Elsevier

Chaos, Solitons & Fractals

Volume 76, July 2015, Pages 98-110
Chaos, Solitons & Fractals

Adaptive fuzzy observer-based stabilization of a class of uncertain time-delayed chaotic systems with actuator nonlinearities

https://doi.org/10.1016/j.chaos.2015.03.006Get rights and content

Abstract

An observer-based output feedback adaptive fuzzy controller is proposed to stabilize a class of uncertain chaotic systems with unknown time-varying time delays, unknown actuator nonlinearities and unknown external disturbances. The actuator nonlinearity can be backlash-like hysteresis or dead-zone. Based on universal approximation property of fuzzy systems the unknown nonlinear functions are approximated by fuzzy systems, where the consequent parts of fuzzy rules are tuned with adaptive schemes. The proposed method does not need the availability of the states and an observer based output feedback approach is proposed to estimate the states. To have more robustness and at the same time to alleviate chattering an adaptive discontinuous structure is suggested. Semi-global asymptotic stability of the overall system is ensured by proposing a suitable Lyapunov–Krasovskii functional candidate. The approach is applied to stabilize the time-delayed Lorenz chaotic system with uncertain dynamics amid significant disturbances. Analysis of simulations reveals the effectiveness of the proposed method in terms of coping well with the modeling uncertainties, nonlinearities in actuators, unknown time-varying time-delays and unknown external disturbances while maintaining asymptotic convergence.

Introduction

The control and synchronization of chaotic systems has always been a challenging and yet rewarding problem. This is because the chaotic systems are very sensitive to initial conditions and parameter variations which make them to be unpredictable [1]. On the other hand, the chaotic behavior can be seen in many physical and industrial systems like, biological and ecological systems, chemical processes, mechanical and electrical systems, etc. Chaos also has many applications such as in secure communication, information processing and optimization[1], [2], [3].

Time delay mostly happens in practical systems. It degrades the performance of systems and even usually becomes the source of instability [4]. Therefore, the stability analysis and controller synthesis for nonlinear time-delay systems should be considerably received much attentions [5], [6], [7]. Besides, time-delayed systems are infinite-dimensional [8]. For chaotic systems this gives hyper-chaotic systems which are complicated and difficult to be controlled but more useful in applications like secure communication [8], [9]. The control and synchronization of time-delayed chaotic systems become more challenging when we have time-varying time-delayed chaotic systems [10]. Another highly important practical issue is the unavailability of states. In most practical situations either the states are not available for measurement or due to the expensive sensors and transducers it is not economical to measure all the states. Therefore, the states are fully or partially unknown. In order to cope with the states unavailability observer-based methods are vastly studied [11], [12], [13], [14], [15].

Nonlinearities in actuators can be seen in a wide range of practical systems and devices [16], [17]. Thus, the control of nonlinear systems with actuator nonlinearities is of utmost important [18]. One of the actuator nonlinearities may cause through hysteresis which can be seen in ferromagnetic, magnetostrictive, and piezoelectric actuators [19]. Since, the hysteresis nonlinearity is non-differentiable and non-memoryless, the system performance is severely limited and usually exhibits undesirable inaccuracies or oscillations and even instability [20], [21]. Another most important non-smooth actuator nonlinearity is dead-zone. Dead-zone is a memory-less nonlinearity which can be seen in actuators such as hydraulic servo valves and electric servomotors which can jeopardize the system stability and performance [22]. Due to uncertainty in its parameters the control of nonlinear systems preceded by dead-zone is a challenging task [23].

