ADAPTIVE FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS WITH UNKNOWN DISTRIBUTED TIME-VARYING DELAYS AND UNKNOWN CONTROL DIRECTIONS

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1 Iranian Journal of Fuzzy Systems Vol. 11, No. 1, 14 pp ADAPTIVE FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS WITH UNKNOWN DISTRIBUTED TIME-VARYING DELAYS AND UNKNOWN CONTROL DIRECTIONS H. Y. YUE AND J. M. LI Abstract. In this paper, an adaptive fuzzy control scheme is proposed for a class of perturbed strict-feedback nonlinear systems with unknown discrete and distributed time-varying delays, and the proposed design method does not require a priori knowledge of the signs of the control gains. Based on the backstepping technique, the adaptive fuzzy controller is constructed. The main contributions of the paper are that i by constructing appropriate Lyapunov functionals and using the Nussbaum functions, the adaptive tracking control problem is solved for the strict-feedback unknown nonlinear systems with the unknown discrete and distributed time-varying delays and the unknown control directions ii the number of adaptive parameters is independent of the number of rules of fuzzy logic systems and system state variables, which reduces the computation burden greatly. It is proven that the proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output converges to a small neighborhood of the desired reference signal. Finally, an example is used to show the effectiveness of the proposed approach. 1. Introduction Over the past few decades, considerable attention has been paid to fuzzy control and it has emerged as a promising way to handle nonlinear control problems and some fuzzy control techniques have been developed, for examples, [11, 1] and the references therein. Moreover, approximation-based adaptive fuzzy control has emerged as a popular and convenient tool in analysis and synthesis of complex and ill-defined systems, to which the application of conventional control techniques is not straightforward or feasible. The main idea of such a control methodology is to employ fuzzy logic systems to approximate the unknown nonlinearities in dynamic systems and the adaptive backstepping technique to construct the controllers. Following such an idea, some adaptive fuzzy control schemes were proposed for nonlinear systems [5, 8,, 3, 5, 6, 7, 35]. Moreover, some adaptive neural control methods have been developed for nonlinear systems, such as [6, 18, ]. Received: February 1; Revised: May 1; Accepted: June 13 Key words and phrases: Fuzzy adaptive control, Nonlinear systems, Discrete and distributed time-varying delays, Backstepping, Lyapunov-Krasovskii functionals. This work is partially supported by NNSFC, and partially supported by the Fundamental Research Funds for the Central Universities.

2 H. Y. Yue and J. M. Li It is well known that time delays are frequently encountered in real engineering systems, and the existence of time delays usually becomes the source of instability and degrading performance of systems. Therefore, the stability analysis and controller synthesis of nonlinear systems with time-delay are important. Recently, several fuzzy adaptive control schemes have been reported that combined the Lyapunov- Krasovskii functional and the FLS with the backstepping technique for nonlinear systems with constant or time-varying delays [, 4, 3], in which Lyapunov- Krasovskii functionals are employed to compensate the time-delay terms, the FLS are used to approximate the unknown functions, and an adaptive fuzzy controller is constructed recursively, and in [7, 13], the systems with time delays were investigated by using the neural network control approach. However, it is worth noting that the results in [, 4, 7, 13, 3] were obtained in the context of continuous systems with constant or time-varying delays. When the number of summands in a system equation is increased and the differences between neighboring argument values are decreased, systems with distributed delays will arise. Therefore, distributed delay systems have been an attractive research topic in the past years. For example, via different approaches, the authors in [5, 7] have investigated the robust H state feedback and output feedback controller design problem for uncertain distributed delay systems, respectively. The H filtering for uncertain distributed delay systems, either delay-independent or delay-dependent, have been considered in [33, 34, 36]. Nevertheless, the problem of adaptive fuzzy tracking control by using Lyapunov- Krasovskii functionals and the backstepping technique for uncertain systems with distributed time-varying delays is still open and remains unsolved, which motivates the present study. On the other hand, a common weakness of the early fuzzy control methods is that the number of adaptation parameters depends on the number of the fuzzy rule bases. With an increase of fuzzy rules, the number of parameters to be estimated will increase significantly. As a result, the online learning time becomes prohibitively large. To avoid this disadvantage, authors in [4, 1, 9, 3] considered the norm of the ideal weighting vector in fuzzy logic systems as the estimation parameter instead of the elements of weighting vector. Therefore, the number of the online adaptive parameters is not more than the order of the original system. Thus, it is worth studying to devise the controller containing less adaptive parameters for nonlinear system with unknown discrete and distributed time-varying delays. Another challenging problem for practical systems is that the control direction may be unknown. When the signs of control directions are unknown, the control problem becomes much more difficult, because in this case, we cannot decide the direction along which the control operates. In this case, Nussbaum-gain technique is an effective way to deal with it, for details see [1, 3, 9, 14, 15, 19, 37, 38], and the references therein. However, if the system is with time-varying delays, especially distributed time-varying delays, how to design the controller is a challenging problem. To the authors knowledge, up to now, there is no result reported on this problem. Motivated by the above observations, in this paper, the problem of output tracking is investigated for unknown strict-feedback nonlinear systems with unknown

3 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed... 3 control directions via fuzzy control method, in which the discrete and distributed time-varying delays appear in states. Under approximate Assumptions, the appropriate Lyapunov- Krasovskii functionals are constructed to compensate the unknown time-varying delays terms. Then, the FLS are employed to approximate the unknown nonlinear functions and Nussbaum-type functions are used to detect the high-frequency gain signs. Finally, the adaptive backstepping approach is utilized to construct the fuzzy controller. The main contributions of the paper are listed as follows. i By combining the apppropriate Lyapunov- Krasovskii functionals and the Nussbaum-type functions, the unknown control direction problem is solved for the strict-feedback nonlinear system with unknown discrete and distributed timevarying delays. ii The number of adaptive parameters is independent of the number of rules of fuzzy logic systems and system state variables, which reduces the computation burden greatly. The proposed controller guarantees the boundedness of all the signals in the closed loop system and gets a good tracking performance. Simulation results are provided to show the effectiveness of the proposed approach.. Problem Formulation and Preliminaries.1. Problem Formulation. Consider the following uncertain system with timevarying distributed delays ẋ i = g i x i x i+1 + f i x i + h i x i t, x i t d i1 t + t d m it i x i s ds + ω i t, x t, ẋ n = g n x u + f n x + h n x t, x t d n1 t + t d n t m n x s ds + ω n t, x t, y t = x 1 t, 1 where x = [x 1,..., x n ] T R n, u R and y R denote state variable, the control input and the output of the system 1, respectively. For 1 i n 1, x i t d i1 t = [x 1 t d i1 t,..., x i t d i1 t] T and x i = [x 1,..., x i ] T. Functions f i, g i, m i and h i are unknown smooth functions. d i1 t and d i t represent unknown discrete and unknown distributed delays of system, respectively. ω i t, xt stand for unknown external disturbance inputs. Control objectives: Utilizing the fuzzy logic systems and the backstepping technique to determine a fuzzy controller and parameter adaptive laws for system 1 such that the system output y t tracks a desired reference signal y d t, while all [ the signals ] remain bounded. To the end, define a vector function as ȳdi T = y d,..., y i d, i = 1,..., n, where y i d is the ith time derivative of y d. Now, the following Assumptions are introduced. Assumption.1. d i1 t d 1 and d i t d. Furthermore, their derivations satisfy d i1 t d 1 < 1 and d i t d < 1, respectively.

