Adaptive Fuzzy Dynamic Surface Control for a Class of Perturbed Nonlinear Time-varying Delay Systems with Unknown Dead-zone

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1 Intenatonal Jounal of Automaton and Computng 95, Octobe 0, DOI: 0.007/s Adaptve Fuzzy Dynamc Suface Contol fo a Class of Petubed Nonlnea Tme-vayng Delay Systems wth Unknown Dead-zone Hong-Yun Yue Jun-Mn L Depatment of Appled Mathematcs, Xdan Unvesty, X an 7007, Chna Abstact: In ths pape, adaptve dynamc suface contol DSC s developed fo a class of nonlnea systems wth unknown dscete and dstbuted tme-vayng delays and unknown dead-zone. Fuzzy logc systems ae used to appoxmate the unknown nonlnea functons. Then, by combnng the backsteppng technque and the appopate Lyapunov-Kasovsk functonals wth the dynamc suface contol appoach, the adaptve fuzzy tackng contolle s desgned. Ou development s able to elmnate the poblem of exploson of complexty nheent n the exstng backsteppng-based methods. The man advantages of ou appoach nclude: fo the n-th-ode nonlnea systems, only one paamete needs to be adjusted onlne n the contolle desgn pocedue, whch educes the computaton buden geatly. Moeove, the nput of the dead-zone wth only one adjusted paamete s much smple than the ones n the exstng esults; the poposed contol scheme does not need to know the tme delays and the uppe bounds. It s poven that the poposed desgn method s able to guaantee that all the sgnals n the closed-loop system ae bounded and the tackng eo s smalle than a pescbed eo bound, Fnally, smulaton esults demonstate the effectveness of the poposed appoach. Keywods: Adaptve fuzzy contol, dynamc suface contol DSC, dscete and dstbuted tme-vayng delays, Lyapunov- Kasovsk functonals, dead-zone. Intoducton Snce adaptve backsteppng technque was poposed n [], a lot of extensons on ths technque have been epoted, and t has been poven to be a poweful tool fo dealng wth unmatched and lnealy paametezed uncetantes n lowe-tangula-stuctued systems. In ode to cope wth unmatched and nonlnealy paametezed uncetantes, many adaptve backsteppng contol methods have been developed. Fo example, adaptve backsteppng neual netwok contol [, 3] and adaptve backsteppng fuzzy contol [4,5]. Howeve, compaed wth neual netwok NN, the man advantage of fuzzy logc system FLS s that t can combne some expeence and knowledge fom desgnes o expets [6]. Thus, as a unvesal appoxmato, FLS s supeo to NN. Recently, many mpotant esults on adaptve backsteppng fuzzy contol have been also epoted [7,8]. Howeve, a dawback wth the backsteppng technque s the poblem of exploson of complexty [9]. That s, the complexty of the contolle gows dastcally as the ode n of the system nceases. Ths exploson of complexty s caused by the epeated dffeentatons of cetan nonlnea functons. To ovecome the exploson of complexty, dynamc suface contol DSC was poposed n many exstng esults [9]. In [9], a DSC technque was poposed to elmnate ths poblem by ntoducng a fst-ode flteng of the synthetc nput at each step of the tadtonal backsteppng appoach. In [0], DSC fo the tackng poblem of non-lpschtz systems was consdeed. The contol scheme was developed by usng the dynamc suface contol ap- Manuscpt eceved August 7, 0; evsed Decembe 3, 0 Ths wok was patally suppoted by Natonal Natual Scence Foundaton of Chna Nos and and Fundamental Reseach Funds fo the Cental Unvestes No poach fo pue-feedback nonlnea systems wth unknown dead zone []. Wang et al. [] constucted an adaptve neual contolle fo a class of pue-feedback nonlnea tmedelay systems va DSC technque. Howeve, the weakness of the above mentoned adaptve dynamc suface contol methods s that the numbe of the adjustable paametes depends on the ode of the subsystem. If the ode of subsystem s added onlne o taken offlne, the numbe of the adjustable paametes wll be nceased. Fom the vew pont of the engneeng applcaton, the onlne computaton buden s vey heavy. Thus, by combnng the FLS wth the DSC technque, how to desgn the adaptve fuzzy contolle contanng fewe adjusted paametes fo uncetan nonlnea systems s a challenge. Dead-zone s one of the mpotant non-smooth nonlneates n many ndustal pocesses, whch