To this point, the control and synchronization of time-delayed chaotic systems have been the focus of the numerous studies [10], [24], [25], [26], [27], [28], [29], [30]. For all of these schemes the actuators should be linear. Moreover, in some of them the states of the system should be known and/or the time-delay should be a known constant. For instance, an adaptive neural network based synchronization is proposed for a class of uncertain time-delayed chaotic systems in [29], but the time-delay is a known constant and the actuators are assumed to be linear. An adaptive sliding mode fault-tolerant control against network fault and time-delay for a class of coupled chaotic systems is suggested in [24]. The method can cope with time-varying time-delays but all the states of the system should be available for measurement, the actuators are assumed to be linear and the input gains should be constant. In [30], the exponential H synchronization of a general time-delayed discrete-time chaotic neural networks is studied. However, the method considers known constant time-delays and linear actuators. The impulsive control is used to synchronize two Lur’s chaotic systems with time-delay and parameter mismatches in [10]. Although the method can cope with time-varying time-delays, but the actuators should be linear. A time-delay dependent feedback control is proposed in [25] to stabilize general chaotic systems. For this method the time-delay is constant and the actuator is linear with constant gains, meanwhile, the states of the system should be available for measurement.

In order to address the aforementioned deficiencies caused by actuator nonlinearities, some papers studied the control and synchronization of chaotic systems in the presence of dead-zone actuator nonlinearity [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. These methods can only be applied to dead-zone nonlinearity. Meanwhile, some of these methods should know the states and/or cannot cope with time-delays. For example, in [40], the exponential synchronization of Genesio–Tesi chaotic systems have been studied. Although the proposed method can handle unknown dead-zone nonlinearities but it can be only applied to Genesio–Tesi chaotic systems without time-delay and the uncertainties should be partially known. An adaptive sliding mode control for a class of chaotic systems preceded by dead-zone actuators is proposed in [41]. The dead-zone parameters and the states of the chaotic system should be known in advance. Meanwhile, the dead-zone gain is constant and the chaotic systems should be modeled in Brunovsky canonical form without time-delay. In [31], H adaptive fuzzy approach is used to control a class of time-delayed chaotic systems with dead-zone actuator nonlinearity. Although, the method can cope with unknown dead-zone parameters and unknown time-varying time-delays, but all the states of the chaotic system should be available for measurement and the control law depends on the upper bound of time-delay derivative. Meanwhile, the gain for actuator nonlinearity is constant. A projective synchronization is studied for a class of master–slave chaotic systems with dead-zone nonlinearity in [36]. The slope of dead-zone should be known and the method cannot handle time-delay chaotic systems.

In this paper, an output feedback based adaptive fuzzy controller is proposed for a class of uncertain MIMO time-delayed chaotic systems with unknown actuators and unknown but bounded disturbances. The actuator nonlinearities are unknown and can be backlash-like hysteresis or dead-zone, meanwhile, the gains for actuators can be nonlinear. The time-delay is unknown and can be time-varying. The unknown nonlinear functions are approximated by fuzzy systems based on universal approximation lemma, where the consequent parts of the fuzzy rules are tuned with adaptive schemes. The proposed approach does not need the availability of the states and uses an observer based output feedback approach to estimate the states. An adaptive discontinuous structure is used as a robust control term which attenuates chattering in control effort. The control input does not have the singularity problem by proposing a suitable control law. The proposed approach has the added advantage that for external disturbances it only requires a bound to exist, without needing to know the magnitude of this bound. The semi-global asymptotic stability of the closed-loop system is proved by proposing a suitable Lyapunov–Krasovskii functional candidate. The proposed controller is applied to an uncertain time-delayed Lorenz chaotic system with unknown actuator nonlinearities and unknown significant disturbances.

This paper is organized as follows. Section 2 formulates the class of chaotic systems under consideration and briefly reviews fuzzy logic systems and universal approximation lemma. In Section 3, the proposed output feedback based adaptive fuzzy controller is presented and the semi-global asymptotic stability of the proposed method is proved. To show the effectiveness of the proposed method, in Section 4 it is applied to an uncertain time-varying time-delayed Lorenz system with unknown nonlinear actuators with both backlash-like hysteresis and dead-zone nonlinearities amid unknown disturbances. Throughout this paper, for xRn,x=max1inxi denotes the infinity norm of the vector. For notation conciseness, x is shortened to x.