4 4 H. Y. Yue and J. M. Li Remark.. In this paper, it should be emphasized that the constants d 1, d, d 1 and d are introduced only for the stability analysis and they are not used in the controller design, thus they can be unknown in the controller design procedure. Assumption.3. For i = 1,..., n, the following inequality holds h i x i, x i t d i1 t h i1 x i + h i x i t d i1 t, where h i1 x i and h i x i t d i1 t are unknown smooth functions. Assumption.4. For 1 i n, there exist unknown positive smooth functions ρ i x i such that ω i t, x ρ i x i. Assumption.5. The desired trajectory vectors ȳ di are known and continuous, and ȳ di Ω di R i+1 with Ω di being known compact sets. Assumption.6. For 1 i n, the sign of g i x i is unknown, and there exist b and c such that < b g i x i c <. Remark.7. The existence of nonlinear distributed time-varying delays terms t t d i t m i x i s ds makes controller design more difficult than the case without distributed time-varying delays terms [14, 15, 1, 9, 3]. During the controller design process, the discrete and distributed time-varying delays are required to satisfy Assumption.1. That is to say they are bounded and their derivations are less than constant one. Practically speaking, Assumption.1 is common and relaxed, for example [1,, 7, 36] and the references therein. On the other hand, Assumption.3 is imposed on functions h i x i, x i t d i1 t, which means that they satisfy separation principle [17], and the Assumption is very common. It should be emphasized that in Assumption.6 the constants b and c are introduced only for the purpose of analysis and are not used to design controllers. Therefore, they can be unknown as well... Preliminaries. Before proposing our main results, we introduce the knowledge about Nussbaum-type function and fuzzy logic system. Definition.8. [1] Any continuous even function N ζ : R R is called a Nussbaum-type if it has the following properties: lim s sup 1 s s N ζdζ = +, lim s inf 1 s s N ζdζ =. Many functions such as ζ cos ζ, exp ζ cos π ζ can serve as Nussbaum-type functions. Throughout this paper, we will choose even function exp ζ cos π ζ as Nussbaum-type function to carry out the controller design and the simulation. Lemma.9. [1] If N ζ is a Nussbaum-type function, then: 1 Given an arbitrary bounded function g : R n R, and g [ε, g in which ε is an arbitrary positive constant and g is an unknown positive constant with ε < g < +, then, g N ζ is also a Nussbaum-type function. Given an arbitrary function c [ ε, ε ] R, then, N ζ + c is also a Nussbaum-type function.

5 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed... 5 Lemma.1. [1] Let V t and ζ i t, i = 1,... p be smooth functions defined on [, t f, and N ζ i t, i = 1,... p be even smooth Nussbaum-type functions. If the following inequality holds: p V t c + e Dt N ζ i τ ζ i τ e Dτ dτ, 3 i=1 where p is bounded integer, D is positive constant and c represents some suitable constant, then, V t, ζ i t, i = 1,..., p and N ζ i τ ζ i τ dτ must be bounded on [, t f. Lemma.9 has been proven in [1] See pp. 51 and lemma.1 has been proven in [1] See pp , Appendix A. Lemma.1 can be easily extended to case where t f = + due to Proposition.11 given below. Consider [3, 14] ẋt = F xt, 4 with x = x, where z F z R N is upper semicontinuous on R N with nonempty convex and compact values. It is well known that the initial-value problem has a solution and that every solution can be maximally extended. Proposition.11. [4] If x : [, t f R N is a bounded maximal solution of 4, then t f = +. In this paper, the following rules are used to develop the adaptive fuzzy controller R l : if x 1 is F1 l and x is F l and x n is Fn l, then y is G l, l = 1,..., N, where x = [ ] T x 1,..., x n and y are the FLS input and output, respectively. Fuzzy sets Fi l and Gl, associated with the membership function µ F l i x i and µ G l y, respectively. N is the rules number. Through singleton function, center average defuzzification, the FLS can be expressed as follows N l=1 y x = Φ lπ n i=1 µ F x i i l N l=1 Πn i=1 µ F x i, 5 i l where Φ l = arg sup y R µ G l y. Let fuzzy basis function Π n i=1 ξ l = µ F x i i l N l=1 Πn i=1 µ F x i. 6 i l Denoting ϕ = [Φ 1,..., Φ N ] T and ξ x = [ξ 1 x,..., ξ N x] T. Then, the fuzzy logic system 5 can be rewritten as y x = ϕ T ξ x. 7 Our first choice for the membership function is the Gaussian function µ F l i x i = exp 1 xi ai l /σ l i, where σi l and al i are fixed parameters. It has been proven that when the membership functions are chosen as Gaussian functions, the above fuzzy logic system is capable of uniformly approximating any continuous nonlinear function over a compact set with any degree of accuracy. This property is shown by the following lemma.

