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Adaptive Fuzzy Control of Strict-Feedback Nonlinear Time-Delay Systems With Unmodeled Dynamics Shen Yin, Senior Member, IEEE, Peng Shi, Fellow, IEEE, and Hongyan Yang

Abstract—In this paper, an approximated-based adaptive fuzzy control approach with only one adaptive parameter is presented for a class of single input single output strict-feedback nonlinear systems in order to deal with phenomena like nonlinear uncertainties, unmodeled dynamics, dynamic disturbances, and unknown time delays. Lyapunov–Krasovskii function approach is employed to compensate the unknown time delays in the design procedure. By combining the advances of the hyperbolic tangent function with adaptive fuzzy backstepping technique, the proposed controller guarantees the semi-globally uniformly ultimately boundedness of all the signals in the closed-loop system from the mean square point of view. Two simulation examples are finally provided to show the superior effectiveness of the proposed scheme. Index Terms—Fuzzy control, nonlinear systems, time-delays, unmodeled dynamics.

I. I NTRODUCTION VER past decades, controller design of complex systems has played an important role for nonlinearity issue. Several superior control techniques have been proposed in this area, such as fuzzy control [1]–[5], fault tolerant control [6], and adaptive backstepping technique [7], [8], etc. In practice, fuzzy control has performed satisfactory performance with many successful applications. To mention a few, Gao et al. [1] developed H∞ fuzzy control scheme for nonlinear systems under unreliable communication links. In [9], with immeasurable variables, a robust H∞ state estimator was designed for a class of continuous-time nonlinear systems via Takagi–Sugeno (T–S) fuzzy affine dynamic model. Moreover, a general observer-based fault detection filter was used as a residual generator and the robust fault detection was formulated as a filtering problem as illustrated in [10]. While Kanellakopoulos et al. [11] firstly proposed a backstepping-based adaptive control scheme for a class

O

Manuscript received March 3, 2015; revised May 15, 2015; accepted July 12, 2015. This work was supported by the National Natural Science Foundation of China under Grant 60774010, Grant 61304102, and Grant 61472104. This paper was recommended by Associate Editor M. J. Er. S. Yin and H. Yang are with the Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]). P. Shi is with the Department of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA 5005, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2457894

of strict-feedback nonlinear systems, systematic control approaches were further provided and several removable results have been obtained to solve the control problems from the uncertainty point of view. In order to handle highly uncertain, complex and ill-defined systems, approximation-based adaptive fuzzy control (or neural networks) strategies have been popular candidates for the last decade [12]–[18]. Further works mainly consider the time-delay which is often a source of instability and frequently exists in various practical systems such as mechanical, biological, and economical processes. In [19], based on a new time-delay model, a novel approach to network-based control was presented. Furthermore, global asymptotic stability conditions for nonlinear stochastic systems with multiple state delays were obtained in [20]. Remarkable works have been done in the framework of linear system [21], [22]. For instance, Basin et al. [21] focused on the issue of the optimal regulator with state delay via maximum principle. While Basin et al. [22] further solved the joint state filtering and parameter estimation issues for linear stochastic time-delay systems. More efforts have also been paid on discrete-time fuzzy systems with time delay. In [23], a delay-dependent stabilization approach was developed for both state feedback and observer-based output feedback cases. A delay-dependent robust H∞ filtering design scheme was proposed by Qiu et al. [24], in which the state delay is assumed to be time-varying and of an interval-like type. Su et al. [25] further focused on the problem of analyzing model transformation of discrete-time T–S fuzzy systems in case of time-varying delays. Recently, the problem of dissipativity analysis and synthesis for discrete-time T–S fuzzy systems with stochastic perturbation and time-varying delay was solved as indicated in [26]. Further problem of adaptive fuzzy tracking control for a class of stochastic nonlinear time-delay systems was solved in [27]. Several most recent works have extended time-delay to fault diagnosis and H∞ control issue (see [28]–[33]). It should be pointed out that the fault diagnosis technique in [34]–[36] are mainly based on data-driven. At the meanwhile, the nonlinear systems with the unmodeled dynamics and dynamic disturbances which are caused by external disturbances, modeling errors, measurement noise, modeling simplifications, etc., have received more attention both from academic and industrial domains. To deal with these difficulties, several initial results have been obtained.

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In [37], a backstepping-based adaptive output feedback controller was proposed for a class of strict-feedback nonlinear systems to deal with dynamics uncertainties combined via small-gain theory. By introducing a dynamic signal in [38], an adaptive fuzzy control scheme was presented for a class of nonlinear systems with unmodeled dynamics. It is worth noting that the number of adaptation laws and fuzzy base functions are based on steps. The more fuzzy rules are applied to improve approximation accuracy, the more adaptive parameters will be necessary and consequently, the online learning might become problematic in practice. Therefore, reduction of the number of adjustable parameters and complexity of computation load become more important from both academic and practice points of view. Motivated by the aforementioned observations, this paper mainly focuses on approximation-based adaptive fuzzy control scheme for nonlinear strict-feedback systems with unmodeled dynamics. A dynamic signal is firstly introduced to handle the unmodeled dynamics. Then, fuzzy logic systems are further utilized to approximate the unknown nonlinearities, while appropriate Lyapunov–Krasovskii functionals are chosen to deal with the time-delay terms. By combining adaptive backstepping control technique with hyperbolic tangent function, a novel approximated-based adaptive fuzzy control scheme is presented. The proposed controller could guarantee that all the signals in the closed-loop system are semi-globally uniformly bounded. The highlights of the proposed method could be briefly summarized as follows. 1) A efficient adaptive fuzzy controller is developed for a class of strict-feedback nonlinear systems with unmodeled dynamics and time-delay. 2) Only one adaptive parameter is involved for online computation to significantly decrease the computational load. II. P RELIMINARIES Consider a class of single input single output nonlinear system in the following form: ⎧ z˙(t) = q(z(t), x1 (t)) ⎪ ⎪ ⎪ ⎪ ⎪ x˙ i (t) = gi xi+1 (t) + fi (¯xi (t)) + hi (¯xi (t − τi )) ⎪ ⎪ ⎪ ⎨ + i (x, z, t), i = 1, . . . , n − 1 (1) ⎪ x˙ n (t) = gn u(t) + fn (¯xn (t)) + hn (¯xn (t − τn )) ⎪ ⎪ ⎪ ⎪ ⎪ + n (x, z, t) ⎪ ⎪ ⎩ y(t) = x1 (t) where x(t) = [x1 (t), x2 (t), . . . , xn (t)]T ∈ Rn , u(t) ∈ R, and y(t) ∈ R represent the state vector, control input, and output, respectively. x¯ i (t) = [x1 (t), x2 (t), . . . , xi (t)]T ∈ Ri ; gi (.) is an unknown constant; fi (.) and hi (.) are unknown smooth functions; i (i = 1, 2, . . . , n) denotes the nonlinear dynamic disturbance; z(t) ∈ R0 is the unmeasured portion of the state which is referred as the unmodeled dynamics. It is assumed that i and q are uncertain Lipschitz continuous functions. For simplicity, the variable t is omitted from function expressions except in the delayed state variables x¯ i (t − τi ), i = 1, 2, . . . , n during the controller design. To proceed, the following assumptions are introduced.

Assumption 1: For the dynamic disturbances i (i = 1, 2, . . . , n), there exist unknown nonnegative smooth functions φi1 (·) and φi2 (·), such that |i | ≤ φi1 (|(x1 , . . . , xi )|) + φi2 (|z|).

