Adaptive Backstepping Output Feedback Control for SISO Nonlinear System Using Fuzzy Neural Networks

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1 Intenational Jounal of Automation and Computing 6(), May 009, DOI: 0.007/s Adaptive Backstepping Output Feedback Contol fo SISO Nonlinea System Using Fuzzy Neual Netwoks Shao-Cheng Tong Yong-Ming Li Depatment of Basic Mathematics, Liaoning Univesity of Technology, Jinzhou 00, PRC Abstact: In this pape, a new fuzzy-neual adaptive contol appoach is developed fo a class of single-input and single-output (SISO) nonlinea systems with unmeasued states. Using fuzzy neual netwoks to appoximate the unknown nonlinea functions, a fuzzyneual adaptive obseve is intoduced fo state estimation as well as system identification. Unde the famewok of the backstepping design, fuzzy-neual adaptive output feedback contol is constucted ecusively. It is poven that the poposed fuzzy adaptive contol appoach guaantees the global boundedness popety fo all the signals, diving the tacking eo to a small neighbodhood of the oigin. Simulation example is included to illustate the effectiveness of the poposed appoach. Keywods: Nonlinea systems, backstepping contol, adaptive fuzzy neual netwoks contol, state obseve, output feedback contol. Intoduction Fuzzy contol methodologies have emeged in ecent yeas as pomising ways to solve nonlinea contol poblems. Fuzzy contol, in paticula, has had an impact on the contol community because a fuzzy contolle can use expet s knowledge in contol design. In vey complicated situations whee the plant paametes ae subject to petubations o whee the dynamics of the systems ae too complex to be descibed by a mathematical model, adaptive schemes have to be used online to gathe data and adjust the contol paametes automatically. Based on the univesal appoximation theoem and by incopoating fuzzy logic systems into adaptive contol schemes, a stable fuzzy adaptive contolle was fist developed to contol unknown nonlinea systems [. Aftewads, vaious adaptive fuzzy contol appoaches have been intoduced fo contolling nonlinea systems wee intoduced in [ 6. Refeences [ 6 studied the single-input and single-output (SISO) nonlinea systems; the efeences [7 0 addessed multiple-input and multipleoutput (MIMO) nonlinea systems; and efeences [ 6 discussed nonlinea systems with states unmeasued. Geneally, the adaptive fuzzy contol appoaches have good pefomance. Howeve, these appoaches ae only applicable to a elatively simple class of nonlinea systems. The key equiement is that the unknown nonlineaities appea on the same equation as the contol input in a state space epesentation. Such estictions on the location of the uncetain nonlinea functions ae usually efeed to as matching conditions. If physical systems ae subject to some unknown nonlinea functions that do not satisfy the matching conditions, the adaptive fuzzy contol appoaches mentioned above cannot be implemented. In ecent yeas, with the development of adaptive and obust backstepping designs in nonlinea systems [7, many Manuscipt eceived Mach, 008; evised Octobe 0, 008 This wok was suppoted by National Natual Science Foundation of China (No ) and National Key Basic Reseach and Development Pogam of China (No. 00CB300), Outstanding Youth Funds of Liaoning Povince (No ), and Educational Depatment of