can seveely lmt system pefomance, and ts study has been dawn much nteest n the contol communty fo a long tme [3,, 35]. To handle systems wth unknown dead-zone, adaptve deadzone nveses wee poposed [3]. In [4], unde the condton that the dead-zone slopes n the postve and negatve egons must be the same, obust adaptve contol was developed fo a class of specal nonlnea systems wthout constuctng the nvese of the dead zone. Moeove, adaptve neual contol was developed fo a class of uncetan mult-nput, mult-output nonlnea systems wth unknown nonlnea dead-zones and unknown gan sgns [3,5], and n [], the contol poblem fo the nonlnea systems wth unknown dead-zone n pue feedback fom has been solved by usng adaptve DSC appoach. Tme delays ae often encounteed n vaous systems. The exstence may degade the contol pefomance and make the stablzaton poblem moe dffcult. Recently,

2 546 Intenatonal Jounal of Automaton and Computng 95, Octobe 0 seveal fuzzy adaptve contol schemes have been epoted whch combne the Lyapunov-Kasovsk functonal wth the adaptve backsteppng fuzzy contol fo nonlnea systems wth tme delays [4,8,5]. Howeve, t s woth notng that the esults wee obtaned n the context of contnuous fuzzy systems wth constant o tme-vayng delays. When the numbe of summands n a system equaton s nceased and the dffeences between neghbong agument values ae deceased, systems wth dstbuted delays wll ase. Thus, the topc of dstbuted delay systems has been an attactve eseach topc n the past yeas [68]. Motvated by the above obsevatons, n ths pape, the adaptve tackng contol s constucted fo uncetan nonlnea tme-vayng delay systems wth unknown dead-zone va fuzzy DSC appoach, whee the dead-zone slopes n postve and negatve egons ae dffeent. The appopate Lyapunov-Kasovsk functonals ae used to deal wth the unknown tme-vayng delays tems. Fnally, FLS s employed to appoxmate the nonlnea functons, and the adaptve backsteppng appoach s utlzed to constuct the fuzzy contolle. The man contbutons of the pape ae lsted as follows. By combnng the DSC appoach wth FLS, the tackng contol poblem s nvestgated fo a class of nonlnea tme-vayng delay systems wth unknown dead-zone. The poposed contol scheme can elmnate the poblem of exploson of complexty nheent n the backsteppng desgn method. Moeove, compaed wth [, ], n whch the esults wee obtaned by usng the DSC technque, n ths pape, only one paamete needs to be adjusted onlne dung the contolle desgn pocess, whch geatly educes the computaton buden onlne. In addton, the nput of the dead-zone wth only one adjusted paamete s much smple than the ones n the exstng esults [3, 5], thus, t s ease to be mplemented n applcatons. The man dffculty encounteed n the contolle desgn pocess s how to deal wth the unknown tmedelay tems. To ovecome ths dffculty, the appopate Lyapunov-Kasovsk functonals ae used fo stablty analyss and synthess. In addton, the constucted contolle dose not eque the tme-vayng delays and the uppe bounds. Ths s moe elaxed than that of the exstng lteatue [7,5,9]. It can be poven that all the sgnals n the closed-loop system ae bounded and the tackng eos convege to a small neghbohood aound zeo. Effectveness of the developed scheme s llustated by the smulaton example. System descpton and pelmnaes. System descpton Consde the followng nonlnea tme-delay system ẋ = g x x f x h x t d t,, x td t td t m x s ds ω t, x ẋ n = g n x u t f n x h n x t d t,, x t d n t yt = x t x t = ϕ t max d, d } t 0 td n t m n x s ds ω n t, x wth dead-zone Q υ, f υ t b u = D υ = 0, f b l < υ t < b Q l υ, f υ t b l whee x = [x,, x n] T R n and y R denote the state vecto and the output of the system, espectvely. u R and υ R ae the output and the nput of the dead zone, espectvely. b l and b ae the unknown paametes of the dead zone. Q l v and Q v ae contnuous functons. Fo n, x = [x,, x ] T, f, g, m and h ae unknown smooth functons. d t and d t epesent unknown dscete and dstbuted tme-vayng delays, espectvely. ω t, x stands fo unknown extenal dstubance. ϕ t s a known contnuous ntal state functon. Then, the followng assumptons ae ntoduced. Assumpton. 