(2)

Remark 1: This assumption relaxes the restriction in [38] in which φi1 (·) and φi2 (·) are known. Assumption 2 [38]: The unmodeled dynamics in (1) is exponentially input-to-state practically stable (exp-ISpS); i.e., for the system z˙ = q(z, x), there exists an exp-ISpS Lyapunov function V(z) such that α1 (|z|) ≤ V(z) ≤ α2 (|z|) ∂V(z) q(z, x1 ) ≤ −c0 (|z|) + μ0 (|x|) + d0 ∂z

(3) (4)

where α1 , α2 , and μ are of class K∞ -functions, c0 and d0 are known positive constants. Assumption 3: For 1 ≤ i ≤ n, with known signs of gi , there exists unknown positive constant b and c such that 0 < b ≤ |gi | ≤ c < ∞, ∀¯xi ∈ Ri .

(5)

Remark 2: Without loss of generality, it is supposed that 0 < b ≤ gi . Since b is only for analysis purpose, thus, its value is not trivial in the design procedure. To clarify our controller design, the following lemmas are introduced. Lemma 1 [38]: If V is an exp-ISpS Lyapunov function for a control system, that is, (3) holds, then for any constant c¯ in (0, c0 ), with initial condition x0 = x0 (0), and function μ(x ¯ 1 ) ≥ μ(|x1 |), there exists a finite time T0 = T0 (¯c, r0 , z0 ), a non-negative function D(t) defined for all t ≥ 0 and a signal described by r˙ = −¯cr + μ(x ¯ 1 (t)) + d0 , r(0) = r0

(6)

such that D(t) = 0 for all t ≥ T0 V(z(t)) ≤ r(t) + D(t).

(7)

For all t ≥ 0, the solutions are defined. Without losing of generality, we makes μ(.) ¯ as μ(s) ¯ = s2 μ0 (s2 ), where μ(.) ¯ is a non-negative smooth function. Thus, the dynamical r defined by (6) follows:     r˙ = −¯cr + x12 μ0 x12  + d0 , r(0) = r0 (8) where μ0 is a non-negative smooth function. Lemma 2 [13]: For any variable η ∈ R and constant  > 0, the following inequality holds: η ≤ δ, δ = 0.2785. (9) 0 ≤ |η| − η tanh  Throughout this paper, the following rules are employed to develop the adaptive fuzzy controller. Ri : if x1 is F1i and x2 is F2i and . . . and xn is Fni , then y is Bi (i = 1, 2, . . . , N), where x = [x1 , . . . , xn ] ∈ Rn and y ∈ R are the input and output of the fuzzy system, respectively. Fji and Bi are fuzzy sets in R. By use of center-average defuzzification

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and product inference, the fuzzy logical system (FLS) can b formulated as

 N n i=1 i j=1 μFji xj y(x) =

 N n i=1 j=1 μF i xj j

where i = maxy∈R μBi (y). n Let ξi (x) = (nj=1 μFi (xj ))/( N i=1 [i=1 μF i (xj )]), ξ(x) = j

j

[ξ1 (x), . . . , ξN (x)]T , and φ = [ 1 , . . . , N ]T , it follows that: y(x) = φ T ξ(x).

(10)

If all the fuzzy membership functions are chosen as Gaussian functions, the following lemma holds. Lemma 3 [12]: Let f (x) be a continuous function defined on a compact set , then for any given constant ε > 0, there exists a fuzzy logic system (10) such that   (11) sup  f (x) − φ T ξ(x) ≤ ε. x∈

III. A DAPTIVE F UZZY C ONTROLLER D ESIGN In this section, the approximation-based adaptive fuzzy backstepping controller design for system (1) is proposed in detail. For the ith subsystem, a virtual control signal αi will be designed as   zi ξi  ˆ (12) αi (Xi ) = −θ ξi  tanh ai where Xi = [¯xi , r, θˆ ]T , i = 1, 2, . . . , n will be omitted from function expressions for simplicity. ai is a positive design parameter. ξi denotes a fuzzy basis function vector. θˆ represents an estimation of unknown constant θ , which is defined as θ = max{φi  : i = 1, 2, . . . , n}

(13)

where φi will be specified in the context later. The estimation error is θ˜ = θˆ − θ and the adaptive law follows:   n ˙θˆ =  γ z ξ  tanh zi ξi  − k θˆ (14) i i 0 ai i=1

where γ and k0 are positive design parameters. The actual controller u is designed at step n as   zn ξn  ˆ (15) u = −θ ξn  tanh an where an is a design parameter. The detailed steps for controller design is listed as follows. Step 1: Consider a Lyapunov–Krasovskii function as  t 1 P1 (x1 (τ1 ))dτ (16) VP1 = z21 + 2 t−τ1 where the unknown positive function P1 will be chosen later. Then, differentiating VP1 along with z1 = x1 is V˙ P1 ≤ z1 { f1 (x1 ) + g1 x2 } + z1 h1 (x1 (t − τ1 )) +|z1 ||1 | + P1 (x1 ) − P1 (x1 (t − τ1 )).

(17)

3

Based on Assumption 1 and Lemma 2, we know that |z1 1 | ≤ |z1 |(φ11 (|x1 |) + φ12 (|z|))   z1 φ11 (|x1 |) ≤ z1 φ11 (|x1 |) tanh + 11 11   z1 φ¯ 12 (r) 1 + 12 + z1 φ¯ 12 (r) tanh + z21 + d1 (t) 12 4 ˆ ˆ ≤ z1 φ11 (x1 ) + 11 + z1 φ12 (x1 , r) + 12 1 + z21 + d1 (t) (18) 4 = 0.2785 ,  = 0.2785 , and φˆ (x ) = where 11 11 12 11 1 12 φ11 (|x1 |) tanh((z1 φ11 (|x1 |))/(11 )) is a smooth function, φ¯ 12 (r) = φ12 ◦ α1 −1 (2r), d1 (t) = (φ12 ◦ α1 −1 (2D(t)))2 , and φˆ 12 (x1 , r) = φ¯ 12 (r) tanh((z1 φ¯ 12 (r))/(12 )). With Young’s inequality, the term z1 h1 (x1 (t − τ )) can be rewritten as

z1 h1 (x1 (t − τ1 )) ≤

1 2 1 2 z + h (x1 (t − τ1 )). 2 1 2 1

(19)

Substituting (18) and (19) into (17) results in   3 ˆ ˆ ˙ VP1 ≤ z1 f1 (x1 ) + g1 x2 + φ11 (x1 ) + φ12 (x1 , r) + z1 4 1 + 12 + d1 + h21 (x1 (t − τ1 )) + 11 2 + P1 (x1 ) − P1 (x1 (t − τ1 )). (20) To cancel the unknown time-delay term in (20), P1 (x1 ) can be chosen as P1 (x1 ) =

1 2 h (x1 ). 2 1

(21)

After taking (21) into account, V˙ P1 can be expressed as  V˙ P1 ≤ z1 f1 (x1 ) + g1 x2 + φˆ 11 (x1 ) + φˆ 12 (x1 , r) h2 (x1 ) 3 + z1 + 1 4 2z1

 + 12 + d1 . + 11

(22)

Noting that (h21 (x1 ))/(2z1 ) is discontinuous at z1 = 0, we employ function tanh(z1 /υ1 ) to compensate it. Then, it can be approximated by FLS, (22) becomes  V˙ P1 ≤ z1 f1 (x1 ) + g1 x2 + φˆ 11 (x1 ) + φˆ 12 (x1 , r)    z1 16 3 H1 + 11 tanh2 + 12 + d1 + z1 + 4 z1 υ1    z1 H1 + 1 − 16 tanh2 (23) υ1 where H1 = (h21 (x1 ))/2. To construct the virtual control law α1 , consider a Lyapunov function Vz1 as Vz1 = VP1 +

b 2 1 θ˜ + r. 2γ λ0

(24)