6 6 H. Y. Yue and J. M. Li Lemma.1. [8] Let f x be a continuous function defined on compact set Ω. Then, for any constant ε >, there exists an FLS 7 such that f x ϕ T ξ x ε. 8 sup x Ω Lemma.13. [16] For any constant matrix M >, any scalars c and d with c < d, and a vector function ωs : [c, d] R n such that the integrals concerned as well defined, then the following holds d d ω T sds M ω sds d c c c 3. Controller Design d c ω T smω s ds. 9 In this section, we will use the recursive backstepping technique to develop the adaptive fuzzy tracking control laws as follows α i = N ζ i M i, ζi = e i M i, M i = k i e i + ϑ ˆθ i i ξi T Z i ξ i Z i e i ηi, 1 ˆθ i = ϑ i ξi T Z i ξ i Z i e i σ i ˆθi, 11 where 1 i n, ϑ i > and σ i > are designed parameters. θ i = ϕ i and ϕ i is an unknown weight parameter vector and will be specified later. ˆθi is the estimation of θ i and the estimation error is θ i = ˆθ i θ i. e i = x i α i 1, α is equal to y d, the control gain k i satisfy k i > 1 λ i with λ i being a positive design parameter, ξ i Z i is a fuzzy basis function vector with Z i being the input vector. Note that when i = n, α n is the true control input ut. Now, we propose the following backstepping-based design procedure. Step1: Define tracking error as e 1 = x 1 y d. Then, its time derivative is given by ė 1 = ẋ 1 ẏ d = f 1 x 1 + h 1 x 1, x 1 t d 11 t + g 1 x 1 x + t d 1 t m 1 x 1 sds + ω 1 t, x t ẏ d. 1 Choose Lyapunov-Krasovskii functional candidate as where Λ 1 = d e γt d 1 d + e γt d 1 1 d 1 V e1 = 1 e 1 + Λ 1, 13 t d 1t τ eγs m 1 x 1 s dsdτ t d 11 t eγs h 1 x 1 sds and γ is a positive constant. Differentiating V e1 and then using 1 give that V e1 = e 1 ė 1 + eγd 1 1 d 1 h 1 x 1 + d 1td e γd m 1 d 1 x 1 1 d 1t 1 d d t d 1 t e γt d s m 1 x 1 s ds 1 d 11t d 11 1 d 1 t h eγd1 1 x 1 t d 11 t γλ 1. 14

7 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed... 7 By Assumptions.1,.3-.4 and the triangular inequality, the following inequalities can be obtained e 1 h 1 x 1, x 1 t d 11 t 1 e h 1 x 1 t d 11 t + 1 h 11 x 1, 1 d 1t + a a 11 11, 1 d 11 t 1 d 1 eγd1 d11t 1, e γt d s 1, s [t d 1 t, t], 15 we can also get the following inequalities by using Assumption.1, the triangular 1 d 1, e 1ω 1 t, x e 1 φ 1 x1 inequality and Lemma.13 1 d 1td 1 d t d 1 t e γt d s m 1 x 1 s ds d t d 1t m 1 x 1 s ds, [ t ] e 1 t d 1 t m 1 x 1 s ds 1 e t t d 1 t m 1 x 1 s ds [ ] 1 e d t t d 1 t m 1 x 1 s ds, 16 where a 11 > is a design constant. Then, substituting x = e + α 1 into 14 and using yield that V e1 e 1 [f 1 x 1 + g 1 x 1 e + α 1 ẏ d + e 1 + e 1φ 1 x 1 a 11 where H 11 = 1 h 1 11 x 1 + h 1 d 1 eγd1 1 x 1 + d1tdeγd furthermore, it follows from d 1 t d that H 11 H 1 = 1 h 11 x 1 + Then, we have 1 d ] + H11 e 1 + a 11 γλ 1, m 1 x 1, 1 1 d 1 eγd1 h 1 x 1 + d e γd 1 d m 1 x V e1 e 1 [f 1 x 1 + g 1 x 1 e + α 1 ẏ d + e 1 + e1φ 1 x1 a 11 ] + H 1 e 1 + a 11 γλ Notice that in 18 H 1 e 1 is discontinuous at e 1 =. Therefore, it can not be approximated by the FLS. Similarly to [3], we introduce hyperbolic tangent function tanh e1 ν 1 to deal with the term. Define f 1 Z 1 = f 1 x 1 ẏ d + e 1 + e 1φ 1 x 1 a + 16 tanh e1 H 1, 11 e 1 ν 19 1 where ν 1 is a positive design parameter, Z 1 = [ e 1, ȳ T d1] T ΩZ1 R 3, and Ω Z1 being some known compact set in R 3. Note that lim e1 16 e 1 tanh e 1 ν 1 H 1 exists, thus, the nonlinear function f 1 Z 1 can be approximated by an FLS ϕ T 1 ξ 1 Z 1 such that f 1 Z 1 = ϕ T 1 ξ 1 Z 1 + δ 1 Z 1. Consequently, substituting 19 and into 18 produces that [ V e1 g 1 x 1 e + α 1 e 1 + e 1 ϕ T 1 ξ 1 Z 1 + δ 1 Z 1 ] tanh e 1 ν 1 H 1 + a 11 γλ 1. 1

8 8 H. Y. Yue and J. M. Li Let ε 1 be the upper bound of the fuzzy approximation error δ 1 Z 1 and according to the definition of θ 1, the following inequality can be obtained e 1 ϕ T 1 ξ 1 Z 1 + e 1 δ 1 Z 1 ϑ 1θ 1 ξ T η1 1 Z 1 ξ 1 Z 1 e 1 + η 1 ϑ 1 + e 1 λ 1 + λ 1ε 1. Then, combining 1 with yields that V e1 g 1 x 1 e + α 1 e 1 + ϑ 1θ 1 ξ T η1 1 1 ξ 1e 1 + e 1 λ 1 +q tanh e1 ν 1 H 1 γλ 1 3 where q 1 = a 11 + λ1ε 1 + η 1 ϑ 1 with η 1 and λ 1 being positive design parameters. Now, choosing the virtual control α 1 in 1 and substituting it into 3 yield that V e1 g 1 x 1 e e 1 + g 1 x 1 N ζ ζ 1 k 1e 1 + e 1 ϑ 1 ˆθ η1 1 ξ1 T Z 1 ξ 1 Z 1 e 1 + ϑ 1 θ η1 1 ξ1 T ξ 1 e 1 + according to the definition of θ 1, 4 becomes V e1 g 1 x 1 e e 1 + g 1 x 1 N ζ ζ 1 e 1 ϑ 1 θ η1 1 ξ1 T Z 1 ξ 1 Z 1 e 1 + where g 1 x 1 e e 1 will be canceled in the next step. Remark 3.1. In 13, we introduce the appropriate Lyapunov- Krasovskii function- t d eγs 11t h 1 x 1 sds and e γt d 1 d d t d 1t τ eγs m 1 x 1 s dsdτ als e γt d 1 1 d 1 λ 1 + q 1 γλ tanh e1 ν 1 H 1, 4 k 1 1 λ tanh e1 ν 1 H 1 + q 1 γλ 1, 5 to deal with the discrete and distributed time-varying delays terms, moreover, by employing them, we successfully construct the memoryless controller, and the stability analysis will be given later. In addition, it is worth pointing out that after disposing the time-delay terms we can get the equation 17 which is handled by the FLS in, thus, in this paper the time delays and the upper bounds of them do not need to be known. Step i 1 i n: Considering e i = x i α i 1, where α i 1 defined in 1, the dynamics of e i -subsystem is given by ė i = ẋ i α i 1 = f i x i + g i x i x i+1 + h i x i, x i t d i1 t + m t d i t i x i sds + ω i t i 1 h j x j, x j t d j1 t where W i = i 1 α i 1 + t d j t mj xj sds + ωj t, x W i, α i 1 f j x j + g j x j x j+1 + i 1 choose the Lyapunov-Krasovskii functional as where Λ i = d e γt d 1 d + e γt d 1 1 d 1 V ei = 1 e i + Λ i, i i t d jt τ eγs m j x j s dsdτ t d j1 t eγs h j x j sds. α i 1 ˆθ j ˆθj + αi 1 ȳ di 1 ȳ di 1. We

9 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed... 9 The time derivative of V ei is given by V ei = e i ė i + eγd 1 i h 1 d j x j + d e γd i d 1 1 d j t m j x j i 1 d j1 t 1 d 1 eγd 1 d j1 t hj x j t d j1 t d 1 d j t t d j t i e 1 d γt d s m j x j s ds γλ i. 6 By applying Assumptions.1,.3-.4 and the triangular inequality, we have e i h i x i, x i t d i1 t e i + 1 h i x i t d i1 t + 1 h i1 x i, α i 1 ω j t, x e i 1 i αi 1 φ i 1 j a a ij + ij, i 1 e i i 1 e i α i 1 h j x j, x j t d j1 t e i 1 i + 1 i 1 h j x j t d j1 t + 1 i 1 h j1 x j, αi 1 1 d j te γ d d j t 1 d 1, e γt d s 1, s [t d jt, t], e i ω i t, x e i φ i x i + a a ii, 1 d j1 te γ d 1 d j1 t < ii 1 d 1 1, 7 then, by using Assumption.1, the triangular inequality and Lemma.13, we have 1 d j td 1 d t d j t e γt d s m j x j s ds 1 t d t d j t m j x j s ds, [ t e i m t d i t i x i s ds 1 e i + 1 t m t d i t i x i s ds [ ] 1 e i + 1 d i 1 t t d i t m i x i s ds, e i e i i 1 αi 1 i α i 1 ] t d j t m j x j sds t d j t m j x j sds, 8 where a ij > are design parameters. Consequently, substituting 7and 8 into 6 produces that αi 1 V ei g i x i e i x i+1 + e i + e i αi 1 where H i1 = 1 f i x i + e i i 1 +e i + g i 1e i 1 W i + H i1 e i + e iφ i x i + i a ii i 1 h j1 x j + 1 d 1 eγd 1 i h j x j + d e γd 1 d i 1 Furthermore, in the light of d j t d, we get i i H i1 H i = 1 1 h j1 x j + 1 d 1 eγd 1 h j x j + d eγd 1 d Then, similar to step 1, 9 can be rewritten as i φ j a ij a ij gi 1ei 1ei γλi, 9 i d j t m j x j i m j x j. a V ei g i x i e i x i+1 + e i fi Z i + ij g i 1 x i 1 e i 1 e i tanh ei ν i H i γλ i, 3

10 1 H. Y. Yue and J. M. Li where f i Z i is defined as i 1 αi 1 + f i Z i = f i x i + g i 1 x i 1 e i 1 + e i + e i i 1 e i αi 1 φ j a ij Wi + e iφ i x i + 16 a e ii i tanh ei ν i H i, next, we will approximate f i Z i by using the FLS, now define ϕ T i ξ i + δ i Z i = f Z i, 31 where Z i = [ ē T i, ȳt di] T ΩZi R i+1 with ē i = [e 1, e,... e i ] T and Ω Zi being some known compact set. Consequently, by substituting 31 into 3, one can get V ei g i x i e i x i+1 + e i ϕ T i ξ i + δ i Z i + i g i 1 x i 1 e i 1 e i tanh e i ν i a ij H i γλ i, 3 according to the inequalities similar to, the above formula becomes V ei where q i = g i x i e i+1 e i + g i x i e i α i + ϑiθi ξ T ηi i Z i ξ i Z i e i + q i + e i λ i tanh e i ν i H i g i 1 x i 1 e i 1 e i γλ i, 33 i 1 into 33 yields V ei a ij + λ iε i + η i ϑ i. Now, substituting the virtual control defined α i in g i x i e i+1 e i + g i x i N ζ i + 1 ζ i ϑ i θ i e i k i 1 λ i tanh e i ν i ξ T ηi i Zi ξi Zi e i + q i H i g i 1 x i 1 e i 1 e i γλ i. 34 Step n: In this step, the true control ut will be constructed. e n = x n α n 1, the dynamics of e n -subsystem is given by Considering ė n = ẋ n α n 1 = f n x n + g n x n u + h n x n, x n t d n1 t n 1 α n 1 h j x j, x j t d j1 t + t t d m jt j x j sds + ω j t, x where W n = n 1 α n 1 + t d n t m n x n sds + ω n t, x t W n, 35 f j x j + g j x j x j+1 + α n 1 ȳ dn 1 ȳ dn 1 + n 1 Then, choose the following Lyapunov-Krasovskii functional α n 1 ˆθ j ˆθj. where V en = 1 e n + Λ n, 36 Λ n = e γt d 1 d d n t d j t τ eγs m j x j s dsdτ + e γt d 1 n t 1 d 1 t d j1 t eγs h j x j sds.

11 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed Differentiating V en yields that V en = e n ė n + eγd 1 1 d n n 1 d j1 t n h j x j + d e γd 1 d n d j t m j x j 1 d 1 eγd 1 d j1 t h j x j t d j1 t d 1 d j t t d j t 1 d Moreover, similar to step i, we have where V en e γt d s m j x j s ds γλ n. 37 g n x n e n u + e n f n x n + g n 1 x n 1 e n 1 n 1 n 1 +e n + e αn 1 n + e n αn 1 W n + H n e n H n = eγd 1 1 d 1 Now define + enφ n xn a nn n h j x j + 1 ϕ T n ξ n + δ n Z n = f n Z n + n n φ j a nj a nj g n 1e n 1 e n γλ n, 38 h j1 x j + d eγd 1 d = f n x n + g n 1 x n 1 e n 1 + e n + e nφ n x n a nn + e n 1 n αn 1 φ j x j a nj + 16 e n tanh e n νn H n, n m j x j. n 1 W n + e n a nj νn αn 1 where Z n = [ e T, ȳdn] T T ΩZn R n+1 with e = [e 1, e,... e n ] T and Ω Zn being some known compact set. Consequently, 38 becomes V en g n x n e n u + e n ϕ T n ξ n + δ n Z n + n g n 1 x n 1 e n 1 e n tanh e n H n γλ n. 39 Combining 39 with similar to produces V en g n x n e n u g n 1 x n 1 e n 1 e n + e n λ n tanh e n νn H n γe γt d 1 n 1 d 1 Now, choosing the virtual control as we have + ϑ nθ n ηn ξn T ξ n e n + q n γλ n t d eγτ j1t h j x j sds. 4 u = N ζ n M n, ζn = e n M n, M n = k n e n + ϑ n ˆθ n ξ T n Z nξ n Z n e n η, n V en g n x n N ζ n + 1 ζ n ϑ n θ n η ξ T n n ξ n e n e n tanh e n νn Finally, choosing a Lyapunov function k n 1 λ n H n g n 1 x n 1 e n 1 e n + q n γλ n. 41 V t = n V ej + n θ j 4ηj,