Liaoning Povince (No. 006R9 and No. 007T80). *Coesponding autho. addess: jztsc@sohu.com fuzzy adaptive contol schemes have been epoted that combined the backstepping technique with adaptive fuzzy logic systems fo unknown nonlinea systems [8 5. Fuzzy adaptive backstepping contol schemes can povide a systematic famewok fo the design of tacking o egulation stategies, in which the fuzzy logic systems ae used to appoximate the unknown nonlinea functions, and an adaptive fuzzy contolle is constucted ecusively. These appoaches ae suitable fo some uncetain nonlinea systems that do not satisfy the equiement of matching conditions, no equie the lineaity in the paamete assumption of nonlinea systems. Fo this eason, fuzzy adaptive backstepping contol techniques have attacted moe and moe attention in the fuzzy contol fields. Howeve, the existing fuzzy adaptive backstepping contol methods ae all based on the assumption that the states of the systems ae diectly measued. Recently, the adaptive obseve backstepping contol appoach of using neual netwoks was poposed fo a class of nonlinea systems not satisfying the matching conditions [6, in which the sepaation pinciple is utilized to design a state obseve and output feedback contolle; that is, by assuming that the obseve eos ae bounded, the stability of contol systems is given. As pointed out by efeences [4, 7, the sepaation pinciple does not hold fo nonlinea systems; theefoe, the adaptive neual contolle cannot ensue the stability of the whole closed-loop systems. In addition, adaptive fuzzy output feedback contolles based on the state obseves have been developed fo the SISO uncetain systems, and the stability of the closedloop systems ae given [ 6, but these contol appoaches equie that the nonlinea systems must satisfy the matching conditions, and they ae not capable of dealing with the nonlinea systems with the matched conditions. In this pape, a new fuzzy neual adaptive contol appoach is developed fo a class of SISO nonlinea systems with the matched conditions and unmeasued states. Using fuzzy neual netwoks to appoximate the unknown nonlinea functions, a fuzzy neual adaptive obseve is intoduced fo state estimation as well as system identification. Combining backstepping design with fuzzy logic systems,

2 46 Intenational Jounal of Automation and Computing 6(), May 009 a fuzzy neual adaptive output feedback contol scheme is constucted ecusively, and the adjusting paamete vectos ae deived fom the Lyapunov functions. It is poven that the poposed fuzzy neual adaptive output contol appoach guaantees the stability of the whole closed system; that is, it has the global boundedness popety fo all the signals, and the tacking eo conveges to a small neighbodhood of the oigin. Peliminaies. Nonlinea contol poblem Conside the SISO nonlinea system diffeomophic to the output feedback fom [6 ẋ = x + f 0(y) + f (y) ẋ = x 3 + f 0(y) + f (y). () ẋ n = x n + f (n )0 (y) + f n (y) ẋ n = u + f n0(y) + f n(y) y = x whee the input u and the output y in R, (x, x,, x n) R n of the state, and only x is measued. f i0(y) is a known function, and f i(y) is an unknown smooth function, i =,,, n. Rewite () in the following fom ẋ = Ax + Ky + B i[f i0(y) + f i(y) + Bu () i= whee k A =. I k n [ B =, Bi =., K = k. k n 0 0 and K is chosen such that A is a stict Huwitz