae wih LAIL, CNRS UPRESA 8, Ecole Cenale de Lille, BP48, 5965 Villeneuve d Ascq CEDEX-FRANCE. Moivaed by echnological advances, he inees ecenly emeged in exending he sliding mode conol appoach o infinie-dimensional dynamic sysems such as disibued paamee sysems and ime-delay sysems. he ealie woks [9, 7] on exensions of sliding mode conol algoihms o infinie-dimensional sysems an ino a majo difficuly, caused by he pesence of an unbounded infiniesimal opeao in he plan equaion, and called fo fuhe heoeical invesigaions. Pesenly, he sliding mode conol synhesis in he infinie-dimensional seing is well documened [4,5,7,8]. his synhesis eains obusness feaues, simila o hose possessed by is counepa in he finie-dimensional case, and being complemenay o he H -design, i consiues a moe pacical appoach o infinie-dimensional sysems han he ones of high compuaional complexiy oulined in [3,6,8,]. he exising esuls [,7,9,,,5] on applicaion of sliding mode conol algoihms o ime-delay sysems, govened by funcional diffeenial equaions (FDE), have cooboaed hei uiliy fo his class of sysems as well. Howeve, hese esuls, being inheied fom he finie-dimensional eamen, conained no agumens suppoing he well-posedness of FDE soluions o subsaniae he sabiliy analysis. Even he soluion concep fo disconinuous ime-delay sysems has no ye been addessed. Resolving he fundamenal issue on he pecise meaning of FDE wih disconinuous igh-hand side consiues one of he conibuions of he pesen pape. Subsanial sabiliy analysis and sliding mode conol synhesis of ime-delay sysems wih small ime lags and

2 Y. Olov e al.: Sliding Mode Conol Synhesis of Unceain ime-delay Sysems 569 weak nonlineaiies is anohe conibuion of he pape. In his egad, i is woh noicing ha allowing he weak nonlineaiies, inoduced in Secion by means of Assumpions A and A, admis unmached disubances heeby making a sep beyond a sandad sliding mode conol eamen. Specifically, he ciical delay value when he closed-loop sysem, coesponding o his value, becomes asympoically unsable is explicily calculaed as a funcion of linea gowh consans of he unmached disubances. he ouline of he pape is as follows. Plan descipion and fomal poblem saemen appea in Secion. Afe giving backgound maeials on disconinuous ime-delay sysems in Secion 3, Secion 4 pesens a new delay/disubance-dependen sabiliy cieion fo a class of ime-delay sysems wih nonlinea unceainies and unknown delay value whose unpeubed dynamics is linea. Based on his cieion, in Secion 5 we synhesize a sliding mode conolle which asympoically sabilizes he sysem fo any value of he delay less han is ciical value obained via solving a convex opimizaion poblem. Simulaion esuls ae discussed in Secion 6. Finally, Secion 7 pesens some conclusions.. Noaion he noaion is faily sandad. he maix inequaliy M > (M < ) means ha M is posiive (especively, negaive) definie; I denoes he ideniy maix of an appopiae dimension. he veco nom e sands fo he Euclidean nom of a veco e wheeas he maix nom M = sup Me e = n sands fo he induced nom of a maix M; L [,] τ denoes he Hilbe space of squae inegable (n)-veco funcions defined on he segmen [ τ, ]. II. PROBLEM SAEMEN In his pape, we focus on ime-delay sysems ha ae epesenable by means of a nonlinea change of sae coodinaes and a feedback ansfomaion, in he fom dz () = ( Az i i( ) + Bz i i( τ )) d i= + p (, z()) + p (, z( τ )), dz () = + d + u+ f(, z ), z () = φ()fo [ τ,]. ( Aizi( ) Bizi( τ )) () i= Heeinafe, z() = (z, z ) is he sae veco wih componens