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Along with (8) and (23), the time derivative of Vz1 is  ˙ Vz1 ≤ z1 f1 (x1 ) + g1 x2 + φˆ 11 (x1 ) + φˆ 12 (x1 , r)    1 16 3 2 2 z1 H1 tanh + z1 + x1 μ0 (x1 ) + 4 λ0 z1 υ  1  z1 H1 + 11 + 12 + d1 + 1 − 16 tanh2 υ1 1 c¯ b ˙ ˆ (25) + d0 − r + θ˜ θ. λ0 λ0 γ When f1 (x1 ), φˆ 11 (x1 ), and φˆ 12 (x1 , r) are known nonlinear functions, we can take the intermediate control signal as  −1 αˆ 1 = −g1 k1 z1 + f1 (x1 ) + φˆ 11 (x1 )   3 1 + φˆ 12 (x1 , r) + z1 + x1 μ0 x12 4 λ0    z 16 1 1 H1 + g21 z1 + tanh2 z1 υ1 2

(26)

where k1 > 0 is a design parameter. Furthermore, by adding and subtracting g1 z1 αˆ 1 and via defining z2 = x2 − α1 , we obtain

 V˙ z1 ≤ −k1 z21 + g1 z1 z2 + g1 z1 α1 − αˆ 1 1 1 + 12 + d1 + d0 − g21 z21 + 11 2 λ0    c¯ b ˙ 2 z1 ˆ (27) H1 − r + θ˜ θ. + 1 − 16 tanh υ1 λ0 γ

Step i (2 ≤ i ≤ n − 1): Based on zi = xi − αi−1 , it holds z˙i = gi xi+1 + fi (¯xi ) −

j=1

+ hi (¯xi (t − τi )) −

j=1

k=i+1

Then, Lyapunov–Krasovskii function is designed as i  t  1 Pj (xj (τj ))dτ. VPi = z2i + 2 t−τj

− g1 z1 αˆ 1 ≤ |g1 z1 |φ1 ξ1  + |g1 z1 δ1 |   z1 ξ1  + δθ a1 g1 ≤ g1 z1 θ ξ1  tanh a1 1 1 2 + g21 z21 + 13 (29) 2 2 where θ has been defined in (14). Then, considering the virtual control signal α1 in (12), we can further get   z1 ξ1  ˆ . (30) g1 z1 α1 = −g1 z1 θξ1  tanh a1 Substituting (29) and (30) into (27) shows    z1 c¯ H1 − r V˙ z1 ≤ −k1 z21 + g1 z1 z2 + 1 − 16 tanh2 υ1 λ0    z1 ξ1  b + D1 (31) + θ˜ θ˙ˆ − γ z1 ξ1  tanh γ a1 +  + d + δθ a g + where D1 = C1 + (1/λ0 )d0 , C1 = 11 1 1 1 12 2 (1/2)13 .

(34)

j=1

Differentiating VPi along (32) and (33), it yields that ⎧ i−1 ⎨  ∂αi−1 V˙ Pi ≤ zi fi (¯xi ) + gi xi+1 + hi (¯xi (t − τi )) − ⎩ ∂xj j=1

i−1

   ∂αi−1

gj xj+1 + fj x¯ j × hj xj (t − τj ) − ∂xj j=1   i   ∂αi−1  zk ξk  ˆ − k0 θ − γ zk ξk  tanh ak ∂ θˆ k=1 ⎫  i−1  zi ξi   ∂αj−1 ∂αi−1 ⎬ − γ ξi  tanh r˙ zj − ai ∂r ⎭ ∂ θˆ

(28)

where δ1 is an approximation error, and 13 > 0. Following the completion of squares, Young’s inequality and the similar inequality showed in (9), we know:

i−1   ∂αi−1

hj x¯ j t − τj ∂xj

∂α ∂α ¯ i − i−1 θ˙ˆ − i−1 r˙ (32) + ˆ ∂r ∂θ ¯ i = i − i−1 (∂αi−1 )/(∂xj )j . where  j=1 From (14), we have  i    zk ξk  ∂αi−1 ˙ ∂αi−1  γ zk ξk  tanh − k0 θˆ θˆ = ak ∂ θˆ ∂ θˆ k=1 ⎛ ⎞   n ∂αi−1 ⎝  zk ξk  ⎠ + γ zk ξk  tanh . (33) ak ∂ θˆ

However, f1 (x1 ), φˆ 11 (x1 ), and φˆ 12 (x1 , r) are unknown smooth functions, αˆ 1 can not be implemented in practice. Thus, apply the FLS φ1T ξ1 to approximate αˆ 1 such that αˆ 1 = φ1T ξ1 + δ1 , |δ1 | ≤ 13

i−1 

 ∂αi−1

gj xj+1 + fj x¯ j ∂xj

j=2

  ¯ i − + z i 

i  j=2

+

zj

∂αj−1 ∂ θˆ

i 

n 

γ zk ξk  tanh

k=i+1





  Pj xj (τj ) − Pj xj t − τj .



zk ξk  ak



(35)

j=1

Similarly based on Assumption 1 and Lemma 2, it follows that:     z i  ¯ i  ≤ zi φˆ i1 x¯ i , θˆ , r + i1 + zi φˆ i2 (¯xi , θˆ , r) ⎤ ⎡ 2 i−1   z2i ∂α i−1 ⎦ + i2 + ⎣1 + + di (t) (36) 4 ∂xj j=1

i−1 where φˆ i1 (¯xi , θˆ , r) = (φi1 + j=1 |(∂αi−1 )/(∂xj )|φj1 ) × i−1 = 0.2785 tanh((zi (φi1 + j=1 |(∂αi−1 )/(∂xj )|φj1 ))/(i1 )), i1 , r) tanh((zi φ¯ i2 (¯xi , θˆ , r))/(i2 )), i1 , φˆ i2 (¯xi , θˆ , r) = φ¯ i2 (¯xi , θˆ i−1 −1 ˆ ¯ |(∂i−1 )/(∂xj )|φj2 α1−1 (2r), φi2 (¯xi , θ , r) = φi2 ◦ α1 (2r) + j=1 i i2 = 0.2785i2 , di (t) = j=1 (φj2 ◦ α1−1 (2D(t)))2 , noting that di (t) ≥ 0 for all t ≥ 0.

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And then following Young’s inequality, we reach: h2 (¯xi (t − τi )) z2i + i 2 2    z2i ∂αi−1 2 ∂αi−1

−zi hj x¯ j t − τj ≤ ∂xj 2 ∂xj



 2 hj x¯ j t − τj + . 2 Substituting (36)–(38) into (35), it holds that ⎧   i−1 ⎨ zi  zi ∂αi−1 2 V˙ Pi ≤ zi fi (¯xi ) + gi xi+1 + + ⎩ 2 2 ∂xj zi hi (¯xi (t − τi )) ≤

(37)

 

 ∂αi−1

gj xj+1 + fj x¯ j + φˆ i1 x¯ i , θˆ , r ∂xj j=1 ⎞ ⎛ 2 i−1    z  ∂α i i−1 ⎠ + φˆ i2 x¯ i , θˆ , r + ⎝1 + 4 ∂xj j=1   i   ∂αi−1  zk ξk  − − k0 θˆ γ zk ξk  tanh ak ∂ θˆ

(38)



− γ ξi  tanh

zi ξi  ai

 i−1 j=2

zj

⎫ ⎬

(39)