0 d t d and 0 d t d. Moeove, the devatves satsfy d t and d t d d wth d d < and d d <, espectvely. Assumpton. Nonlnea functon h yelds h x t d t,, x t d t p k x k t d k t 3 k= whee p k x k t d k t s an unknown contnuous functon. Assumpton 3. Fo n, thee exsts an unknown contnuous functon ρ x such that ω t, x ρ x. Assumpton 4. Fo n, the sgn of g x s known, and thee exst unknown constants g 0 and g such that 0 < g 0 g x g, x R. Wthout loss of genealty, t s assumed that g g x g 0 > 0. Assumpton 5. The dead-zone output ut s not avalable. Moeove, the dead-zone paametes b and b l ae unknown bounded constants, but the sgns ae known,.e., b > 0 and b l < 0. Assumpton 6. The desed tackng tajectoy y d s contnuous and avalable, moeove, y d, ẏ d, ÿ d ae bounded,.e., thee exsts a postve constant } B 0, such that Π := yd, ẏ d, ÿ d : yd ẏd ÿd B 0. Assumpton 7. Fo contnuous functons Q l v and Q v, thee exst unknown postve constants k l0, k l, k 0 and k such that 0 < k 0 Q υ k, υt [b,, 0 < k l0 Q l υ k l, υt, b l ], whee Q l υ = dq lz dz z=υ and Q υ = dqz dz z=υ. Based on Assumpton 7, smla to [3], the dead zone can be ewtten as follows: u = D υ = K T t Φ t υt d υ 4

3 H. Y. Yue and J. M. L / Adaptve Fuzzy Dynamc Suface Contol fo 547 whee Φ t = [ϕ t, ϕ l t] T, f υ t > b l ϕ t = 0, f υ t b l, f υ t < b l ϕ l t = 0, f υ t b l K t = [K υ, K l υ] T 0, f υ t b l K υ t = Q ξ υ, f b l < υ t < Q l ξ l υ, f < υ t < b l K l υ t = 0, f υ t b l Q ξ υ b, f υ t b d υ= [Q l ξ l υ Q ξ υ] υ, f b l < υ t < b l Q l ξ l υ b l, f υ t b l. 5 In 5, f υ < b l, ξ l υ υ, b l ; f b l υ < b, ξ l υ b l, υ; f b < υ, ξ υ b, υ; f b l < υ b, ξ υ υ, b. Moeove, fom 5, we can get d υ p, whee p s an unknown postve constant wth p = k l k max b, b l }. Remak. In Assumpton, d, d, d and d ae ntoduced only fo the pupose of analyss and ae not used to desgn contolles, thus, they can be unknown as well. Remak. Assumpton s mposed on the tme-delay functon h. It means that the functon h satsfes the sepaaton pncple [0]. As a matte of fact, ths assumpton can be seen n [, 7]. Wth Assumpton, we futhe obtan h x t d t,, x t d t 6 p k x k t d k t p k x k t d k t. k= k= Remak 3. Assumpton 7 s geneally adopted n [3, 5] and s easonable fo ndustal systems. As shown n [3, 5], we know that K T t Φ t [mn k l0, k 0}, k l k ] 0,. Fo smplcty, denote a constant β 0 wth 0 < β 0 mn k l0, k 0}, then, we have K T t Φ t β 0.. Pelmnaes Thoughout ths pape, the followng IF-THEN ules ae used to develop the adaptve fuzzy contolle: R l : If x s F l, x s F l,, and x n s F ln, then y s G l, l =,, N, whee x = [x,, x n] T and y ae the FLS nput and output, espectvely. Fuzzy sets F l and G l ae assocated wth the membeshp functon µ Fl x and µ Gl y, espectvely. N s the ule numbe. Wth the help of sngleton functon and cente aveage defuzzfcaton, FLS can be expessed as follows: y x = N Φ l l= N l= n µ Fl x n µ Fl x 7 whee Φ l = ag sup y R µ Gl y. Let 8 be the fuzzy bass functon n µ Fl x ξ l =. 8 N n µ Fl x l= Denotng ξ x = [ξ x,, ξ N x] T and φ = [Φ,, Φ N ] T, the fuzzy logc system 7 can be ewtten as y x = φ T ξ x. 9 It has been poven that the above fuzzy logc system s capable of unfomly appoxmatng any contnuous nonlnea functon ove a compact set Ω x wth any degee of accuacy. Ths popety s shown by the followng lemma. Lemma [6]. Let f x be a contnuous functon defned on compact set Ω. Then, fo any constant ε > 0, thee exsts an FLS 9 such that sup f x φ T ξ x ε. 0 tc x Ω Lemma [6]. Gven a postve-defnte matx W R n n and two scalas c > d 0, fo any vecto ωt R n, we have d d ω T sds W ω sds d c d tc 3 Contolle desgn tc ω T sw ω s ds. In ths secton, based on the dynamc suface contol appoach, the fuzzy adaptve contol scheme s pesented fo the n-th-ode nonlnea system descbed by. Smla to the tadtonal backsteppng, the desgn of the adaptve contol laws s based on the followng change of coodnates: z = x y d, z = x α, =,, n, whee α s the output of a fst ode flte wth α d as the nput, and α d s an ntemedate contol whch wll be developed fo the coespondng -th subsystem. At step n, an nput of the dead-zone υt s constucted. To develop a backsteppngbased desgn pocedue, we fst defne a constant θ = max θ,, θ, θ n}, whee θ = g 0 φ and θ n = g 0 β 0 φ n wth =,, n. φ s an unknown weght paamete vecto and wll be specfed late. ˆθ s the estmaton of θ and the paamete estmaton eo s θ = θ ˆθ. In ths secton, we wll use the ecusve backsteppng technque to develop the adaptve fuzzy tackng contol law as follows: α,d = k z ˆθ Z ξ Z z η 3 whee n, k and η ae postve desgn paametes, moeove, k > g 0 and kn > λ g 0 βλ g 0 fo =,, β wth λ beng a desgn paamete. Note that f = n, α n,d s the nput sgnal υt of the dead-zone. Futhemoe, n

