Adaptive fuzzy control for full states constrained systems with nonstrict-feedback form and unknown nonlinear dead zone

Accepted Manuscript

Adaptive Fuzzy Control for Full States Constrained Systems with Nonstrict-Feedback Form and Unknown Nonlinear Dead Zone Jian Wu, Benyue Su, Jing Li, Xu Zhang, Xiaobo Li, Weisheng Chen PII: DOI: Reference:

S0020-0255(16)31210-5 10.1016/j.ins.2016.10.016 INS 12575

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Information Sciences

Received date: Revised date: Accepted date:

4 May 2016 7 July 2016 6 October 2016

Please cite this article as: Jian Wu, Benyue Su, Jing Li, Xu Zhang, Xiaobo Li, Weisheng Chen, Adaptive Fuzzy Control for Full States Constrained Systems with Nonstrict-Feedback Form and Unknown Nonlinear Dead Zone, Information Sciences (2016), doi: 10.1016/j.ins.2016.10.016

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Adaptive Fuzzy Control for Full States Constrained Systems with Nonstrict-Feedback Form and Unknown Nonlinear Dead Zone

Abstract

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Jian Wua,b , Benyue Sua,b , Jing Li‡,c , Xu Zhanga , Xiaobo Lid , and Weisheng Chene

This paper addresses the problem of direct adaptive fuzzy tracking control design for a class of uncertain nonstrict-feedback systems with nonlinear dead zone and full state constraints. Fuzzy logic systems are used to approximate some unknown nonlinear functions and less adjustable parameters are adopted in each bacestpping design process. This advantage is first to take into account the full

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state constrained nonstrict-feedback systems with input dead zone nonlinearity. To guarantee that the full state constraints are not violated, a novel adaptive fuzzy controller is developed by introducing Barrier Lyapunov Function with the error variables. Furthermore, it is proved that all the closed-loop signals remain semi-globally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin. Two simulation examples are provided to verify the effectiveness of the

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proposed control method.

Index Terms

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Nonstrict-feedback systems, adaptive fuzzy control, full state constraints, backspepping design.

I. I NTRODUCTION

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In the past decades, adaptive backstepping fuzzy control has become one of the most popular

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design methods to solve the regulation and tracking control problems for various nonlinear This work is supported by National Natural Science Foundation of China (61603003, 61673014, 61673308, 61203074), Natural Science Foundation of Anhui Province (1608085QF131), the Fundamental Research Funds for the Central Universities

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(K5051370014), the Key Program of Baoji University of Arts and Sciences (ZK14062), and the Foundation of University Research and Innovation Platform Team for Intelligent Perception and Computing of Anhui Province. a

School of Computer and Information, Anqing Normal University, Anqing 246011, China;

b

The University Key Laboratory

of Intelligent Perception and Computing of Anhui Province, Anqing Normal University, China; Statistics, Xidian University, Xi’an 710071, China; 721013, China;

e

d

c

School of Mathematics and

Department of Mathematics, Baoji University of Arts and Sciences, Baoji

School of Aerospace Science and Technology, Xidian University, Xi’an 710071, China.

‡ Corresponding author. E-mail addresses: [email protected](J. Wu), [email protected](W. Chen), [email protected](J.

Li).

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systems with completely unknown system functions, e.g., see [1]–[5], [12], [13], [17], [21]–[26], [38], [48], [49], [51], [53] and references therein. In the beginning, some adaptive backstepping fuzzy control schemes have been proposed for uncertain nonlinear systems with strict-feedback forms. A direct adaptive backstepping fuzzy control method for nonlinear strict-feedback systems has been developed in [3]. The advantage of this control strategy is that only a fuzzy logic

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system is needed to approximate all the unknown functions in each bacestepping design. For the controlled systems with unmeasured states, observer-based adaptive backstepping fuzzy control methods have been proposed for strict-feedback systems under different conditions [4], [12], [13], [17], [21]–[23], [38]. In [12], for the controlled systems with unknown control direction and time delays, adaptive fuzzy output-feedback controller has been developed. By constructing

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some appropriate observers, adaptive backstepping fuzzy control schemes have been proposed for nonlinear systems with periodic disturbances [4] and unknown dead zone [38]. Since the pure feedback system can be transformed to a uncertain system with strict-feedback structure by using the mean value theorem, adaptive fuzzy control of uncertain pure feedback systems has been addressed in [53] based on input-to-state stability. Recently, by adopting fuzzy logic

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systems as feedforward compensators, the problem of global adaptive fuzzy control for outputfeedback systems with unknown high-frequency gain sign has been solved in [5]. Subsequently,

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adaptive backstepping fuzzy control has been employed to solve the control problems of uncertain discrete-time systems, e.g., see [48] and [26]. Different from the classical adaptive backstepping control, in the adaptive fuzzy control design, fuzzy logic systems are mainly used to approximate

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the unknown nonlinearities in the controlled systems, and then stable fuzzy controllers have been constructed by combining the classical adaptive control method and the backstepping technique.

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In addition, lots of adaptive fuzzy control schemes have been adopted to solve the control problems of some practical systems. In [34], for the cooperating robotic manipulators moving

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an object with impedance interaction, a decentralized adaptive fuzzy control has been designed. For manipulation in complex perturbation environments, novel hybrid adaptive controller has been developed by combining the advantages of task-space and joint-space control in [37]. It is emphasized that the aforementioned adaptive fuzzy control strategies are feasible when the

controlled systems have strict-feedback form or can be transformed to a strict-feedback structure. For the nonlinear systems with the whole state variables in each subsystem function, these control methodologies may be invalidated [6]. To control this class of systems, as mentioned in [6]–[9], October 8, 2016

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[27], the main difficulties come from that i) in each backstepping design, the virtual control signal αi must be only the function of state vector [x1 , · · · , xi ]T ; and ii) how to deal with the

functions of xi bequeathed from the previous design step to the current step. Very recently, by utilizing the monotonously increasing property of the bounding functions and the variable separation technique, feasible adaptive fuzzy control scheme has been developed for a class of

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nonlinear systems with nonstrict-feedback form in [6]. Subsequently, for the nonlinear systems with nonlinear unknown dead zone and without a strict-feedback, the problem of adaptive fuzzy tracking control has been addressed in [7]. The problem of adaptive fuzzy tracking control for a class of nonstrict-feedback systems with time delays and stochastic disturbances has been solved in [27]. In [9], for a class of nonlinear time-delay systems in nonstrict-feedback form with

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unmeasured states, the problem of observer-based adaptive fuzzy control has been addressed. It is generally that many practical systems suffer from the effect of the constraints, such as the temperature of chemical reactor and physical stoppages. Thus, it is significative to study the control problems of the constrained plants. For linear discrete-time systems with input constraints subject to actuator saturation, a feasible control method has been proposed in [54]. In [10], for

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a class of uncertain multi-input and multi-output nonlinear systems with non-symmetric input constraints, adaptive tracking problem has been addressed. In [36], [39], [40], by introducing

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the so-called Barrier Lyapunov Functions (BLFs), for several classes of nonlinear systems subject to output constraints, feasible control schemes have been designed, and the proposed control methods have been applied to solve the control problems of some practical systems,

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such as electrostatic micro actuators [41], flexible marine riser [14], flexible crane system [15], and thruster assisted position mooring system [16]. Very recently, for uncertain pure-feedback

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systems with full state constraints, by adopting the mean value theorem, adaptive control methods have been developed based on BLFs by transforming the controlled systems into the strict-

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feedback structures [19] and [28]. It should be noted that the aforementioned control schemes are feasible for the uncertain systems with strict-feedback structures. For the controlled systems with nonstrict-feedback structure and full state constraints, these control strategies may be invalidated. In this paper, we focus on the problem of adaptive fuzzy tracking control of uncertain systems

with nonstrict-feedback form, unknown nonlinear dead zone and full state constraints. The main contributions of this paper can be summarized as follows. (i) Different from the existing results reported in [9], [10], [14]–[16], [19], [28], [36], [39]– October 8, 2016

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[41], [52], [54], where the control schemes have been developed for nonlinear strictfeedback or pure-feedback systems with state or output constraints, this paper proposes a generalization of the results for a class of nonstrict-feedback systems with the full state constraints and unknown nonlinear dead zone. As far as we know, it is first to develop an nonstrict-feedback form and full state constraints.