Similar to the procedures in step 1, choosing Pj (¯xj ) = (h2j (¯xj ))/2 to cancel time-delay terms in (39) and introducing a function tanh(zi /υi ) to compensate the discontinuous term (Hi )/(zi ), it yields ⎧   i−1 ⎨ zi  zi ∂αi−1 2 V˙ Pi ≤ zi fi (¯xi ) + gi xi+1 + + ⎩ 2 2 ∂xj j=1

 

 ∂αi−1

gj xj+1 + fj x¯ j + φˆ i1 x¯ i , θˆ , r ∂xj j=1 ⎛ ⎞ 2 i−1    z  ∂α i i−1 ⎠ − ∂αi−1 r˙ + φˆ i2 x¯ i , θˆ , r + ⎝1 + 4 ∂xj ∂r j=1   i   ∂αi−1  zk ξk  − γ zk ξk  tanh − k0 θˆ − γ ξi  ak ∂ θˆ k=1   ⎫   i−1 ⎬ 16 tanh2 υzii zi ξi   ∂αj−1 × tanh zj Hi + ⎭ ai zi ∂ θˆ −

i−1 

j=2

  ∂αj−1 zk ξk  γ zk ξk  tanh + i1 ˆ ak ∂ θ j=2 k=i+1    2 zi + i2 + di + 1 − 16 tanh Hi υi −

i 

zj

∂xj

 

 gj xj+1 + fj x¯ j + φˆ i1 x¯ i , θˆ , r

⎛ ⎞ 2 i−1    z  ∂α i i−1 ⎠ − ∂αi−1 r˙ + φˆ i2 x¯ i , θˆ , r + ⎝1 + 4 ∂xj ∂r j=1   i   ∂αi−1  zk ξk  − − k0 θˆ γ zk ξk  tanh ak ∂ θˆ k=1  i−1  zi ξi   ∂αj−1 − γ ξi  tanh zj ai ∂ θˆ j=2   ⎫ ⎬ 16 tanh2 υzii 1 + Hi + g2i zi + gi−1 zi−1 (41) ⎭ zi 2

∂αj−1 ∂αi−1 r˙ − ˆ ∂r ⎭ ∂θ

  i n  ∂αj−1  zk ξk  − zj γ zk ξk  tanh ak ∂ θˆ k=i+1 j=2



 i i   h2j x¯ j t − τj

 Pj xj τj + + 2 j=1 j=1



 − Pj xj t − τj + i1 + i2 + di .

−

i−1  ∂αi−1

j=1

i−1 

k=1

where Hi = ( ij=1 h2j (¯xj ))/2. If we choose intermediate control signal as ⎧   i−1 ⎨ zi  zi ∂αi−1 2 −1 fi (¯xi ) + ki zi + + αˆ i = −gi ⎩ 2 2 ∂xj j=1

j=1

−

5

n 

(40)

where ki > 0 is a design parameter. Adding and subtracting gi zi αˆ i in (40), it shows V˙ Pi ≤ −ki z2i + gi zi zi+1 + gi zi (αi − αˆ i )   i n  ∂αj−1  zk ξk  − zj γ zk ξk  tanh ak ∂ θˆ k=i+1 j=2    zi Hi + i1 + i2 + di + 1 − 16 tanh2 υi 1 − g2i z2i − gi−1 zi−1 zi . (42) 2 Then, the fuzzy logic system is employed to approximate αˆ i such that for any given positive constant i3 αˆ i = φiT ξi + δi , |δi | ≤ i3

(43)

where δi is an approximation error. Following the same procedures (29) and (30), it follows that:   zi ξi  + δθ ai gi − gi zi αˆ i ≤ gi zi θ ξi  tanh ai 1 1 2 + g2i z2i + i3 (44) 2 2   z ξ  ˆ i  tanh i i . (45) gi zi αi = −gi zi θξ ai Taking (44) and (45) into account, it is obviously that   z ξ  ˜ i  tanh i i V˙ Pi ≤ −ki z2i + gi zi zi+1 − bzi θξ ai   i n  ∂αj−1  zk ξk  + Ci zj γ zk ξk  tanh − ak ∂ θˆ k=i+1 j=2    zi + 1 − 16 tanh2 (46) Hi − gi−1 zi−1 zi υi 2 +  +  + d . where Ci = δθ ai gi + (1/2)i3 i i1 i2

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Choose a Lyapunov function candidate as Vzi = Vzi−1 + VPi .

(47)

Following the same procedures in the former steps, we can easily obtain V˙ zi−1 . Differentiating (47) along with V˙ zi−1 and (46), it results V˙ zi ≤ −

i  j=1

⎞ ⎛   i  # # z ξ  bθ˜ ⎝ ˙ j j ⎠ − c¯ r + Di θˆ − γ + zj #ξj # tanh γ aj λ0 j=1 ⎛ ⎞   i n  ∂αj−1  ξ  z k k ⎝ ⎠ − zj γ zk ξk  tanh ak ∂ θˆ +

i  

k=i+1



1 − 16 tanh2

j=1

 zj Hj υj

(48)

where Di = Di−1 + Ci . Step n: In this step, the actual control law u is designed. Since zn = xn − αn−1 , we have z˙n = gn u + fn (¯xn ) −

n−1  ∂αn−1

∂xi

i=1

+ hn (¯xn (t − τn )) −

(gi xi+1 + fi (¯xi ))

n−1  ∂αn−1 i=1

∂xi

hi (¯xi (t − τi )) (49)

¯ n = n − n−1 (∂αn−1 )/(∂xi )i . where  i=1 Choose the Lyapunov–Krasovskii function as 1 2 z + 2 n

n   i=1

t

h2 (¯xn (t − τn )) z2n + n 2 2   ∂αn−1 z2n ∂αn−1 2 −zn hi (¯xi (t − τi )) ≤ ∂xi 2 ∂xi h2i (¯xi (t − τi )) . + 2 zn hn (¯xn (t − τn )) ≤

Pi (xi (τi ))dτ.

−

Differentiating VPn and considering same idea in (33), it yields

   n−1   ∂αn−1 2 zn + φˆ n2 (¯xn , θˆ , r) + 1+ 4 ∂xi i=1   n−1 ∂αn−1 ˙  ∂αi−1 zn ξn  ˆ − zi γ ξn  tanh θ− an ∂ θˆ ∂ θˆ i=2  n n  h2i (¯xi (t − τi ))  ∂αn−1 r˙ + + − (Pi (xi (τi )) ∂r 2

V˙ Pn = zn fn (¯xn ) + gn u + hn (¯xn (t − τn )) n−1  ∂αn−1 i=1 n−1 

∂xi

i=2

n  + (Pi (xi (τi )) − Pi (xi (t − τi ))). i=1

∂xi

(51)

  (gi xi+1 + fi (¯xi )) + φˆ n1 x¯ n , θˆ , r

   n−1   ∂αn−1 2 zn + φˆ n2 x¯ n , θˆ , r + 1+ 4 ∂xi i=1   n−1 ∂αn−1 ˙  ∂αi−1 zn ξn  − zi θˆ − γ ξn  tanh an ∂ θˆ ∂ θˆ i=2   ⎫ ⎬ 16 tanh2 υznn ∂αn−1 − Hn + n1 + n2 r˙ + ⎭ ∂r zn    zn Hn + dn + 1 − 16 tanh2 (56) υn where Hn = ( ni=1 h2i (¯xi ))/2. 