matix. Thus, given Q > 0, thee exists P > 0 satisfying T A T P + P A = Q. (3) Let y be a efeence signal containing finite deivatives up to the n-th ode. Define the output tacking eo as χ = y y. Contol objectives. Use fuzzy neual netwoks, efeence signal y, the output of the system y and the state estimates ˆx i to detemine a fuzzy neual contolle and paametes adaptive laws such that all the signals involved in the closed-loop systems ae bounded and the tacking eo χ is as small as possible.. Desciption of fuzzy neual netwoks The basic configuation of fuzzy neual netwoks consists of some fuzzy IF-THEN ules and a fuzzy infeence engine. The fuzzy infeence engine uses the fuzzy IF-THEN ules to pefom a mapping fom an input linguistic vecto x T = [x x x n R n to an output linguistic vaiable y R The i-th fuzzy IF-THEN ule is witten as R i : if x is F i and x is F i x n is F i n, then y is B i, and i =,,, N whee F, i F, i, Fn, i and B i ae fuzzy sets, and N is the numbe of the fuzzy IF-THEN ules. By using poduct infeence, cente-aveage and singleton fuzzifie, the output of the fuzzy logic system can be expessed as y(x) = N ȳ l [ n Fi l (x i) l= N l= i= (4) [ n Fi l(xi) i= whee the membeship function value of the fuzzy vaiable x i is Fi l (x i) = a l i exp[ ( xi xl i ). (5) σi l Denote ϕ(x) = [ϕ (x),, ϕ N (x) T, in which ϕ l (x) is defined as ϕ l (x) = N l= n Fi l (x i) i= (6) [ n Fi l(xi) i= whee µ F i(ȳ l ) = and θ = (ȳ,, ȳ N ) T is an adjustable j paamete vecto. When the inputs ae cast into the fuzzy neual netwoks shown in Fig., the tuth value of ϕ i (laye III) of the antecedent pat of the i-th implication is calculated by (6). Among the commonly used defuzzification stategies, the output (laye IV) of fuzzy neual netwoks is expessed as (4). The fuzzy logic appoximato based on neual netwoks can be established. Fig. shows the configuation of an fuzzy neual function appoximato. The appoximato has fou layes. At laye I, nodes that ae input ones stand fo input linguistic vecto x T = [x x x n R n. At Laye II, it pefoms a membeship function value. At laye III, notes ae the values of the fuzzy basis vecto ϕ(x). Each node of laye III pefoms a fuzzy ule. Laye III and laye IV ae fully connected by the weighting factos θ = (y,, y N ) T, i.e., the adjusted paametes. At laye IV, the outputs stand fo the values of output y(x). Accoding to efeences [,, a fuzzy logic system (4) can be ewitten as y(x) = θ T ξ(x). (7) Lemma. [ Let f(x) be a continuous function defined on a compact set Ω. Then, fo any constant ε > 0, thee exists a fuzzy logic system () such that sup f(x) θ T ϕ(x) ε. (8) x Ω

3 S. C. Tong and Y. M. Li / Adaptive Backstepping Output Feedback Contol fo SISO Nonlinea System 47 Fig. Fuzzy neual netwoks By Lemma, fuzzy logic systems ae univesal appoximatos, i.e., they can appoximate any smooth function on a compact space. With this appoximation capability, we can assume that the nonlinea tems in () can be appoximated as f i(y) = ˆf i(y θ i ) + ( ˆf i(y θ i ) ˆf i(y θ i )) + (f i(y) ˆf i(y θ i )) = θ T i ϕ i(y) + θ T i ϕ i(y) + ε i(y), i n (9) whee θ i = θ i θ i and the optimal paamete vecto of θ i is defined as θ i = ag min [sup θ i Ωi y U ˆf i(y θ i ) f i(y) whee Ω i and U ae compact fo θ i and y, espectively. Define fuzzy-neual minimum appoximation and fuzzyneual appoximation eos as ε i (y) = f i(y) ˆf i(y θ i ) δ i(y, θ i) = f i(y) ˆf i(y θ i ), i =,,, n. Assumption. Appoximation eo ε i and fuzzy neual appoximation eo δ i(y, θ i) ae bounded by ε i(y) ε i0 and δ i(y, θ i) δ i0, i =,,, n fo some constants ε i0 and δ i0. 