12 1 H. Y. Yue and J. M. Li and using the adaptive laws defined in 11 give that V t n g j x j N ζ j + 1 ζ j n n e j where q j is defined as q j = theorem. k j 1 λ j + n j k=1 σ j θj η j ˆθ j + n q j 1 16 tanh e j ν i H i γλ n, 4 a jk + λjε j + η j ϑ j. Next, we will give the following Theorem 3.. Consider the closed-loop system consisting of system 1 under Assumptions.1,.3-.6, the control laws 1 and the adaptive laws 11. Suppose that the packaged uncertain functions f i Z i, i = 1,,..., n, can be approximated by fuzzy logic systems in the sense that approximation errors are bounded. Then, with bounded initial conditions ˆθ t and ζ i t, for 1 i n, the following properties are guaranteed: i all the signals in the closed-loop system are bounded; ii the error signal e, in the mean square sense, eventually converges to the following set: Ξ χ := {e R n e rχ Π χ }, where e = [e 1,... e n ] T, e rχ = 1 t e τ dτ and Π χ will be given later, see 69. Remark 3.3. In [9, 3], the strict-feedback adaptive fuzzy tracking control problem was investigated by the control directions known. In [1] the tracking control was addressed for nonlinear delay-free systems with unknown control directions. However, in this paper, the existence of the unknown discrete and distributed time-varying delays terms makes the controller design much more complex and difficult. For i = 1,... n, by employing the Lyapunov-Krasovskii functionals terms e γt d 1 1 d 1 i t d j1 t eγτ h j x j τdτ and e γt d 1 d d i t d j t s eγτ m j x j s dτ ds, we successfully construct the memoryless controller such that the system output y tracks a desired reference signal y d, while all the signals in the closed-loop system remain bounded. Before proving of this theorem, we first introduce the following lemma. Lemma 3.4. [3] For 1 j n and ν j >, consider the set Θ νj given by Θ νj := {e j e j.554ν j }. Then, for e j / Θ νj, the inequalities 1 16tanh e j ν j < are satisfied. Proof. Now, we assume that ν j = ν, for j = 1,...,n, where ν > is an arbitrary small constant. Then, the set Θ νj can be written as Θ ν. Firstly, we need to prove that all the signals in the closed-loop system are boundedness. i Boundedness of all signals in the closed-loop system can be proved by the following three cases. Case 1: For j = 1,... n, e j Θ ν. In this case, e j.554ν.

13 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed According to 1, it is obvious that ˆθ j is bounded for bounded e j. Further, θ j is bounded as θ j is a constant. Next, we need to prove the boundedness of all other signals in the closed-loop systems step by step. Firstly, according to Assumption.5 one gets the boundedness of y d. Since e 1 = x 1 y d and y d are bounded, the boundedness of x 1 is obtained. In addition, since H 1 is a continuous function of x 1, we can conclude that H 1 is also bounded. Let H 1 be the upper bound of H 1, that is, to say H 1 H 1, where H 1 is a positive constant. Now, considering the following function candidate by using 5 and 11, one gets V 1 = V e1 + θ 1 4η1, V 1 g 1 x 1 e e 1 + g 1 x 1 N ζ ζ 1 + C 1 σ 1 θ 1 4η1 k 1 1 λ 1 e tanh e 1 ν H1 γλ 1, 43 where C 1 = σ 1θ 1 + q 4η1 1. Moreover, according to e j.554ν and Lemma 3.4, we have g 1 x 1 e e 1 e 1 λ λ 1c.554ν, 1 16 tanh e 1 ν Following in view of H 1 H 1, we have 1 16tanh e 1 /ν H 1 H Therefore, combining 43, 44 with 45 yields that V 1 g 1 x 1 N ζ ζ 1 k 1 1λ1 e 1 + K 1 σ θ 1 1 4η1 γλ 1, 46 where K 1 = H 1 +λ 1 c.554ν /+C 1 and k 1 > 1 λ 1, so we can deduce k 1 1 λ 1 >. Now, denoting ψ 1 = min k 1 /λ 1, σ 1, γ, one gets V 1 ψ 1 V 1 + g 1 x 1 N ζ ζ 1 + K 1, 47 multiplying both side by e ψ1t, 47 can be expressed as dv 1 t e ψ 1t dt K 1 e ψ1t + e ψ1t g 1 x 1 N ζ ζ Integrating 48 over [, t shows that V 1 t K 1 [ ψ 1 ] + V 1 K 1 ψ 1 e ψ1t + e ψ 1t t g 1 x 1 N ζ ζ 1 e ψ1τ dτ. 49 [ ] Note that < e ψ1t < 1, we have V 1 K 1 ψ 1 e ψ1t V 1. Then, 49 becomes V 1 K 1 ψ 1 + V + e ψ1t τ g 1 x 1 N ζ ζ 1 dτ. 5