∂αn−1 ∂αn−1 ˙ θˆ (gi xi+1 + fi (¯xi )) − ˆ ∂xi ∂ θ i=1    n−1  ∂αn−1 ∂αi−1 zn ξn  − r˙ zi γ ξn  tanh − an ∂r ∂ θˆ

−

(55)

i=1

n−1  ∂αn−1 i=1

¯n hi (xi (t − τi )) + 

i=1

Choosing Pi (¯xi ) = (h2i (¯xi ))/2 and introducing a function tanh(zn /υn ), it yields that    n−1 zn  zn ∂αn−1 2 ˙ + VPn ≤ zn fn (¯xn ) + gn u + 2 2 ∂xi −



−

∂xi

  (gi xi+1 + fi (¯xi )) + φˆ n1 x¯ n , θˆ , r

i=1

(50)

(54)

i=1

n−1  ∂αn−1

+ n2 + dn . − Pi (xi (t − τi ))) + n1

t−τi

(53)

Substituting (52)–(54) into (51), then    n−1 zn  zn ∂αn−1 2 + V˙ Pn ≤ zn fn (¯xn ) + gn u + 2 2 ∂xi

i=1

∂α ∂α ¯ n − n−1 θ˙ˆ − n−1 r˙ + ∂r ∂ θˆ

VPn =

j=1

, φˆ (¯ where φˆ n1 (¯xn , θˆ , r), n1 n2 xn , θˆ , r), n2 , and dn (t) are defined in (36) with i = n. Similarly to (37) and (38), it follows that:

kj z2j + gi zi zi+1

j=2

Using the same estimation methods in (36), we have     ¯ n | ≤ zn φˆ n1 x¯ n , θˆ , r + n1 + zn φˆ n2 x¯ n , θˆ , r |zn  ⎤ ⎡  n−1   ∂αn−1 2 ⎦ z2n ⎣ + + dn (t) (52) + n2 1+ 4 ∂xj



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Then, we choose the intermediate control signal as    n−1 zn  zn ∂αn−1 2 −1 + uˆ = −gn fn (¯xn ) + kn zn + 2 2 ∂xi i=1

−

n−1  ∂αn−1 i=1

∂xi

  (gi xi+1 + fi (¯xi )) + φˆ n1 x¯ n , θˆ , r 

2 

n−1   ∂αn−1 zn 1+ 4 ∂xi i=1   n−1 zn ξn  ∂αn−1 ˙  ∂αi−1 − zi θˆ − γ ξn  tanh an ∂ θˆ ∂ θˆ i=2   16 tanh2 υznn ∂αn−1 1 − Hn + g2n zn r˙ + ∂r zn 2 

+ φˆ n2 (¯xn , θˆ , r) +

+ gn−1 zn−1

(57)

where kn > 0 is a design parameter. Similarly to (42), the following inequality holds: + n2 + dn V˙ Pn ≤ −kn z2n + gn zn (u − uˆ ) + n1    z 1 n Hn − g2n z2n + 1 − 16 tanh2 υn 2 − gn−1 zn−1 zn .

i=1

Considering the adaptive law (14), the following expression can be obtained: n  bk0 θ˜ θˆ c¯ ki z2i − − r + Dn−1 + Cn V˙ zn ≤ − γ λ0 i=1   n   2 zi 1 − 16 tanh Hi + υi i=1

n 

bk0 2 c¯ θ˜ − r + Dn 2γ λ0 i=1    n  zi 1 − 16 tanh2 Hi + υi

≤−

ki z2i −

(64)

i=1

IV. S TABILITY A NALYSIS OF THE C LOSED -L OOP S YSTEM (58)

(59)

where δn is an approximation error, and n3 denotes a positive constant. Following the same procedures (44) and (45), the following equations hold:   zn ξn  + δθ an gn − gn zn uˆ ≤ gn zn θ ξn  tanh an 1 1 2 + g2n z2n + n3 (60) 2 2   z ξ  ˆ n  tanh n n . (61) gn zn u = −gn zn θξ an By combining (60) with (58) and (61), the inequality below can be easily verified   zn ξn  + Cn V˙ Pn ≤ −kn z2n − bzn θ˜ ξn  tanh a   n  zn Hn − gn−1 zn−1 zn + 1 − 16 tanh2 (62) υn

The main result and the stability analysis are summarized and proved in this section. Theorem 1: Consider the system (1) consisting of the adaptive laws (14), the feasible virtual control signal (12), and the actual controller (15). Under Assumptions 1–3, by using the above design procedures, for the bounded initial conditions, the boundedness of all the signals in the closed-loop system can be guaranteed by the proposed controller design procedures. The following lemma are necessary for the proof of Theorem 1. Lemma 4 [39]: For 1 ≤ j ≤ n, consider the set υj by / υj , the inequality

υj := {zj ||zj | < 0.2554υj }. Then, for zj ∈ 1 − 16 tanh2 (zj /υj ) ≤ 0 is satisfied. Proof: To prove the boundedness of all the signals, we process it by the following three cases with υj = υ, j = 1, . . . , n, where υ > 0 denotes an arbitrary small positive constant and  it follows that the set υj can be expressed as υ . / υ , j = 1, 2, . . . , n. Under this Case 1: For all zj ∈ situation, we can easily obtain that Hj is non-negative as j Hj = ( l=1 h2l (¯xl ))/2. Following Lemma 2, it follows that:   2 zj 1 − 16 tanh Hj ≤ 0. (65) υj With the help of (65), (64) becomes V˙ n ≤ −

n  i=1

2 +  +  + d . where Cn = δθ an gn + (1/2)n3 n n1 n2 Choose a Lyapunov function as

Vzn = Vzn−1 + VPn .

Differentiating (63) results    n n   bθ˜ ˙ zi ξi  2 θˆ − γ ki zi + zi ξi  tanh V˙ zn ≤ − γ ai i=1 i=1    n  c¯ 2 zi 1 − 16 tanh Hi . − r + Dn−1 + Cn + λ0 υi

where the Young’s inequality has been employed to deal with the term −(bk0 /γ )θ˜ θˆ as −(bk0 /γ )θ˜ θˆ ≤ −(bk0 /2γ )θ˜ 2 + (bk0 /2γ )θ 2 , Dn = Dn−1 + Cn + (bk0 /2γ )θ 2 .

While uˆ cannot be implemented because the functions fn (¯xn )(n = 1, . . . , n) and φˆ n1 (¯xn , θˆ , r), φˆ n2 (¯xn , θˆ , r) are unknown. Thus the fuzzy logic system φnT ξn is employed to model the desired and unknown control signal uˆ such that uˆ = φnT ξn + δn , |δn | ≤ n3

7

(63)

ki z2i −

bk0 2 c¯ ¯n θ˜ − r + D 2γ λ0

(66)

¯ n = Dn + ni=1 (1 − 16 tanh2 (zi /υi ))Hi is a constant. where D Since the signal r is bounded, z(t) is bounded. Thus (66) presents the boundedness of all the signals in the system.