3 Fuzzy neual adaptive contol design In this section, it is assumed that the states of system () ae not available fo feedback. In this situation, a state obseve should be established to estimate the states, and to this end, the fuzzy neual adaptive output feedback contol scheme is investigated. To estimate the states of the system, the following fuzzy obseve is designed fo () ˆx = Aˆx + Ky + B i(f i0(y) + ˆf i(y θ i )) + Bu i= ŷ = C ˆx (0) whee C = [ 0 0 T. Let e = x ˆx be obseve eos. eos equation is ė = Ae + Then, the obseve B i(f i(y) ˆf i(y θ i )) = Ae + δ () i= [ whee δ T = δ (y, θ ) δ n(y, θ n) Fo convenience, in the following, ε i (y) and δ i(y, θ i) ae denoted by ε i and δ i, and w i = ε i δ i. Fom Assumption, it is known that w i ε i0 + δ i0 = w i0. The systematic design pocedue involves, at each step, augmenting an integato and designing a vitual contol law to stabilize the augmented system, until the actual contol appeas at the n-th step. The detailed design pocedues of the fuzzy adaptive output feedback contolle include the following steps: Step. Define the tacking eo fo the system as χ = x y. Expessing x in tems of its estimate x = ˆx + e, we obtain χ = ẋ ẏ = x + f 0(y) + f (y) ẏ =. ˆx + f 0(y) + f (y) ẏ + e = ˆx + f 0(y) + θ T ϕ (y) + θ T ϕ (y) + e + ε ẏ. () Take ˆx as a vitual contol, and define Then, we have χ = ˆx α (ˆx, θ, y, y ) ẏ. χ = χ + α + f 0(y) + θ T ϕ (y) + θ T ϕ (y) + e + ε. Conside the following Lyapunov function (3) V = et P e + χ + γ θt θ (4) whee P = P T > 0, and γ > 0 is a design constant. Fom () and (3), the time deivative of V is V = ėt P e + et P ė + χ χ + γ θt θ = et [P A T + AP e + e T P δ + χ χ + γ θt θ = e T Qe + e T P δ + χ [χ + α + f 0(y) + θ T ϕ (y) + ε + e χ + γ θt (γ ϕ (y)χ θ ). (5) By using the inequality ab a + b, we have e T P δ + e χ e + P δ + e + χ Substituting (6) into (5) yields e + χ + P δ. (6) V (λ min(q) ) e + χ [χ + χ + α +

4 48 Intenational Jounal of Automation and Computing 6(), May 009 f 0(y) + θ T ϕ (y) + ε + P δ + γ θt (γ ϕ (y)χ θ ). (7) Since vaiables χ and x ae available, the intemediate contol function α (ˆx, θ, y, y ) and the adaptation function τ ae chosen as α = c χ χ f0(y) θt ϕ (y) ε 0 tanh(ε 0 χ κ ) (8) τ = γ ϕ (y)χ σ(θ θ 0) (9) whee c > 0, κ > 0, and θ 0 ae design paametes. In view of (8), we use the following nice popety with egad to function tanh( ): χ χ ε χ ε 0 tanh(ε 0 ) 0.785κ = κ κ whee κ > 0 is a given small constant. Substituting (8) and (9) into (7) esults in V (λ min(q) ) e c χ + χ χ + θt (τ γ θ ) + σ θt (θ θ 0) + γ P δ + κ. (0) Step. Diffeentiating χ yields χ = ˆx α (ˆx, θ, y, y ) ÿ = ˆx α ˆx ˆx α θ θ α y ẏ α y ẏ ÿ = ẋ ė α (ẋ ė ) α θ α ˆx θ y ẋ α ẏ ÿ. () y Using (), (0), and (), we have ẋ ė = ˆx + k e + f 0(y) + θ T ϕ (y) + θ T ϕ (y) + w () ẋ ė = ˆx 3 + k e + f 0(y) + θ T ϕ (y) + θ T ϕ (y) + w (3) ẋ = x + f 0(y) + f (y) = ˆx + e + f 0(y) + θ T ϕ (y) + δ. (4) Substituting () (4) into () gives χ = ˆx 3 + k e + f 0(y) + θ T ϕ (y) + θ T ϕ (y) + w α ˆx [ˆx + f 0(y) + θ T ϕ (y) + k e + θ T ϕ (y) + w α y [ˆx + f0(y) + θt ϕ (y) + e + δ α θ α ẏ ÿ = θ y whee ˆx 3 + H (ˆx, θ, y, y, ẏ ) ÿ α θ θ + θ T ϕ (y) + w α [ θ T ϕ (y) + w α (e + δ) (5) ˆx y H = k e + f 0(y) + θ T ϕ (y) α [ˆx + f 0(y) + θ T ϕ (y) + k e ˆx α ẏ α y y [ˆx + f0(y) + θt ϕ (y). Conside the following Lyapunov function V = V + χ + γ θt θ (6) whee γ > 0 is a constant. By (5), the time deivative of V is V = V + χ χ + γ θt θ (λ min(q) ) e c χ + χ χ + γ θt (τ θ ) + χ [ˆx 3 + H ÿ α θ + w α ( θ T ϕ (y) + w ) θ ˆx α y (e + δ) + θt (γ χ ϕ (y) γ θ ) + σ θt (θ θ 0) + γ P δ + κ. (7) Again using the inequality ab a + b, we have α χ (e + δ) α y χ y e + α χ y δ e + χ ( α y ) + δ Substituting (8) into (6) yields e + χ ( α y ) + δ. (8) V (λ min(q) ) e c χ + χ χ + χ [ˆx 3 + H ÿ + ( α y ) χ α θ + w θ α w + α θt (τ γ χ ϕ (y) ˆx γ ˆx θ ) + σ θt (θ θ 0) + θt (γ χ ϕ (y) γ γ θ ) + δ + P δ + κ. (9) Take ˆx 3 as a vitual contol, and intoduce the vaiable χ 3 = ˆx 3 α (ˆx, ˆx, θ, θ, y, y, ẏ ) ÿ. It is noted that vaiables χ, ˆx, and ˆx ae available; theefoe, intemediate contol function α, and the adaptation functions τ and τ ae chosen as α = χ c χ H ( α y ) χ + α θ τ

5 S. C. Tong and Y. M. Li / Adaptive Backstepping Output Feedback Contol fo SISO Nonlinea System 49 χ w 0 tanh(w 0 κ ) α α w 0 χ ˆx tanh( w 0 ˆx κ ) (30) τ = τ γ χ α ˆx ϕ (y) (3) τ = γ χ ϕ (y) σ(θ θ 0). (3) In (30) and (3), c > 0, and θ 0 is a design paamete vecto. It is noted that χ χ w χ w 0 tanh(w 0 κ ) α χ w ˆx α α χ w 0 χ ˆx tanh( w 0 ˆx κ )) κ. (33) Fom (30) (33), we have V (λ min(q) ) e c k χ k + χ χ 3 + θt (τ γ θ ) + ( α θt + χ )(τ γ θ θ ) + σ θt k (θ k θ k0 ) + γ k (δ + P δ ) + 3κ. (34) Step i. (3 i n ) A simila pocedue is employed ecusively at each step. By defining we have Since = ˆx i α i (ˆx, ˆx,, ˆx i, θ, θ,, θ i, y, y,, y (i ) ) y (i ) = ˆx i α i = ẋ i ė i j= y (k ) y (k) y (i) ˆxj θk αi ẋ. (35) y ẋ i ė i = ˆx i+ + k ie + f i0(y) + θ T i ϕi(y) + θ T i ϕ i(y) + w i, i n. (36) Substituting (4) and (36) into (35) esults in = ˆx i+ + H i y (i) θk + θ θ i T ϕ i(y) + w i) k whee j= [ θ j T ϕ j(y) + w j αi y (e + δ) (37) H i = k ie + f i0(y) + θi T ϕ i(y) j= [ˆx j+ + θj T ϕ j(y) + f j0(y) + k je y [ˆx + f0(y) + θt ϕ (y). Conside the following Lyapunov function: y (k ) y (k) V i = V i + + γ i θt i θi. (38) The time deivative of V i is V i = V i + + γ i θt i j= θi V i + [ˆx i+ + H i y (i) + w i ( θ j T ϕ j(y) + w j) θk y (e + δ) + γ i θt i (γ iϕ i(y) θ i). (39) Via mathematical induction, we can obtain V i (λ min(q) (i )) e c k χ k + + ( i α l θt k + χ l+ )(τ k(i ) γ k θ θ k )+ k σ γ k θt k (θ k θ k0 ) + P δ + Substituting (40) into (39) gives (i ) δ + (i )i κ. (40) V i (λ min(q) i + ) e c k χ k + + ( i α l θt k + χ l+ )(τ k(i ) γ k θ θ k ) + k [ˆx i+ j= + H i y (i) θk + w i [ θ j T ϕ j(y) + w j [e + δ + y θt i (γ iϕ i(y) γ θ σ i) + θt i (θ k θ k0 ) + i γ k (i ) δ + P δ + (i )i κ. (4) In the same way, we have the following inequalities y [e + δ e + ( αi y ) χ i + δ (4) w i w i0 tanh(w i0 κ ) χi j= w j

