Adaptive fuzzy control of MIMO nonlinear systems with fuzzy dead zones

Applied Soft Computing Journal 80 (2019) 700–711

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

Adaptive fuzzy control of MIMO nonlinear systems with fuzzy dead zones Hang Su, Weihai Zhang

∗

College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China

highlights • • • •

Fuzzy dead zones are researched for more general MIMO nonlinear systems. The slopes of the dead zones are fuzzy values rather than deterministic ones. Only one adaptive parameter needs to be updated online for each subsystem. The ‘‘explosion of complexity’’ issue can be avoided by employing DSC technique.

article

info

Article history: Received 16 May 2018 Received in revised form 2 April 2019 Accepted 26 April 2019 Available online 3 May 2019 Keywords: Multi-input multi-output nonlinear systems Fuzzy dead zones Fuzzy logic systems Backstepping Dynamic surface control

a b s t r a c t In this paper, the problem of adaptive fuzzy tracking control is investigated for a class of multi-input multi-output nonlinear systems with fuzzy dead zones. The virtual control gain functions and uncertain functions considered in the studied system are all unknown. Fuzzy logic systems are employed to approximate the unknown functions. With the combination of adaptive backstepping design technique and dynamic surface control method, the problem caused by differentiating nonlinear functions repeatedly is avoided. Furthermore, only one adaptive parameter needs to be updated online for each subsystem, which reduces the computation burden considerably. The presented controller not only guarantees the desired control performance, but also guarantees the boundedness of all closed-loop signals. Simulation results are shown to demonstrate the effectiveness of the proposed algorithm. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In the past several decades, adaptive backstepping control of nonlinear systems has drawn considerable attention and some important results have been acquired, such as [1–3]. In the meantime, due to the effective approximation ability, fuzzy logic systems or neural networks (NNs) have been utilized for adaptive control of nonlinear systems. Therefore, with the help of fuzzy or neural techniques, many adaptive backstepping control schemes have been widely researched for nonlinear uncertain systems [4– 12]. In [4,5], fuzzy logic systems are employed to approximate the unknown nonlinear functions, and state observers are designed for time-delay systems [4] and uncertain switched nonlinear systems [5]. Adaptive fuzzy output-feedback control approaches are proposed for nonlinear systems with immeasurable states and input saturation in [6,7]. Adaptive neural control was developed in [8] for perturbed pure-feedback nonlinear systems, in [9] for uncertain nonlinear strict-feedback systems with full-state constraints, and in [10] for nonstrict-feedback nonlinear systems, ∗ Corresponding author. E-mail address: [email protected] (W. Zhang). https://doi.org/10.1016/j.asoc.2019.04.040 1568-4946/© 2019 Elsevier B.V. All rights reserved.

respectively. By employing the backstepping recursive design technique and the approximation property of NNs, stochastic nonlinear nonstrict-feedback systems and stochastic nonlinear time-delay systems were considered in [11] and [12], respectively. In short, many works have been carried out by relying on adaptive fuzzy or neural backstepping techniques [13–19]. The past decades have been witnessed a growing interest in control synthesis of nonlinear systems with nonsmooth nonlinearities, and dead zone is one of the significant nonsmooth nonlinearities. Dead zone nonlinearity is frequently encountered in many industrial processes, and this kind of unfavorable factor usually destroys the system performances and even becomes the source of instability. Therefore, intensive researches have been made on nonlinear systems with this issue such as [20–25]. By resorting to adaptive dead zone inverses, measurable dead zone inputs were considered in nonlinear systems [20,21]. In [22], the controlled system preceded by a non-symmetric dead-zone input was represented as an uncertain nonlinear system subject to a linear input. An inverse to the dead-zone nonlinearity is constructed for a class of nonlinear systems with unknown functions and immeasurable states in [23]. Dead-zone input nonlinearity was studied for flexible model of hypersonic flight vehicle in [24].

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

Based on the information of the dead-zone slopes as well as treating the unknown inputs coefficients as a system uncertainty, reference [25] developed a new adaptive fuzzy output-feedback control method. By utilizing the information of the bounds of the dead-zone slopes, adaptive fuzzy control problems were researched in [26] for nonstrict-feedback nonlinear system and in [27] for strict-feedback nonlinear system, respectively. In [28], the dead-zone was represented as a time-varying system with a bounded disturbance. By combining a sector-bounded property of a hysteretic quantizer with a simplified dead zone model, reference [29] established a novel connection between control signal and system input. In addition, unknown dead zones were developed for large-scale time delay systems [30] and multi-input multi-output (MIMO) nonlinear systems [31,32]. Recently, fuzzy dead zone was proposed in [33], where the dead zone input of the nonlinear systems is fuzzy. However, the control algorithm in [33] was valid for single-input single-output (SISO) nonlinear system. To the best of the authors’ knowledge, the control issue of MIMO nonlinear systems with fuzzy dead zones is still an open problem. It is well known that with the use of the approximation-based adaptive control algorithm, many corresponding adaptive parameters will be generated. To cope with this problem, a powerful approximation-based adaptive fuzzy or neural control algorithm was proposed [34–36]. Moreover, in the backstepping design process, virtual controllers are differentiated time and again. As a result, increasing the order of the system increases the complexity of the controller. In order to tackle this problem, the dynamic surface control (DSC) technique was proposed in [37], in which the design procedure was simplified. In [38,39], the problem of ‘explosion of complexity’ in traditional backstepping design is avoided by utilizing DSC technique, and adaptive DSC algorithms were further extended to more general classes of pure-feedback systems [39]. In [40], an adaptive fuzzy backstepping DSC approach was rendered for a class of MIMO nonlinear systems with immeasurable states. In addition, the adaptive DSC technique was also developed for kinds of practical systems such as magnetic levitation systems [41] and mechanical systems [42], etc. Therefore, DSC method is of fundamental importance from a theoretic point of view and is of great practical interest for applications. Although some efforts have been put on MIMO nonlinear systems, fuzzy dead zones and adaptive DSC method have not been simultaneously studied in MIMO nonlinear systems and there are few works on adaptive fuzzy control for such systems. Inspired by the aforementioned discussion, the problem of approximation-based adaptive fuzzy tracking control for MIMO nonlinear systems with fuzzy dead zones is investigated in this paper. Compared with the existing results, the main contributions of this paper are highlighted as follows. (1) For the first time, the control problem of more general MIMO nonlinear systems with fuzzy dead zones is researched. The primary technical difficulty arising from fuzzy dead zones is overcome by defuzzifying the fuzzy slopes of the dead zones and applying the integrated design method in [33]. However, reference [33] only considered the case of SISO systems. (2) This work is different from the previous works for MIMO nonlinear systems in existing researches, where the problems of dead zones were ignored in [13–15], and the control methods in [31,32] were limited to the issue that the slopes of the considered dead zones are deterministic values rather than fuzzy ones. Therefore, it deserves further investigation for MIMO systems with fuzzy dead zones.

701

(3) The computation burden is significantly reduced. On one hand, by exploiting the maximum norm of the weighting vectors as the estimation parameter, there is only one adaptive parameter that needs to be updated online for each subsystem. On the other hand, based on DSC technique, the ‘explosion of complexity’ issue can be avoided and the whole controller design process is simplified. The rest of this paper is organized as follows. In Section 2, we introduce the control problem of MIMO nonlinear system with fuzzy dead zones, as well as some preliminaries. In Section 3, we present an adaptive fuzzy DSC scheme and the stability analysis for the developed MIMO nonlinear control systems. In Section 4, simulation results are depicted to demonstrate the effectiveness of the proposed method. The final section is a conclusion for the whole paper. 2. Preliminaries and problem formulation 2.1. Nonlinear control problem Consider the following MIMO nonlinear system:

⎧ ⎨x˙ j,pj = fj,pj (x¯ j,pj ) + ϑj,pj (x¯ j,pj )xj,pj +1 x˙ = fj,mj (X , u¯ j−1 ) + ϑj,mj (X , u¯ j−1 )Ξ˜ j (uj ) ⎩ j,mj yj = xj,1 , j = 1, . . . , n; pj = 1, . . . , mj − 1,

(1)

where x¯ j,pj = [xj,1 , xj,2 , . . . , xj,pj ]T ∈ Rpj stands for the state variable for the first pj differential equations of the jth subsystem, X = [xT1 , xT2 , . . . , xTn ]T with xj = [xj,1 , xj,2 , . . . , xj,mj ]T . y = [y1 , y2 , . . . , yn ]T ∈ Rn is the output. u¯ j = [u1 , u2 , . . . , uj ]T is the input for the first j subsystems. fj,pj (·) and ϑj,pj (·) are the unknown smooth nonlinear functions of the system, and ϑj,pj (·) is called the ˜ j (uj ) stands for the fuzzy dead zone virtual control gain function. Ξ output of the actuator, which is defined as follows:

⎧ ⎨ℓ˜ j (uj − bjr ), Ξ˜ j (uj ) = 0, ⎩˜ ℓj (uj − bjl ),

uj ≥ bjr , bjl < uj < bjr , uj ≤ bjl ,

(2)

where bjr and bjl are the breakpoints of the fuzzy dead zone. The slope of the dead zone ℓ˜ j is a fuzzy value and is defined as follows