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adaptive fuzzy control method with less adjustable parameters for uncertain systems with (ii) To achieve the control objective of this paper, under some reasonable assumptions on system functions, by using the variable separation theorem, the difficult from the system functions containing whole system states is overcome. Then, by using the modified backstepping design and Lyapunov stability theorem, an adaptive fuzzy controller is designed

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to prevent the violation of the full state constraints based on some BLFs. It can be proved that all the closed-loop signals are semi-globally bounded and the system output converges to a small neighborhood of the reference signal.

The rest of this paper is organized as follows. In Section II, we present some preliminaries including system stability and some important lemmas. Section III gives the control design

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process and the main result on the control performance and the closed-loop stability. In Section IV, two simulation examples are provided to verify the effectiveness of the proposed control

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approach. We conclude the work of this paper in Section V. Notation: In this paper, R denotes a set of real numbers. R+ denotes a set of nonnegative real numbers. Rn×n denotes a set of n × n real matrices, and Rn denotes a set of n-dimensional

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real vectors. sup(·) denotes the least upper bound. ||Ξ|| denotes a norm of a vector or matrix

Ξ. |x| denotes a absolute value of a real number x. tanh(·) is a hyperbolic tangent function, and

II. P RELIMINARIES

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exp(·) denotes a exponential function.

This section introduces some basic definitions on the system stability, fuzzy logic system and

some useful lemmas which play an important role in the controller design and the closed-loop stability analysis. Definition 1 (UUB) [20]: For a general nonlinear system x˙ = f (x, t),

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where x(t) ∈ Rn is the system state, f : Rn × [t0 , ∞] → Rn is a continuous vector-valued

function, t0 and x0 ∈ Rn denote the initial time and the initial state vector, respectively. If there

exists a compact set U ∈ Rn such that for all x0 ∈ U , there exists a δ > 0 and a number

T (δ, x0 ) such that kx(t)k < δ for all t ≥ t0 + T (δ, x0 ), we say that the solution of this system is uniformly ultimately bounded (UUB).

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Lemma 1 [43]: Let H(Z) be a continuous function that is defined on a compact set Ω. Then, for a given desired level of accuracy ε > 0, there exists a fuzzy logic system W T S(Z) such that sup |H(Z) − W T S(Z)| ≤ ε,

Z∈Ω

where W = [w1 , · · · , wN ] is the ideal weight vector, and S(Z) = [s1 (Z), · · · , sN (Z)]T /Σni=1 si (Z) T

is the basis function vector, with N > 1 being the number of the fuzzy rules, and si (Z) are

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chosen as Gaussian functions, i.e., for i = 1, · · · , N, h −(Z − µ )T (Z − µ ) i i i , si (Z) = exp 2 ηi where µi = [µi1 , · · · , µiN ]T is the center vector, and ηi is the width of the Gaussian function.

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Lemma 2 (Young’s Inequality) [35]: For ∀(x, y) ∈ R2 , the following inequality holds: ιp 1 xy ≤ |x|p + q |y|q , p qι where ι > 0, p > 1, q > 1, and (p − 1)(q − 1) = 1.

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Lemma 3 [35]: For any Υ ∈ R and ε > 0, the following holds Υ 0 ≤ |Υ| − Υ tanh ≤ δε ε with ε ≈ 0.2785.

Lemma 4: For a strictly increasing function Θ(·) : R+ → R+ with Θ(0) = 0, and ai ≥ 0, i =

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1, · · · , n, there exist some smooth functions ~(·) such that n n X  X Θ ai ≤ nai ~(nai ). i=1

i=1

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Remark 1: For Lemma 4, it can be easily obtained according to Remark 1 presented in [7].

Thus, the detailed proof is omitted here. III. P ROBLEM F ORMULATION , C ONTROL D ESIGN AND S TABILITY A NALYSIS

This section gives the problem description, then the design process of the desired adaptive fuzzy controller is presented by using the backstepping technique with some BLFs. Finally, we analyze the tracking performance and the closed-loop stability of the whole control system. October 8, 2016

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A. Problem Description

x˙ n = fn (x) + gn (¯ xn )u     y = x1

(1)

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Consider a class of nonstrict-feedback systems described by    xi )xi+1 , i = 1, · · · , n − 1   x˙ i = fi (x) + gi (¯

where x = [x1 , x2 , · · · , xn ]T ∈ Rn is the state vector, y ∈ R is the system output, and u ∈ R

is the control input; all the states are constrained in the compact sets, i.e., xi is required to remain in the set |xi | ≤ Bi with Bi being a positive constat; let x¯i =: [x1 , · · · , xi ]T ∈ Ri ;

fi : Ri → R and gi : Ri → R are unknown continuous functions with fi (0) = 0. Here, assume

that gi (¯ xi )xi+1 + fi (x) 6= 0.

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This paper considers the case that the controlled system (1) is with dead zone in the following

form [7]

(2)

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     hr (µ), µ ≥ br u = Z(µ) = 0, bl < µ < b r     hl (µ), µ ≤ bl

where bl < 0 and br > 0 are unknown design parameters, µ(t) ∈ R is the input to the dead

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zone, hl (·) and hr (·) are smooth nonlinear functions.

The control objective of this paper is presented as follows. For a given reference signal yr (t),

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to develop an adaptive fuzzy controller u such that (i) the tracking error z1 = y − yr converges to a bounded compact set;

(ii) all the closed-loop signals remain uniformly bounded;

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(iii) the full state constraints are not violated. Remark 2: For system (1), it is called the nonstrict-feedback system considered in [6]–

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[9], [27]. Obviously, strict-feedback and pure-feedback structure can be as its special cases. The proposed control methods in [9], [10], [14]–[16], [19], [28], [36], [39]–[41], [52], [54] just can solve the tracking control problems for strict-feedback/pure-feedback systems with the constraints. These approaches cannot be directly applied to system (1) due to its nonstrictfeedback structure. To design the desired adaptive fuzzy controller, the following assumptions are needed.

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Assumption 1 [7]: For 1 ≤ i ≤ n, the signs of the functions gi (x(t)) are known, and there

exits some unknown positive constants gi0 such that |gi (x(t))| ≥ gi0 , which implies that gi (x(t)) is strictly either positive or negative. Without loss of generally, this paper assumes that gi (x(t)) > 0. Assumption 2 [7]: The reference signal yr (t) is continuous and bounded. The p-order deriva(p)

tives yr (t) of yr (t) are also continuous and bounded for p = 1, · · · , n. For simplicity, this paper (p)

(p)

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assumes that yr (t) and yr (t) satisfy |yr (t)| ≤ d and |yr (t)| ≤ d, where d > 0 is a constant. Assumption 3 [6], [7]: Assume that functions fi (x), i = 1, · · · , n satisfy |fi (x)| ≤ Ψi (||x||),

where Ψi (·) ≥ 0 are strictly increasing smooth functions with Ψi (0) = 0.

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Remark 3: Assumption 3 directly comes from References [6] and [7]. Based on this as-

sumption, the variable separation technique is adopted to deal with the system function fi (·) containing whole system states. Thus, the modified backstepping design method can be used to develop the desired adaptive fuzzy controller.