4 548 Intenatonal Jounal of Automaton and Computng 95, Octobe 0 ode to obtan the flteed vtual contolle α, we pass α,d though a fst-ode flte wth tme constant τ > 0. τ α α = α,d α 0 = α,d 0 4 whee =,, n. Now, we popose the desgn pocedue. Step. The fst eo suface s defned as z = x y d, then, ts tme devatve s gven by ż = ẋ ẏ d = g x x h x t d t f x td t m x sds ω t, x t ẏ d. 5 Choose Lyapunov-Kasovsk functonal canddate as V z = z Λ, whee Λ = eγtd t td t eγτ h x τ dτ e γtd d d τ eγs m s dsdτ wth γ beng a pos- gves that d td t tve desgn constant. Dffeentatng V z V z = z ż γλ eγd h d x d e γd m d x e γd d t d d t h x t d t d t t d d e γtds m x s ds 6 td t Usng Assumptons 3 and yelds z h x t d t z h x t d t d k t d k, k =, z ω t, x z ϕ a e γd d t a e γtds, s [t, t d t] d t t d d e γtds m x s ds td t z d td t m x s ds m x ds td t z z t d m x s ds td t m x ds td t 7 whee a > 0 s a desgn constant. Fom 6 and 7, we have V z z g x x f x ẏ d z z ϕ x a a H x γλ 8 whee H = eγd d h x d eγd d m x. Addng and subtactng ν g x z at the ght hand sde of 8 gves V z g x x z ν g x z z f Z a γλ whee f Z = f x ν g x z ẏ d z z ϕ x a H x z, wth Z = [x, y d, ẏ d ] T Ω Z3 R 3, and Ω Z3 s some known compact set n R 3. Notce that H x z s dscontnuous at z = 0. Theefoe, t cannot be appoxmated by FLS. Smlaly to [], we ntoduce tanh z ϑ to deal wth the tem H x z. Defne f Z = f Z H x z z tanh z ϑ H x, whee ϑ s a postve desgn paamete. Note that lm z 0 z tanh z ϑ H exsts, thus, f Z can be appoxmated by an FLS φ T ξ Z such that f Z = φ T ξ Z δ Z. Theefoe, we have V z g x x z z φ T ξ Z δ Z ν g x z a γλ tanh z ϑ 9 H. Fo smplcty, let ε be the uppe bound of δ Z. Appaently, we have z φ T ξ z δ Z g0θ η ξ z η z λ λε 0 whee θ = g 0 φ, λ and η ae postve desgn paametes. Usng 9, 0, and x = z α yelds V z g x z z g x α z ν g x z z λ q g 0θ η ξ z γλ tanh z H ϑ whee q = η λε a and g x z z wll be canceled n the next step. Remak 4. In ths pape, the adaptve law fo ˆθ wll be gven n the last step. Moeove, the esdual tem ν g x z n wll play an mpotant ole n the stablty analyss pocess, ths pont wll be seen late. Step. Consdeng the second eo suface z = x α, the dynamcs of z s gven by ż = g x x 3 h x t d t, x t d t f x m x td sds ω t, x t α. Choose t a Lyapunov-Kasovsk functonal as V z = z Λ, whee Λ = d e γtd td t τ eγs m x s dsdτ e γtd d gven by d k= td k t eγτ p k x k τ dτ, ts devatve s V z = z ż γλ eγd e γd d k t d k= d t t d d d k= d k t td t p k x k d e γd m d x p k x k t d k t e γtd s m x s ds.

5 H. Y. Yue and J. M. L / Adaptve Fuzzy Dynamc Suface Contol fo 549 Smla to 9, fom Assumptons 3, 6 and, we have z h x t d t, x t d t z z 4 h x t d t, x t d t z td t p k x k t d k t k= m x ds z d td t m x s ds d k t e γd d k t d, k =, d t d, eγtd s, s [t, t d t] z ω t, x t z ϕ x a Usng and 3 yelds a. 3 V z g x x 3z z f x 3 z 4 z ϕ x a α H x a z γλ whee H x = d eγd d m x eγd d p k x k. In ode k= to cancel the esdual tem g x z z of, we add and subtact g x z z at the ght hand sde of 4, futhemoe, smla to, we can get V g x x 3z ν gz g x z z z f Z tanh z H a γλ. 5 ϑ The functon f Z s defned as f Z = f x g x z ν gz 3 z z ϕ x α a z tanh z ϑ H, whee Z = [x, x, y d, z, α ] T Ω Z R 5, and Ω Z s some known compact set n R 5. Smla to Step, we have f Z = φ T ξ Z δ Z. 6 Usng 5, 6 and the nequalty smla to 0 gves that V g x x 3z ν gz g x z z z q λ g 0θ η ξ z tanh z H γλ 7 ϑ whee q = η ϑ λε a, δ Z ε and η s a postve desgn paamete. Combnng x 3 = e 3 α wth 7 yelds V g x z 3z g x z α ν g x z g x z z g0θ η ξ z z λ tanh z H q γλ. ϑ Step. Smla to Step, fo each step 3, we have V g x z z g x z α ν g x z g x z z g0θ Z ξ Z z 8 η z λ tanh z H q γλ. ϑ Step n. Consdeng z n = x n α and 4, the dynamcs