6 50 Intenational Jounal of Automation and Computing 6(), May 009 αi αi wj0 tanh( j= Intoduce the vaiable w j0 κ ) iκ. (43) + = ˆx i+ α i(ˆx, ˆx,, ˆx i, θ, θ,, θ i, y, y,, y (i ) ) y (i). Since vaiables and ˆx i ae available, intemediate contol function α i and adaptation functions τ ki, k =,,, i, ae chosen as α i = c i H i + τ ki α l ( χ l+ )γ k ϕ k (y) χ k w i0 tanh(w i0 κ ) αi j= wj0 tanh( αi w j0 κ ) (44) τ ki = τ k(i ) γ i χ k ϕ k (y), k i (45) τ ii = γ iϕ i(y) σ(θ i θ i0) (46) whee γ i > 0 and θ i0 ae design constants. By (4) (47), (4) becomes V i (λ min(q) i ) e i c k χ k i ( α l θt k + χ l+ )(τ ki γ k θ θ k ) + k i σ γ k θt k (θ k θ k0 ) + (i ) δ + P δ + (i + )i κ. (47) Step n. The actual contol input u appeas. We conside the oveall Lyapunov function as V n = V n + χ n + γ n θt n θn. (48) Set i = n, the contol u and adaptation functions τ kn descibed by u = χ n c nχ n H n n α n τ ki ( αi ) χ n + ˆx k n n α l α n ( χ l+ )γ k ϕ k (y) ˆx k w n0 tanh(w n0 χ n κ ) n Then, we have αn αi wj0 tanh( ˆx k w j0 χ n κ ) (49) θ k = τ kn, k n. (50) V n (λ min(q) n ) e c k χ k + σ θt k (θ k θ k0 ) + n δ + γ k P δ + By Assumption, (5) can be witten as Let V n (λ min(q) n) e σ θt k θk + σ γ k δk0 + P λ = σ n(n + ) κ. (5) λ min(q) n > 0 c k χ k θk θ k0 + n δ0 + n(n + ) κ. (5) c = min{ λmin(q) n, λ min(p ) c i, σ; i =,, n} Then, (5) becomes P θk θ k0 + n δ0 + δk0 + n(n + ) κ. V n cv n + λ. (53) Fom (53), it can be seen that fo each i =,, n, the signals x i(t), ˆx i(t), e(t), θ i, and u(t) ae globally unifomly ultimately bounded, and thee exists a time T such that fo all t T, the tacking eo satisfies y(t) y (t) µ, whee µ > (λ/c). In ode to make the tacking eo convege to a small neighbohood aound zeo, the paametes c i, σ, and Q should be chosen appopiately to guaantee that it is possible to make µ as small as desied. The above design and analysis pocedue is summaized in the following theoem. Theoem. Suppose that Assumption holds. Then, the fuzzy adaptive output tacking design descibed by state obseve (), contol law (4), and paamete adaptive laws (3), (3), (39), and (43) guaantees that all the signals ae unifomly ultimately bounded and the output tacking eo conveges to a small neighbohood of the oigin.

7 S. C. Tong and Y. M. Li / Adaptive Backstepping Output Feedback Contol fo SISO Nonlinea System 5 4 Simulation example In this section, the feasibility of the poposed method and the contol pefomances ae illustated by an example. Conside a system govened by the following fom: ẋ (t) = x (t) + (x x3 ) 0.0 sin ( + x 4 x ) ẋ (t) = u(t) e x sin(5x ) cos x (54) y(t) = x (t) y ; Fig. 3 shows the tajectoies of state x and its estimate ˆx ; Fig. 4 shows the tajectoies of statex and its estimate ˆx ; Fig. 5 shows the tajectoy of input u; Figs. 6 and 7 show the tajectoies of paametes of θ and θ in fuzzy logic systems. Fom Figs. 7, we can conclude that the poposed fuzzy contol appoach can guaantee the boundedness of the signals x, ˆx, x, ˆx, u, θ, and θ. Especially, the tacking eo χ = x y conveges to a small neighbodhood of the oigin. whee f (x ) = (x x 3 )/( + x 4 ) and f (x ) = e x sin(5x ) ae unknown functions. f 0(x ) = 0.0 sin x, and f 0(x ) = 0.0 cos x ae known functions. The given tacking efeence signal is y = sin(t). Choose fuzzy membeship functions as µ F l (x ) = exp[ (x 3 + l), l =,, 5 6 µ F l (x ) = exp[ Define fuzzy basis functions as ϕ j(x ) = (x 3 + l), l =,, 5. 6 exp[ (x 3+j) 6 5 n= exp[ (x 3+n) 6, j =,, 5 Fig. The tajectoies of x and y ϕ j(x ) = exp[ (x 3+j) 6 5 n= exp[ (x 3+n) 6, j =,, 5. The fuzzy logic systems can be expessed in the fom θ T ϕ (x ) = 5 θjϕ T j(x ) j= 5 θ T ϕ (x ) = θjϕ T j(x ) j= whee θ T = [θ, θ, θ 3, θ 4, θ 5 Fig. 3 The tajectoies of x and ˆx θ T = [θ, θ, θ 3, θ 4, θ 5 ϕ (x ) = [ϕ (x ), ϕ (x ), ϕ 3(x ), ϕ 4(x ), ϕ 5(x ) T ϕ (x ) = [ϕ (x ), ϕ (x ), ϕ 3(x ), ϕ 4(x ), ϕ 5(x ) T. Design paametes in contolle and in adaptive laws ae chosen as k =, k =, γ = γ = 0., σ = 0., ε 0 = ε 0 = 0.0, c = 5, and c = 3. The initial conditions ae chosen as x (0) = 0, x (0) = 0, ˆx (0) = 0., x (0) = 0, θ (0) = [0.9, 0.7, 0.5, 0.3, 0., and θ (0) = [0.4, 0.3, 0., 0., 0.. Then, we obtain the simulation esults, which ae shown by Figs. 7. Fig. shows the tajectoies of state x and Fig. 4 The tajectoies of x and ˆx