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Case 2: For 1 ≤ j ≤ n, zj ∈ υ . In this case, |zj | ≤ 0.2554υ, thus zj is bounded. Therefore, for bounded zj , it is obvious that θˆ in (14) is bounded. Further, as θ is a constant, we know that θ˜ is bounded. Following |zj | ≤ 0.2554υj , the definition of Hj and Lemma 4, we have nj=1 (1−16 tanh2 (zj /υj ))Hj > 0. That is to say, we cannot deal with the term nj=1 (1−16 tanh2 (zj /υj ))Hj in this case as indicated in Case 1. The boundedness of all the other signals in the system will be analyzed step by step. At first, combining |z1 | ≤ 0.2554υ1 with z1 = x1 , then x1 is bounded. Moreover, following H1 = (h21 (x1 ))/2 and the continuity of h21 (x1 ), it follows that H1 is also bounded. For ¯ 1 . Now, it is follows ¯ 1 and 0 ≤ H1 ≤ H a positive constant H that V1 = Vz1 . As the derivative of Vz1 has been shown in (31), by taking (14) into consideration, then    c¯ 2 2 z1 ˙ H1 − r V1 ≤ −k1 z1 + g1 z1 z2 + 1 − 16 tanh υ1 λ0   bθ˜ ˙ z1 ξ1  ˜ − bz1 θξ1  tanh + θˆ + D1 a1 γ    z1 c¯ H1 − r ≤ −k1 z21 + g1 z1 z2 + 1 − 16 tanh2 υ1 λ0   n  bk0 θ˜ θˆ zi ξi  ˜ − + D1 . (67) + bθ zi ξi  tanh ai γ i=2

With 0 ≤ ξi  ≤ 1 and the boundedness of zi , we can see that ni=2 zi ξi  tanh((zi ξi )/ai ) is bounded. To be convenient for analysis, M1 is considered as the upper bound of n i=2 zi ξi  tanh((zi ξi )/ai ). Then, it holds bθ˜ M1 −

bk0 2 bk0 2 bk0 2 bk0 θ˜ θˆ ≤− θ˜ + θ + θ˜ + bk0 γ M12 γ 2γ 2γ 4γ bk0 2 θ˜ + K1 ≤− (68) 4γ

where K1 = (bk0 /2γ )θ 2 + bk0 γ M12 . ¯ 1 and Furthermore, considering |zi | ≤ 0.2554υ, 0 ≤ H1 ≤ H Lemma 4, it follows that: (0.2554υ)2 1 2 z1 + λc2 2λ   2  2 z1 ¯ 1. 0 ≤ 1 − 16 tanh H1 ≤ H υ1

g1 z1 z2 ≤

From (67) to (70), it can be concluded that   1 2 bk0 2 c¯ ˙ z − θ˜ − r + K¯ 1 V1 ≤ − k1 − 2λ 1 4γ λ0

(69)

b 2 θ˜ . 2γ

bk0 θ˜ θˆ γ   n  zi ξi  ˜ zi ξi  tanh + bθ + Cl ai

V˙ l ≤ V˙ l−1 − kl z2l + gl zl zl+1 −

i=1,i=l

  n ∂αj−1  zk ξk  γ zk ξk  tanh ak ∂ θˆ k=l+1 j=2    zl Hl − gl−1 zl−1 zl + 1 − 16 tanh2 υl

−

l 

zj

(73)

2 +  +  + d . where Cl = δθ al gl + (1/2)l3 l l1 l2 Similarly, applying the same techniques presented in pre vious steps, let ni=1,i=l zi ξi  tanh((zi ξi )/(ai )) ≤ Ml and ¯ l , where Ml and H ¯ l are positive constants. 0 ≤ Hl ≤ H It follows:   n  zi ξi  bk0 θ˜ θˆ ˜ − zi ξi  tanh + bθ γ ai i=1,i=l

bk0 2 bk0 θ˜ θˆ ≤− θ˜ + Kl γ 4γ 1 2 z2 + 2λc2 (0.2554υ)2 − gl−1 zl−1 zl + gl zl zl+1 ≤ 2λ    zl ¯l Hl ≤ H 0 ≤ 1 − 16 tanh2 υl ≤ bθ˜ Ml −

(74) (75) (76)

where Kl = (bk0 /2γ )θ 2 + bk0 γ Ml2 . Similarly, by employing the smoothness of αl−1 and ˆ the boundedness of the boundedness of zl−1 and θ, ˆ ˆ (∂α n l−1 )/(∂ θ) can be obtained. Let |(∂αl−1 )/(∂ θ)| ≤ Nl−1 , k=l+1 γ zk ξk  tanh((zk ξk )/ak ) ≤ γ Mγ l , it is obviously that   l n  ∂αj−1  zk ξk  zj γ zk ξk  tanh − ak ∂ θˆ k=l+1 j=2      l ∂αj−1   zj ≤ γ Mγ l ∂ θˆ   j=2 ≤ 0.2554(l − 2)

(70)

l 

Nj−1 γ Mγ l .

(77)

j=2

Repeating the procedures in the former steps, V˙ l−1 follows: (71)

¯ 1 + λc2 ((0.2554υ)2 )/2 + D1 . where K¯ 1 = K1 + H To this end, assuming that for k = 2, . . . , l − 1, the ˆ can be guaranteed. boundedness of Vk (t), xk and (∂αk )/(∂ θ) , it follows that xl is also bounded. Considering xl = zl + α l−1 Furthermore, as Hl = ( li=1 h2i (¯xi ))/2 is a positive continuous function of [x1 , . . . , xl ]T , the boundedness of Hl also can be obtained. The Lyapunov function is Vl = Vl−1 + VPl +

Differentiating Vl , it presents that

(72)

V˙ l−1 ≤ −

 l−1   1 2 (l − 1)bk0 2 c¯ ki − zi − θ˜ − r + K¯ l−1 . 2λ 4γ λ0 i=1

(78) From (74) to (78), then V˙ l ≤ −

l   i=1

ki −

 1 2 lbk0 2 c¯ z − θ˜ − r + K¯ l 2λ i 4γ λ0

(79)

2 2 ¯ ¯ where K¯ l = l Kl−1 + Kl + Hl + 2λc (0.2554υ) + 0.2554(l − 2) j=2 Nj−1 γ Mγ l + Cl .

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Since ki > (1/2λ), it holds that ki − (1/2λ) > 0. At the same time, z(t) is bounded as the signal r is bounded. Then, the boundedness of V˙ l (t) and αl can be obtained by similar discussion previously. Furthermore, following xl+1 = zl+1 + αl , xl+1 is bounded. To this end, it follows the same way that αj , u and xj+1 , j = l + 1, . . . , n − 1 are all bounded in Case 2. / υ , define I Case 3: For the case zi ∈ υ , while zj ∈ ˆ as the subsystem consisting of zi ∈ υ , and then zi , θ and ˜θ can be proven to be bounded for i ∈ I as shown in / υ , define Case 2. While for those zj ∈ J as the sub/ υ and consider the Lyapunov system consisting of zj ∈ function as V J = Vz1 +

 j∈ J ,j=1

b 2 θ˜ . 2γ

VPj +

(80)

All the zj , j = 1, 2 satisfy zj ∈ / υ . Based on the controller design procedure and the expression of V˙ z1 , the derivative of V J can be calculated as V˙ J ≤ −



$ % gj zj zj+1 − gj−1 zj−1 zj

kj z2j +

j∈ J

j∈ J

⎛

j∈

J

j+1∈ J j∈ J

+

gj zj zj+1 −



j+1∈ I j∈ J

j−1∈ J j∈ J

gj zj zj+1 −

gj−1 zj−1 zj



j−1∈ I j∈ J

gj−1 zj−1 zj .

i=2 j+1∈ J j∈ J

k=j+1

i=2

k=j+1

(82)

I

The first term also can be canceled in the back≤ Ni−1 , stepping, by denoting |(∂αi−1 )/(∂ θˆ )| n γ z ξ  tanh((z ξ )/a ) ≤ γ M . Similar to (77), k k k k k γj k=j+1 it holds ⎛ ⎞   j n   ∂αi−1 ⎝  zk ξk  ⎠ − zi γ zk ξk  tanh ak ∂ θˆ j+1∈ j∈ J

I

i=2

k=j+1

The first two terms in (82) can be canceled during backstepping. Furthermore, the last two terms are −

 j−1∈ I j∈ J

gj−1 zj−1 zj +

≤

 j+1∈ I j∈ J

 1 z2j + 2λ

j∈

J

≤

gj zj zj+1 

j−1∈ I j+1∈ J

2λc2 (0.2554υ)2 .