z IR n, z IR, A ij, B ij i, j =, ae consan maices of coesponding dimensions, u IR is he inpu veco; p, p IR n and f IR ae exenal disubances, φ is he iniial piecewise coninuous funcion defined on [ τ, ], z (θ) is he funcion associaed wih z and defined on [ τ, ] by z (θ) = z( + θ). Since he delay value τ is ypically unknown a pioi, in he sequel we shall also use he noaion, ( θ ) fo he funcion z τ z, τ ( θ) = z( + θ) defined on some lage ineval [ τ, ] wih τ > τ. Seing A A B B A =, B = A A B B pi Pi =, i =,, C =, F = I f sysem () can be simplified o z () = Az () + Bz ( τ ) + Cu+ Pz (, ()) + P (, z( τ )) + F(, z ). Alhough ou invesigaion is confined o ime-delay sysems wih delayed saes, howeve, he exension o he case of delayed inpus is saighfowad. Indeed, le a sysem wih an inpu delay be govened by x () = Ax () + Bx ( τ ) + Du ( ), > x () = ψ(), [ τ,], u () = ψ (), [ τ,] whee x IR n ; A, B ae consan n n maices, D is a n maix, u IR is he inpu veco, φ and φ ae piecewise coninuous funcions. hen, an addiional inpu inegao ansfoms he sysem ino he above fom dz () = Az( ) + Bz( τ) + Dz( τ), d dz () = υ d wih he conol inpu υ IR. houghou he pape, he following assumpions ae made fo echnical easons: A) p (, z()) and p (, z( τ )) ae Lipschiz coninuous and saisfy he linea gowh condiions p(, z ()) α z (), p(, z ( τ)) α z ( τ) wih some posiive consans α, α. A) f is Lipschiz coninuous and i is bounded f(, z ) Ψ (, z, τ ) by a coninuous funcional () (3) (4) (5)

3 57 Asian Jounal of Conol, Vol. 5, No. 4, Decembe 3 Ψ (, z, τ ) known a pioi. A3) (A + B, A + B ) is conollable. Assumpions A and A, coupled ogehe, guaanee he unfoced FDE () wih u = o have a unique soluion fo all [3]. Apa fom his, Assumpion A allows us o ejec he unceain disubance f by a bounded-gain sliding mode feedback. Assumpion A3 implies he conollabiliy of he delay/disubance-fee sysem () subjec p = p =, τ =, f = (see [3, p.] fo deails). hus, he above assumpions ensue ha a delay-fee sysem () wih τ = uns ou o be sabilizable unde sufficienly small exenal disubances p, p, f. Due o consevaism, ime-delay sysems of he fom () emain sabilizable fo all τ τ wih some posiive, sufficienly small τ. he goal of he pesen invesigaion is o consuc a sabilizing sliding mode conolle of () ha makes he value of τ as lage as possible. III. BACKGROUND MAERIALS ON DIS- CONINUOUS IME-DELAY SYSEMS Since he closed-loop sysem, diven by a sliding mode conolle, is govened by a FDE wih disconinuous igh-hand side, he pecise meaning of such an equaion should fis be defined. Fo his pupose, le us epesen he ime-delay sysem (3) as a dynamic sysem dζ d = Aζ + Cu+ P(, µ ()) + P (, ν( τ)) + F (, ν ) (6) n+ n+ n+ evolving in he Hilbe space M = IR L [ τ, ] whose sae ζ = (µ(), ν ( )) a a ime momen consiss of he componens () () IR n + µ = z and ν (θ) = z (θ) n+ L [ τ,], and he linea opeaos A: D( A) M M, C :IR M and nonlinea opeaos n+ n+ n+ Pi :IR M, i =,, F : M M ae given by + n+ n+ n+ Aµ + B ν( τ) µ A = dν, νθ ( ) dθ Cu Pi F C =, Pi =, F =. Appaenly, C is a bounded opeao on IR wheeas he nonlinea opeaos P i, i =, and F ae Lipschiz coninuous on hei domains due o Assumpions A and A. In un, i is well-known [6], ha he infiniesimal opeao A geneaes a (songly coninuous) C -semigoup (7) n on M + and is domain n+ dν n+ DA ( ) = { ζ = ( µ, ν) M : L [ τ,] dθ and ν () = µ } n is dense in M +. hus, soluions of he Hilbe space-valued dynamic sysem (6) wih disconinuous igh-hand side can igoously be defined in he sense of [8]. Fo