Assumption 4 [7]: The dead zone parameters bl < 0 and br > 0 are unknown bounded

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constants. Furthermore, the smooth functions hl (µ) and hr (µ) satisfy hl (µ) = φl (µ)(µ − bl ) and

hr (µ) = φr (µ)(µ − br ), and there exist unknown positive constants φr0 , φr1 , φl0 , and φl1 such

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that

0 < φl0 ≤ φl (µ) ≤ φl1 , ∀µ ∈ (−∞, bl ],

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0 < φr0 ≤ φr (µ) ≤ φr1 , ∀µ ∈ (−∞, br ],

and ϑ0 ≤ min{φl0 , φr0 } is an unknown positive constant.

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where

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According to Assumption 3 and [7], the dead zone nonlinearity (2) can be rewritten as u = Z(µ) = ΦT (t)Ψ(t)µ + d(µ),

Ψ(t) = [ϕr (t), ϕl (t)]T , Φ(t) = [Φr (µ(t)), Φl (µ(t))]T ,      −φr (µ)br ,

d(µ) =

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(3)

µ ≥ br

−[φl (bl ) + φr (br )]µ, bl < µ < br     −φl (µ)bl , µ ≤ bl

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  1, ϕr (t) =  0,   1, ϕl (t) =  0,      0,

µ(t) > bl µ(t) ≤ bl µ(t) < br µ(t) ≥ br

Φr (µ(t)) =

bl < µ(t) < br

bl < µ(t) < +∞ µ(t) ≥ br

bl < µ(t) < br

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φr (br ),     φr (µ(t)),      0, Φl (µ(t)) = φl (bl ),     φl (µ(t)),

µ(t) ≤ bl

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with

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−∞ < µ(t) < br .

It is noted that from the above presentation, it can be clearly seen that there exists an unknown constant κ∗ = (φr1 + φl1 ) max{br , bl } such that |d(µ)| ≤ κ∗ holds.

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Based on (3), system (1) can be expressed as the following form    xi ) + gi (¯ xi )xi+1 , i = 1, · · · , n − 1   x˙ i = fi (¯

(4)

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x˙ n = fn (¯ xn ) + gn (¯ xn )d(µ) + gn (¯ xn )ΦT (t)Ψ(t)µ     y = x1 .

In the following subsection, we will combine the direct adaptive control method and the

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backstepping design technique to develop the real controller µ for system (4).

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B. Adaptive Fuzzy Control Scheme Design

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To design the desired controller and adaptive laws, we define n error variables as follows (5)

z1 = y − yr , zi = xi − αi−1 ,

i = 2, · · · , n,

(6)

where αi denotes the virtual control variables and will be assigned below.

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Based on the above coordinate transformations (5) and (6), the error system of system (4) can

where

     z˙1 = f1 (x) + g1 (x1 )(z2 + α1 ) − y˙ r , z˙i = fi (x) + gi (¯ xi )(zi+1 + αi ) − α˙ i−1 ,     z˙n = fn (x) + gn (¯ xn )u − α˙ n−1 , i = 2, · · · , n − 1,

α˙ i−1 =

i−1 X ∂αi−1 j=1

∂xj

[fj (x) + gj (¯ xj )xj+1 ] +

i−1 X ∂αi−1 j=1

∂$ ˆj

(7)

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be expressed as

$ ˆ˙ j +

i−1 X ∂αi−1 (j) j=0 ∂yr

y˙ r(j+1)

and $ ˆ j denote the estimations of unknown constants $j which will be defined later.

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Define the following positive definite Lyapunov function n  b2  1X V = log 2 i 2 , 2 i=1 bi − zi

m where log(ω) denotes the natural logarithm of ω, b1 = B1 − d and bi = Bi − αi−1 , i = 2, · · · , n m are positive constants, Bi and αi−1 will be defined in Subsection C, define Ωzi = {|zi | < bi }, i =

1, · · · , n and bi will be specified later on. It can be seen that V is continuous when zi ∈ Ωzi .

=

n X

b2 i=1 i

1 zi z˙i − zi2

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dV dt

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Along the solutions of (7), the time derivative of V is

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= Λ1 (z1 )z1 [f1 (x) + g1 (x1 )(z2 + α1 ) − y˙ r ] + +Λn (zn )zn [fn (x) + gn (¯ xn )u − α˙ n−1 ]

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= Λ1 (z1 )z1 [f1 (x) + g1 (x1 )(z2 + α1 ) − y˙ r ] +

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+

n−1 X i=2

where φi−1 =

p = 1, · · · , n.

n−1 X

Λi (zi )zi [fi (x) + gi (¯ xi )(zi+1 + αi ) − α˙ i−1 ]

n X

i−1 h i X ∂αi−1 Λi (zi )zi fi (x) − fj (x) ∂xj j=1

i=2

i=2

Λi (zi )zi gi (¯ xi )(zi+1 + αi ) + Λn (zn )zn gn (¯ xn )u −

Pi−1

∂αi−1 ˙ $ ˆj j=1 ∂ $ ˆj

+

Pi−1

∂αi−1 (j+1) j=0 ∂y (j) y˙ r r

n X

(8)

Λi (zi )zi φi−1 ,

i=2

with i = 2, · · · , n and Λp (zp ) =

1 b2p −zp2

with

To estimate the term Λi (zi )zi fi (x) in the above equality, the following lemma needs to be

introduced.

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Lemma 5: For i = 1, · · · , n, the error variables zi = xi − αi−1 defined as (5) and (6), there

exist some smooth functions ψi (·) ≥ 0 such that ||x|| ≤

n X i=1

|zi |ψi (zi , $ ˆ i ) + d,

where define α0 = yr and αj are designed as (21) for j = 1, · · · , n − 1.

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Remark 4: Based on Assumption 2, the fact SiT (Zi )Si (Zi ) ≤ 1, and the structure of virtual

control variables designed as (21), by using the similar derivations presented in [6] and [7], Lemma 5 is obtained, where Si (Zi ) will be defined in (17).

Remark 5: As mentioned in [6] and [7], Lemma 5 plays an important role in the developed control design process because it sets up the relation between the norm of system state x and the

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norm of error variables zi . Later, by employing the variable separation method, system function fi (x) can be estimated by a sum of some functions with respect to zi . Therefore, the desired adaptive fuzzy controller can be developed.

Based on Assumption 3, by using Lemmas 4 and 5, we can have

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Λ1 (z1 )z1 f1 (x) ≤ Λ1 (z1 )|z1 |Ψ1 (||x||) n X |zk |ψk (zk , $ ˆ k )ρ1 (|zk |ψk (zk , $ ˆ k )) + Λ1 (z1 )|z1 |ρ1 (d) ≤ Λ1 (z1 )|z1 | k=1

|zk |¯ ρ1 (|zk |ψk ) + Λ1 (z1 )|z1 |ρ1 (d)

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= Λ1 (z1 )|z1 |

n X k=1

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n X 1 2 2 1 2 2 Λ1 (z1 )z1 + z ρ¯ (|zk |ψk ) + Λ1 (z1 )|z1 |ρ1 (d), ≤ 2 2 k 1 k=1

(9)

where define ρ¯1 (|zk |ψk ) = ψk (zk , $ ˆ k )ρ1 (|zk |ψk (zk , $ ˆ k )) with a smooth function ρ1 (·) ≥ 0.