ℓj,qj ℓ j,1 ℓ j,2 ℓ˜ j = + + ··· + , oj,1

oj,2

(3)

oj , q j

where ℓj,1 , ℓj,2 , . . . , ℓj,qj are all the possible values of ℓ˜ j . oj,ij is the fuzzy grade of ℓj,ij , ℓj,ij ̸ = 0(ij = 1, 2, . . . , qj ). We suppose that ℓj,i

ℓj,ij > 0 and ℓj,1 < ℓj,2 < · · · < ℓj,qj . ‘ o j ’ expresses relationship j,ij between the mapping of ℓj,ij and oj,ij , ‘+ ’ denotes as a collection in the universe of discourse Uj = {ℓj,1 , ℓj,2 , . . . , ℓj,qj }. As the value ˜ j (uj ) is uncertain, the controller design of the dead zone output Ξ ˜ j (uj ) needs to be defuzzified cannot be accomplished directly. Ξ by the following center-of-gravity method [33]:

⎧ ⎨ℓ¯ j (uj − bjr ), ˜ j (uj )) = 0, Dj (Ξ ⎩¯ ℓj (uj − bjl ),

uj ≥ bjr , bjl < uj < bjr , uj ≤ bjl ,

(4)

where Dj (·) is a center-of-gravity defuzzification operation of Ξ˜ j (uj ), The defuzzified value ℓ¯ j is given as

∑qj ℓ¯ j =

ij =1

∑qj

oj,ij ℓj,ij

ij =1

oj,ij

.

(5)

˜ j (uj )) in (4) can be further expressed as Then, Dj (Ξ ˜ j (uj )) = ℓ¯ j uj + z¯j (t) Dj (Ξ

(6)

702

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

sets Fjl and Gl are associate with the fuzzy membership functions µF l (xj ) and µGl (y), respectively. N is the number of rules. The

with

⎧ ⎨−ℓ¯ j bjr , z¯j (t) = −ℓ¯ j uj , ⎩ ¯ −ℓj bjl ,

uj ≥ bjr , bjl < uj < bjr , u ≤ bjl ,

j

output of the fuzzy system can be expressed as [43] l=1

where z¯j (t) is a bounded function and satisfies |¯zj (t)| ≤ Z¯j , Z¯j = max{ℓj,qj |bjr |, ℓj,qj |bjl |}. Remark 1. Compared with previous MIMO nonlinear systems with deterministic dead zone inputs in [30–32], fuzzy dead zones are taken into account in the MIMO nonlinear system (1), in which the slopes of the investigated dead zones are fuzzy values. Therefore, the controller design for the system (1) is different from the controller construction for the existing MIMO nonlinear systems with deterministic dead zone models. Remark 2. Although there exists an adaptive fuzzy control method for the nonlinear system with fuzzy dead zone in [33], the presented algorithm in [33] was developed in the sense of SISO systems, they cannot guarantee the tracking performances for MIMO systems in theory. Remark 3. In the networked industrial control systems, the devices are connected to each other via communication cables with limited capacity. Because of the effect of the physical properties of the actuators and external interference, the parameters of the dead zone cannot guaranteed to always be precise and certain. It is very difficult to describe the complex actuator output by conventional certain dead zone model due to the vague information existing in the actuators. In addition, under the influence of these uncertainties, the stability of the practical systems such as mechanical and electrical components decrease seriously. Therefore, the effects of fuzzy dead zones should be taken into consideration. The investigated system (1) has not only important theoretical significance but also great practical interest for the synthesis of practical networked industrial control system. Assumption 1. The reference signal ydj (t) is a sufficiently smooth function of t, and ydj , y˙ dj and y¨ dj are bounded. Assumption 2 ([35]). There exist unknown constants ϑj,0 and ϑ¯ j0 such that 0 < ϑj0 ≤ |ϑj,pj (·)| ≤ ϑ¯ j0 < ∞.

∏ Θl nj=1 µF l (xj ) j , y(x) = ∑N ∏n j=1 µF l (xj )] l=1 [ ∑N

(7)

Without loss of generality, we further assume ϑj,pj (·) ≥ ϑj0 > 0.

j

where Θl = maxy∈R µGl (y). Simultaneously, we define the fuzzy basis function as

∏n

µF l (xj ) j , ∏n j=1 µF l (xj )] l=1 [

Ψl (x) = ∑N

j=1

j

then the fuzzy logic system can be designed as y(x) = Θ T Ψ (x),

where Θ = [Θ1 , Θ2 , . . . , ΘN ] and Ψ (x) = [Ψ1 (x), Ψ2 (x), . . . , ΨN (x)]T . Lemma 2 ([43]). f (x) is a continuous function defined on a compact set Ω . Then, for any constant σ > 0, there exists a fuzzy logic system (8) such that sup |f (x) − Θ T Ψ (x)| ≤ σ .

if the constant η > 0, ι > 0 and the function θ (t) > 0, then φˆ (t0 ) ≥ 0 implies φˆ (t) ≥ 0 for t ≥ t0 . The control objective of this paper is to design an adaptive fuzzy tracking controller to guarantee that: (1) the system output can track the reference output signal ydj , (2) all the signals of the closed-loop system are semi-globally uniformly ultimately bounded. 2.2. Fuzzy Logic systems To approximate the continuous function f (x) defined on a compact set Ω , a fuzzy logic system needs to be designed. Define the following If-Then fuzzy rules: Rl : If x1 is F1l and . . . and xn is F ln , then y is Gl , l = 1, 2, . . . , N, where x = [x1 , . . . , xn ]T and y are the input and output of the fuzzy system, respectively. Fuzzy

(9)

x∈Ω

3. Adaptive fuzzy controller design In the subsequent section, by combining the dynamic surface control with backstepping technique, adaptive fuzzy controllers will be designed for the slope of the dead zone ℓj,ij and the fuzzy value ℓ˜ j , respectively.

3.1. Controller design for the slopes of the dead zones ℓj,ij (j = 1, 2, . . . , n; ij = 1, 2, . . . , qj ) For convenience, we define the following constant as

δj = max

{

∥Θj,pj ∥2 ϑj0

, pj = 1, 2, . . . , mj

} (10)

with ϑj0 being defined in Assumption 2, and the estimate of δj is denoted by δˆ j and let δ˜ j = δj − δˆ j . The virtual control signal υj,pj is designed as follows:

υj,pj = −ςj,pj ωj,pj −

Lemma 1 ([29]). Consider the following differential equation:

φ˙ˆ (t) = −ηφˆ (t) + ιθ (t),

(8) T

δˆj ωj,pj ΨjT,pj (Xj,pj )Ψj,pj (Xj,pj ), 2℘j2,p j

j = 1, 2, . . . , n; pj = 1, 2, . . . , mj − 1,

(11)

where ςj,pj and ℘j,pj are positive design parameters, Ψj,pj (Xj,pj ) is the basis function vector, and ωj,pj is defined on the following change of coordinates:

ωj,1 = xj,1 − ydj , ωj,pj = xj,pj − ρj,pj j = 1, 2, . . . , n; pj = 2, . . . , mj ,

(12)

where ρj,pj is the output of a first-order filter with the intermediate control signal υj,pj −1 in (11) as the input. ωj,pj stands for the pj th surface error of the jth subsystem. In addition, for 1 ≤ j ≤ n, the adaptive laws are expressed as mj

∑ ηj ωj2,g ΨjT,g (Xj,g )Ψj,g (Xj,g ) − λj δˆj , δˆj (0) ≥ 0, δ˙ˆj = 2℘j2,g g =1 where ηj and λj are positive design constants.

(13)

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

Theorem 1. Consider the MIMO nonlinear system with fuzzy dead zones in (1), under Assumptions 1–2, if we choose the control law uj as uj = −

[

1

ςj,mj ωj,mj

ℓj,ij

] δˆj T + 2 ωj,mj Ψj,mj (Xj,mj )Ψj,mj (Xj,mj ) , ℘j,mj

(14)

system (1) are semi-globally ultimately uniformly bounded. Proof. Step j, 1(1 ≤ j ≤ n). Consider the following Lyapunov function: 1 2

ϑj0

ωj2,1 +

2ηj

δ˜j2

(15)

with ηj being given in (13) and ϑj0 being defined in Assumption 2. Then, we calculate the time derivative of Vj,1 as V˙ j,1 = ωj,1

= ωj,1

(

ϑj0 ˙ δ˜j δˆj ηj ) ϑj0 2 ϑj0 − δ˜j δˆ˙j − ω , ηj 2 j,1

fj,1 + ϑj,1 xj,2 − y˙ dj

(

f¯j,1 + ϑj,1 xj,2

)

−

(16)

where f¯j,1 = fj,1 − y˙ dj + ((ϑj0 ωj,1 )/2). According to Lemma 2, the unknown smooth function f¯j,1 can be approximated by a fuzzy logic system as f¯j,1 = ΘjT,1 Ψj,1 (Xj,1 ) + ϖj,1 (Xj,1 ),

|ϖj,1 (Xj,1 )| ≤ σj,1

(17)

with ϖj,1 being the approximation error, σj,1 being a positive constant, and Xj,1 = [xj,1 , ydj , y˙ dj ]T . By combining with (17) and utilizing Young’s inequality, we have