14 14 H. Y. Yue and J. M. Li From Assumption.6, we know that g 1 x 1 is a bounded function. With the help of Lemma.9, it can be shown that g 1 x 1 N ζ is Nussbaum-type function. Then, using Lemma.1, it can easily conclude that V 1 t, ζ 1 t and g 1 x 1 N ζ ζ 1 dτ must be bounded on [, t f. Further, since N ζ 1 is the continuous function of ζ 1 t, the boundedness of N ζ 1 can be obtained. According to the definition of V 1 t, boundedness of e 1 and ˆθ 1 on [, t f is obtained. Moreover, θ 1 is bounded as θ 1 is a constant. Furthermore, according to the boundedness of x 1, e 1, ξ 1 Z 1 and ˆθ 1, we can get that M 1 and α 1 are also bounded for any t [, t f. Following in view of e = x α 1, the boundedness of x is ensured for t [, t f. Secondly, considering the following function candidate it can easily be obtained that V = V e + θ, 4η where C = q + σθ 4η V g x e 3 e g 1 x 1 e 1 e + g x N ζ + 1 ζ tanh e H ν σ θ + C 4η H = d eγd 1 d and m j x j + 1 h j1 x j + eγd 1 1 d 1 k 1 λ e γλ, 51 h j x j. According to the boundedness of x j, 1 j and the the continuity of H, one gets that H is also bounded. Then, the terms of g x e 3 e g 1 x 1 e 1 e and 1 16 tanh e /ν H can be dealt with similar to 44 and 45. Similarly, let H be the upper bound of H, where H is a positive constant. Then, we have g x e 3 e g 1 x 1 e 1 e e 4λ + λ c.554ν + e 4λ + λ c.554ν = e λ + λ c.554ν, 1 16tanh e H ν H. 5 Combining 51 with 5 yields that V g x N ζ + 1 ζ σ θ 4η + K e k 1 λ γλ, 53 where K = H + λ c.554ν + C. Then, similar to the process 46-5, we can prove that V t, ˆθ, ζ t, θ, M, α, g x N ζ + 1 ζ dτ and x 3 are bounded on [, t f. Furthermore, in view of e 3 = x 3 α we get the boundedness of x 3. Similarly, we can prove that all other signals in the closed-loop are bounded on [, t f. Case : For j = 1,... n, e j / Θ ν. In this case, e j >.554ν. It follows from the definitions of H j and Lemma 3.4 that n 1 16 tanh e j Hj ν <, then, combining 4 with the following inequality n σ j θj n σ j θ j n σ jθj ˆθ ηj j + 4ηj 4ηj

15 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed yields that where Λ = n denoting and C = n V t n gj xj N ζj + 1 ζj n σ j θ j 4η j + n σ j θ j 4η j + n q j n e j k j 1 γλ, λ j 54 Λ j and k j satisfy k j > 1 λ j, so we can deduce that k j 1 λ j σ jθ j 4η j ψ = min k 1 1/λ 1,..., k n 1/λ n, σ 1,... σ n, γ + n q j, we have Similar to 46-5, we have >. Now, n V ψv + gj N ζ j + 1 ζj + C, 55 V t [ ] C ψ + V C e ψt + ψ e ψt n gj xj N ζj + 1 ζj e ψτ dτ C ψ + V + e ψt n gj xj N ζj + 1 ζj e ψτ dτ. 56 Next, according to the boundedness of g j x j, 1 j n and Lemma.9, we known that g j x j N ζ j + 1 are Nussbaum-type functions. Then, using Lemma.1, it can easily conclude that V t, ζ j t and g j x j N ζ j + 1 ζ j dτ, 1 j n must be bounded on [, t f. Further, the boundedness of N ζ j can be obtained. Then, according to the definition of V t, boundedness of e j, j = 1,..., n and ˆθ j on [, t f is obtained. In addition, θ j is bounded as θ j is a constant. According to Assumption.5, y d, ẏ d,..., y n d are all bounded. Consequently, it follows from e 1 = x 1 y d that x 1 is also bounded, then, combining with 1 one gets that M 1 and α 1 are also bounded for any t [, t f. Furthermore, since e = x α 1, the boundedness of x is ensured for t [, t f. Similarly, we can conclude that all the state variables in the closed-loop system are bounded on [, t f. Case 3: Some e m Θ ν, while some e j / Θ ν. Define Σ M and Σ J as the index sets of subsystems consisting of e m Θ ν and e j / Θ ν, respectively, and then, for j Σ J, choose the Lyapunov function candidate as V ΣJ = V ej + 1 j Σ J j Σ 4η J j By the controller design process, we have V ΣJ gj xj N ζj + 1 ζj σ j θ j j Σ J j Σ 4η J j + q j e j k j 1 + λ j Σ J j Σ j J j Σ J γ Λ j + j Σ J θ j σ j θ j j Σ J 4η j 1 16tanh e j ν H j j Σ J gj xj ej+1 e j g j 1 xj 1 ej 1 e j. 58 It follows from the definitions of H j and Lemma 3.4 that 1 16 tanh e j ν j Σ J H j <. Further, similar to the proof of Theorem 1 in [3], the last term of 58 can be expressed as g j x j e j+1 e j g j 1 x j 1 e j 1 e j j Σ J e j + λ λ j c.554ν. j j 1 Σ M,j+1 Σ M j Σ J

16 16 H. Y. Yue and J. M. Li Now, let C ΣJ = λ j c.554ν + j 1 Σ M,j+1 Σ M σ j θj 4η j Σ j J Based on the above discussion, 58 can be rewritten as + j Σ J q j. Now, denoting V ΣJ since k j > 1 λ j, we get k j 1 λ j Σ J g j x j N ζ j + 1 ζ j Σ J σ j θ j Σ J e j k j 1 + C λ ΣJ γ j ψ ΣJ = min k j /λ j, σ j, γ, >, then, we have 4η j j Σ J Λ j. 59 V ΣJ ψ ΣJ V ΣJ + Σ J g j x j N ζ j + 1 ζ j + C ΣJ. 6 Similar to case, it can be shown that V ΣJ, e j, ζ j t, ˆθ j, gj xj N ζj + 1 ζ jdτ and θ j are bounded on [, t f for j Σ J. For m Σ M, we know that e m are bounded. Then, by combining the conclusions of j Σ J and the boundedness of e m, similar to the proof process of case 1, we can obtain that x m, g m x m N ζ m + 1 ζ m dτ and ζ m t are bounded on [, t f. In the light of the discussion for cases 1-3, we can conclude that all the signals in the closed-loop system are bounded on [, t f. Furthermore, owing to the smoothness of the proposed controller, the closed-loop system admits a solution on its maximum interval of existence [, t f. Therefore, according to Proposition.11, no finite time escape phenomenon may occur and thus t f can be extended to + [3, 14, 37]. As an immediate result, all signals in the closed-loop system are bounded on [, +. This ends the proof of i. ii Similar to the proof of i, the proof of ii is also divided into the following three cases. Case 1: For j = 1,... n, e j Θ ν. In this case, e j.554ν. The process of proof is similar to that of Theorem 1 in [3]. Let ν = [ν,..., ν] T, we can obtain e rχ = 1 t e τ dτ.554 ν. 61 Case : For j = 1,... n, e j / Θ ν. In this case, e j >.554ν, by using the following inequalities and 54 γλ, 1 16 tanh e j H ν j, n σ j θ j 4η j, we can deduce V n k j 1 e λ j j + n g j x j N ζ j + 1 ζ j + C. 6 Integrating 6 from to t shows that 1 t V t V n 1 k t j 1 t λ j e j τdτ n g j x j N ζ j τ + 1 ζ j τ dτ + C, + 1 t