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Adopting the same way presented in the above inequality, we can obtain the following

inequality

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n X i=2

i−1 h i X ∂αi−1 fj (x) Λi (zi )zi fi (x) − ∂xj j=1

=−

≤

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n X

Λi (zi )zi

i=2

j=1

∂xj

fj (x)

n X i ∂α ∂α X i−1 i−1 Λi (zi )|zi ||zk | Λi (zi )|zi | ρ¯j (|zk |ψk ) + ρj (d) ∂xj ∂xj i=2 j=1 k=1

n X i X n X i=2 j=1

i X ∂αi−1

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n X i X n X 1 i=2 j=1 k=1

∂α i−1 Λi (zi )|zi | ρj (d), ∂x j j=1

n X i X i=2

∂αi−1 ∂xi

where define

2

∂αi−1 zi2 Λ2i (zi ) 2 ∂xj

functions ρj (·) ≥ 0.

+

n X i X n X 1 i=2 j=1 k=1

2

zk2 ρ¯2j (|zk |ψk ) (10)

= −1 and ρ¯j (|zk |ψk ) = ψk (zk , $ ˆ k )ρj (|zk |ψk (zk , $ ˆ k )) with some smooth

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≤

11

Substituting (9) and (10) into (8) yields

dV dt

n−1 X

≤

i=1

Λi (zi )zi gi (¯ xi )(zi+1 + αi ) + Λn (zn )zn gn (¯ xn )u − Λ1 (z1 )z1 y˙ r −

n X

Λi (zi )zi φi−1

i=2

+

n X i X n X 1 i=2 j=1 k=1

2

zk2 ρ¯2j (|zk |ψk ) +

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n n X i X n ∂α 2 X X 1 2 1 2 2 1 2 2 i−1 2 + Λ1 (z1 )z1 + zk ρ¯1 (|zk |ψk ) + Λ1 (z1 )|z1 |ρ1 (d) + zi Λi (zi ) 2 2 2 ∂xj i=2 j=1 k=1 k=1

∂α i−1 Λi (zi )|zi | ρj (d). ∂x j j=1

n X i X i=2

(11)

Note that some terms in inequality (11) can be rearranged as

k=1

2

zk2 ρ¯21 (|zk |ψk ) +

n X i X n X 1

2

zk2 ρ¯2j (|zk |ψk ) =

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n X 1

i=2 j=1 k=1

n X i=1

zi2

n X n−k+1 k=1

2

ρ¯2k (|zi |ψi ).

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By using Lemma 3, we have the following inequalities: z L  1 1 + δε1 Λ1 (z1 )|z1 |ρ1 (d) ≤ z1 L1 tanh ε1 and

(12)

(13)

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n h ∂α z L  i X i−1 i i Λi (zi )|zi | ρ (d) ≤ z L tanh + δε (14) j i i i , ∂x ε j i i=2 j=1 i=2 Pi ∂αi−1 where L1 = Λ1 (z1 )ρ1 (d), Li = j=1 Λi (zi ) ∂xj ρj (d), and εi > 0 are design parameters with

CE

n X i X

i = 1, · · · , n.

AC

Recalling u = Z(µ) = ΦT (t)Ψ(t)µ + d(µ) expressed in (3) and applying Lemma 2, we get Λn (zn )zn gn (¯ xn )u = Λn (zn )zn gn (¯ xn )[ΦT (t)Ψ(t)µ + d(µ)] 1 1 ≤ Λn (zn )zn gn (¯ xn )ΦT (t)Ψ(t)µ + Λ2n (zn )zn2 gn2 (¯ xn ) + κ∗2 . 2 2

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Substituting (12)-(17) into (11) becomes

i=1

1 xn ) Λi (zi )zi gi (¯ xi )(zi+1 + αi ) + Λn (zn )zn gn (¯ xn )ΦT (t)Ψ(t)µ + Λ2n (zn )zn2 gn2 (¯ 2

−Λ1 (z1 )z1 y˙ r − + +

n X

zi2

n X

i=1 k=1 n h X

i=2

n−1 X

n X i X n ∂α 2 X 1 2 1 2 2 i−1 2 Λi (zi )zi φi−1 + Λ1 (z1 )z1 + zi Λi (zi ) 2 2 ∂xj i=2 j=1 k=1

z L  n−k+1 2 1 1 ρ¯k (|zi |ψi ) + z1 L1 tanh + δε1 2 ε1

zi Li tanh

i=2

=

n X

z L  i

εi

Λi (zi )zi gi (¯ xi )αi +

i=1

i

i

CR IP T

≤

n−1 X

1 + δεi + κ∗2 2

n X

zi Fi +

i=1

n X i=1

1 δεi + Λn (zn )zn gn (¯ xn )ΦT (t)Ψ(t)µ + κ∗2 , 2

AN US

dV dt

(16)

M

where define some unknown functions as follows n z L  X n−k+1 2 1 2 1 1 + z1 ρ¯k (|z1 |ψ1 ), F1 = −Λ1 (z1 )y˙ r + Λ1 (z1 )z1 + L1 tanh 2 ε1 2 k=1

+zi

n X k=1

n−k+1 2 ρ¯k (|zi |ψi ) + Li tanh 2

∂α 2 i−1 zi Λ2i (zi ) 2 ∂xj

i X n X 1 j=1 k=1

z L  i

i

εi

,

i = 2, · · · , n − 1,

PT

and

ED

Fi = Λi−1 (zi−1 )zi−1 gi−1 (¯ xi−1 ) − Λi (zi )zi φi−1 +

CE

1 Fn = Λn−1 (zn−1 )zn−1 gn−1 (¯ xn−1 ) + Λ2n (zn )zn gn2 (¯ xn ) − Λn (zn )zn φn−1 2 n n X n ∂α 2 z L  X X n−k+1 2 1 n−1 n n 2 zn Λn (zn ) ρ¯k (|zn |ψn ) + Ln tanh . + + zn 2 ∂x 2 εn j j=1 k=1 k=1

AC

Based on Lemma 1, for i = 1, · · · , n, some fuzzy logic systems WiT Si (Zi ) are used to

approximate unknown functions Fi (Zi ) as follows

Fi (Zi ) = WiT Si (Zi ) + εi (Zi ),

(17)

where Wi denotes the ideal constant weight vector and Si (Zi ) is the basis vector function of the ith fuzzy logic system which is defined as in Lemma 1, and εi (Zi ) denotes the approximation error satisfying |εi (Zi )| ≤ i with for any given constant i > 0. October 8, 2016

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Note that for i = 1, · · · , n, the ideal weight vector Wi are unknown, thus the adaptive technique

is used to estimate them online. To reduce the number of adaptive parameters and learning time

online, here the norm of Wi is estimated online. By using Lemma 2, we have the following inequality

CR IP T

zi Fi = zi WiT Si (Zi ) + zi εi (Zi ) 1 ι2i 1 1 2 2 T ≤ ||W || z S (Z )S (Z ) + + gi0 zi2 + 0 2i i i i i i i 2 2ιi 2 2 2gi ι2i 1 1 gi0 2 T $ z S (Z )S (Z ) + + gi0 zi2 + 0 2i = i i i i i i 2 2ιi 2 2 2gi and

where define $i =

||Wi ||2 gi0

and $n =

||Wn ||2 , 0ϑ gn 0

AN US

zn Fn = zn WnT Sn (Zn ) + zn εn (Zn ) g 0 ϑ0 g 0 ϑ0 2 ι2 ≤ n 2 $n zn2 SnT (Zn )Sn (Zn ) + n + n zn2 + 0n , 2ιi 2 2 2gn ϑ0

(18)

(19)

and ιi > 0 are design parameters with i = 1, · · · , n.