ωj,1 f¯j,1 = ωj,1

ΘjT,1

+

℘

2

+

ϑj0 2

ωj2,1 +

σ

+

℘

2

+

ϑj0 2

ωj2,1 +

ϵ˙j,2 = ρ˙ j,2 − υ˙ j,1 = −

2ϑj0

δ˙ˆj δˆj T Ψ + ω ˙ Ψ ωj,1 ΨjT,1 Ψj,1 j,1 j,1 j,1 2℘j2,1 2℘j2,1 ( δˆj ∂ Ψj , 1 ∂ Ψj , 1 ∂ Ψj , 1 T + 2 ωj,1 Ψj,1 x˙ j,1 + y˙ dj + y¨ dj ∂ xj,1 ∂ ydj ∂ y˙ dj ℘j,1 ) ∂ Ψj,1 ˙ ˆ + δj . ∂ δˆj

According to (21)–(23) and (12) with (j = 1, . . . , n; pj = 2), we have V˙ j,1 ≤ −ςj,1 ϑj0 ωj2,1 + ϑj,1 ωj,1 ωj,2 + ϑj,1 ωj,1 ϵj,2 + τj,1

ϑj0 + δ˜j ηj

+

(18)

,

(24)

1

ωj2,pj −1 + ϵj2,pj −1

(25)

2

∑

ςj,g ϑj0 ωj2,g

g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

+ τj,pj −1 +

∑

)

pj −1

ϑj,g ωj,g ωj,g +1 +

∑

ϑj,g ωj,g ϵj,g +1

g =1

pj −1

ϑj0 δˆj ωj2,1 ΨjT,1 (Xj,1 )Ψj,1 (Xj,1 ). 2℘j2,1

V˙ j,1 ≤ −ςj,1 ϑj0 ωj2,1 + ωj,1 ϑj,1 (xj,2 − υj,1 ) +

( p∑ j −1

ϑj0 δ˜j ηj

g =1

(20)

℘j2,1

) ∑ ( ωj,g ϵj,g ϵj2,g − + ϵj,g Pj,g , κj,g κj,g ∑pj −1

2 2 where τj,pj −1 = g =1 [(℘j,g /2) + (σj,g /(2ϑj0 ))]. Similarly to (23), the time derivative of the output error of pj th filter is

ϵ˙j,pj = ρ˙ j,pj − υ˙ j,pj −1 = −

ϵj,pj κj,pj

+ Pj,pj (ωj,1 , . . . , ωj,pj , ϵj,2 , . . . , ϵj,pj , δˆj , ydj , y˙ dj , y¨ dj ),

. (21)

(27)

where Pj,pj (·) is a continuous function of variables including ωj,1 ,

δˆj 2℘j2,p −1

ω˙ j,pj −1 ΨjT,pj −1 Ψj,pj −1

j

)

(26)

g =2

Pj,pj (·) = ςj,pj −1 ϑj0 ω ˙ j,pj −1 +

2

ηj 2 T ωj,1 Ψj,1 (Xj,1 )Ψj,1 (Xj,1 ) − δ˙ˆj 2℘j2,1

+

. . ., ωj,pj , ϵj,2 , . . ., ϵj,pj , δˆj , ydj , y˙ dj , y¨ dj , and is expressed as

Then, substituting (20) into (19) results in

(

1 2

pj −1

.

Based on the virtual control signal υj,1 in (11) and Assumption 2, we have

ϑj0 + + δ˜j 2ϑj0 ηj

)

g =1

℘ ϑj0 2 ωj,1 δˆj ΨjT,1 (Xj,1 )Ψj,1 (Xj,1 ) + ϑj,1 ωj,1 xj,2 + 2 2 2℘j,1 ( ) 2 σj,1 ϑj0 ηj 2 T ˙ ˜ ˆ + + δj ωj,1 Ψj,1 (Xj,1 )Ψj,1 (Xj,1 ) − δj . (19) 2ϑj0 ηj 2℘j2,1

σj2,1

ηj 2 T ωj,1 Ψj,1 (Xj,1 )Ψj,1 (Xj,1 ) − δ˙ˆj 2℘j2,1

pj −1

2 j,1

ωj,1 ϑj,1 υj,1 ≤ −ςj,1 ϑj0 ωj2,1 −

(

where τj,1 = ℘j2,1 /2 + (σj2,1 /(2ϑj0 )). Step j, pj (1 ≤ j ≤ n, 2 ≤ pj ≤ mj − 1). It is assumed that there exist υj,g and ρj,g +1 (j = 1, . . . , n; g = 1, . . . , pj − 1) such that the Lyapunov function

Then, V˙ j,1 in (16) can be rewritten as V˙ j,1 ≤

ϵj,2 + Pj,2 (ωj,1 , ωj,2 , ϵj,2 , δˆj , ydj , y˙ dj , y¨ dj ), (23) κj,2

Pj,2 (·) = ςj,1 ϑj0 ω ˙ j,1 +

2ϑj0

σ

(22)

where Pj,2 (·) is a continuous function of variables including ωj,1 , ωj,2 , ϵj,2 , δˆj , ydj , y˙ dj and y¨ dj , with the following expression

V˙ j,pj −1 ≤ −

2 j,1

2 j,1

ρj,2 (0) = υj,1 (0).

satisfies the following inequality

ϑj0 2 ≤ ωj,1 δj ΨjT,1 (Xj,1 )Ψj,1 (Xj,1 ) 2℘j2,1 2 j,1

κj,2 ρ˙ j,2 + ρj,2 = υj,1 ,

Vj,pj −1 = Vj,pj −2 +

∥Θj,1 ∥Ψj,1 (Xj,1 ) + ωj,1 ϖj,1 (Xj,1 )

∥ Θj , 1 ∥ ϑj0 2 ∥Θj,1 ∥2 T ≤ ωj,1 Ψj,1 (Xj,1 )Ψj,1 (Xj,1 ) ϑj0 2℘j2,1 2 j,1

In order to avoid differentiating υj,1 repeatedly, let υj,1 pass through a first-order filter with a time constant κj,2 to get the state variable ρj,2 as

Define ϵj,2 = ρj,2 − υj,1 , then we obtain ρ˙ j,2 = −(ϵj,2 )/(κj,2 ) and

˙ and the adaptive laws δˆ j (j = 1, 2, . . . , n) in (13), where ςj,mj and ℘j,mj are positive design parameters, then all the signals of the

Vj,1 =

703

+

δ˙ˆj 2℘j2,p −1 j

ωj,pj −1 ΨjT,pj −1 Ψj,pj −1

704

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

+

δˆj ℘

2 j,pj −1

ωj,pj −1 ΨjT,pj −1

∂ Ψj,pj −1

+··· +

∂ xj,pj −1

(

∂ Ψj,pj −1 ∂ xj,1

x˙ j,pj −1 +

pj −1

x˙ j,1

∂ Ψj,pj −1

∂ ydj ) ∂ Ψj,pj −1 ∂ Ψj,pj −1 ˙ + δˆj . y¨ dj + ∂ y˙ dj ∂ δˆj

+

1 2

+

1

(28)

The time derivative of Vj,pj is calculated as

−

ϵj2,pj κj,pj

f¯j,pj + ϑj,pj xj,pj +1 +

+ ϵj,pj Pj,pj −

ϑj0 2

ϑj,g ωj,g ϵj,g +1

g =1

g =2

2

V˙ j,pj = V˙ j,pj −1 + ωj,pj

∑

pj ( ) ∑ ϵj2,g ωj,g ϵj,g + − + ϵj,g Pj,g κj,g κj,g

y˙ dj

ωj2,pj + ϵj2,pj .

(

ϑj,g ωj,g ωj,g +1 +

g =1

Then, Consider the following Lyapunov function Vj,pj = Vj,pj −1 +

pj −1

∑

ϵj,pj

ϑj0 δ˜j ηj

pj (∑ g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

+τj,pj .(34)

Subsequently, we adopt the similar procedure as the above step and introduce the following first-order filter

κj,pj +1 ρ˙ j,pj +1 + ρj,pj +1 = υj,pj ,

)

)

ρj,pj +1 (0) = υj,pj (0),

(35)

where κj,pj +1 is a time constant, and ρj,pj +1 is a state variable. Let ϵj,pj +1 = ρj,pj +1 − υj,pj , then we obtain ρ˙ j,pj +1 = −(ϵj,pj +1 )/(κj,pj +1 )

κj,pj

ωj2,pj ,

(29)

and

ϵ˙j,pj +1 = ρ˙ j,pj +1 − υ˙ j,pj = −

where f¯j,pj = fj,pj + ((ϑj0 ωj,pj )/2). The unknown nonlinear function f¯j,pj can be approximated by the following fuzzy logic system as f¯j,pj = ΘjT,pj Ψj,pj (Xj,pj ) + ϖj,pj (Xj,pj ), |ϖj,pj (Xj,pj )| ≤ σj,pj ,

ϵj,pj +1 κj,pj +1

+ Pj,pj +1 (ωj,1 , . . . , ωj,pj +1 , ϵj,2 , . . . , ϵj,pj +1 , δˆj , ydj , y˙ dj , y¨ dj ), (36)

