Elsevier

Information Sciences

Volume 376, 10 January 2017, Pages 233-247
Information Sciences

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

https://doi.org/10.1016/j.ins.2016.10.016Get rights and content

Abstract

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 backstepping design process. This advantage is first to take into account the full 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 proposed control method.

Introduction

In the past decades, adaptive backstepping fuzzy control has become one of the most popular design methods to solve the regulation and tracking control problems for various nonlinear systems with completely unknown system functions, e.g., see [1], [2], [3], [4], [5], [12], [13], [17], [21], [22], [23], [24], [25], [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 system is needed to approximate all the unknown functions in each backstepping 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], [22], [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 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 systems as feedforward compensators, the problem of global adaptive fuzzy control for output-feedback systems with unknown high-frequency gain sign has been solved in [5]. Subsequently, 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 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. 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 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], [7], [8], [9], [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 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 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 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 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, 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 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-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], [15], [16], [19], [28], [36], [39], [40], [41], [52], [54], where the control schemes have been developed for nonlinear strict-feedback 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 adaptive fuzzy control method with less adjustable parameters for uncertain systems with nonstrict-feedback form and full state constraints.

  • (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 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 2, we present some preliminaries including system stability and some important lemmas. Section 3 gives the control design process and the main result on the control performance and the closed-loop stability. In Section 4, two simulation examples are provided to verify the effectiveness of the proposed control approach. We conclude the work of this paper in Section 5.

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 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 exp(·) denotes a exponential function.

Section snippets

Preliminaries

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),x(t0)=x0,where x(t) ∈ Rn is the system state, f: Rn × [t0, ∞] → Rn is a continuous vector-valued function, t0 and x0Rn denote the initial time and the initial state vector, respectively. If there exists a compact set URn such that for all

Problem formulation, control design and stability analysis

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.

Simulation examples

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] {x˙1=x1x22+x12sin(x2)+(1.5+0.5sin(x1))x2x˙2=x1x2ex2+x1cos(x1x2)+(1.5+sin(x1x2))uy=x1,where u=Z(μ)={(10.2sinμ)(μ2.5),μ>2.50,1.5μ2.5(0.80.1cosμ)(μ+1.5),μ<1.5.Apparently, this system is without the strict-feedback structure and it can be verified that Assumption 1, Assumption 3 and 4 are satisfied. In

Conclusions

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 signals remain semi-globally

References (55)

  • LiY.X. et al.

    Robust adaptive fuzzy control of a class of uncertain switched nonlinear systems with mismatched uncertainties

    Inf. Sci.

    (2016)
  • LiY.M. et al.

    Prescribed performance adaptive fuzzy output-feedback dynamic surface control for nonlinear large-scale systems with time delays

    Inf. Sci.

    (2015)
  • K.P. Tee et al.

    Barrier lyapunov functions for the control of output-constrained nonlinear systems

    Automatica

    (2009)
  • K.P. Tee et al.

    Control of nonlinear systems with time-varying output constraints

    Automatica.

    (2011)
  • WangH. et al.

    Robust adaptive fuzzy fault-tolerant control for a class of non-lower-triangular nonlinear systems with actuator failures

    Inf. Sci.

    (2016)
  • J. Wu et al.

    Fuzzy-approximation-based global adaptive control for uncertain strict-feedback systems with a priori known tracking accuracy

    Fuzzy Sets Syst.

    (2015)
  • WuJ. et al.

    Global adaptive neural control for strict-feedback time-delay systems with predefined output accuracy

    Inf. Sci.

    (2015)
  • ZhouB. et al.

    Discrete-time L and L2 norm vanishment and low gain feedback with their applications in constrained control

    Automatica.

    (2013)
  • ZhangT.P. et al.

    Decentralized adaptive fuzzy output feedback control of stochastic nonlinear large-scale systems with dynamic uncertainties

    Inf. Sci.

    (2015)
  • O.A. Arqub

    Adaptation of reproducing kernel algorithm for solving fuzzy fredholm-volterra integrodifferential equations

    Neural Comput. Appl.

    (2015)
  • O.A. Arqub et al.

    Numerical solutions of fuzzy differential equations using reproducing kernel hilbert space method

    Soft Comput.

    (2015)
  • ChenW. et al.

    Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances

    IEEE Trans. Fuzzy Syst.

    (2010)
  • ChenW. et al.

    Globally stable adaptive backstepping neural network control for uncertain strict-feedback systems with tracking accuracy known a priori

    IEEE Trans. Neural Networks Learn.Syst.

    (2015)
  • ChenB. et al.

    Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach

    IEEE Trans. Fuzzy Syst..

    (2012)
  • ChenB. et al.

    Fuzzy approximation-based adaptive control of nonlinear delayed systems with unknown dead zone

    IEEE Trans. Fuzzy Syst.

    (2014)
  • ChenB. et al.

    Adaptive fuzzy tracking control for a class of MIMO nonlinear systems in nonstrict-feedback form

    IEEE Trans. Cybern.

    (2015)
  • ChenB. et al.

    Observer-based adaptive fuzzy control for a class of nonlinear delayed systems

    IEEE Trans. Syst., Man Cybern.

    (2016)
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    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 (JBG160709), 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.

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