Consider (17)-(19), then inequality (16) can be represented as

i=1

+

n−1 0 X gi Λi (zi )zi gi (¯ xi )αi + Λn (zn )zn gn (¯ xn )Φ (t)Ψ(t)µ + $ z 2 S T (Zi )Si (Zi ) 2 i i i 2ιi i=1

n−1 X 1 i=1

where define %0 =

2

T

M

≤

n−1 X

gi0 zi2 +

Pn

gn0 ϑ0 2 gn0 ϑ0 2 T $ z S z + %0 , (Z )S (Z ) + n n n n n n 2ι2i 2 n

ED

dV dt

ι2i i=1 2

+ 12 κ∗2 +

Pn−1

1 2 i=1 2gi0 i

+

2n 0ϑ . 2gn 0

AC

CE

PT

Based on (20), we design the virtual controls and the actual control input as follows:  0.5  1 αi = − ci + zi − 2 $ ˆ i zi SiT (Zi )Si (Zi ), i = 1, · · · , n − 1, Λi (zi ) 2ιi Λi (zi )

and

 µ = − cn +

(20)

0.5  $ ˆ n zn zn − 2 SnT (Zn )Sn (Zn ), Λi (zn ) 2ιn Λn (zn )

(21)

(22)

where for i = 1, · · · , n, ci > 0 are design parameters, and the adaptive laws of estimations $ ˆi are designed as

σi $ ˆ˙ i = 2 zi2 SiT (Zi )Si (Zi ) − σi $ ˆi 2ιi

(23)

with σi being positive design parameters.

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Note that according to the discussions presented in [6] and [7], for (23), when we choose the initial condition $ ˆ i (t0 ) ≥ 0, it can be guaranteed that the solution $ ˆ i (t) ≥ 0 for t ≥ t0 .

Therefore, in this paper, we assume that $ ˆ i (t) ≥ 0 holds.

Based the above assumption and Assumptions 1 and 4, considering (21) and (22), we have

the following inequalities gi0 $ ˆ i zi2 SiT (Zi )Si (Zi ) 2ι2i

and Λn (zn )zn gn (¯ xn )ΦT (t)Ψ(t)µ ≤ −(cn Λn (zn ) + 0.5)gn0 ϑ0 zn2 − Substituting (24) and (25) into (20) leads to ≤ −

n−1 X

ci gi0 Λi (zi )zi2

i=1

gi0 ϑ0 $ ˆ i zi2 SiT (Zi )Si (Zi ). 2ι2i

(24)

(25)

n−1 0 X gi 2 − z $ ˆ i SiT (Zi )Si (Zi ) − cn gn0 ϑ0 Λn (zn )zn2 2 i 2ι i i=1

AN US

dV dt

CR IP T

Λi (zi )zi gi (¯ xi )αi ≤ −(ci Λi (zi ) + 0.5)gi0 zi2 −

n−1

X g0 g 0 ϑ0 i 2 − n 2 zn2 $ ˆ n SnT (Zn )Sn (Zn ) + z $i SiT (Zi )Si (Zi ) 2 i 2ιn 2ι i i=1 gn0 ϑ0 2 z $n SnT (Zn )Sn (Zn ) + %0 . 2ι2n n

(26)

M

+

Remark 6: Different from the work presented in [6]–[9], [27], this paper deals with the

ED

adaptive tracking control problem for a more general class of nonlinear uncertain systems with nonstrict-feedback structure and the full state constrains. Obviously, aforementioned control

PT

schemes cannot be directly employed to solve the control problem considered in this paper. In addition, in [19] and [28], the adaptive tracking control problems with full state constrains

CE

have been addressed for pure-feedback systems. However, this paper considers the uncertain systems with the nonstrict-feedback structure which takes the work reported in [19] and [28] as

AC

the special cases.

C. Stability Analysis of the Closed-loop System The closed-loop system stability and the performance of the whole control system are presented

by the following theorem. Theorem 1: Under Assumptions 1-4, consider the closed-loop system consisting of the plant (1), the control law (22) with the virtual control variable (21), and the adaptation laws (23). If

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the design parameters are appropriately chosen to satisfy m Bi > bi + αi−1 , (j)

m := max |αi (¯ xi , $ ˆ j , yr , yr )|, j = 1, · · · , i. Then, for bounded initial where α0m := d, αi−1

conditions on a compact set, we have the following statements.

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(i) All the signals of the closed-loop system remain UUB. (ii) The tracking error can converge to a bounded compact set; (iii) The full state constraints are not violated.

Proof. Choose the overall Lyapunov function candidate as follows n−1 X gi0 2 gn0 ϑ0 2 $ ˜ + $ ˜ 2σi i 2σn n i=1

AN US

V¯ = V +

with $ ˜ i = $i − $ ˆ i for i = 1, · · · , n. Considering (26) and differentiating V¯ yields n−1

X g0 g 0 ϑ0 dV i $ ˜ i$ ˆ˙ i − n $ ˜ n$ ˆ˙ n − dt σ σ i n i=1

≤ − +

n−1 X i=1

ci gi0 Λi (zi )zi2 − cn gn0 ϑ0 Λn (zn )zn2 +

M

=

By using (24) and $ ˜ i$ ˆ i ≤ − 2σ1 i $ ˜ i2 + ≤ −

n−1 X

ci gi0 Λi (zi )zi2

PT

dV¯ dt

i=1

n−1 X

with %¯0 =

Pn−1 i=1

i=1

−

σi 2 $i , 2

gi0 σi 2 $i 2

+

0ϑ σ gn 0 n $n2 2

cn gn0 ϑ0 Λn (zn )zn2

+

n X

AC − log

zi2 − b2 −z 2. i i

≥

dV¯ dt

≤ −

n−1 X

gi0 $ ˜ i$ ˆ˙ i + gn0 ϑ0 $ ˜ n$ ˆ˙ n + %0

i=1

n−1 X i=1

gi0 2 gn0 ϑ0 2 $ ˜ − $ ˜ + %¯0 2σi i 2σn n

(28)

+ %0 . b2

i As mentioned in [28], it is a fact that log b2 −z 2 ≤

b2i 2 bi −zi2

(27)

inequality (27) can be re-expressed as

ci gi0 Λi (zi )zi2 − cn gn0 ϑ0 Λn (zn )zn2 −

CE

≤ −

n−1

n−1 0 X gi 2 z $ ˜i SiT (Zi )Si (Zi ) 2 i 2ι i i=1

X g0 gn0 ϑ0 gn0 ϑ0 2 i T ˙ $ ˆ − ˆ˙ n . z $ ˜ S (Z )S (Z ) + % − $ ˜ $ ˜ n$ n n n 0 n n i i 2ι2n n σ σ i n i=1

ED

dV¯ dt

i

Recall Λi (zi ) =

1 , b2i −zi2

i

zi2 2 bi −zi2

in the interval |zi | ≤ bi , i.e.,

then inequality (28) becomes n−1

ci gi0 log

i=1

≤ −λV¯ + %¯0 ,

X g0 gn0 ϑ0 2 b2i b2n i 2 0 ϑ log − $ ˜ − $ ˜ n + %¯0 − c g n n 0 i b2i − zi2 b2n − zn2 2σ 2σ i n i=1

(29)

0 where define λ = min{2c1 g10 , · · · , 2cn−1 gn−1 , 2cn gn0 ϑ0 , 1}. October 8, 2016

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Multiplying both sides by eλt , (29) can be rewritten as %¯0 %¯0 V¯ (t) ≤ [V¯ (0) − ]e−λt + λ λ % ¯ 0 (30) ≤ V¯ (0) + . λ From inequality (30) and the definition of V¯ , we can conclude that for i = 1, · · · , n, all the b2

i

CR IP T

i signals log b2 −z ˜ i remain bounded. Due to the boundedness of $i , it can be seen that 2 and $ i

$ ˆ i = $i − $ ˜ i is also bounded.