17 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed defining k [ = min k 1 1 λ 1,..., k n 1 λ n ], we have e rχ = 1 t e τ dτ V + t 1 t Using the result of i, let be the upper bound of ζ j τ dτ, then one can get the following inequality n g j x jnζ j τ+1 ζ j τdτ+c. 63 k n g j x j N ζ j τ + 1 e rχ V /t+ /t+c k. 64 Case 3: Some e j / Θ ν, while some e m Θ ν. Considering the index set Σ M of subsystem consisting of e m Θ ν and according to case 3 in the proof of i, we have e rχ ΣM = 1 t t eσ M τ dτ.554 ν ΣM, 65 where e ΣM = Σ m ΣM e m and ν ΣM = Σ m ΣM ν. Further, considering the index set Σ J of subsystem consisting of e j / Θ ν. Using 59 and the following inequalities γ Λ j, σ j θ j, 4η 1 16 tanh e j Hj, ν j Σ J Σ j J j Σ J 66 gives that V ΣJ k j 1 λ j e j + g j x j N ζ j + 1 ζ j + C ΣJ. j Σ J j Σ J Defining k ] 1 = min [k j 1λj with j Σ J and similar to the procedures from 6 to 64 yield that e rχ ΣJ = e Σ J τ dτ V Σ /t+ J J /t+c ΣJ, 67 k 1 where J is the upper bounded of g j x j N ζ j + 1 ζ j τ dτ. j Σ J Because no finite-time escape phenomenon may happen, we can get V ΣJ /t+ lim J /t+c ΣJ t + k lim t + = C Σ J k, V n/t+ /t+c k 1 = C k1, 68 which means that e eventually converges to the following set: where Ξ χ := {e R n e rχ Π χ }, Π χ = max.554 ν, C k,.554 ν ΣM + C Σ J. k 1 69 This completes the proof of Theorem 3.. Remark 3.5. The boundedness e i, i = 1,..., n have been proved in the first part of the Theorem 3.. Now, we assume that e i M i are established, where M i are the upper bounds of e i, and denote the set Ω Z i as Ω Z i := { [ȳt di, e 1,... e i ] T ȳ T di Ω di, e i M i } R i+1. In the light of the requirement in Lemma.1, we know that if we choose Ω Zi large enough such that Ω Z i Ω Zi is satisfied, then the above FLS which have been implemented are true.

18 18 H. Y. Yue and J. M. Li 4. Simulation In this section, two numerical simulation examples are given to demonstrate the effectiveness of the proposed control method. One is a mathematically constructive system, and the other is a physically system Brusselator model. In addition, in Remark 4.3, authors illustrate the advantages of the results in the paper compared with existing ones in literature. Example 4.1. Consider the following unknown time-varying delay system: ẋ 1 = g 1 x 1 t x + f 1 x 1 t + ω 1 t, x t +h 1 x 1 t, x 1 t d 11 t + t d 1 t m 1 x 1 s ds, ẋ = g x t u + f x t + ω t, x t +h x t, x t d 1 t + t d t m x s ds, y = x 1, 7 where h 1 = cosx 1 sin x 1 t d 11 t, g 1 x 1 = + x 1, f 1 x 1 = e.1x 1 x 1, ω 1 t, x =.7x 1 cos1.5t, m 1 x 1 = sin x 1, g x = 3 + cosx 1 x, f x = x 1x, h =. sinx 1 t d 1 t cosx x t d 1 t, m x = x 1 exp x and ω t, x t = x 1 + x sin 3 t. In this simulation, we choose d 11 t =.1 + sint, d 1 t = 1.5cost, d 1 t =.1 sint and d t = cost, the upper bounds of them are d 1 =.4 and d = 1.5, and the derivations are given by d 11 t =.cost, d 1 t =.5sint, d 1 t =.cost and d t =.5sint. Thus, Assumption.1 is also satisfied. The initial states are [ chosen as x 1, x, ζ 1, ζ, ˆθ 1, ˆθ T ] = [3, 1, 1,.5,.,.1] T. The simulation objective is to apply the developed adaptive fuzzy controller such that 1the boundedness of all the signals in the closed-loop system is guaranteed and the system output y follows the reference signal y d to a small neighborhood of zero where y d = 3.5 sin.5t +.5 sint. Then, select the design parameters as ϑ 1 = 5, ϑ =, η 1 =, η = 1, σ 1 =., σ =.1, k = 6 and k 1 =.5. When t [ d i, ], for i = 1,, choose x 1 t = 3 and x t = 1. Membership functions are specified as µ F l i x i = exp.5 x i + 6 l 1, l = 1,..., 7. The control laws and the online adaptation laws for the system 7 are constructed as follows: α 1 = N ζ 1 M i, ζ1 = e 1 M 1, M 1 = k 1 e 1 + ϑ1 ˆθ 1ξ T 1 Z1ξ1Z1e1, η1 ˆθ 1 = ϑ 1 ξ1 T Z 1 ξ 1 Z 1 e 1 σ 1 ˆθ1. u = N ζ M, ζ = e M, M = k e + ϑ ˆθ ξ T Z ξ Z e, η ˆθ = ϑ ξ T Z ξ Z e σ ˆθ. Simulation results in Figures 1-6 show the effectiveness of the developed adaptive fuzzy control schemes for the system 7. From Figures 1 and it can be seen that good tracking performance is obtained. The boundedness of u is illustrated in Figure 3. It can be concluded that the adaptive parameters ˆθ 1 and ˆθ are also bounded from Figure 4. The variable x is also bounded by Figure 5. We can deduce that parameters ζ 1 and ζ are also bounded from Figure 6.

19 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed the signals x 1 and y d time s Figure 1. The output ydashed line and the Reference Signal y d solid line 1.5 the tracking error e time s Figure. The Tracking Error e 1 the controller ut time s Figure 3. The Trajectory of Controller ut.5 the estimations of θ 1 and θ time s Figure 4. The curves of ˆθ 1 dashed line and ˆθ solid line Example 4.. In [3], control problem for a controlled Brusselator model with external disturbances and constant delays is investigated via fuzzy approach with the control directions known. In this paper, the system will also be considered with the control directions unknown as in 71. Moreover, unlike [3], we will consider