convenience of he eade, we ecall he basic idea of ha definiion. Le he conol inpu u(ζ) undego disconinuiies on some linea manifold Gζ = in he sae space and le u(ζ) be coninuously diffeeniable beyond his manifold. hen ajecoies of (6) ae well-defined in he convenional sense wheneve hey ae beyond he disconinuiy manifold wheeas in a viciniy of his manifold he oiginal sysem is eplaced by a elaed sysem, whose soluions also exis in he convenional sense. A moion along he disconinuiy manifold, if any, is hen obained by making he chaaceisics of he new sysem appoach hose of he oiginal one. Such a moion is fuhe efeed o as a sliding mode ha has become sandad in he lieaue. I is woh noing ha given an iniial condiion n+ ζ = ( φ(), φ()) M, he Cauchy poblem, hus saed fo sysem (6) wih he assumpions above, poves o have a unique soluion wheneve he opeao GC is coninuously inveible. Jus in case, he sliding moion of he sysem on he disconinuiy manifold is govened by he so-called sliding mode equaion deived hough he equivalen conol mehod. he validiy of he equivalen conol mehod in infinie-dimensional seing has been jusified in [9] fo paabolic sysems and in [7,8] fo moe geneal sysems, including hypebolic and, paiculaly, ime delay sysems. Accoding o his mehod, he sliding mode equaion is obained by subsiuing he soluion ueq = + + P (, ν( τ)) + F(, ν )] [ GC] G[ Aζ P (, µ ( )) of he equaion G ζ = ino (6) fo u. he sliding mode equaion plays an impoan ole in he subsequen sabiliy analysis of he disconinuous ime delay sysem in quesion. Since his equaion conains no disconinuiies in he igh-hand side, is sabiliy is esablished via sandad echniques. IV. DELAY/DISURBANCE-DEPENDEN SABILIY CRIERION Fo lae use, we deive delay/disubance-dependen sabiliy condiions fo he ime-delay sysem

4 Y. Olov e al.: Sliding Mode Conol Synhesis of Unceain ime-delay Sysems 57 dx() = Ax () + Bx ( τ) + p(, x ()) + p(, x ( τ)) d x IR n, x() = ζ() fo [, ] (8) wih a piece-wise coninuous iniial funcion ζ(), some consan maices A, B, and he same nonlineaiies as befoe. Ou objecive is o find an uppebound τ of admissible delay values τ such ha he above sysem, while being asympoically sable wih τ =, is so fo all delays τ < τ. his uppebound τ = τ (α, α ) depends on he linea gowh consans α, α of he disubances p, p and elaes o a posiive soluion of he following opimizaion poblem τ ( α, α) = supτ subjec o he consains H ( τ) P P τpb τpb τpba τpb P γ I P γ I τb P γ3τi < τb P γ 4τI τabp τr τb P τr ove all posiive consans γ i, i =,, 3, 4 and symmeic posiive definie maices P, R, R IR n n, and H(τ) = (E P + PE) + (γ α + γ α )I + τ(γ 3 α + γ 4 α )I + τ(r + R ) wih E = A + B. heoem. Le he maix E = A + B be Huwiz. Fo α, α > sufficienly small, he above opimizaion poblem has a posiive soluion τ (α, α ) and sysem (8) is globally asympoically sable fo each delay value τ [, τ (α, α )]. (9) Poof. Conside he following Lyapunov-Kasovskii funcional V( z ) = V ( z ) + V ( z ) + V ( z ) 3 ( ) = () () = γα τ + s 3 n τ + γα τ + s 4 n 3( ) = γα ( ) ( ) τ V z z Pz V ( z ) z ( w)( R I ) z( w) dwds z ( w)( R I ) z( w) dwds V z z w z w dwds () wih P, R, R symmeic posiive definie maices, and posiive γ i, i =,, 3, 4. he funcional V(z ) is posiive definie and adially unbounded. Using z( τ) = z ( θ) dθ + z( ), we can ewie he sysem in he τ fom dz() = ( A+ B) z( ) d B [ Az( w) + Bz( w τ ) + p ( w, z( w)) τ + p ( w, z( w τ))] dw+ p (, z( )) + p (, z( τ)). Le us now calculae he deivaive of V(z ) along he ajecoies of he sysem (8): V = z ( E P + PE) z + z Pp + z Pp τ z PBA z( w) dw z PB z( w τ ) dw τ z PB p ( w, z( w)) dw τ z PB p ( w, z( w τ )) dw, τ = τ + γ3α + τ + γ4α V z ( R I) z z ( R I) z τ τ z ( w)( R + γα I) z( w) dw 3 z ( w τ)( R + γ 4αI) z( w τ ) dw, V = γ α z () z() γ α z ( τ) z( τ). 