Because x1 = z1 + yr and |yr | ≤ d, we have |x1 | ≤ b1 + d. Let B1 > b1 + d, and then we

know |xi | ≤ B1 . According to the definition of α1 in (21) which is a function of some bounded

signals z1 , x1 , $ ˆ 1 and y˙ r , and using a fact S1T (Z1 )S1 (Z1 ) ≤ 1, we can see that α1 is bounded

AN US

and satisfies |α1 | ≤ α1m . Then, |x2 | ≤ |α1 | + |z2 | ≤ α1m + b2 with defining B2 > α1m + b2 , which m implies that |x2 | ≤ B2 . By the same way, we can in turn prove that |xi | ≤ Bi with Bi = bi +αi−1

for i = 3, · · · , n. Following the definition of µ in (22), we also can conclude the controller µ is bounded.

Based on the above discussions, it can be concluded that all the closed-loop signals including

M

zi , xi , $ ˆ i , αi and µ are bounded and the system states are not violated. By using (30), we obtain

%¯0 2¯ %0 b21 ≤ 2[V¯ (0) − ]e−λt + . b21 − z12 λ λ

ED

log

That is,

CE

PT

% ¯ 2% ¯ b21 2[V¯ (0)− λ0 ]e−λt + λ0 ≤ e . b21 − z12 q q % ¯ 2% ¯ 2% ¯0 −2[V¯ (0)− λ0 ]e−λt − λ0 Thus, it is easy to get |z1 | ≤ b1 1 − e . Define ∆ = b1 1 − e− λ . As

analyzed in [28], we can have |z1 | ≤ ∆ as t → ∞. It can be seen that by choosing the

AC

design parameters, z1 can be arbitrarily small. The proof of Theorem 1 is completed.

Remark 7: From the above analysis, the advantages of the proposed control scheme can

be summarized as follows. First, for the uncertain system with nonstrict-feedback form, the proposed fuzzy controller ensures that the full state constraints are not violated. In addition, for the uncertain nonstrict-feedback system with dead zone uncertainty, a feasible adaptive fuzzy controller has been developed. Compared with the existing results, the of advantage of our October 8, 2016

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control method is obviously, but it is not perfect. There are some drawbacks needed to be further improved. For example, the proposed controller is obtained by using the conventional backstepping technique, thus the burdensome calculations cannot be avoided. In addition, the proposed controller just ensure the semi-global boundedness of the closed-loop control system.

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IV. S IMULATION E XAMPLES This section introduces two simulation examples to illustrate the effectiveness of the proposed adaptive fuzzy control approach.

Example 1: Consider the following second-order nonlinear system [6]   2 2    x˙ 1 = x1 x2 + x1 sin(x2 ) + (1.5 + 0.5 sin(x1 ))x2

AN US

x˙ 2 = x1 x2 ex2 + x1 cos(x1 x2 ) + (1.5 + sin(x1 x2 ))u     y = x1 ,

where

(31)

M

   µ > 2.5   (1 − 0.2 sin µ)(µ − 2.5), u = Z(µ) = 0, −1.5 ≥ µ ≤ 2.5     (0.8 − 0.1 cos µ)(µ + 1.5), µ < −1.5.

Apparently, this system is without the strict-feedback structure and it can be verified that

ED

Assumptions 1, 3 and 4 are satisfied. In addition, the states are constrained in |x1 | < 1.5 and |x2 | < 2.

PT

For the given reference signal yr = 0.5(sin(t) + sin(0.5t)), nine fuzzy sets are defined over

[-7,7] for all state variables by choosing the partitioning points as -7,-5,-3,-1,0,1,3,5,7, and the

CE

fuzzy membership functions are presented as follows: µFi2 (xi ) = exp(−0.5(xi + 5)2 ),

µFi3 (xi ) = exp(−0.5(xi + 3)2 ),

µFi3 (xi ) = exp(−0.5(xi + 1)2 ),

µFi4 (xi ) = exp(−0.5(xi + 0)2 ),

µFi1 (xi ) = exp(−0.5(xi − 7)2 ),

µFi2 (xi ) = exp(−0.5(xi − 5)2 ),

µFi5 (xi ) = exp(−0.5(xi − 3)2 ),

AC

µFi1 (xi ) = exp(−0.5(xi + 7)2 ),

µFi7 (xi ) = exp(−0.5(xi − 1)2 ). According to Lemma 1, Si can be constructed for i = 1, 2.

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According to Theorem 1, adaptive fuzzy controller (22) with virtual control α1 defined as in (21) and adaptive laws $ ˆ˙ 1 and $ ˆ˙ 2 in form (23) to control system (31). In simulation, the Matlab “ode45” method has been used to obtain the required results. The control parameters are selected as c1 = c2 = 0.7, ι1 = ι2 = 1, and σ1 = σ2 = 0.1. The initial conditions are chosen as x1 (0) = x2 (0) = 0.3 and $ ˆ 1 (0) = $ ˆ 2 (0) = 0. Similar to the

CR IP T

methods presented as [27] and [34], by using the Matlab routing, it can be obtained b1 = 0.3 and b2 = 1.107.

AN US

1

y, yr

0.5

0

0

10

20

30

Time(s)

40

50

60

70

ED

−1

M

−0.5

Fig. 1. Trajectories of system output signal y(t) (dash-dotted line) and reference signal yr (t) (solid line).

PT

The simulation results are shown in Figs. 1-4. It can be seen from Fig. 1 that a good tracking performance is obtained. Fig. 2 presents the trajectories of system states x1 and x2 , from which

CE

we can see that they are not overstepped. The boundedness of µ, $ ˆ 1 , and $ ˆ 2 are shown in Figs. 3 and 4. The simulation results are shown in Figs. 1-4 accord with the statement in Theorem 1.

AC

Example 2: Consider the following Brusselator model without time delays in [7], which can be described by

  2    x˙ 1 = C − (D + 1)x1 + x1 x2 + ∆1 (x, t) x˙ 2 = Dx1 − x21 x2 + (2 + cos x1 )u + (1.5 + ∆2 (x, t))     y = x1 ,

(32)

where x1 and x2 denote the concentrations of the reaction intermediates, which are constrained in |x1 | < 3.7, |x2 | < 2, C and D are positive parameters that describe the supply of reservoir October 8, 2016

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1.5

1

x1 , x2

0.5

−0.5

−1

0

10

20

30

Time(s)

40

50

CR IP T

0

60

70

AN US

Fig. 2. Trajectories of system states x1 (t) (solid line) and x2 (t) (dash-dotted line).

4 3 2

M

µ

1 0

ED

−1 −2 −3

PT

−4

0

10

20

30

Time(s)

40

50

60

70

CE

Fig. 3. Trajectory of control input signal µ.

AC

chemical. u denotes the system input. As mentioned in [6], it is assumed that x1 6= 0 and to

consider the practical case, the modeling errors ∆1 (x, t) and ∆2 (x, t) are taken as ∆1 (x, t) = 0.7x21 cos(1.5t) and ∆2 (x, t) = 0.5(x21 + x22 ) sin3 (t). Obviously, the controlled system (32) is in a nonstrict-feedback structure, and it can be verified that it satisfies Assumptions 1-4. Here,

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0.7

0.6

0.4

0.3

0.2

0.1

0

0

10

20

30

Time(s)

40

50

consider the dead zone nonlinearity is      (1 − 0.2 sin µ)(µ − 2.5),

70

µ > 2.5

0, −1.5 ≥ µ ≤ 2.5     (0.8 − 0.1 cos µ)(µ + 1.5), µ < −1.5.

M

u = Z(µ) =

60

AN US

Fig. 4. Trajectories of $ ˆ 1 (t) (solid line) and $ ˆ 2 (t) (dash-dotted line).

CR IP T

̟ ˆ 1, ̟ ˆ2

0.5

ED

For the simulation, the reference is selected as yr = 2 + cos(t). We still use the fuzzy logic systems defined in Example 1 to construct the adaptive fuzzy controller. Similar to [6], let C = 1 and D = 3. The system initial conditions are chosen as x1 (0) = x2 (0) = 1.0 and $ ˆ1 = $ ˆ 2 = 0.