of z n s gven by ż n = g n x K T Φυ d υ f n x h n x t d t,, x n t d n t td n t mn x s ds ωn t, x t α. We choose the Lyapunov-Kasovsk functon as V zn = z n Λ n, wth Λ n = d d e γtd t td n t τ eγs m n x n s dsdτ e γtd n d k= td k t eγτ p k x k τ dτ. Then, the devatve of V zn s gven by V zn = z nż n eγd d n e γd d k t d k= d n t t d d n k= p k x k d e γd m d n x n γλ n d k t td n t p k x k t d k t e γtd s m n x n s ds. Moeove, usng d υ p poduces g n x d υ z n z n gn x p. Smla to Step, we have V zn g n x K T Φυz n g x z z n z n fn Z n a nn tanh zn H n γλ n 9 ϑ n whee f n Z n = f n x n g x z z n α z nϕ n xn a nn z n tanh z n ϑ n H n wth H = eγd d d m n x n n p k x k gn x p, Z n = [ α, z, z n, e γd d k= x] T Ω Zn R n3, and Ω Zn s some known compact set n R n3. Then, we have f n Z n = φ T nξ n Z n δ n Z n. Employng the tangula nequalty yelds z n φ T nξ n δ n g0β0θn ηn n ξ nzn η n z n λ λε n 30 whee θ n = g 0 β 0 φ T n. Usng 9, 30 and yelds V zn g n x K T Φυz n ḡ x z z n z n λ z n g 0β 0θ ηn n ξ nzn q n tanh zn H n γλ n ϑ n whee q n = a nn η n λε n. Substtutng υ t defned n 3 nto t and usng g nxk T Φ g 0β 0 yelds V zn k nzn g x z z n ηn n ξ nzn γλ n q n tanh zn H n 3 ϑ n whee k n = g 0β 0k n λ. g0β0 θ

6 550 Intenatonal Jounal of Automaton and Computng 95, Octobe 0 In the followng secton, we wll desgn the adaptve law fo ˆθ. Choose the Lyapunov functon as V = n V z g 0 θ, the devatve of V s gven by n g0 θ V = V ˆθ g x z α ν g z 3 g 0 θ g 0θ η ξ z z λ k nz n n n n ˆθ q γλ g0β0 θ ηn n ξ nzn tanh z ϑ H whee θ = θ ˆθ. Now, defne the bounday laye eos as y = α α,d, =,, n 33 By usng τ α α = α,d, we obtan that ẏ = y τ α,d, =,, n. 34 By usng 3, 33 and α,d defned n 3, we have V g x z y ν g x z g x k z z n λ g x ˆθ ξ z η g 0θ η ξ z k nzn g0β0 θ ηn n Z n ξ n Z n zn g0 θ n ˆθ q n n γλ tanh z H 35 ϑ Then, usng g x z y ν g x z, g 0θ η g x ˆθξ ξ z η yelds ξ ξ z = g 0 θ ξ η ξ z V g 0 θ η 4ν y g 0 ˆθξ ξ z, g η 0 ˆθξ ξ z η y g 0k z z 4ν λ k nzn ˆθ ξ z β0 ξn T ξ nzn 36 n n n γλ q η n tanh z ϑ H. Accodng to 36, the paamete adaptve law s chosen as ˆθ = ξ η T Z ξ Z z β0 ηn n ξ nzn σˆθ 37 whee σ and ae postve desgn paametes. Substtutng 37 nto 36 poduces that V n g 0k z γλ g0σ θˆθ z λ k nzn n 4ν y n q tanh z H. 38 ϑ The followng wll show the stablty of the esultng closed-loop system. 4 Stablty analyss Befoe poposng the man theoem, we fst ntoduce the followng lemma. Lemma 3 []. Consde the set Ω ϑj defned by Ω ϑj := z j z j < 0.884ϑ j} fo j n. Then, fo any z j / Ω ϑj, y g 0 θ the nequalty tanh zj ϑ j 0 s satsfed. The man esults ae summazed as follows. Theoem. Consde the closed-loop system consstng of system unde Assumptons 7, the contol law 3 and the paamete adaptve law 37. Suppose that the packaged uncetan functons f Z, =,,, n, can be appoxmated by fuzzy logc systems n the sense that appoxmaton eos ae bounded. Then, fo any ntal condton satsfyng n Vz p, whee p s any postve constant, thee exst k, τ and σ such that the soluton of the closed-loop system s sem-globally unfomly ultmately bounded, and the steady-state tackng eo s smalle than a pescbed eo bound. Poof. Fstly, we need to pove that all the sgnals n the closed-loop system ae bounded. The poof pocedue s dvded nto the followng thee cases: Case. Fo j n, z j Ω ϑj. In ths case, z j < 0.884ϑ j. Accodng to 37, t s obvous that ˆθ s bounded fo bounded z j. Futhe, θ s bounded as θ s a constant. Accodng to Assumpton, y d s also bounded. Snce z = x y d, we can get that x s also bounded. Usng 3 and notng that z, ˆθ and ξ Z ae all bounded fo =,, n, we can conclude that fo =,, n, α,d s bounded. Futhemoe, fom 4, the boundedness of α s obtaned fo =,, n, theefoe, accodng to y = α α,d, =,, n, the boundedness of y s also ensued. Consequently, fo =,, n, by usng x = z α and the boundedness of z and α, the boundedness of x s obtaned. Fnally, accodng to 4 and 3, we have u = D υ = K T t Φ t υ t d υ = K T t Φ t k nz n ˆθ η n n ξ nz n d υ p β 0k nz n β 0 ˆθ ηn n Z n ξ n Z n z n, thus, ut s bounded. Hence, all the sgnals n the closed-loop system ae bounded n case. Case. Fo j =, n, z j / Ω ϑj. In ths case, z j 0.884ϑ j. Choose the Lyapunov functon canddate as V = n Vz y g 0 θ. By usng 38 and 34, we have V n k z y g0σ θˆθ 4ν y y α,d τ n n q γλ tanh z H ϑ whee k = g 0k and k λ n = g 0β 0k n. Fom the λ defnton of H, we get that H 0. Then, usng Lemma 3 yelds n tanh z ϑ H 0. Moeove, accodng to the defntons of θ and α,d, we have σg 0 θ ˆθ σg 0 θ θ θ σg 0 θ σg 0θ, α,d = k ż ˆθ ξ z η ˆθ ξ T ξ z = χ z,, z n, y,, y n, ˆθ, y d, ẏ d, ÿ d, η ˆθ ξ ż η