3 () Due o he well-known inequaliy u υ αu Ru + α υ R υ, valid fo any vecos u, υ IR n, symmeic posiive definie maix R IR n n and posiive consan α, he following six inequaliies ae in foce: z Pp z ( γ P + γ α I) z, z Pp γ z P z+ γ α z( τ) z( τ), τ τ τ τ τ τ z ( w τ) R z( w τ) dw, τ τ τγ z ( wzwdw )( ), τ τγ τ z ( w ) z( w ) dw τ z PBA z( w) dw z Z z+ z ( w) R z( w) dw z PB z( w ) dw z Z z + z PB p ( w, z( w)) dw z Z z + γα z PB p ( w, z( w h)) dw z Z z + γα τ τ wih I ideniy maix, Z = PBAR A B P, = Z PB R B P, Z3 = PBB P, γ >, i =,,3, 4. i

5 57 Asian Jounal of Conol, Vol. 5, No. 4, Decembe 3 Now i follows fom () ha he inequaliy holds fo V ( z ) z M( τ ) z M( τ) = ( E P+ PE) + ( γ + γ ) P + ( γα + γα + τγα + τγα ) + τ( + ) + τpbar A B P + PB R B P + τγ I R R τ 3 PBB P τγ 4 PBB P. he maix E is Huwiz and hence M() = (E P + PE) + ( γ + γ )P + ( γα + γα )I is negaive definie if P is a symmeic posiive definie soluion P of he Lyapunov equaion E P + PE = I and posiive consans γ, γ, ae sufficienly small. By coninuiy M(τ) emains negaive definie fo sufficienly small posiive τ. Since by Schu s lemma [], M(τ) < if and only if inequaliy (9) holds, i follows ha he afoe-given opimizaion poblem has a posiive soluion τ (α, α ). hus, he maix M(τ) is negaive definie fo all τ [, τ (α, α )] and sysem (8) is heefoe globally asympoically sable fo all delays τ < τ (α, α ). heoem is poven. Remak. he inequaliy (9) is a linea maix inequaliy in P, R, R, γ, γ, γ 3, γ 4 and can efficienly be solved by convex opimizaion algoihms (see []). Example 3. (Compaison wih ohe cieia) Le sysem (8) be specified by A=, B = () and le p and p be such ha p. z and p. z ( τ ). Using heoem, we find τ (.,.) =.63 wheeas he lages uppebound of ime-delay available in he lieaue ( see [6,,]) is τ =.448. I should also be noed ha neihe he disubance-fee cieia fom [9,] no he delay-independen cieion fom [6] is capable of poviding any conclusion. hus, he example shows ha he poposed cieion makes a sep beyond he exising esuls. V. SLIDING MODE SAE FEEDBACK CONROLLER he following sae feedback conol law ε sz ( ( )) u= Ω [ (, z, τ, α, α ) +Ψ (, z, ) g s ], τ + sz ( ( )) g >, ε [,) (3) is poposed o dive he unceain ime-delay sysem () o a linea manifold s(z) = wih s(z) = z + Kz (4) in finie ime heeby globally asympoically sabilizing he sysem fo each τ [, τ ) Heeinafe, Ψ (, z, τ ) is he same uppebound of he nom of he exenal disubance f(, z ) as i is in Assumpion A, z, τ ( θ ) sands fo he funcion z( θ) wih θ [ τ, ] and τ > τ, he linea funcional i i i i τ i i i= Ω (, z) = { Az() + Bz( ) + KAz [ () + Bz( τ) + p( z, ( )) + p( z, ( τ))]} i i is viewed on he ajecoies of he conolled sysem (), and Ω (, z, α, α ) = sup ( Ω (, z( θ))), τ θ [ τ,] is is uppebound on a ime ineval θ [ τ, ] wheeas τ = τ (α, α ) and K IR n n ae subjec o he opimizaion τ ( α, α) = supτ unde he consains J( τ) S S τs S γ α In S γ α In τs τγ α τ S γ5l( τ) γ6l( τ) τs γ5l( τ) γ6l( τ) <, τγ 4 α In τ Q τ Q 5 ( A AK) S < γ S ( B BK) S < γ 6S S >, Q >, Q >, γ >, i =,,6 i I 3 n (5) Due o Assumpion A, such an uppebound can be chosen o only depend upon he linea gowh consans α, α of he exenal disubances p and p egadless of a concee ealizaion of hese disubances.