PT

The control parameters are selected as c1 = c2 = 1.7, ι1 = ι2 = 3, and σ1 = σ2 = 0.5. Similar to [28], we can obtain b1 = 1.37 and b2 = 1.17 by using the Matlab routine. The simulation

CE

results are shown in Figs. 5-8. From Fig. 5, it can be seen that a good tracking performance is obtained. Fig. 6 presents the trajectories of system states x1 and x2 , from which we can see that

AC

they are not overstepped. The boundedness of µ, $ ˆ 1 , and $ ˆ 2 are shown in Figs. 7 and 8. Remark 8: From the simulation results, it can be clearly seen that the proposed fuzzy

controller is feasible. In addition, the adaptive fuzzy/neural network control schemes proposed in the existing literature [6]–[11], [14]–[16], [18], [19], [27]–[33], [36], [39]–[42], [44]–[47], [50], [55] cannot be directly adopted to solve the control problems considered in the above two controlled systems.

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3.5

3

2

1.5

1

0.5

0

10

20

30

Time(s)

40

50

CR IP T

y, yr

2.5

60

70

AN US

Fig. 5. Trajectories of system output signal y(t) (dash-dotted line) and reference signal yr (t) (solid line).

3.5 3 2.5

M

x1 , x2

2 1.5 1

ED

0.5 0

−0.5

0

PT

−1

10

20

30

Time(s)

40

50

60

70

AC

CE

Fig. 6. Trajectories of system states x1 (t) (solid line) and x2 (t) (dash-dotted line).

V. C ONCLUSIONS

This paper has proposed a new adaptive fuzzy control scheme for a class of uncertain systems

with nonstrict-feedback form, unknown nonlinear dead zone and full state constraints. In the fuzzy controller design process, only less adjustable parameters are used. To guarantee that the full state constraints are not violated, some BLFs with the error variables are employed to construct the desired controller. Furthermore, it has been proved that all the closed-loop

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20 15 10 5

−5 −10 −15 −20

0

10

20

30

Time(s)

40

50

1

70

M

0.8

̟ ˆ 1, ̟ ˆ2

60

AN US

Fig. 7. Trajectory of control input signal µ.

CR IP T

µ

0

0.6

ED

0.4

0.2

0

PT

0

10

20

30

Time(s)

40

50

60

70

CE

Fig. 8. Trajectories of $ ˆ 1 (t) (solid line) and $ ˆ 2 (t) (dash-dotted line).

AC

signals remain semi-globally uniformly bounded and the tracking error converges to a small neighborhood of the origin. Finally, some simulation examples have been provided to verify the effectiveness of the proposed control method. In the future, there are some significative work about this topic to be addressed, for example, how to extend the design method proposed in this paper to more general systems such as nonstrict-feedback systems with stochastic disturbance and full state constraints, how to further simplify the proposed controller by using the dynamic surface control technique to avoid the ‘dimension complexity’ phenomenon, how to determine October 8, 2016

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the range of the tracking error. In addition, to seek the specific applications of the adaptive backstepping fuzzy control scheme proposed in this paper is another interesting topic as well. R EFERENCES [1] O. Abu Arqub, “Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations,” Neural Computing & Applications. (2015), DOI: 10.1007/s00521-015-2110-x.

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[2] O. Abu Arqub, M. AL-Smadi, S. Momani, and T. Hayat, “Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method,” Soft Computing. (2015), DOI: 10.1007/s00500-015-1707-4.

[3] B. Chen, X.P. Liu, K. Liu, and C. Lin, “Direct adaptive fuzzy control of nonlinear strict-feedback systems,” Automatica. 45 (6) (2009) 1530-1535.

[4] W. Chen, L. Jiao, R. Li, and J. Li, “Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances,” IEEE Transactions on Fuzzy Systems. 18 (4) (2010) 674-685.

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[5] W. Chen, S.S. Ge, J. Wu, and M. Gong, “Globally stable adaptive backstepping neural network control for uncertain strict-feedback systems with tracking accuracy known a priori,” IEEE Transactions on Neural Networks and Learning Systems. 26 (9) (2015) 1842-1854.

[6] B. Chen, X.P. Liu, S.S. Ge, and C. Lin, “Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach,” IEEE Transactions on Fuzzy Systems. 20 (6) (2012) 1012-1021.

[7] B. Chen, X.P. Liu, K.F. Liu, and C. Lin, “Fuzzy approximation-based adaptive control of nonlinear delayed systems with unknown dead zone,” IEEE Transactions on Fuzzy Systems. 22 (2) (2014) 237-248.

M

[8] B. Chen, C. Lin, X. P. Liu, and K. F. Liu, “Adaptive fuzzy tracking control for a class of MIMO nonlinear systems in nonstrict-feedback form,” IEEE Transactions on Cybernetics. 45 (12) (2015) 2744-2755. [9] B. Chen, C. Lin, X. P. Liu, and K. F. Liu, “Observer-based adaptive fuzzy control for a class of nonlinear delayed systems,”

ED

IEEE Transactions on Systems, Man, and Cybernetics: Systems. 46 (1) (2016) 27-36. [10] M. Chen, S.S. Ge, and B. Ren, “Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints,” Automatica. 47 (3) (2011) 452-465.

PT

[11] M. Chen P. Shi, and C.C. Lim, “Adaptive neural fault-tolerant control of a 3-DOF model helicopter system,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46 (2) (2016) 260-270. [12] C.C. Hua and Y.F. Li, “Output feedback prescribed performance control for interconnected time-delay systems with

CE

unknown PrandtlCIshlinskii hysteresis,” Journal of the Franklin Institute. 352 (7) (2015) 2750-2764. [13] J.K. Ho, B.K. Geun, B.P. Jin, and H.J. Young, “Decentralized sampled-data H∞ fuzzy filter for nonlinear large-scale systems,” Fuzzy Sets and Systems. 273 (2015) 68-86.

AC

[14] W. He, S.S. Ge, B.V.E. How, Y.S. Choo, and K.S. Hong, “Robust adaptive boundary control of a flexible marine riser with vessel dynamics,” Automatica. 47 (4) (2011) 722-732.

[15] W. He, S. Zhang, and S.S. Ge, “Adaptive control of a flexible crane system with the boundary output constraint,” IEEE Transactions on Industrial Electronics. 61 (8) (2014) 4126-4133.

[16] W. He, S. Zhang, and S.S. Ge, “Robust adaptive control of a thruster assisted position mooring system,” Automatica. 50 (7) (2014) 1843-1851. [17] Y. Jiang, C. Yang, and H. Ma, “A review of fuzzy logic and neural network based intelligent control design for discrete-time systems,” Discrete Dynamics in Nature and Society. (2016), DOI: 10.1155/2016/7217364.

October 8, 2016

DRAFT

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24

[18] S.J. Yoo, “Adaptive neural tracking and obstacle avoidance of uncertain mobile robots with unknown skidding and slipping,” Information Sciences. 238 (2013) 176-189. [19] B.S. Kim and S.J. Yoo, “Approximation-based adaptive control of uncertain non-linear pure-feedback systems with full state constraints,” IET Control Theory & Applications. 8 (17) (2014) 2070-2081. [20] M. Krstic, I. Kanellakopoulos, and P.V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley-Interscience, 1995. [21] Z. Li, C. Yang, C. Su, and W. Ye, “Adaptive fuzzy-based motion generation and control of mobile under-actuated

CR IP T

manipulators,” Engineering Applications of Artificial Intelligence. 30 (2014) 86-95.