7 H. Y. Yue and J. M. L / Adaptve Fuzzy Dynamc Suface Contol fo 55 whee χ s a contnuous functon fo n. Then, V n k z y σg0 θ 4ν y τ n y α,d γλ C 39 whee C = n q σg 0θ. Accodng to k > and g 0 λ k n > g 0 βλ g 0 wth =,, n, we can deduce β k > 0,, k n > 0. Moeove, fo any B 0 > 0 and p > 0, the sets Π = } y, ẏ, ÿ : y ẏ ÿ B 0 and n Π = j= z j γ j= Λj g 0 θ } j= y j p ae compact n R 3 and R n, espectvely. Theefoe, Π Π s also compact n R n3. Thus, χ has a maxmum M on Π Π. Usng y χ τ y χ τ V y τ k z k nzn τ y χ τ y 4ν n yelds σg0 θ γλ C 40 whee τ s a postve desgn constant. Now, choose k = k n = k and τ = 4ν M τ τ wth k and τ beng postve constants, we have V n k z σg0 θ γ n Λ n τ C y M 4ν 4ν τ τ y y M χ M τ = n k z σg0 θ γ τ y χ M n Λ C M y τ whee C = C τ. Accodng to χ < M 4 V α 0 V C 4 whee 0 < α 0 < mn k, σ, } τ, γ. Let α0 > C, then, p V 0 on V 0 = p. Thus, V p s an nvaant set,.e., f V 0 p, then V t p fo all t 0. Thus, 4 holds fo all V 0 p and all t 0. Solvng 4 gves 0 V t C α 0 V 0 C α 0 e α0t, t 0, whch means that V C t eventually s bounded by α 0. Thus, all sgnals of the close-loop system,.e., z t, θ and y ae sem-globally unfomly ultmately bounded. Case 3. Some z m Θ ϑm, whle some z j / Θ ϑj. Defne Σ M and Σ J as the ndex sets of subsystems consstng of z m Θ ϑm and z j / Θ ϑj, espectvely. Then, fo j Σ J, choose the Lyapunov functon canddate as V ΣJ = V zj g0 θ y j ts tme devatve s gven by V ΣJ kjz j mz n β0 ηn n ξ nzn σˆθ tanh zj ϑ j yj g0 θ 4ν j η j Σ j M j ξ jz j y j α j,d γ Λ j yj q j τ j H j g x z z g x z z 43 0, f n Σ J whee mz n =. When, f n Σ M j Σ M, we know that z j s bounded, then, usng the fact that 0 < ξj T ξ j yelds that η j ξj T ξ jzj mz n β 0 ξ η n T ξ n nzn s bounded. If j Σ M j Σ M η j j ξ jzj mz n β 0 ηn n ξ nzn L, we can have σg 0 θˆθ g 0 θ j ΣM η j j ξ jzj mz n β 0 ηn n ξ nzn σg 0 θ σg 0θ g 0 4 σ L. Futhe, smla to the poof of Theoem n [8], the last tem of 43 can be expessed as g j x j z jz j g j x j z jz j zj j Σ M,j Σ M λg0.884 ϑ j ϑj. Moeove, λ smla to case, we can get tanh zj ϑ j H j 0. By usng the above dscusson, one gets V ΣJ kjz j y j α j,d yj yj σg0 θ 4ν j 4 zj λ γ τ j Λ j C 44 whee C ΣJ = j Σ M,j Σ M λg0.884 ϑ j ϑj q j. Snce k j = g 0k j and k λ n = g 0β 0k n λ wth k j > g 0 and kn > λ g 0 βλ g 0 fo j =,, n, β we deduce that k j > 0 fo j =,, n. Applyng the λ smla pocedues as 39 and 4, we have V ΣJ = kσ J zj σg0 4 θ γ Λ j C ΣJ τ j yj χ j M j y j Mj τ whee C ΣJ = τ CΣ, J k Σ J = k j, j ΣJ and λ τ j = 4ν j M j τ τj wth kσ J and τ j beng postve constants. Smla to 4, we have V ΣJ α ΣJ VΣJ C ΣJ 45 whee 0 < α ΣJ < mn kσ J, σ, } 4 τj, γ. Let ασj > C ΣJ, p then, VΣJ 0 on V ΣJ 0 = p. Thus, VΣJ p s an nvaant set,.e., f VΣJ 0 p, then V ΣJ t p fo all t 0. Thus, 45 holds fo all V ΣJ 0 p and all

8 55 Intenatonal Jounal of Automaton and Computng 95, Octobe 0 t 0. Solvng nequalty 45 gves that fo t 0 0 V ΣJ t C ΣJ α ΣJ VΣJ 0 C ΣJ α ΣJ e α Σ t J, whch means that V ΣJ t eventually s bounded by C ΣJ α ΣJ. Thus, we can conclude that sgnals ˆθ, z j, y j ae all bounded fo j Σ J. Now, we consde the boundedness of all the sgnals n the whole closed-loop system. We know that all z m ae bounded fo m Σ M, and fom Assumpton, we have that y d, ẏ d, ÿ d ae bounded. Followng the smla dscusson n case, we can conclude that all the sgnals n the closed-loop system ae bounded unde ths case. In lght of the dscusson fo Cases 3, we can conclude that all the sgnals n the closed-loop system ae bounded. Moeove, by nceasng the values of k j and σ and educng the value of τ j and ϑ j,.e., nceasng the value of α 0 and α ΣJ, whle educng the value of 0.884ϑ j, and C ΣJ α ΣJ C α 0 can be made abtaly small. Thus, the tackng eo e may be made abtaly small. 