6 Y. Olov e al.: Sliding Mode Conol Synhesis of Unceain ime-delay Sysems 573 wih symmeic posiive definie maices S, Q, Q IR (n m) (n m), J( τ ) = [( A + B ) ( A + B ) K] S + S[( A + B ) ( A + B ) K] I Q Q + [ γ + γ + τ( γ + γ ) γ ] + τ( + ), and L(τ) = τ (B B K)S. I is of inees o noe ha if sysem () is only affeced by exenal disubances f(, z ), vanishing in he oigin, hen an uppebound Ψ (, z, τ ), such ha Ψ(, ) = fo all, becomes available so ha he coesponding conolle (3) wih a posiive paamee ε is coninuous in he oigin wih no undesiable effecs of swiching in he seady sae. I should be poined ou ha he conol law (3) appeas o be applicable even if he sysem delay is unknown a pioi. We ae now in a posiion o sae he main esul of his secion. heoem 4. Le Assumpions A-A3 be saisfied. hen fo α, α sufficienly small he opimizaion poblem (5) has a posiive soluion τ (α, α ) and he closedloop sysem (), (3) is globally asympoically sable fo each delay value τ [, τ (α, α )). Poof. We beak up he poof in 3 simple seps. Fisly, we pove ha he disconinuiy manifold s(z) = is eached in finie ime, secondly, we deive he sae equaions along he manifold s(z) =, and hidly, we pove he asympoic sabiliy of he sysem moion on his manifold, heeby compleing he poof of heoem 4. ) Aaciveness of he manifold. aking ino accoun ha he ime deivaive of s(z()) along he soluions of () is given by sz ( ( )) =Ω ( z, ) + u+ fz (, ), diffeeniaing of he Lyapunov-Kasovskii funcional s ( z( )) s( z( ) V () =. (6) on he ajecoies of he closed-loop sysem unde Assumpions A-A3 yields V ( ) = s ( z( )) s ( z( )) = s ( z( )){ Ω (, z) + f(, z) ε s Ω [ ( z,, τ, α, α ) +Ψ ( z,, ) gs ] } τ + (7) s ε + ε + < gsz ( ( )) gv ( ). Since an abiay soluion of he lae inequaliy is well-known o vanish afe a finie ime momen, he finie-ime convegence of he ajecoies of he closedloop sysem o he disconinuiy manifold s(z) = is concluded. In ode o epoduce his conclusion, one should noe ha fo all abiay soluion V() o he lae inequaliy is dominaed V() V () by he soluion ε + o he diffeenial equaion V () = gv(), iniialized wih he same iniial condiion V () = V(). Since ε V () = fo all () g( ε ) V, V() vanishes afe ε he finie ime momen = () g( ε ) V. ) Sliding mode equaion. In ode o descibe he sysem moion on he disconinuiy manifold s(z) = one should follow he exension [7] of he equivalen conol mehod [3] o infinie-dimensional sysems diven in a Hilbe space. Accoding o his mehod, he coninuous soluion u eq = [Ω(, z ) + f(, z )] of he equaion s = is subsiued ino () fo u. hus, on he disconinuiy manifold, he sysem is govened by he diffeenial equaion dz () = ( A AK) z ( ) + ( B BK) z ( τ ) d + p (, z ()) + p (, z ( τ )). (8) 3) Asympoic sabiliy of he educed sysem. Due o Assumpion A3, he maix E = (A A K) + (B B K) is Huwiz by a pope choice of he maix K. Applying heoem o he sliding mode equaion (8), we obain ha he opimizaion poblem (5) has a posiive soluion τ (α, α ). Moeove, as in poving heoem, (8) is asympoically sable if ( SE + ES) + ( γ + γ ) I n 3 4 S SRS SRS BAR A B B R B 3 τγ 4 BB + ( γα + γα + τγα + τγα ) + τ( + ) + τ + τ + ( τγ + ) < whee A = A A K and B = B B K. In un, unde consains Q = SRS, Q = SR S, A S A KS < γ S, 5 B S B KS < γ S 6 wih some posiive γ 5, γ 6, inequaliy (9) esuls fom ( SE + ES) + ( γ + γ ) I + τ( γ + γ ) γ I n n Q Q 3 4 S + τ( + ) + ( γ α + γ α + τγ α + τγ α ) (9) ()