[22] Y.M. Li, S.C. Tong, and T.S. Li, “Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control directions and unknown dead zones,” IEEE Transactions on Fuzzy Systems. 23 (4) (2015) 1228-1241. [23] Y.M. Li, S.C. Tong, and T.S. Li, “Adaptive fuzzy output feedback dynamic surface control of interconnected nonlinear pure-feedback systems,” IEEE Transactions on Cybernetics. 45 (1) (2015) 138-149.

[24] Y.J. Liu and S.C. Tong, “Adaptive fuzzy identification and control for a class of nonlinear pure-feedback MIMO systems with unknown dead-zones,” IEEE Transactions on Fuzzy Systems. 23 (5) (2015) 1387-1398.

AN US

[25] Y.J. Liu and S.C. Tong, “Adaptive fuzzy control for a class of unknown nonlinear dynamical systems,” Fuzzy Sets and Systems. 263 (2015) 49-70.

[26] Y.J. Liu and S.C. Tong, “Adaptive fuzzy control for a class of nonlinear discrete-time systems with backlash,” IEEE Transactions on Fuzzy Systems. 22(5)(2014) 1359-1365.

[27] Y.M. Li and S.C. Tong, “Adaptive fuzzy output-feedback stabilization control for a class of switched non-strict-feedback nonlinear systems,” IEEE Transactions on Cybernetics. (2016), DOI:10.1109/TCYB.2016.2536628.

M

[28] Y.J. Liu and S.C. Tong, “Barrier Lyapunov Functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints,”Automatica. 64 (2) (2016) 70-75. [29] Z. Liu, C. Chen, Y. Zhang, and C.L. Philip Chen, “Coordinated fuzzy control of robotic arms with actuator nonlinearities

ED

and motion constraints,” Information Sciences. 296 (2015) 1-13. [30] Y.X. Li and G.H. Yang, “Robust adaptive fuzzy control of a class of uncertain switched nonlinear systems with mismatched uncertainties,” Information Sciences. 339 (2016) 290-309.

PT

[31] Y.M. Li, S.C. Tong, and T.S. Li, “Composite adaptive fuzzy output feedback control design for uncertain nonlinear strictfeedback systems with input saturation,” IEEE Transactions on Cybernetics. 45 (10) (2015) 2299-2308. [32] Y.M. Li and S.C. Tong, “Prescribed performance adaptive fuzzy output-feedback dynamic surface control for nonlinear

CE

large-scale systems with time delays,” Information Sciences. 292 (2015) 125-142. [33] Y.M. Li, S.C. Tong, and T.S. Li, “ Hybrid fuzzy adaptive output feedback control design for uncertain MIMO nonlinear systems with time-varying delays and input saturation,” IEEE Transactions on Fuzzy Systems. (2015), DOI:

AC

10.1109/TFUZZ.2015.2486811.

[34] Z.J. Li, C.G. Yang, C.Y. Su, S. Deng. F.C. Sun, and W.D. Zhang, “Decentralized fuzzy control of multiple cooperating robotic manipulators with impedance interaction,” IEEE Transactions on Fuzzy Systems. 23 (4) (2015) 1044-1056.

[35] M.M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Transactions on Automatic Control. 41 (3) (1996) 447-451. [36] B. Ren, S.S. Ge, K.P. Tee, and T.H. Lee, “Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function,” IEEE Transactions on Neural Networks. 21 (8) (2010) 1339-1345.

October 8, 2016

DRAFT

ACCEPTED MANUSCRIPT

SUBMITTED TO INFORMATION SCIENCES

25

[37] A. Smith, C. Yang, H. Ma, P. Culverhouse, A. Cangelosi, and E. Burdet, “Novel hybrid adaptive controller for manipulation in complex pertubation environments,” PLoS One. 10 (6) (2015) e0129281. [38] S.C. Tong, L.L. Zhang, and Y.M. Li, “Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46 (1) (2016) 37-47. [39] K.P. Tee, S.S. Ge, and E.H. Tay, “Barrier Lyapunov functions for the control of output-constrained nonlinear systems,” Automatica, 45 (4) (2009) 918-927. (2011) 2511-2516.

CR IP T

[40] K.P. Tee, B. Ren, and S.S. Ge, “Control of nonlinear systems with time-varying output constraints,” Automatica. 47 (11) [41] K.P. Tee, S.S. Ge, and E.H. Tay, “Adaptive control of electrostatic micro actuators with bidirectional drive,” IEEE Transactions on Control Systems Technology. 17 (2) (2009) 340-352.

[42] S.C. Tong, S. Sui, and Y.M. Li, “Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained,” IEEE Transactions on Fuzzy Systems. 23 (4) (2015) 729-742.

[43] L.X. Wang and J.M. Mendel, “Fuzzy basis functions, universal approximation, and orthogonal least-squares learning,”IEEE

AN US

Transactions on Neural Networks. 3 (5) (1992) 807-814.

[44] H. Wang, X. Liu, X. Liu, and S. Li, “Robust adaptive fuzzy fault-tolerant control for a class of non-lower-triangular nonlinear systems with actuator failures,” Information Sciences. 336 (2016) 60-74.

[45] J. Wu, W. Chen, and J. Li, “Fuzzy-approximation-based global adaptive control for uncertain strict-feedback systems with a priori known tracking accuracy,” Fuzzy Sets and Systems. 273 (8) (2015) 1-25.

[46] J. Wu and J. Li, “Adaptive fuzzy control for perturbed strict-feedback nonlinear systems with predefined tracking accuracy,”

M

Nonlinear Dynamics. 83 (3) (2016) 1185-1197.

[47] J. Wu, W.S. Chen, F. Yang, J. Li, and Q. Zhu, “Global adaptive neural control for strict-feedback time-delay systems with predefined output accuracy,”Information Sciences. 301 (2015) 27-43.

ED

[48] Y.Q. Xia, H.J. Yang, P. Shi, and M.Y. Fu, “Constrained infinite-horizon model predictive control for fuzzy discrete-time systems,” IEEE Transactions on Fuzzy Systems. 18 (2) (2010) 429-436. [49] C. Yang, Z. Li, R. Cui, and B. Xu, “Neural network-based motion control of an under-actuated wheeled inverted pendulum

PT

model,” IEEE Transactions on Neural Networks and Learning Systems. 25 (11) (2014) 2004-2016. [50] C. Yang, X. Wang, L. Cheng, and H. Ma, “Neural-learning based telerobot control with guaranteed performance,” IEEE Transactions on Cybernetics. (2016) DOI: 10.1109/TCYB.2016.2573837.

CE

[51] C. Yang, Y. Jiang, Z. Li, W. He, and C.-Y Su, “Neural control of bimanual robots with guaranteed global stability and motion precision,” IEEE Transactions on Industrial Informatics. (2016) DOI: 10.1109/TII.2016.2612646. [52] C. Yang, X. Wang, Z. Li, Y. Li, and C.-Y. Su, “Teleoperation control based on combination of wave variable and neural

AC

networks,” IEEE Transactions on Systems, Man, and Cybernetics: Systems. (2016) DOI: 10.1109/TSMC.2016.2615061.

[53] T.P. Zhang, H. Wen, and Q. Zhu, “Adaptive fuzzy control of nonlinear systems in pure feedback form based on input-to-state stability,” IEEE Transactions on Fuzzy Systems. 18 (1) (2010)80-93.

[54] B. Zhou, Z.Y. Li, and Z. Lin, “Discrete-time L∞ and L2 norm vanishment and low gain feedback with their applications in constrained control,” Automatica. 49 (1) (2013) 111-123. [55] T.P. Zhang and X. Xia, “Decentralized adaptive fuzzy output feedback control of stochastic nonlinear large-scale systems with dynamic uncertainties,” Information Sciences. 315 (2015) 17-38.

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