(30)

where ϖj,pj denotes the approximation error, σj,pj is a positive constant, and Xj,pj = [¯xTj,p , ρj,pj , ydj , y˙ dj , y¨ dj ]T with x¯ j,pj = j

where Pj,pj +1 (·) is a continuous function of variables including

ωj,1 , . . ., ωj,pj +1 , ϵj,2 , . . ., ϵj,pj +1 , δˆj , ydj , y˙ dj , y¨ dj , and is expressed in the following expression

[xj,1 , . . . , xj,pj ]T . Similarly to the aforementioned procedures, we have

δ˙ˆj δˆj T + Ψ Ψ ω ˙ ωj,pj ΨjT,pj Ψj,pj j , p j , p j j j,pj 2℘j2,p 2℘j2,p j j ( ∂ Ψ ∂ Ψ δˆj j , p j , pj j + 2 ωj,pj ΨjT,pj x˙ j,1 + · · · + x˙ j,pj ∂ xj,1 ∂ xj,pj ℘j,pj

Pj,pj +1 (·) = ςj,pj ϑj0 ω ˙ j,pj +

ωj,pj f¯j,pj ≤

℘j2,pj ϑj0 σj2,pj ϑj0 2 T 2 ) + (X ) Ψ ω δ Ψ (X + ω + . j,pj j,pj j,pj j,pj j j,pj 2 2 j,pj 2ϑj0 2℘j2,p j (31)

+

Combining (29) with (26) and (31), V˙ j,pj can be rewritten as

∂ Ψj,pj ∂ ydj ∂ Ψj,pj

follows

+

pj −1

V˙ j,pj ≤ −

∑

ςj,g ϑj0 ω

2 j,g

+ ϑj,pj ωj,pj xj,pj +1

pj −1

∑

pj −1

ϑj,g ωj,g ωj,g +1 +

g =1

∑

ϑj,g ωj,g ϵj,g +1

V˙ j,pj ≤ −

g =1

ϑj0 δ˜j ηj

+ τj,pj +

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

g =1

ϑj0 2℘j2,p

ωj2,pj δˆj ΨjT,pj (Xj,pj )Ψj,pj (Xj,pj ),

) + (32)

∑pj

ϑj0 δˆj ωj2,pj ΨjT,pj (Xj,pj )Ψj,pj (Xj,pj ). 2℘j2,p j (33)

Then, combining with (33), the inequality (32) can be rewritten as

+

g =1

pj ∑

pj (∑ g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g pj ∑

ϑj,g ωj,g ωj,g +1 +

)

ϑj,g ωj,g ϵj,g +1

g =1

pj ( ) ∑ ϵj2,g ωj,g ϵj,g − + ϵj,g Pj,g . κj,g κj,g

(37)

Step j, mj (1 ≤ j ≤ n). Take the following Lyapunov function as

Vj,mj = Vj,mj −1 +

ςj,g ϑj0 ωj2,g + ϑj,pj ωj,pj (xj,pj +1 − υj,pj )

1 2

1

ωj2,mj + ϵj2,mj .

(38)

2

Differentiating Vj,mj results in V˙ j,mj = V˙ j,mj −1 + ωj,mj

pj

∑

.

g =2

2 2 where τj,pj = g =1 [(℘j,g /2) + (σj,g /(2ϑj0 ))]. Combining the virtual control signal υj,pj in (11) with Assumption 2, we have

V˙ j,pj ≤ −

)

ςj,g ϑj0 ωj2,g + τj,pj

g =1

j

ωj,pj ϑj,pj υj,pj ≤ −ςj,pj ϑj0 ωj2,pj −

pj ∑

ϑj0 + δ˜j ηj

g =2

+

∂ Ψj,pj ˙ δˆj ∂ δˆj

g =1

pj ( ) ∑ ϵj2,g ωj,g ϵj,g + − + ϵj,g Pj,g κj,g κj,g pj (∑

y¨ dj +

Combining (34)–(36) with (12), we have

g =1

+

∂ y˙ dj

y˙ dj

−

ϵ

2 j,mj

κj,mj

(

f¯j,mj + ϑj,mj ℓj,ij uj + ϑj,mj zj (t) +

+ ϵj,mj Pj,mj −

ϑj0 2

ωj2,mj ,

ϵj,mj

)

κj,mj (39)

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

705

where f¯j,mj = fj,mj + ((ϑj0 ωj,mj )/2). As f¯j,mj is an unknown nonlinear function, there exists a fuzzy logic system such that

According to Young’s inequality, we have the following inequalities:

f¯j,mj = ΘjT,mj Ψj,mj (Xj,mj ) + ϖj,mj (Xj,mj ), |ϖj,mj (Xj,mj )| ≤ σj,mj

mj −1

with σj,mj being a positive constant, and Xj,mj = [X , ¯ ydj , y˙ dj , y¨ dj ]T . Similarly to the analysis in (31), we have T

ωj,mj f¯j,mj

uTj−1

(40)

, ρj,mj ,

+

ϑj0 2

ωj2,mj +

σ

2ϑj0

.

(41)

mj −1

∑

+

mj −1

∑

∑

ϑj,g ωj,g ωj,g +1 +

g =1

+

∑( g =2

+

ϑj,g ωj,g ϵj,g +1

ϑj0 δ˜j ηj

ϵ ωj,g ϵj,g − + ϵj,g Pj,g κj,g κj,g 2 j,g

mj (∑ g =1

+ τj,mj +

g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

δˆj ≤ − 2 ωj2,mj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ). 2℘j,m j

δˆj ωj2,mj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ).

g =1

∑

(

2

ϵj2,g Pj2,g 2νj

ωj2,mj + Z¯j2

+

)

νj

)

2

,

(48)

,

(49) (50)

mj ( ∑

1

2

2κj,g

ςj,g ϑj0 − ϑ¯ j0 −

mj ( ∑

1 2κj,g

)

3

1

Pj2,g

2

2νj

− ϑ¯ j0 −

)

ωj2,g

ϵj2,g −

ϑj0 λj 2 δ˜ + τ¯j,mj , 2ηj j (51)

V =

∑mj

g =1

n ∑

Vj,mj ,

(52)

j=1

V˙ ≤ (43)

n [ ∑

−

j=1

−

mj ( ∑ g =1

mj ( ∑ g =2

1 2κj,g

3

1

2

2κj,g

ςj,g ϑj0 − ϑ¯ j0 − 1

Pj2,g

2

2νj

− ϑ¯ j0 −

)

ϵj2,g −

)

ωj2,g

ϑj0 λj 2 δ˜ 2ηj j

]

+τ¯ ,

(53)

∑n

(44)

λj ϑj0 δ˜j δˆj + τj,mj + ωj,mj ϑj,mj zj (t) ηj

where τ¯ = ¯j,mj . Define Cj = min{2(ςj,g ϑj0 − ((3ϑ¯ j0 )/2) − j=1 τ (1/(2κj,g )))(1 ≤ g ≤ mj ), 2((1/(2κj,g )) − (ϑ¯ j0 /2) − (Pj2,g /(2νj )))(2 ≤ g ≤ mj ), λj } and C = min{C1 , . . . , Cn }. Then, (53) can be rewritten as V˙ ≤ −CV + τ¯ .

(54)

Integrating (54) over [0, t ] yields 0 ≤ V (t) ≤ V (0)e−Ct +

τ¯ C

(1 − e−Ct ),

which means that

ϑj,g ωj,g ωj,g +1

0 ≤ V (t) ≤ V (0)e−Ct +

ϑj,g ωj,g ϵj,g +1

g =1 mj ( ) ∑ ϵj2,g ωj,g ϵj,g + − + ϵj,g Pj,g , κ j,g κj,g g =2

ϑ¯ j0

+

then, we obtain the following inequality

mj −1

∑

(47)

[(℘j2,g /2) + (σj2,g /(2ϑj0 ))] + (ϑj0 λj /(2ηj ))δj2 + ¯2 ¯ g =2 (νj /2) + (ϑj0 /2)Zj . Take the whole Lyapunov function as

g =1

+

2κj,g

2κj,g

g =2

ϵj2,g

−

∑mj

mj −1

+

mj ( ∑ ωj2,g

where τ¯j,mj =

˙ Substituting (44) and the adaptive law δˆ j (13) into (42), we have ςj,g ϑj0 ωj2,g +

≤

g =2

j

mj ∑

,

(42)

ωj,mj ϑj,mj ℓj,ij uj ≤ −ςj,mj ϑj0 ωj2,mj

V˙ j,mj ≤ −

ωj2,g + ϵj2,g +1

)

g =2

−

Combining the actual control signal uj in (14) with Assumption 2 and (43), we have

ϑj0

(

2

g =1

[(℘j2,g /2) + (σj2,g /(2ϑj0 ))].