7 574 Asian Jounal of Conol, Vol. 5, No. 4, Decembe 3 z ime z ime z ime suface ime zoom of he suface ime conol u ime Fig.. Response of he closed-loop sysem (3), (3), () wih τ =.63 and ε =. + γτ( B S B W) Q ( B S B W) γτ( B S B W) Q ( B S B W) <. () Since () is equivalen o (5) by Schu s Lemma [], he global asympoic sabiliy of he sliding mode equaion (8) is hen concluded fo each delay value τ [, τ (α, α )]. heoem 4 is compleely poven. Remak 5. While γ i >, i =,, 6 being fixed, (5) is a genealized eigenvalue poblem which can efficienly be solved by convex opimizaion. A subopimal uppebound τ (α, α ) is hus found by elaxaion ype algoihms. VI. EXAMPLE Conside sysem (3) wih A =.4.5.8, B =..5.,. 4 5 C =, F =, P = P = sin( z ( τ )) and he iniial condiion () z ( ) =.5 fo [ τ,]. 3 (3) Le he sliding suface (4) be specified wih K = (.37, 3.979). hen by using semidefinie pogamming, sysem (3), (), diven by he sliding mode conolle (3) subjec o ε =, Ψ (, z, τ ) = [, ] z () θ τ, g =., is globally asympoically sable fo each delay value τ [, τ ]) whee τ =.63. he simulaion esuls depiced in Fig. ae obained wih a 5 h ode inegaion scheme of sep 3. By seing ε =, he conol inpu becomes con- inuous in he oigin and he chaeing phenomenon is hus dasically educed. As shown in Fig., he ampliudes of he inpu oscillaions, coming in his simulaion un, ae educed fom o. wihou appaen changes in he convegence ae. Remak 6. Compaison wih pevious esuls: he sliding mode conolles poposed in [4,] canno sabilize sysem () since he pai (A, C) is no conollable in his case. Wih ou appoach, (A + B, C)-conollabiliy elaxes his equiemen. he sandad linea sae feedback, developed in [6,6], can only aenuae vanishing disubances. Conolles fom [5], which ae based on linea models ove ings and ejec nonvanishing disu-

8 Y. Olov e al.: Sliding Mode Conol Synhesis of Unceain ime-delay Sysems 575 z ime z ime z ime suface ime zoom of he suface ime conol u ime Fig.. Response of he closed-loop sysem (3), (3), () wih =.63 and e = bances, equie he delay value o be known a pioi. A las, conolles fom [9,] ae no applicable o he peubed sysem (3), () when P, P heeby uning ou o be exemely sensiive o he exenal disubances P and P. VII. CONCLUSION Disconinuous ime-delay conol sysems ae unde sudy. he sysem behavio on he manifold whee he conolle undegoes disconinuiies is shown o be govened by he equaion obained hough he equivalen conol mehod exended o Hilbe spacevalued dynamic sysems. Based on his obsevaion, sliding mode conol synhesis, obus agains sufficienly small delay vaiaions and exenal disubances, is developed fo a class of unceain nonlinea imedelay sysems. he synhesis consiss of wo seps. Fis, a manifold, such ha if confined o his manifold he ime delay sysem has desied dynamic popeies, is designed. Second, a disconinuous conol law, which dives he sysem o his manifold in finie ime, is synhesized. An admissible uppebound such ha he coesponding closed-loop sysem emains globally asympoically sable fo all delay values less han his uppebound is deived. heoeical esuls ae suppoed by a numeical example. REFERENCES. Basin, M.V., L. Fidman, and M. Sklia, Opimal and Robus Inegal Sliding Mode File Design fo Sysems wih Coninuous and Delayed Measuemens, Poc. he 4s IEEE Conf. Decis. Con., Las Vegas, USA, pp ().. Boyd, S., L. El Ghaoui, E. Feon, and V. Balakishan, Linea Maix Inequaliies in Sysem and Conol heoy, SIAM, Philadelphia (994). 