bk0 θ˜ θˆ bk0 2 θ˜ + K (85) ≤− γ 4γ

where K = (bk0 /2γ )θ 2 + bk0 γ M 2 . Furthermore, the fourth term in (81) gives ⎛ ⎞   j n  ∂αi−1 ⎝  zk ξk  ⎠ − zi γ zk ξk  tanh ak ∂ θˆ k=j+1 j∈ J i=2 ⎛ ⎞   j n   ∂αi−1 ⎝  zk ξk  ⎠ zi γ zk ξk  tanh =− ak ∂ θˆ

j+1∈ j∈ J

$ % gj zj zj+1 − gj−1 zj−1 zj =

J

⎛ ⎞   j n    ∂αi−1 ⎝ zk ξk  ⎠ − zi γ zk ξk  tanh . ak ∂ θˆ

The second term of (81) can be rewritten as



I

Similarly, with the similar techniques indicated in the previous procedure, let j∈ zj ξj  tanh((zj ξj )/aj ) ≤ M, where I M is a positive constant. It holds that ⎛ ⎞   # # z ξ  bθ˜ ⎝ ˙  j j ⎠ θˆ − γ zj #ξj # tanh γ aj ≤ bθ˜ M −

j∈ J



j∈



  j n  zk ξk  ∂αi−1  zi γ zk |ξk  tanh − ak ∂ θˆ k=j+1 j∈ J i=2    zj c¯ 1 − 16 tanh2 Hj + D1 − r + λ0 υj j∈ J  + Cj . (81)

j∈ J

By taking (14) into consideration, the third term in (81) can be expressed as ⎞ ⎛   # # z ξ  bθ˜ ⎝ ˙  j j ⎠ θˆ − γ zj #ξj # tanh γ aj j∈ J ⎛   n zj ξj  bθ˜ ⎝ − k0 θˆ γ zj ξj  tanh = γ aj j=1 ⎞    zj ξj  ⎠ − γ zj ξj  tanh aj j∈ J    zj ξj  bk0 θ˜ θˆ . (84) + bθ˜ zj ξj  tanh =− γ aj

j∈

⎞  zj ξj  bθ˜ ⎝ ˙  ⎠ + γ zj ξj  tanh θˆ − γ aj

9

(83)

 j+1∈ I j∈ J

0.2554( j − 2)

j 

Ni−1 γ Mγ j . (86)

i=2

Following Lemma 4 and the definition of Hj , it is obviously that:    zj 1 − 16 tanh2 Hj ≤ 0. (87) υj j∈

J

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Example 1: Consider the following second-order nonlinear system as: ⎧ z˙ = −z + x12 + 3 ⎪ ⎪ ⎪ ⎪ ⎪ x˙1 = −x2 + x12 e−0.5x1 + x1 sin (x1 )z ⎪ ⎪

 ⎪ ⎨ + x1 (t − τ1 ) sin x22 (t − τ2 ) (89) ⎪ x˙2 = u + x1 x22 + x12 + x1 x2 z ⎪ ⎪

 ⎪ ⎪ ⎪ + x2 (t − τ2 ) ln 1 + x12 (t − τ1 ) ⎪ ⎪ ⎩ y = x1 where f1 (x1 ) = x12 e−0.5x1 , f2 (x1 , x2 ) = x1 x22 + x12 , 1 = x1 sin (x1 )z, 2 = x1 x2 z, h1 = x1 (t − τ1 ) sin(x22 (t − τ2 )), h2 = x2 (t − τ2 ) ln(1 + x12 (t − τ1 )), and q(z, x1 ) = −z + x12 + 3 is the unmodeled dynamics. Note that Assumption 1 can be easily verified to be satisfied. In order to check Assumption 2 for z-subsystem in (96), consider Vz (z) = z2 , then   V˙ z (z) = 2z −z + x12 + 3 ≤ −2z2 + Fig. 1.

Block diagram of control scheme.

Finally, from (74) to (77), it gives   1 2 bk0 2 c¯ ˙ kj − z − θ˜ − r + K¯ (88) V J ≤− 2λ j 4γ λ0 J 2 ¯ where K = K + j∈ J (1 − 16 tanh (zj /υj ))Hj + j 0.2554(j − 2) 2 2 j−1∈ I 2λc (0.2554υ) + j+1∈ i=2 I j+1∈ J j∈ J Ni−1 γ Mγ j + j∈ Cj + D1 . J Since kj > (1/2λ), it is known that kj − (1/2λ) > 0. Then, z(t) is bounded as the boundedness of the signal r. Similarly ˆ as the discussion in Case 1, the boundedness of αj , u, zj , θ , θ˜ , and z(t) are ensured for j ∈ J in this case. To this end, according to the discussion in Cases 1–3, it can be concluded that the boundedness of all the closed-loop signals can be guaranteed. The overall scheme for the control algorithm through step 1 to step n is shown by Fig. 1. Remark 3: Note that several similar results on nonlinear systems with time-delay and unmodeled dynamics based on backstepping have been reported in [8] and [38]. The main differences between the proposed results with the previous work are summarized as follows. 1) Both unmodeled dynamics and time-delay of the controlled systems are not considered simultaneously in the existing works. 2) θ = max{φi  : i = 1, 2, . . . , n} is employed as the estimated parameter in this paper. Therefore, only one adaptive law is necessary to be updated online for nth order nonlinear systems. V. S IMULATION E XAMPLE Two simulation examples are showed in this section to illustrate the effectiveness of the proposed method.

1 z2 (2z)2 + εx14 + 9ε + . 4ε ε

By choosing ε = 2.5, it follows that: V˙ z (z) ≤ −1.2z2 + 2.5x14 + 22.5. Then, by defining α1 (|z|) = 0.5z2 , α2 (|z|) = 2z2 , c0 = 1.2, d0 = 22.5, and μ(|x1 |) = 2.5x14 , Assumption 2 is satisfied. Taking c¯ = 1 ∈ (0, c0 ) and defining the dynamic signal as r˙ = −r + 2.5x14 + 22.5.

(90)

In this simulation, choose the time-delays τ1 = τ2 = 3 s. To construct fuzzy controller, the fuzzy membership functions are chosen as follows: μF2 (xi ) = e−0.5∗(xi +2)

2

μF4 (xi ) = e−0.5∗(xi −0)

2

i

2

μF5 (xi ) = e−0.5∗(xi −1)

2

i

i

i

μF6 (xi ) = e−0.5∗(xi −2) . 2

i

μF3 (xi ) = e−0.5∗(xi +1)

(91)

Furthermore, the virtual and real control signals and the adaptive laws are constructed as follows:   z1 ξ1  (92) α1 (X1 ) = −θˆ ξ1  tanh a1   z2 ξ2  (93) u = −θˆ ξ2  tanh a2   2  zi ξi  ˆθ˙ = − k0 θˆ . (94) γ zi ξi  tanh ai i=1

Given the initial conditions [x1 (0), x2 (0), z(0), θˆ ]T = [1, 1, 1, 0], the simulation can be carried out by choosing the design constants as γ = 2, k0 = 20 and a1 = 0.2, a2 = 0.2. Then, Figs. 2–4 show the simulation results. Apparently, it is obvious that the proposed control approach can guarantee ˆ the boundedness of the variables x1 , x2 , u, and θ.

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Fig. 2.

State variables x1 and x2 .

Fig. 5.

State variables x1 and x2 .

Fig. 3.

ˆ Adaptive parameter θ.

Fig. 6.

ˆ Adaptive parameter θ.

Fig. 4.

Actual control input u.

Fig. 7.

Actual control input u.