2℘j2,m

ϑ¯ j0 ωj2,g , (46)

mj ( ) ∑ ϵj2,g ωj,g ϵj,g − + ϵj,g Pj,g κj,g κj,g

)

δˆj 2 ω Ψ T (Xj,mj )Ψj,mj (Xj,mj ) ℘j2,mj j,mj j,mj

−

≤

mj ∑ g =1

∑ ϑ¯ j0 g =1

V˙ j,mj ≤ −

Remark 4. From Lemma 1 and (13), we know that δˆ j (0) ≥ 0 implies that δˆ j ≥ 0 for all t > 0. Therefore, the following inequality holds

−

ϑj,g ωj,g ϵj,g +1 ≤

)

ϑj0 2 ωj,mj δˆj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ), 2℘j2,m j

∑mj

where τj,mj =

g =1 mj −1

2

)

where νj is a positive constant. Then, substituting (46)–(50) into (45), we have

g =1

mj

g =1

+ω

ω

2 j,g +1

2 j,g

λj ϑj0 λj ϑj0 2 λj ϑj0 2 δ˜j δˆj ≤ − δ˜ + δ , ηj 2ηj j 2ηj j

g =1 mj −1

ϑj,g ωj,g ωj,g +1 ≤

ωj,mj ϑj,mj zj (t) ≤

+ ωj,mj (ϑj,mj ℓj,ij uj + ϑj,mj zj (t))

ςj,g ϑj0 ω

2 j,g

∑ ϑ¯ j0 (

g =1

Then, substituting (37) with (pj = mj − 1) and (41) into (39), one has V˙ j,mj ≤ −

mj −1

mj −1

∑

℘j2,mj ϑj0 2 T (X ) Ψ (X ) + ≤ δ Ψ ω j , m j , m j , m j j,mj j,mj j j j 2 2℘j2,m j 2 j,mj

∑

(45)

τ¯ C

.

(55)

From (55), we can see that all the signals of the closed-loop system are semi-globally uniformly √ ultimately bounded. At the same time, we get |ωj,1 (t)| ≤ 2(V (0) exp(−Ct) + (τ¯ /√ C )). As limt →∞ exp(−Ct) = 0, we have limt →∞ |ωj,1 (t)| ≤ 2τ¯ /C . Therefore, the tracking error can be made arbitrarily small by choosing the design parameters approximately.

706

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

3.2. Controller design for the fuzzy values ℓ˜ j (j = 1, 2, . . . , n)

+

mj ( ) ∑ ϵj2,g ωj,g ϵj,g − + ϵj,g Pj,g κj,g κj,g g =2

From the above analysis, we can see that the controller uj in (14) can make all the signals in the closed-loop system semiglobally uniformly ultimately bounded for ℓj = ℓj,ij (ij = 1, 2,

ϑj0 + δ˜j ηj

. . . , qj ). Subsequently, for a fuzzy value ℓ˜ j , an integrated fuzzy

mj (∑ g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

mj −1

controller will be designed to guarantee the boundedness of the closed-loop system and the desired tracking performance.

+

∑

mj −1

ϑj,g ωj,g ωj,g +1 +

g =1

Theorem 2. For the nonlinear MIMO system (1), if the adaptive laws in (13) and the following controller (56) are chosen, then (1) for the defuzzified value ℓ¯ j shown in (5), all the signals of the closed-loop system are semi-globally uniformly ultimately bounded and the system outputs converge to a small neighborhood of the desired outputs; (2) for a fuzzy value ℓ˜ j ∈ [ℓ¯ j − ιj , ℓ¯ j + ιj ] of the MIMO nonlinear system, where ιj is a disturbance, if 0 < ιj ≤ (ℓj,1 /2), then the boundedness of the closed-loop system and the desired tracking performance can be certified. uj = −

1

ℓ¯ j

[

ςj,mj ωj,mj

δˆj + 2 ωj,mj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ) ℘j,mj

]

.

V˙ j,mj ≤ −

ςj,g ϑj0 ω

mj −1

V˙ j,mj ≤ −

∑

ςj,g ϑj0 ωj2,g + ωj,mj

∑

+

g =1

+

∑

δˆj ωj,mj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ) ℘j2,mj

(56)

+ τj,mj

+

ϑj0 δ˜j ηj

V˙ j,mj ≤ −

mj (∑ g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

∑

ϑj,g ωj,g ωj,g +1

+

∑

ϑj,g ωj,g ϵj,g +1 + τj,mj + ωj,mj ϑj,mj z¯j (t).

As ℓ¯ j ∈ [ℓj,1 , ℓj,qj ] and 0 < ιj ≤ (ℓj,1 /2), the following inequality holds:

) ℓ˜ j ℓ¯ j − ιj ℓj,1 1 ≥ ≥1− ≥ . 2 ℓ¯ j ℓ¯ j 2ℓ¯ j Combining (43) with (60), we obtain

−

ℓ˜ j δˆj 2 ωj,mj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ) ℓ¯ j ℘j2,mj ≤−

δˆj ωj2,mj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ). 2℘j2,m j

ςj,g ϑj0 ω

+ ωj,mj

mj −1

V˙ j,mj ≤ −

∑

ςj,g ϑj0 ωj2,g −

g =1

ςj,mj 2

ϑj0 + 2 ωj,mj δˆj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ) 2℘j,m j

)

ϑj0 ωj2,mj +

λj ϑj0 δ˜j δˆj + τj,mj ηj

+ωj,mj ϑj,mj zj (t)

∑ g =1

g =1

(61)

˙ Substituting (61) and the adaptive law δˆ j (13) into (59) results in

+

ϑj,mj ℓ˜ j uj

(60)

(57)

mj −1 2 j,g

(59)

g =1

ϑj0 2 + ωj,mj δˆj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ), 2℘j2,m j

(

)

mj −1

ϑj,g ωj,g ϵj,g +1

Substituting (56) into (57) and repeating the similar analysis in (43)–(50), we can obtain that V˙ j,mj satisfies (51). Then, define the whole Lyapunov function as the form in (52), then V˙ can be deduced to satisfy (54). Therefore, based on the above analysis, for the defuzzified value ℓ¯ j , the controller (56) together with the adaptive law (13), guarantees that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded. (2) For a fuzzy value ℓ˜ j ∈ [ℓ¯ j − ιj , ℓ¯ j + ιj ] in MIMO nonlinear system (1), the similar analysis in Step j, pj (1 ≤ j ≤ n; 1 ≤ pj ≤ mj − 1) is left out. Step j, mj (1 ≤ j ≤ n). Consider Vj,mj as the form in (38). Similarly to the analysis (39)–(41), the time derivative of Vj,mj satisfies

∑

]

g =1

ηj 2 T ωj,g Ψj,g (Xj,g )Ψj,g (Xj,g ) − δ˙ˆj 2℘j2,g

mj −1

ςj,mj ωj,mj

g =2

+

g =2

g =1

(

mj −1

mj ( ) ∑ ϵj2,g ωj,g ϵj,g − + ϵj,g Pj,g κj,g κj,g mj (∑

ℓ˜ j ℓ¯ j )

ϑj0 ωj,mj δˆj ΨjT,mj (Xj,mj )Ψj,mj (Xj,mj ) 2℘j2,m j mj ( ) ∑ ϵj2,g ωj,g ϵj,g + − + ϵ j,g P j,g κj,g κj,g

g =1

ϑj0 + δ˜j ηj

−ϑj,mj

+

mj −1

ϑj,g ωj,g ωj,g +1 +

[

g =1

+ ωj,mj (ϑj,mj ℓ¯ j uj + ϑj,mj z¯j (t))

mj −1

(58)

Then, substituting the control law (56) into (58), we have

g =1

+

ϑj,g ωj,g ϵj,g +1

g =1

mj −1 2 j,g

∑

+τj,mj + ωj,mj ϑj,mj z¯j (t).

Proof. (1) For the defuzzified value ℓ¯ j , the stability analysis in Step j, pj (1 ≤ j ≤ n; 1 ≤ pj ≤ mj − 1) is similar to (15)–(37). Step j, mj (1 ≤ j ≤ n). Consider Vj,mj as the form in (38). Repeating the analysis in (39)–(41), we have

∑

)

mj −1

ϑj,g ωj,g ωj,g +1 +

∑

ϑj,g ωj,g ϵj,g +1

g =1

mj ( ) ∑ ϵj2,g ωj,g ϵj,g + − + ϵ j,g P j,g . κj,g κj,g g =2

(62)

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

Combining (46)–(50) with (62), we obtain the following inequality: mj −1

V˙ j,mj ≤ −

∑(

ςj,g ϑj0 −

3ϑ¯ j0 2

g =1

[ −

ςj,mj 2

mj

−

∑( g =2

where τ¯j,mj =

ϑj0 − 1 2κj,g

1 2κj,mj

−

ϑ¯ j0 2

−

)

1 2κj,g

−

3ϑ¯ j0

−

Pj2,g

]

2 2νj

)