20 H. Y. Yue and J. M. Li 1.5 the state x time s Figure 5. The Trajectory of State x the trajectories of ζ 1 and ζ Figure 6. The Parameters Curves of ζ 1 dashed line and ζ solid line the following system with discrete and distributed time-varying delays: ẋ 1 = f 1 x 1 + g 1 x 1 x + h 1 x 1, x 1 t d 1 t + m t d 1 t 1 x 1 s ds + ω 1 t, x, ẋ = f x + g x u + h x, x t d t + t d m t x s ds + ω t, x, y = x 1, 71 where x 1 and x denote the concentrations of the reaction intermediates, u is the control input, g 1 x 1 = x 1, g x = + cosx 1, f 1 x 1 = E F + 1x 1 and f x = Ex 1 x 1 x with E, F > being parameters which describe the supply of reservoir chemicals. h 1 and h are the unknown time-varying delay functions, ω 1 t, x and ω t, x are the external disturbance terms, which come from the modeling errors and other types of unknown nonlinearities in the practical chemical reactions. As stated in [3], it is assumed that x 1. In the simulation, we choose E = 1, F = 3, ω 1 t, x =.7x 1 cos1.5t, ω t, x = x 1 + x sin 3 t, h 1 = cosx 1 sinx 1 t d 1 t, h =.e.5xt dt cosx 1 cosx 1 t d t, m 1 x 1 = exp x 1, m x = x exp x 1, d 11 t =.1 sint, d 1 t = 1.5cost, d 1 = cost and d =.41 cost, the upper bounds of them are d 1 = 1.5 and d = 1.6. The control objective is to design an adaptive fuzzy controller such that all the signals in the closed-loop system remain bounded and the system output y follows the given reference signal y d = sint +.5 sin1.5t. Then, we define fuzzy membership functions as follows: µ F l i x i = exp.5x i + 6 l 1, l = 1,..., 7. The control laws and the online adaptation laws for the system 71 are constructed as follows:

21 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed... 1 α 1 = N ζ 1 M i, ζ1 = e 1 M 1, M 1 = k 1 e 1 + ϑ 1 ˆθ 1 ξ1 T Z 1ξ 1 Z 1 e 1, η1 ˆθ 1 = ϑ 1 ξ1 T Z 1 ξ 1 Z 1 e 1 σ 1 ˆθ1. u = N ζ M, ζ = e M, M = k e + ϑ ˆθ ξ T Z ξ Z e, η ˆθ = ϑ ξ T Z ξ Z e σ ˆθ. In the simulation, the initial values are chosen as [ x 1, x, ζ 1, ζ, ˆθ 1, ˆθ ] T = [3, 1, 1, 1.8,.1,.1] T and when t [ d i, ], for i = 1,, choose x 1 t = 3 and x t = 1. Select the design parameters as ϑ 1 =ϑ =, η 1 =η =, σ 1 = σ =.1, k 1 =.5 and k =.1. The simulation results are shown by Figures 7-1. Apparently, the simulation results show that under the action of the suggested controller, a good tracking performance has been achieved. 4 the signals x 1 and y d time s Figure 7. The Output ydashed line and the Reference Signal y d solid line.5 the tracking error e time s Figure 8. The Tracking Error e 1 Remark 4.3. Similar results have been proposed in [35] for SISO nonlinear delayfree systems, [3] for a class of perturbed strict-feedback nonlinear time-delay systems, [9] nonlinear time-delay systems with unknown virtual control coefficients and [5] for MIMO nonlinear delay-free systems. Unlike [5, 9, 3, 35], this paper mainly addresses the tracking control problem for nonlinear distributed timevarying delays systems with unknown control direction. Although the adaptive fuzzy controller has been constructed based on the similar design idea and the backstepping technique, the existence of nonlinear distributed time-varying delay terms and the unknown control directions make controller design more difficult, as

22 H. Y. Yue and J. M. Li 4 the controller ut time s Figure 9. The Trajectory of Controller ut the estimations of θ 1 and θ time s Figure 1. The Curves of ˆθ 1 dashed line and ˆθ solid line 4 3 the state x time s Figure 11. The Trajectory of State x the trajectories of ζ 1 and ζ time s Figure 1. The Parameters Curves of ζ 1 dashed line and ζ solid line

23 Adaptive Fuzzy Tracking Control for a Class of Nonlinear Systems with Unknown Distributed... 3 a result, the stability analysis of the closed-loop system in this paper is also much more complex than ones in [5, 9, 3, 35]. 5. Conclusion The stable adaptive fuzzy control scheme has been proposed for a class of perturbed strict-feedback nonlinear systems. Based on the appropriate Lyapunov- Krasovskii functionals, the Nussbaum-type functions, the FLS and the backstepping approach, the adaptive tracking controller is constructed. By using the Lyapunov stability analysis, we have proved that all the signals of the resulting closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin. However, there are many further works to be studied in the future, such as how relax the Assumption.1, i.e. if d i1 and d i do not satisfy d i1 t d 1 < 1 and d i t d < 1, how to design the controller for the nonlinear system 1 is a challenging problem and needs to be further investigated in the future. References [1] A. Boulkroune, M. Msaad and H. Chekireb, Design of a fuzzy adaptive controller for MIMO nonlinear time-delay systems with unknown actuator nonlinearities and unknown control direction, Information Sciences, 184 1, [] A. Boulkroune, M. Msaad and M. Farza, Adaptive fuzzy controller for multivariable nonlinear state time-varying delay systems subject to input nonlinearities, Fuzzy Sets and Systems, , [3] A. Boulkroune, M. Msaad and M. Farza, On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign, Fuzzy Sets and Systems, 116 1, [4] B. Chen, X. P. Liu, K. F. Liu, P. Shi and C. Lin, Direct adaptive fuzzy control for nonlinear systems with time-varying delays, Information Sciences, 185 1, [5] B. Chen, S. C. Tong and X. P. Liu, Fuzzy approximate disturbance decoupling of MIMO nonlinear systems by backstepping approach, Fuzzy Sets and Systems, , [6] W. S. Chen and L. C. Jiao, Adaptive tracking for periodically time-varying and nonlinearly parameterized systems using multilayer neural networks, IEEE Trans. Neural Networks, 1 1, [7] W. S. Chen, L. C. Jiao, J. Li and R. H. Li, Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays, IEEE Trans. Systems, Man and Cybernetics-Part B: Cybernetics, 43 1, [8] W. S. Chen, L. C. Jiao, R. H. Li and J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Trans. Fuzzy Systems, 184 1, [9] W. S. Chen and Z. Q. Zhang, Globally stable adaptive backstepping fuzzy control for outputfeedback systems with unknown high-frequency gain sign, Fuzzy Sets and Systems, , [1] H. B. Du, H. H. Shao and P. J. Yao, Adaptive neural network control for a class of lowtriangular-structured nonlinear systems, IEEE Trans. Neural Networks, 17 6, [11] G. Feng, S. G. Cao, N. W. Rees and etc., Design of fuzzy control systems with guaranteed stability, Fuzzy Sets and Systems, , 1-1. [1] S. Filali, S. Bououden, M. L. Fas and A. Djebebla,Robust adaptive fuzzy model predictive control and its application to an industrial surge tank problem, ICIC Express Letters, 1 7, 197-.

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