3. Bynes, C.I., I.G. Lauko, D.S. Gilliam, and V.I. Shubov, Oupu Regulaion fo Linea Disibued Paamee Sysems, IEEE ans. Auoma. Con., Vol. 45, pp (). 4. Choi, H.H., An LMI Appoach o Sliding Mode Conol Design fo a Class of Unceain ime-delay Sysems, Poc. Eu. Con. Conf., Kalsuhe, Gemany (999). 5. Cone, G. and A.-M. Pedon, Sysems Ove Rings: heoy and Applicaions, Plenay Lecue, Poc. IFAC Wokshop Linea ime Delay Sys., Genoble, Fance, p (998). 6. Cuain, R.F. and H.J. Zwa, An Inoducion o Infinie-Dimensional Linea Sysems heoy, Spinge- Velag, New Yok (995). 7. El-Khazali, R., Vaiable Sucue Robus Conol of Unceain ime-delay Sysems, Auomaica, Vol. 34, No. 3, pp (998).

9 576 Asian Jounal of Conol, Vol. 5, No. 4, Decembe 3 8. Foias, C., H. Özbay, and A. annenbaum, Robus Conol of Infinie Dimensional Sysems: Fequency Domain Mehods, Spinge-Velag, London (996). 9. Gouaisbau, F., Y. Blanco, and J.P. Richad, Robus Sliding Mode Conol of Nonlinea Sysems wih Delay : A Design via Polyopic Fomulaion, In. J. Con., (o be published) ().. Gouaisbau, F., M. Dambine, and J.P. Richad, Robus Conol of Delay Sysems: A Sliding Mode Conol Design via LMI, Sys. Con. Le., Vol. 46, No. 4, pp. 9-3 ().. Keulen, B. van, H -Conol fo Disibued Paamee Sysems: A Sae-Space Appoach, Bikhause, Boson (993).. Kein, S.G., Linea Diffeenial Equaions in Banach Space. Ameican Mahemaical Sociey, Povidence (97). 3. Kolmanovskii, V.B. and A. Myshkis, Inoducion o he heoy and Applicaions of Funcional Diffeenial Equaions, Kluwe Academic Publishes, Dodech (999). 4. Levaggi, L., Sliding Modes in Banach Spaces, Diffeenial Inegal Equa., Vol. 5, pp (). 5. Levaggi, L., Infinie Dimensional Sysems Sliding Moions, Eu. J. Con., Vol. 8, pp (). 6. Li, X., and C.E. De Souza, Delay Dependen Robus Sabiliy and Sabilizaion of Unceain Linea Delay Sysem: A Linea Maix Inequaliy Appoach, IEEE ans. Auoma. Con., Vol. 4, No 8, pp (997). 7. Olov, Y., Disconinuous Uni Feedback Conol of Uunceain Infiniedimensional Sysems, IEEE ans. Auoma. Con., Vol. 45, pp (). 8. Olov, Y., and D. Dochain, Disconinuous Feedback Sabilizaion of Minimum-Phase Semilinea Infinie-Dimensional Sysems wih Applicaion o Chemical ubula Reaco, IEEE ans. Auoma. Con., Vol. 47, pp (). 9. Olov, Y. and V.I. Ukin, Sliding Mode Conol in Infinie-Dimensional Sysems, Auomaica, Vol. 3, pp (987).. Shyu, K.K. and J.J. Yan Robus Sabiliy of Unceain ime-delay Sysems and Is Sabilizaion by Vaiable Sucue Conol, In. J. Con., Vol. 57, No., pp (993).. Su, J.H., Fuhe Resuls on he Robus Sabiliy of Delay Dependence fo Linea Unceain Sysems, Sys. Con. Le., Vol. 3, pp (994).. Su, J.H. and C.G. Huang, Robus Sabiliy on Delay Dependence fo Linea Unceain Sysems, IEEE ans. Auama. Con., Vol. 37, pp (99). 