ωj2,g

ωj2,mj

ϵj2,g −

ϑj0 λj 2 δ˜ + τ¯j,mj , (63) 2ηj j

∑mj

2 2 2 g =1 [(℘j,g /2) + (σj,g /(2ϑj0 ))] + (ϑj0 λj /(2ηj ))δj + ∑mj 2 ¯ ¯ ¯ g =2 (νj /2) + (ϑj0 /2)Zj . Let Cj = min{2(ςj,g ϑj0 − (3ϑj0 /2) − (1/(2κj,g )))(1 ≤ g ≤ mj − 1), 2((ςj,mj /2)ϑj0 − (1/(2κj,mj )) − (3ϑ¯ j0 )/2), 2(1/(2κj,g ) − (ϑ¯ j0 /2) − (Pj2,g /(2νj )))(2 ≤ g ≤ mj ), λj } and C = min{C1 , . . . , Cn }. Repeating the analysis in (52)–(53), we

can rewrite (63) as the form in (54), and then we obtain (55). That is, under the controller (56) and the adaptive law (13), for the fuzzy value ℓ˜ j ∈ [ℓ¯ j − ιj , ℓ¯ j + ιj ] with 0 < ιj ≤ (ℓj,1 /2), the tracking performance and boundedness of all the signals can be guaranteed. Remark 5. According to the inequality 0 ≤ V (t) ≤ V (0)e−Ct + (τ¯ /C√) in (55), we can see that the tracking error limt →∞ |ωj,1 (t)| ≤ 2τ¯ /C relies on the corresponding constants C and τ¯ . As the similar statements in [32], increasing C and decreasing τ¯ can decrease the tracking errors. Accordingly, smaller tracking errors can be obtained by selecting smaller ℘j,g and larger ςj,g , ηj . However, it is noted that if the value of C is too big, the control energy becomes larger. Therefore, approximate choice of design parameters should be carried out by balancing the relationship between the control action and transient performance. Remark 6. It should be pointed out that several control strategies have been proposed in [15,32,40,42] for MIMO nonlinear systems. The main differences between our result and the ones in [15,32, 40,42] are summarized as follows: (1) from the controlled MIMO nonlinear systems, the results are obtained in [32,42] for MIMO systems without virtual control gain function ϑj,pj , the dead zone problems are ignored in [15,40], and in this paper, we consider a class of MIMO nonlinear systems with virtual control gain functions and fuzzy dead zones, which are in more general forms. (2) The dead zone models developed in [32] are deterministic, i.e., the effect of the vague information is not taken into account, and in this paper, fuzzy dead zone models are presented to describe the uncertain of the dead zone. (3) For each subsystem, in [15,32,40,42], all the elements of weight vectors in fuzzy logic systems need to be estimated online. To alleviate the computation burden, we take δj = max{(∥Θj,pj ∥2 /ϑj0 ), pj = 1, 2, . . . , mj } as the estimated parameter in this paper, which means that only one adaptive parameter is required to be estimated online, and the computation burden can be reduced significantly. 4. Simulation examples In this section, the following two examples are depicted to illustrate the feasibility of the proposed algorithm. Example 1. Consider the following MIMO nonlinear systems with fuzzy dead zones:

⎧ ⎪ x˙ = −x1,1 + (1 + sin2 (x1,1 ))x1,2 , ⎪ ⎨ 1,1 x˙ 1,2 = x1,1, x1,2 + x2,1 + x2,2 + [1 + sin2 (x1,1 ) ⎪ +0.5 cos2 (x2,2 )]Ξ˜ 1 (u1 ), ⎪ ⎩ y1 = x1,1 ,

707

⎧ x˙ = −x2,1 + x2,2 , ⎪ ⎨ 2,1 x˙ 2,2 = (x1,2 + x2,1 )x2,2 − x1,1 u1 + [1 + 0.5 sin2 (u1 ) −0.5 sin(x2,1 x2,2 − x1,1 )]Ξ˜ 2 (u2 ), ⎪ ⎩ y2 = x2,1 .

(64)

˜ j (uj )(j = 1, 2) represent the inputs and where uj (j = 1, 2) and Ξ ˜ j (uj ) is given as follows: outputs of the dead zones, respectively. Ξ

⎧ ⎨ℓ˜ 1 (u1 − b1r ), Ξ˜ 1 (u1 ) = 0, ⎩˜ ℓ1 (u1 − b1l ), ⎧ ⎨ℓ˜ 2 (u2 − b2r ), Ξ˜ 2 (u2 ) = 0, ⎩˜ ℓ2 (u2 − b2l ),

u1 ≥ b1r , b1l < u1 < b1r , u1 ≤ b1l , u2 ≥ b2r , b2l < u2 < b2r , u2 ≤ b2l .

The parameters in the above dead zones are b1r = 0.8, b1l = −0.8, b2r = 1, b2l = −1. The nonlinear functions in system (64) are f1,1 (x1,1 ) = −x1,1 , ϑ1,1 (x1,1 ) = 1 + sin2 (x1,1 ), f1,2 (X ) = x1,1, x1,2 + x2,1 + x2,2 , ϑ1,2 (X ) = 1 + sin2 (x1,1 ) + 0.5 cos2 (x2,2 ), f2,1 (x¯ 2,1 ) = −x2,1 , ϑ2,1 (x¯ 2,1 ) = 1, f2,2 (X , u1 ) = (x1,2 + x2,1 )x2,2 − x1,1 u1 , ϑ2,2 (X , u1 ) = 1 + 0.5 sin2 (u1 ) − 0.5 sin(x2,1 x2,2 − x1,1 ). The reference signals are chosen as yd1 = yd2 = 0.5 sin(t) + sin(1.5t). Choose the following fuzzy membership functions:

−(x1,1 − 3 + l)2

] , 1 ,1 4 [ ] −(x2,1 − 3 + l)2 µF l (x2,1 ) = exp , µF l (x1,1 ) = exp

[

2 ,1

4

i=2,j=2

µF l (X ) = µF l (X ) = 1 ,2

2,2

∏

[ exp

i=1,j=1

−(xi,j − 3 + l)2 4

] ,

l = 1, 2, . . . , 5.

(65)

The fuzzy basis functions are Ψ1,1,l (x1,1 ) = µF l

∑ / 5i=1 µF i ,

1,1 1,1 ∑ 5 µ , Ψ (x ) = µ / µ and i l i 2 , 1 , l 2 , 1 i=1 F1,2 i=1 F2,1 F2 ,1 1,2 ∑ Ψ2,2,l (X ) = µF l / 5i=1 µF i , l = 1, 2, . . . , 5. Subsequently, we

Ψ1,2,l (X ) = µF l / 2 ,2

∑5

2,2

show the effectiveness of the presented approach under two cases, in which controller (14) constructed in Theorem 1 and controller (56) designed in Theorem 2 are applied in Case 1 and Case 2, respectively. Case 1. The slopes of dead zones ℓ˜ j = ℓj,ij (j, ij = 1, 2) are certain values. When ℓ˜ j = ℓj,ij is a certain value, we design ℓ˜ 1 = ℓ1,ij =

1.2, ℓ˜ 2 = ℓ2,ij = 1.1 and apply the controller design scheme in Theorem 1. To check the effects of the main design parameters ςj,pj and ℘j,pj on the control performances, we take the following two different groups of design parameters into consideration. Case 1.1 ς1,1 = 52, ς1,2 = 54, ς2,1 = 48, ς2,2 = 50, ℘1,1 = 1, ℘1,2 = 1, ℘2,1 = 2, ℘2,2 = 3, η1 = 10, η2 = 10, λ1 = 1, λ2 = 1, κ1,2 = 0.1, κ2,2 = 0.12. Case 1.2 ς1,1 = 32, ς1,2 = 34, ς2,1 = 28, ς2,2 = 30, ℘1,1 = 3, ℘1,2 = 3, ℘2,1 = 4, ℘2,2 = 4.5. The other design parameters are selected similarly as Case 1.1. The following simulation runs in the initial conditions [x1,1 (0), x1,2 (0), x2,1 (0), x2,2 (0)]T = [0.15, 0.1, 0.1, 0.2]T and [δˆ 1 (0), δˆ 2 (0)]T = [0.1, 0.05]T . Figs. 1–3 demonstrate the related simulation results. Fig. 1 displays the tracking performances under Case 1.1 and Case 1.2. Fig. 2 illustrates the system tracking errors yj − ydj (j = 1, 2) and control signals uj (j = 1, 2) under two cases. Fig. 3 provides the trajectories of the state variables xj,2 (j = 1, 2) and adaptive parameters δˆ j (j = 1, 2) under two cases. The simulation results shown in Figs. 1–3 reveal that the system output yj follows the given reference signal ydj in a bounded set, and all the signals of the closed-loop system are bounded.

708

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

Fig. 3. State variables xj,2 (j = 1, 2) and adaptive parameters δˆ j (j = 1, 2) for Case 1.1 and Case 1.2. Fig. 1. Tracking performances for Case 1.1 and Case 1.2.

Fig. 2. Tracking errors yj − ydj (j = 1, 2) and control signals uj (j = 1, 2) for Case 1.1 and Case 1.2.