8 5 Intenational Jounal of Automation and Computing 6(), May 009 SISO stict-feedback nonlinea systems. The main contibution of this pape is that by designing a fuzzy neual adaptive state obseve, the application of adaptive fuzzy backstepping contol is extended to a new class of nonlinea systems with unmeasued states. In addition, the stability of the closed-loop has been poved by using Lyapunov method, i.e., the poposed adaptive fuzzy contol scheme can guaantee that all the signals of closed-loop ae bounded and the tacking eo of the system conveges to a small neighbohood of the oigin. Refeences [ L. X. Wang. Adaptive Fuzzy Systems and Contol: Design and Stability Analysis, Englewood Cliffs, Pentice- Hall, New Jesey, USA, 994. Fig. 5 The tajectoy of u [ C. Y. Su, Y. Stepanenko. Adaptive Contol of a Class of Nonlinea Systems with Fuzzy Logic. IEEE Tansactions on Fuzzy Systems, vol., no. 4, pp , 994. [3 B. S. Chen, C. H. Lee, Y. C. Chang. Tacking Design of Uncetain Nonlinea SISO Systems: Adaptive Fuzzy Appoach. IEEE Tansactions on Fuzzy Systems, vol. 4, no., pp. 3 43, 996. [4 J. T. Spoone, K. M. Passino. Stable Adaptive Contol Using Fuzzy Systems and Neual Netwoks. IEEE Tansactions on Fuzzy Systems, vol. 4, no. 3, pp , 996. [5 S. C. Tong, Q. Li, T. Y. Chai. Fuzzy Adaptive Contol fo a Class of Nonlinea Systems. Fuzzy Sets and Systems, vol. 0, no., pp. 3 39, 999. [6 T. Y. Chai, S. C. Tong. Fuzzy Diect Adaptive Contol fo a Class of Nonlinea Systems. Fuzzy Sets and Systems, vol. 03, no. 3, pp , 999. Fig. 6 The tajectoies of adaptive paametes θ, θ, θ 3, θ 4, and θ 5. [7 R. Odones, K. M. Passino. Stable Multi-input Multioutput Adaptive Fuzzy/Neual Contol. IEEE Tansactions on Fuzzy Systems, vol. 7, no. 3, pp , 999. [8 Y. C. Yang. Robust Tacking Contol of Nonlinea MIMO Systems via Fuzzy Appoaches. Automatica, vol. 36, no. 0, pp , 000. [9 H. X. Li, S. C. Tong. A Hybid Adaptive Fuzzy Contol fo a Class of Nonlinea MIMO Systems. IEEE Tansactions on Fuzzy Systems, vol., no., pp. 4 34, 003. [0 C. S. Chiu. Mixed Feedfowad/Feedback Based Adaptive Fuzzy Contol fo a Class of MIMO Nonlinea Systems. IEEE Tansactions on Fuzzy Systems, vol. 4, no. 6, pp , 006. [ Y. G. Leu, W. Y. Wang. Obseve-based Adaptive Fuzzyneual Contol fo Unknown Nonlinea Dynamical Systems. IEEE Tansactions on Systems, Man, and Cybenetics Pat B, vol. 9, no. 5, pp , 999. Fig. 7 The tajectoies of adaptive paametes θ, θ, θ 3, θ 4, and θ 5 5 Conclusions In this pape, a fuzzy neual adaptive obust contol appoach based on backstepping design has been poposed fo [ N. Golea, A. Golea, K. Benmahammed. Stable Indiect Fuzzy Adaptive contol. Fuzzy Sets and Systems, vol. 37, no. 3, pp , 003. [3 W. Y. Wang, Y. G. Leu, T. T. Lee. Output-feedback Contol of Nonlinea Systems Using Diect Adaptive Fuzzyneual Contolle. Fuzzy Sets and Systems, vol. 40, no., pp , 003.