3. Ukin, V.I., Sliding Modes in Conol Opimizaion. Spinge-Velag, Belin (99). 4. Ukin, V.I., J. Guldne, and J. Shi, Sliding Modes in Elecomechanical Sysems, aylo and Fancis, London (999). 5. Zheng, F. and P. M. Fank, Finie Dimensional Vaiable Sucue Conol Design fo Disibued Delay Sysems, In. J. Con., Vol. 74, No. 4, pp (). 6. Xie, L. and C.E. De Souza C.E, Robus Sabilizaion and Disubance Aenuaion fo Unceain Delay Sysems, Poc. Eu. Con. Conf., Goningen, he Nehelands (993). 7. Zolezzi,., Vaiable Sucue Conol of Semilinea Evoluion Equaions, Paial Diffeenial Equaions and he Calculus of Vaiaions, Essays in hono of Ennio De Giogi, Bikhause, Boson, Vol., pp (989). Yui Olov eceived M.S. degee fom Moscow Sae Univesiy, in 979, he Ph.D. degee fom he Insiue of Conol Science, Moscow, in 984, and he D.Sc. degee fom Moscow Aviaion Insiue, in 99. He is a Pofesso a he Mexican Scienific Reseach Cene CICESE in Ensenada, since 993. He has been wih he Insiue of Conol Sciences of Russian Academy of Sciences since 979. He was also a pa-ime Pofesso a Moscow Aviaion Insiue fom 989 o 99. His eseach ineess include mahemaical mehods in conol, analysis and synhesis of nonlinea sysems, applicaions o elecomechanical sysems and disibued paamee sysems. He auhoed books and abou jounal and inenaional confeence papes in he afoemenioned aeas. Wilfid Peuquei was bon in 968 in Sain Gilles, Fance. In 99, he eceived a M.Sc. in Auomaic Conol and gaduaed fom Insiu Indusiel du Nod (Fench Gande Ecole ). In 994, he obained a Ph.D. in Auomaic Conol, hen joined he Ecole Cenale de Lille (Fench Gande Ecole ) as an Assisan Pofesso in 995. He has published abou 48 book chapes, jounal and confeence papes and is co-edio wih Jean- Piee Babo of he book Sliding Mode Conol in Engineeing, Macel Dekke. He is cuenly woking on sabiliy analysis (including vaious kinds of sabiliy conceps), sabilizaion (in paicula finie-sabilizaion)

10 Y. Olov e al.: Sliding Mode Conol Synhesis of Unceain ime-delay Sysems 577 and sliding mode conol of non linea and delay sysems. Jean-Piee Richad is Pofesso a he Ecole Cenale de Lille, Fance (Fench Gande Ecole ). His majo eseach fields ae delay sysems, sabilizaion and conol of coninuous sysems (linea/nonlinea), wih applicaions o anspoaions and Sciences & echnologies of Infomaion and Communicaion. He is heading he eam Nonlinea and Delay Sysems (hp://syne.fee. f/) of he LAGIS (Lab. of Au. conol, Compue Sc. and Signal, CNRS UMR 8). Bon in 956, he obained his D.Sc. in Physical Sciences in 984, Ph.D. in Auomaic Conol in 98, Dipl. Eng. and DEA in Eleconics in 979. He is a Membe of he IEEE (Senio), of he Russian Academy of Nonlinea Sciences, of he IFAC C. Linea Sysems and of he edioial boad of he In. Jounal of Sysems Science. He is Pesiden of he GRAISyHM (Reseach Goup in Inegaed Auomaion and Man-Machine Sysems, eseaches fom labs of Region Nod - Pas de Calais, Fance) and is in chage of seveal pogams and eseach newoks (CNRS, Fench Minisy of Reseach, Region Nod - Pas de Calais). He belongs o he Advisoy Commiee of he IEEE biennial confeence CIFA, Conféence Inenaionale Fancophone d'auomaique (hp://cifa4.ec-lille.f/).