Moreover, to further illustrate the control performance comparisons between Case 1.1 and Case 1.2, we show the corresponding numerical comparison results the following Table 1. The ∑in M sum of squared tracking errors ydj (m)]2 and the m = ∑M 1 [yj (m) − 2 sum of squared control actions m=1 [uj (m)] are taken as the indexes of the tracking error and control, respectively. M in the index is the number of sampling data. We calculate these related indexes from 0 to 20s with a sampling period of 0.01s. From Fig. 2 and Table 1, it is observed that better tracking performances can be obtained by choosing larger ςj,pj and smaller ℘j,pj . However, it may result in larger control energy at the same time. Therefore, a trade-off should be carefully made between the transient performance and the control energy. In order to illustrate the robustness of the proposed control method to the fuzzy dead zones, we employ the adaptive fuzzy control scheme in [15] to control the system (64). The design parameters and the initial conditions are selected similarly as the ones in Case 1.1. The simulation results are shown in Fig. 4. By comparing the simulation results with the ones in [15], it can be seen that the control algorithm in the work of [15] cannot guarantee the stability of the control system (64) because of the effects of dead zones. In addition, for each subsystem, the number of the adaptive parameters depends on the dimension of the

Fig. 4. Tracking performances and control signals in [15]. Table 1 Performance comparisons between Case 1.1 and Case 1.2. Performance comparisons

Case 1.1

Case 1.2

∑M 2 m=1 [y1 (m) − yd1 (m)] ∑M 2 [ u (m) ] =1 1 ∑m M 2 m=1 [y2 (m) − yd2 (m)] ∑M 2 [ u (m) ] m=1 2

0.0074 1915.9000 0.0146 904.0816

0.0183 466.2182 0.0409 241.4593

subsystem in [15], and in our paper, only one parameter need to be estimated online regardless of the dimension of subsystem. Case 2. The slopes of dead zones ℓ˜ j (j = 1, 2) are fuzzy values. In this case, for the fuzzy slopes ℓ˜ j (j = 1, 2), we adopt the controller (56) designed in Theorem 2. It is assumed that the universe of discourse of ℓ˜ j is Uj = {1, 1.5, 2}, i.e., ℓ1,1 = 1, ℓ1,2 = 1.5, ℓ1,3 = 2, ℓ2,1 = 1, ℓ2,2 = 1.5, ℓ2,3 =(2. The fuzzy)grade of ℓj,ij is represented as o(ℓj,ij ) =

√1 2πσj

exp

−(ℓj,ij −1.5)2 2σj2

with

σj =

1 for j, ij = 1, 2. Apply the control method in Theorem 2. 6 Choose the intermediate virtual signal υj,1 (11), the state variable

ρj,2 (22), and the adaptive parameter law δ˙ˆj (13). The design

parameters and the initial conditions are selected as Case 1.1.

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

709

Fig. 7. Two inverted pendulums connected by a spring and a damper.

Fig. 5. Tracking errors yj − ydj (j = 1, 2) and control signals uj (j = 1, 2) for

and w is distance between the connection points and is given by

ℓ˜ j (j = 1, 2).

√ w=

d2 + dl(sin θ1 − sin θ2 ) +

l2

[1 − cos(θ2 − θ1 )] (68) 2 with d = 0.5 m, k = 150 N/m, b = 1 N sec/m and a = 0.5 m. The relative angular position θ can be defined as θ = tan−1

(

θ − cos θ1 )

l (cos 2 2 l (sin 1 2

)

.

θ − sin θ2 )

d+

(69)

Tfj is assumed to be a LuGre friction model defined as Tfj = ζ0 δ˙ j + ζ1 δ˙ j + ζ2 θ˙j ,

δ˙j = θ˙j −

ζ0 |θ˙j | 2

Tc + (Ts − Tc ) exp(−|θ˙j /θ˙s | )

, j = 1, 2,

ζ0 = 1 Nm, ζ1 = 1 Nm sec, ζ2 = 1 Nm sec, θ˙s = 0.1 rad/sec, Ts = 2 Nm and Tc = 1 Nm. By defining x1,1 = θ1 , x1,2 = θ˙1 , x2,1 = θ2 and x2,2 = θ˙2 , the state space equations of (66) and (67) can be represented as Fig. 6. State variables xj,2 (j = 1, 2) and adaptive parameters δˆ j (j = 1, 2) for

ℓ˜ j (j = 1, 2).

The simulation results are given in Figs. 5–6. Fig. 5 depicts the tracking errors yj − ydj (j = 1, 2) and control signals uj (j = 1, 2) for the fuzzy slopes. Fig. 6 gives the trajectories of state variables xj,2 (j = 1, 2) and adaptive parameters δˆ j (j = 1, 2) for the fuzzy values ℓ˜ j (j = 1, 2). From Figs. 5–6, we can see that, even though ℓ˜ j is a fuzzy value, the desired tracking performances and the boundedness of the signals in the closed-loop system can be guaranteed.

⎧ ⎨x˙ 1,1 = x1,2 , x˙ = f (X ) + ϑ1,2 Ξ˜ 1 (u1 ), ⎩y1,2= x 1,2, 1 1,1 ⎧ ⎨x˙ 2,1 = x2,2 , x˙ = f (X , u1 ) + ϑ2,2 Ξ˜ 2 (u2 ), ⎩y2,2= x 2,2, 2 2,1

where f1,2 (·) = ϑ1,2 [m1 gl sin x1,1 − 0.5Fl cos(x1,1 − θ ) − Tf 1 ], ϑ1,2 = 1/J1 , f2,2 (·) = ϑ2,2 [m2 gl sin x2,1 − 0.5Fl cos(x2,1 − θ ) − Tf 2 ], ϑ2,2 = 1/J2 . The parameters in the above dead zones are b1r = 0.8, b1l = −0.8, b2r = 1, b2l = −1. ℓ˜ is a fuzzy value, of which universe is Uj = {1, 1.5, 2}, i.e., ℓ˜ 1,1 = 1, ℓ˜ 1,2 = 1.5, ℓ˜ 1,3 = 2, ℓ˜ 2,1 = 1, ℓ˜ 2,2 = 1.5, ℓ˜ 2,3 = 2. The fuzzy grade of ℓj,ij is represented as o(ℓj,ij ) =

Example 2. Consider the tracking control problem for two inverted pendulums composed of spring, damper connections, and nonlinear friction with fuzzy dead zone, as shown in Fig. 7. The dynamic equation of the inverted pendulum can be addressed as [42]

˜ 1 (u1 ), J1 θ¨1 = m1 gl sin θ1 − 0.5Fl cos(θ1 − θ ) − Tf 1 + Ξ

(66)

˜ 2 (u2 ), J2 θ¨2 = m2 gl sin θ2 − 0.5Fl cos(θ2 − θ ) − Tf 2 + Ξ

(67)

where θ1 and θ2 are angular positions, J1 = 1 kgm and J2 = 1 kgm2 denote the moments of inertia, m1 = 2 kg and m2 = 2.5 kg are the masses, l = 0.5 m, F = k(w − a) + bw ˙ is the force applied by the spring and damper at the connection points A, B, 2

(70)

√1 2πσj

( exp

−(ℓj,ij −1.5)2 2σj2

)

with σj =

1 6

for

j, ij = 1, 2. The reference signals are designed as yd1 = sin(1.5t) and yd2 = sin(t) + sin(1.5t). The fuzzy membership functions are chosen as similarly as in Example 1. Choose the intermediate control signal υj,1 in (11) with (pj = 1), the controller uj in (56)

˙

and adaptive law δˆ j in (13). In the simulation, the corresponding parameters are selected as ς1,1 = 42, ς1,2 = 44, ς2,1 = 38, ς2,2 = 40, ℘1,1 = 1.5, ℘1,2 = 2, ℘2,1 = 1.8, ℘2,2 = 2.2, η1 = 8, η2 = 8.5, λ1 = 0.8, λ2 = 0.8, κ1,2 = 0.12, κ2,2 = 0.15. The initial values of the variables and the initial conditions of the adaptive parameters are chosen as zero, expect for x1,1 (0) = π/3.6 and x2,1 (0) = π /4. The simulation results are addressed by Figs. 8–9. Fig. 8 provides the

710

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711

proposed control method, the desired tracking performance has been obtained even with the effects of fuzzy dead zones. Although some progress has been achieved for MIMO nonlinear systems with fuzzy dead zones, however, the issue for MIMO stochastic switched nonlinear systems with fuzzy dead zones needs our future attention, which is more interesting and challenging. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 61573227, 61633014, 61703248), the Research Fund for the Taishan Scholar Project of Shandong Province of China, and SDUST Research Fund, China (No. 2015TDJH105). Declaration of competing interest Fig. 8. Tracking errors yj − ydj (j = 1, 2) and control signals uj (j = 1, 2) for Example 2.

No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.04.040. References

Fig. 9. State variables xj,2 (j = 1, 2) and adaptive parameters δˆ j (j = 1, 2) for Example 2.

trajectories of the tracking errors yj − ydj (j = 1, 2) and control signals uj (j = 1, 2). Fig. 9 shows the trajectories of the state variables xj,2 (j = 1, 2) and adaptive parameters δˆ j (j = 1, 2). It can be shown that the proposed control scheme can guarantee the desired tracking performances and the boundedness of the closed-loop control system.