Example 2: To further show the effectiveness of our results, a controlled Brusselator model in [40] with external disturbances was taken into consideration as follows: ⎧ 2 ⎪ ⎨x˙1 = C − (D + 1)x1 + x1 x2 + d1 (t, x¯ 2 ) (95) x˙2 = Dx1 − x12 x2 + (2 + cos(x1 ))u + d2 (t, x¯ 2 ) ⎪ ⎩ y = x1 where x1 and x2 are the concentrations of the reaction intermediates. u denotes the control input. d1 (t, x¯ 2 ) and d2 (t, x¯ 2 ) are the external disturbance terms. C, D > 0 are parameters which describe the supply of “reservoir” chemicals. As a practical chemical reaction, the existences of unmodeled dynamics and time-delay are inevitable for this model. Then, by adding

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the unmodeled dynamics and time-delay terms in this model, we hold ⎧ z˙ = −z + x12 + 3 ⎪ ⎪ ⎪ ⎪ ⎪ x˙1 = C − (D + 1)x1 + x12 x2 + 1 (x, z, t) ⎪ ⎪ ⎪ ⎨ + h1 (x1 (t − τ1 )) (96) ⎪ x˙2 = Dx1 − x12 x2 + (2 + cos(x1 ))u ⎪ ⎪ ⎪ ⎪ ⎪ + 2 (x, z, t) + h2 (x2 (t − τ2 )) ⎪ ⎪ ⎩ y = x1 where q(z, x1 ) = −z + x12 + 3 denotes the unmodeled dynamics. h1 (x1 (t − τ1 )) = x12 (t − τ1 ) and h2 (x2 (t − τ2 )) = x13 (t − τ1 )x2 (t − τ2 ) are the time-delay functions. Furthermore, we choose the dynamic disturbance terms 1 (x, z, t) = 0.1x1 x2 z,

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2 (x, z, t) = 0.1 sin(x2 )z, C = 0.1, D = 3, and τ1 = τ2 = 4. Then, the Assumptions 1 and 2 can be satisfied easily as the same methods in Example 1. In this example, the fuzzy membership functions (91) and the dynamic signal (90) are also employed. By applying the virtual and real control signals and the adaptive laws (92)–(94) with a1 = a2 = 0.76, γ = 20, and k0 = 1.46, this simulation can be carried out under the initial conditions [x1 (0), x2 (0), z(0), θˆ ]T = [1.5, 1.5, 1, 0]. The results are displayed in Figs. 5–7. VI. C ONCLUSION This paper mainly focuses on adaptive fuzzy control for a class of nonlinear time-delay systems in strict-feedback form. By employing the property of hyperbolic tangent function, Lyapunov–Krasovskii functionals as well as the backstepping approach, a novel adaptive fuzzy control scheme contained only one adaptive parameter is proposed. The stability analysis shows that all the signals of the closed-loop system remain semi-globally uniformly ultimately bounded in the sense of mean square. Simulation result finally illustrates the effectiveness of the proposed approach. It should be pointed out that the work in this paper does not consider the problem of timevarying delays. Then, they may occur in practical engineering. So, how to control such a nonlinear system with time-varying delays is our future research direction. R EFERENCES [1] H. Gao, Y. Zhao, and T. Chen, “Fuzzy control of nonlinear systems under unreliable communication links,” IEEE Trans. Fuzzy Syst., vol. 17, no. 2, pp. 265–278, Apr. 2009. [2] H. Gao and T. Chen, “Stabilization of nonlinear systems under variable sampling: A fuzzy control approach,” IEEE Trans. Fuzzy Syst., vol. 15, no. 5, pp. 972–983, Oct. 2007. [3] X. Su, L. Wu, P. Shi, and Y. Song, “Model reduction of Takagi–Sugeno fuzzy stochastic systems,” IEEE Trans. Cybern., vol. 42, no. 6, pp. 1574–1585, Dec. 2012. [4] K. Nina, L. Valery, M. Sergey, and R. Mikhail, “Research of periodic oscillations in control systems with fuzzy controllers,” Int. J. Innov. Comput. Inf. Control, vol. 11, no. 3, pp. 985–997, 2015. [5] S. Yin and Z. Huang, “Performance monitoring for vehicle suspension system via fuzzy positivistic C-means clustering based on accelerometer measurements,” IEEE/ASME. Trans. Mechatronics, DOI: 10.1109/TMECH.2014.2358674. [6] H. Li, H. Gao, P. Shi, and X. Zhao, “Fault tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach,” Automatica, vol. 50, no. 7, pp. 1825–1834, Jul. 2014. [7] Y. Liu, S. Tong, and W. Wang, “Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems,” Fuzzy Sets. Syst., vol. 160, no. 19, pp. 2727–2754, Oct. 2009. [8] Y. Liu, W. Wang, S. Tong, and Y. Liu, “Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters,” IEEE Trans. Cybern., vol. 40, no. 1, pp. 170–184, Jan. 2010. [9] J. Qiu, H. Tian, Q. Lu, and H. Gao, “Non-synchronized robust filtering design for continuous-time T–S fuzzy affine dynamic systems based on piecewise Lyapunov functions,” IEEE Trans. Cybern., vol. 43, no. 6, pp. 1755–1766, Dec. 2013. [10] L. Wu and D. W. C. Ho, “Fuzzy filter design for It o stochastic systems with application to sensor fault detection,” IEEE Trans. Fuzzy Syst., vol. 17, no. 1, pp. 233–242, Feb. 2009. [11] I. Kanellakopoulos, P. Kokotovic, and A. Morse, “Systematic design of adaptive controller for feedback linearizable systems,” IEEE Trans. Autom. Control., vol. 36, no. 11, pp. 1241–1253, Nov. 1991.

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Shen Yin (M’12–SM’15) received the B.E. degree in automation from the Harbin Institute of Technology, Harbin, China, in 2004, the M.Sc. degree in control and information systems, and the Ph.D. degree in electrical engineering and information technology from the University of Duisburg-Essen, Essen, Germany, in 2007 and 2011. His current research interests include model-based and data-driven fault-diagnosis, fault-tolerant control, and big data focused on industrial electronics applications.

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Peng Shi (M’95–SM’98–F’15) received the B.Sc. degree in mathematics from the Harbin Institute of Technology, Harbin, China, in 1982; the M.E. degree in systems engineering from Harbin Engineering University, Harbin, China, in 1985; the Ph.D. degree in electrical engineering from the University of Newcastle, Callaghan, NSW, Australia, in 1994; the Ph.D. degree in mathematics from the University of South Australia, Adelaide, SA, Australia, in 1998; and the D.Sc. degree in Science from the University of Glamorgan, Pontypridd, U.K., in 2006. He was a Senior Scientist with the Defence Science and Technology Organization, Canberra, ACT, Australia, a Lecturer, and a Post-Doctorate with the University of South Australia. He is currently a Professor with the University of Adelaide, Adelaide, Victoria University, Melbourne, VIC, Australia, and University of Glamorgan. His current research interests include system and control theory, computational intelligence, and operational research. He has published widely in the above areas. Dr. Shi has actively served in the editorial board of a number of journals, including Automatica, the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON F UZZY S YSTEMS, the IEEE T RANSACTIONS ON C YBERNETICS, the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS —PART I: R EGULAR PAPERS , and the IEEE ACCESS. He is a fellow of the Institution of Engineering and Technology and the Institute of Mathematics and its Applications.

Hongyan Yang received the B.Sc. degree in mathematics from Bohai University, Jinzhou, China, in 2013, and she is currently pursuing the M.E. degree in automation from Harbin Institute of Technology, Harbin, China. Her current research interests include adaptive fuzzy control, adaptive neural control, and nonlinear systems.