5 Smulaton In ths secton, the followng nonlnea system s consdeed to llustate the valdty of the poposed scheme. The smulaton objectve s to apply the developed adaptve fuzzy contolle such that boundedness of all the sgnals n the closed-loop system s guaanteed and the system output y follows the efeence sgnal y d to a small neghbohood of zeo whee y d = 0.5 sn0.5t snt. Then, select the desgn paametes as =, η = η = η 3 = 0, σ = 0.3, k = 35, k = 70, k 3 = 35, τ = 3 and τ = 0.0. When t [d, 0] wth d = max d, d }, fo =,, 3, choose x t = x t = x 3 t = 0. Membeshp functons ae specfed as µ F l x = exp 0.5 x.5 0.5l, l 7. The contolles and the paamete adaptve law fo the system 46 ae desgned as α,d = k z ˆθξ T ξ z η α 0 = α,d 0, υ = k 3z 3 j=, τ α α = α,d, =,, ˆθ η 3 ξ3 T ξ ˆθ 3z 3, = β 0 η3 3 ξ 3z3 η ξ z σˆθ. Fnally, smulaton esults n Fgs. 4 show the effectveness of the developed contol schemes fo the system 46. Fom Fg., t can be seen that good tackng pefomance s obtaned. The boundedness of υt and ut ae llustated n Fg.. The vaables x and x 3 ae also bounded as gven n Fg. 3. We can deduce that paamete estmaton ˆθ s also bounded fom Fg. 4. ẋ = x x 3 snt d t td t x s ds 0.7x cos.5t ẋ = x 3 0.x 3 x t d t x t d t td t x s x s ds x x sn 3 t ẋ 3 = 3 cosx x x 3 u x x x 3 46 y = x td 3 t wth dead zone x 3 s dsx t d t x t d t Q υ, f υ t b u = D υ = 0, f b l < υ t < b Q l υ, f υ t b l. In ths smulaton, we choose Q υ = υ0.75 snυ.5, Q l υ = 0.8υ0.3 cosυ., b =, b l = 0.5, d t = 0. snt, d t = 0.5 cost, d t = 0. snt, d t = 0. cost and d 3 = 0.0 snt, the uppe bounds of them ae d =.5 and d =.0. Moeove, by the smple manpulaton, we have snx and x x 0.5x 0.5x, thus, Assumptons and ae satsfed. Then, accodng to 0.75 cosυ, f υ u = D υ = 0, f 0.5 < υ < snυ, f υ 0.5 Fg. Fg. The sgnals y, y d, and y y d The sgnals ut and υt we select β 0 = 0.5 and the ntal states ae chosen as x 0 = x 0 = x 3 0 = 0, y = y 0 = 0 and ˆθ 0 = 0. Fg. 3 The states x and x 3

9 H. Y. Yue and J. M. L / Adaptve Fuzzy Dynamc Suface Contol fo 553 systems. IEEE Tansactons on Automatc Contol, vol. 36, no., pp. 4 53, 99. [] S. S. Ge, K. P. Tee. Appoxmaton-based contol of nonlnea MIMO tme-delay systems. Automatca, vol. 43, no., pp. 3 43, 007. Fg. 4 The tajectoy of ˆθ [3] T. P. Zhang, S. S. Ge. Adaptve neual netwok tackng contol of MIMO nonlnea systems wth unknown dead zones and contol dectons. IEEE Tansactons on Neual Netwoks, vol. 0, no. 3, pp , 009. Remak 5. It should be ponted out that the exstng adaptve fuzzy contol appoaches cannot be used to contol system 46 because of the exstence of the unknown dstbuted tme-vayng delays and the unknown dead-zone. Howeve, the adaptve neual contol appoaches suggested n [3,, ] can be appled to desgn the contolles f the tme-vayng delays o the unknown dead-zone dsappeas. A common dawback of these contol methods n [3, ] s that the numbe of adaptaton laws depends on the numbe of the neual netwok nodes. Wth an ncease of neual netwok nodes, the numbe of paametes to be estmated wll ncease sgnfcantly. Fo nstance, n the smulaton studes of [], lots of adaptve paametes ae equed to be tuned onlne to contol the system. To solve ths poblem, n [], the authos consdeed the nom of the deal weghtng vecto n neual netwok as the estmaton paamete nstead of the elements of weghtng vecto. Thus, the numbe of adaptaton laws s educed consdeably. Howeve, the numbe of adjustable paametes depends on the ode of the system. Unlke [3,, ], n ths pape, a key technque s ntoduced to educe the numbe of adaptve paametes,.e., the unknown constant θ = max θ,, θ, θ n} s used as an estmated paamete, whch esults n only one adaptve paamete ˆθ fo the system 46, whch s also the most mpotant advantage of the scheme developed n ths pape. Moeove, by usng the appopate Lyapunov-Kasovsk functonals, the tme delays and the uppe bounds can be unknown. 