5. Conclusion In this paper, the problem of approximation-based adaptive fuzzy tracking control for MIMO nonlinear systems with fuzzy dead zones has been addressed. By combining with backstepping technique, an effective adaptive fuzzy tracking controller has been designed for each order subsystem whether the dead zone slope is certain or fuzzy. By taking the maximum norm of the ideal weighting vectors as the estimation parameter, the number of the adaptive parameters which need to be updated online is only one for each subsystem. Based on DSC scheme, the problem of ‘explosion of complexity’ caused by the derivation of virtual control signal, multi-order derivation and system complexity has been avoided, therefore, the whole controller design process has been simplified. The simulation results show that, under the

[1] W.X. Shi, Adaptive fuzzy control for multi-input multi-output nonlinear systems with unknown dead-zone inputs, Appl. Soft Comput. 30 (c) (2015) 36–47. [2] C.F. Hsu, C.M. Lin, T.T. Lee, Wavelet adaptive backstepping control for a class of nonlinear systems, IEEE Trans. Neural Netw. 17 (5) (2006) 1175–1183. [3] Y.J. Liu, S.M. Lu, S.C. Tong, X.K. Chen, C.L.P. Chen, D.J. Li, Adaptive controlbased barrier lyapunov functions for a class of stochastic nonlinear systems with full state constraints, Automatica 87 (2018) 83–93. [4] B. Chen, C. Lin, X.P. Liu, K.F. Liu, Observer-based adaptive fuzzy control for a class of nonlinear delayed systems, IEEE Trans. Syst. Man Cybern. Syst. 46 (1) (2016) 27–36. [5] S.C. Tong, S. Sui, Y.M. Li, Observer-based adaptive fuzzy tracking control for switched nonlinear systems with dead-zone, IEEE Trans. Cybern. 45 (12) (2015) 2816–2826. [6] Y.M. Li, S.C. Tong, T.S. Li, Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with input saturation, IEEE Trans. Cybern. 45 (10) (2015) 2299–2308. [7] S. Sui, Y.M. Li, S.C. Tong, Adaptive fuzzy control design and applications of uncertain stochastic nonlinear systems with input saturation, Neurocomputing 156 (25) (2015) 42–51. [8] H.Q. Wang, B. Chen, C. Lin, Adaptive neural tracking control for a class of perturbed pure-feedback nonlinear systems, Nonlinear Dynam. 72 (1–2) (2013) 207–220. [9] Y.J. Liu, J. Li, S.C. Tong, C.L.P. Chen, Neural network control-based adaptive learning design for nonlinear systems with full-state constraints, IEEE Trans. Neural Netw. Learn. Syst. 27 (7) (2016) 1562–1571. [10] B. Chen, H.G. Zhang, C. Lin, Observer-based adaptive neural network control for nonlinear systems in nonstrict-feedback form, IEEE Trans. Neural Netw. Learn. Syst. 27 (1) (2016) 89–98. [11] H.Q. Wang, B. Chen, K.F. Liu, X.P. liu, C. Lin, Adaptive neural tracking control for a class of nonstrict-feedback stochastic nonlinear systems with unknown backlash-like hysteresis, IEEE Trans. Neural Netw. Learn. Syst. 25 (5) (2014) 947–958. [12] T. Wang, J.B. Qiu, H.J. Gao, Adaptive neural control of stochastic nonlinear tome-delay systems with multiple constraints, IEEE Trans. Syst. Man Cybern. Syst. 47 (8) (2017) 1875–1883. [13] S.C. Tong, Y.M. Li, P. Shi, Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems, IEEE Trans. Fuzzy Syst. 20 (4) (2012) 771–785. [14] H. Lee, Robust adaptive fuzzy control by backstepping for a class of MIMO nonlinear systems, IEEE Trans. Fuzzy Syst. 19 (2) (2011) 265–275. [15] T.S. Li, S.C. Tong, G. Feng, A novel robust adaptive-fuzzy-tracking control for a class of nonlinear MIMO systems, IEEE Trans. Fuzzy Syst. 18 (1) (2010) 150–160. [16] S.C. Tong, B.Y. Huo, Y.M. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Trans. Fuzzy Syst. 22 (1) (2014) 1–15. [17] M.M. Arefi, J. Zarei, H.R. Karimi, Adaptive output feedback neural network control of uncertain non-affine systems with unknown control direction, J. Franklin. Inst. 351 (8) (2014) 4302–4316.

H. Su and W. Zhang / Applied Soft Computing Journal 80 (2019) 700–711 [18] Y.H. Li, S. Qiang, X.Y. Zhuang, O. Kaynak, Robust and adaptive backstepping control for nonlinear systems using RBF neural networks, IEEE Trans. Neural Netw. 15 (3) (2004) 693–701. [19] F. Wang, B. Chen, C. Lin, X.H. Li, Distributed adaptive neural control for stochastic nonlinear multigagent systems, IEEE Trans. Cybern. 47 (7) (2017) 1795–1803. [20] A. Taware, G. Tao, An adaptive dead-zone inverse controller for systems with sandwiched dead-zones, Internat. J. Control 76 (8) (2003) 755–769. [21] R.R. Selmic, F.L. Lewis, Deadzone compensation in motion control systems using neural networks, IEEE Trans. Automat. Control 45 (4) (2000) 602–613. [22] S. Ibrir, W.F. Xie, C.Y. Su, Adaptive tracking of nonlinear systems with non-symmetric dead-zone input, Automatica 43 (3) (2007) 522–530. [23] S.C. Tong, Y.M. Li, Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE Trans. Fuzzy Syst. 20 (1) (2012) 168–180. [24] B. Xu, Robust adaptive neural control of flexible hypersonic flight vehicle with dead-zone input nonlinearity, Nonlinear Dynam. 80 (3) (2015) 1509–1520. [25] Y.M. Li, S.C. Tong, Adaptive fuzzy output-feedback control of pure-feedback uncertain nonlinear systems with unknown dead zone, IEEE Trans. Fuzzy Syst. 22 (5) (2014) 1341–1347. [26] B. Chen, X.P. Liu, K.F. Liu, C. Lin, Fuzzy approximation-based adaptive control of nonlinear delayed systems with unknown dead zone, IEEE Trans. Fuzzy Syst. 22 (2) (2014) 237–248. [27] Y.M. Li, S.C. Tong, Y.J. Liu, T.S. Li, Adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on a small-gain approach, IEEE Trans. Fuzzy Syst. 22 (1) (2014) 164–176. [28] S.C. Tong, T. Wang, Y.M. Li, H.G. Zhang, Adaptive neural network output feedback control for stochastic nonlinear systems with unknown dead-zone and unmodeled dynamics, IEEE Trans. Cybern. 44 (6) (2014) 910–921. [29] F. Wang, Z. Liu, Y. Zhang, C.L.P. Chen, Adaptive quantized fuzzy control of stochastic nonlinear systems with actuator dead-zone, Inform. Sci. 370–371 (20) (2016) 385–401. [30] J. Zhou, Decentralized adaptive control for large-scale time-delay systems with dead-zone input, Automatica 44 (7) (2008) 1790–1799.

711

[31] Y.M. Li, S.C. Tong, T.S. Li, Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control directions and unknown dead zones, IEEE Trans. Fuzzy Syst. 23 (4) (2015) 1228–1241. [32] S.C. Tong, Y.M. Li, Adaptive fuzzy output feedback of MIMO nonlinear systems with unknown dead zone inputs, IEEE Trans. Fuzzy Syst. 21 (1) (2013) 134–146. [33] F. Wang, Z. Liu, Y. Zhang, X. Chen, C.L.P. Chen, Adaptive fuzzy dynamic surface control for a class of nonlinear systems with fuzzy dead zone and dynamic uncertainties, Nonlinear Dynam. 79 (3) (2015) 1693–1709. [34] L.J. Long, J. Zhao, Adaptive fuzzy output-feedback dynamic surface control of MIMO switched nonlinear systems with unknown gain signs, Fuzzy Sets and Systems 302 (1) (2016) 27–51. [35] Q. Zhou, P. Shi, Y. Tian, M.Y. Wang, Approximation-based adaptive tracking control for MIMO nonlinear systems with input saturation, IEEE Trans. Cybern. 45 (10) (2015) 2119–2128. [36] Y.M. Li, S.C. Tong, T.S. Li, Hybrid fuzzy adaptive output feedback control design for uncertain MIMO nonlinear systems with time-varying delays and input saturation, IEEE Trans. Fuzzy Syst. 24 (4) (2016) 841–853. [37] D. Swaroop, J.K. Hedrick, P.P. Yip, J.C. Gerdes, Dynamic surface control for a class of nonlinear systems, IEEE Trans. Automat. Control 45 (10) (2000) 1893–1899. [38] T.P. Zhang, S.S. Ge, Adaptive dynamic surface control of nonlinear systemms with unknown dead zone in pure feedback form, Automatica 44 (7) (2008) 1895–1903. [39] Y.M. Li, S.C. Tong, T.S. Li, Adaptive fuzzy output feedback dynamic surface control of interconnected nonlinear pure-feedback systems, IEEE Trans. Cybern. 45 (1) (2015) 138–149. [40] S.C. Tong, Y.M. Li, G. Feng, T.S. Li, Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems, IEEE Trans. Syst. Man Cybern. B 41 (4) (2011) 1124–1135. [41] H.J. Shieh, C.H. Hsu, An integrator-backstepping-based dynamic surface control method for a two-axis piezoelectric micropositioning stage, IEEE Trans. Control Syst. Technol. 15 (5) (2007) 916–926. [42] S.C. Tong, S. Sui, Y.M. Li, Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained, IEEE Trans. Fuzzy Syst. 23 (4) (2015) 729–742. [43] L.X. Wang, J.M. Mendel, Fuzzy basis functions universal approximation and orthogonal least squares learning, IEEE Trans. Neural Netw. 3 (5) (1992) 807–814.