9 S. C. Tong and Y. M. Li / Adaptive Backstepping Output Feedback Contol fo SISO Nonlinea System 53 [4 S. C. Tong, H. X. Li, W. Wang. Comments on Diect Adaptive Fuzzy-neual Contol with State Obseve and Supevisoy Contolle fo Unknown Nonlinea Dynamical Systems. IEEE Tansactions on Fuzzy Systems, vol., no. 5, pp , 003. [5 S. C. Tong, H. X. Li, W. Wang. Obseve-based Adaptive Fuzzy Contol fo SISO Nonlinea Systems. Fuzzy Sets and Systems, vol. 48, no. 3, pp , 004. [6 N. Goléa, A. Goléa, K. Baa. Obseve-based Adaptive Contol of Robot Manipulatos: Fuzzy Systems Appoach. Applied Soft Computing, vol. 8, no., pp , 008. [7 M. Kistic, I. Kanellakopoulos, P. V. Kokotovic. Nonlinea and Adaptive Contol Design, Wiley, New Yok, USA, 995. [8 Y. S. Yang, G. Feng, J. S. Ren. A Combined Backstepping and Small-gain Appoach to Robust Adaptive Fuzzy Contol fo Stict-feedback Nonlinea Systems. IEEE Tansactions on Systems, Man, and Cybenetics Pat A, vol. 34, no. 3, pp , 004. [9 Y. S. Yang, C. J. Zhou. Adaptive Fuzzy H Stabilization fo Stict-feedback Canonical Nonlinea Systems via Backstepping and Small-gain Appoach. IEEE Tansactions on Fuzzy Systems, vol. 3, no., pp. 04 4, 005. [0 S. S. Zhou, G. Feng, C. B. Feng. Robust Contol fo a Class of Uncetain Nonlinea Systems: Adaptive Fuzzy Appoach Based on Backstepping. Fuzzy Sets and Systems, vol. 5, no., pp. 0, 005. [ M. Wang, B. Chen, S. L. Dai. Diect Adaptive Fuzzy Tacking Contol fo a Class of Petubed Stict-feedback Nonlinea Systems. Fuzzy Sets and Systems, vol. 58, no. 4, pp , 007. [ S. C. Tong, Y. M. Li. Diect Adaptive Fuzzy Backstepping Contol fo a Class Nonlinea Systems. Intenational Jounal of Innovative Computing, Infomation and Contol, vol. 3, no. 4, pp , 007. [3 S. Jagannnathan, F. L. Lewis. Robust Backstepping Contol of a Class of Nonlinea Systems Using Fuzzy Logic. Infomation Sciences, vol. 3, no. 3 4, pp. 3 40, 000. [4 B. Chen, X. P. Liu. Fuzzy Appoximate Distubance Decoupling of MIMO Nonlinea Systems by Backstepping and Application to Chemical Pocesses. IEEE Tansactions on Fuzzy Systems, vol. 3, no. 6, pp , 005. [5 B. Chen, X. P Liu. Fuzzy Appoximate Distubance Decoupling of MIMO Nonlinea Systems by Backstepping Appoach. Fuzzy Sets and Systems, vol. 58, no. 0, pp , 007. [6 J. Y. Choi, J. A. Faell. Adaptive Obseve Backstepping Contol Using Neual Netwoks. IEEE Tansactions on Neual Netwoks, vol., no. 5, pp. 03, 00. Shao-Cheng Tong gaduated fom Jinzhou Nomal College, PRC, in 98. He eceived the M. Sc. degee fom Dalian Maine Univesity in 988 and the Ph. D. degee fom Notheasten Univesity, PRC, in 997. He is cuently a pofesso at the Depatment of Basic Mathematics, Liaoning Univesity of Technology, China. His eseach inteests include fuzzy contol theoy and nonlinea adaptive contol. Yong-Ming Li gaduated fom Liaoning Univesity of Technology, PRC, in 004. He eceived the M. Sc. degee fom Liaoning Univesity of Technology, PRC, in 007. He is cuently an assistant in the Depatment of Basic Mathematics, Liaoning Univesity of Technology. His eseach inteests include fuzzy contol theoy, nonlinea adaptive contol, and intelligent contol.