6 Conclusons The adaptve fuzzy contol scheme has been poposed fo a class of petubed nonlnea systems wth unknown dead-zone. Based on the appopate Lyapunov-Kasovsk functonals, the FLS and the DSC appoach, the contolle s constucted. The poposed adaptve fuzzy tackng contolle guaantees the boundedness of all the sgnals n the closed-loop system. Moeove, the suggested adaptve fuzzy contolle contans only one adaptve paamete. Ths makes ou desgn scheme ease to be mplemented n pactcal applcatons. Smulaton esults have been gven to llustate the effectveness of the poposed scheme. Refeences [] I. Kanellakopoulos, P. V. Kokotovc, A. S. Mose. Systematc desgn of adaptve contolle fo feedback lneazable [4] S. C. Tong, Y. M. L, Y. Q. Xa, C. L. Lu. Adaptve fuzzy backsteppng output feedback contol of nonlnea tmedelay systems wth unknown hgh-fequency gan sgn. Intenatonal Jounal of Automaton and Computng, vol. 8, no., pp. 4, 0. [5] S. C. Tong, N. Sheng. Adaptve fuzzy obseve backsteppng contol fo a class of uncetan nonlnea systems wth unknown tme-delay. Intenatonal Jounal of Automaton and Computng, vol. 7, no., pp , 00. [6] L. X. Wang. Adaptve Fuzzy Systems and Contol: Desgn and Stablty Analyss, New Jesey, USA: Pentce- Hall, 994. [7] C. C. Hua, Q. G. Wang, X. P. Guan. Adaptve fuzzy outputfeedback contolle desgn fo nonlnea tme-delay systems wth unknown contol decton. IEEE Tansactons on Systems, Man, and Cybenetcs Pat B: Cybenetcs, vol. 39, no., pp , 009. [8] M. Wang, B. Chen, X. P. Lu, P. Sh. Adaptve fuzzy tackng contol fo a class of petubed stct-feedback nonlnea tme-delay systems. Fuzzy Sets and Systems, vol. 59, no. 8, pp , 008. [9] D. Swaoop, J. C. Gedes, P. P. Yp, J. K. Hedck. Dynamc suface contol of nonlnea systems. In Poceedngs of Amecan Contol Confeence, IEEE, Albuqueque, USA, vol. 5, pp , 997. [0] D. Swaoop, J. K. Hedck, P. P. Yp, J. C. Gedes. Dynamc suface contol fo a class of nonlnea systems. IEEE Tansactons on Automatc Contol, vol. 45, no. 0, pp , 000. [] T. P. Zhang, S. S. Ge. Adaptve dynamc suface contol of nonlnea systems wth unknown dead zone n pue feedback fom. Automatca, vol. 44, no. 7, pp , 008. [] M. Wang, X. P. Lu, P. Sh. Adaptve neual contol of puefeedback nonlnea tme-delay systems va dynamc suface

10 554 Intenatonal Jounal of Automaton and Computng 95, Octobe 0 technque. IEEE Tansactons on Systems, Man, and Cybenetcs Pat B: Cybenetcs, vol. 4, no. 6, pp , 0. [3] G. Tao, P. V. Kokotovc. Adaptve contol of plants wth unknown dead-zone. IEEE Tansactons on Automatc Contol, vol. 39, no., pp , 994. [4] X. S. Wang, H. Hong, C. Y. Su. Robust adaptve contol of a class of nonlnea systems wth unknown dead-zone. Automatca, vol. 40, no. 3, pp , 004. [5] T. P. Zhang, S. S. Ge. Adaptve neual contol of MIMO nonlnea state tme-vayng delay systems wth unknown dead-zones and gan sgns. Automatca, vol. 43, no. 6, pp , 007. [6] H. R. Kam. Adaptve H synchonzaton of maste-slave systems wth mxed tme-vayng delays and nonlnea petubatons: An LMI appoach. Intenatonal Jounal of Automaton and Computng, vol. 8, no. 4, pp , 0. [7] S. Xu, T. Chen. An LMI appoach to the H flte desgn fo uncetan systems wth dstbuted delays. IEEE Tansactons on Ccuts and Systems II: Expess Befs, vol. 5, no. 4, pp. 95 0, 004. [8] S. Y. Xu, J. Lam, T. W. Chen, Y. Zou. A delay-dependent appoach to obust H flteng fo uncetan dstbuted delay systems. IEEE Tansactons on Sgnal Pocessng, vol. 53, no. 0, pp , 005. [9] S. S. Ge, F. Hong, T. H. Lee. Adaptve neual contol of nonlnea tme-delay systems wth unknown vtual contol coeffcents. IEEE Tansactons on Systems, Man, and Cybenetcs Pat B: Cybenetcs, vol. 34, no., pp , 004. [0] W. Ln, C. J. Qan. Adaptve contol of nonlnealy paametezed systems: A nonsmooth feedback famewok. IEEE Tansactons on Automatc Contol, vol. 47, no. 5, pp , 00. Hong-Yun Yue gaduated fom Qngdao Technologcal Unvesty, Chna n 008. She eceved the M. Sc. degee fom Xdan Unvesty n 0. She s cuently a ph. D. canddate n Depatment of Appled Mathematcs, Xdan Unvesty. He eseach nteests nclude adaptve fuzzy contol, neual netwok, obust contol, and stochastc contol. E-mal: yuehongyun047@63.com Coespondng autho Jun-Mn L gaduated fom Xdan Unvesty, Chna n 987. He eceved the M. Sc. degee fom Xdan Unvesty n 990 and the Ph. D. degee fom the X an Jaotong Unvesty, Chna n 997. He s cuently a pofesso n Depatment of Appled Mathematcs, Xdan Unvesty. Hs eseach nteests nclude adaptve contol, leanng contol, ntellgent contol, hybd system contol theoy, and the netwoked contol systems. E-mal: jml@mal.xdan.edu.cn