Section snippets

Problem formulation

Consider a class of chaotic systems in the following form:ẋ(t)=(A1+ΔA1(t))x(t)+(A2+ΔA2(t))x(t-τ(t))+B(F(x)+G¯(x)ϕ(v)+d¯(t,x))y=Cx,where F(x)=[f1(x),,fp(x)]T is an unknown continuous nonlinear vector function, G¯(x)=[g¯ij(x)],i,j=1,,p is an unknown continuous nonlinear matrix function. x=[x1,,xn]TRn is the state vector, v=[v1,,vp]TRp and y=[y1,,yp]TRp are the input and the output vector of the system, respectively. A1,A2,B and C are known matrices with appropriate dimensions, x(t-τ(t))

The proposed method

Solving the differential Eq. (2) of the backlash-like hysteresis, the following solution is obtained [20]:ϕj(vj)=cjvj(t)+Bj(vj),Bj(vj)=[ϕj0-cjvj0]e-αj(vj-vj0)sgnv̇j+e-αjvjsgnv̇jvj0vj[βj-cj]eαjηjsgnv̇jdηj,where vj(0)=vj0 and ϕj(vj0)=ϕj0. The dead-zone model (3) can be easily rewritten asϕj(vj)=mjvj+Zj(vj),Zj(vj)=-mjbrj,vjbrj-mjvj,blj<vj<brj-mjblj,vjbljThus, from (8), (9) both backlash-like hysteresis and dead-zone nonlinearities have the same form as follows:ϕj(vj)=kjvj+Dj(vj),where if the

Simulation example

Here, we aim to apply the proposed method to the uncertain time-delayed Lorenz chaotic system amid significant disturbances and with both backlash-like hysteresis and dead-zone actuator nonlinearities.

Consider the time-delayed Lorenz chaotic system as follows [54]ẋ=-1010028-1000-8/3+0000.8sin2t1+0.3cos2t0-1+0.5cost00.4x+-0.100.2-0.510010.3+0000.1cos3t0.20.10.2sint00.4sin3tx(t-0.25sin2t)+001001-x1x3x1x2+2+cosx10.50(1.5+sin3x2)2ϕ1(v1)ϕ2(v2)+d1d2,y=010001x,where x=[x1,x2,x3]T is the state vector.

Conclusions

In this paper, the stabilization of uncertain time-delayed chaotic systems with nonlinear actuators and non-constant nonlinear gains for actuators is investigated. The actuator nonlinearities can be backlash-like hysteresis or dead-zone. The proposed method uses an observer to estimate the states from the output, thus the states does not need to be fully available. The unknown nonlinear functions are approximated via fuzzy systems based on universal approximation lemma. The consequent parts of

References (54)

  • X.Z. Jin et al.

    Adaptive sliding mode fault-tolerant control for nonlinearly chaotic systems against network faults and time-delays

    J Franklin Inst

    (2013)
  • B. Cui et al.

    Synchronization of chaotic recurrent neural networks with time-varying delays using nonlinear feedback control

    Chaos, Solitons Fractals

    (2009)
  • M. Chen et al.

    Robust adaptive neural network synchronization controller design for a class of time delay uncertain chaotic systems

    Chaos, Solitons Fractals

    (2009)
  • W. Li et al.

    Projective synchronization of Chua’s chaotic system with dead-zone in the control input

    Commun Nonlinear Sci Numer Simul

    (2009)
  • T.Y. Chiang et al.

    Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity

    Nonlinear Anal: Theory, Methods Appl

    (2008)
  • H.T. Yau

    Generalized projective chaos synchronization of gyroscope systems subjected to dead-zone nonlinear inputs

    Phys Lett A

    (2008)
  • A. Boulkroune et al.

    A practical projective synchronization approach for uncertain chaotic systems with dead-zone input

    Commun Nonlinear Sci Numer Simul

    (2011)
  • Y.C. Hung et al.

    Adaptive variable structure control for chaos suppression of unified chaotic systems

    Appl Math Comput

    (2009)
  • M.L. Hung et al.

    Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input

    Chaos, Solitons Fractals

    (2008)
  • Y.C. Hung et al.

    Projective synchronization of Chua’s chaotic systems with dead-zone in the control input

    Math Comput Simul

    (2008)
  • Z. Zhang et al.

    Exponential synchronization of Genesio–Tesi chaotic systems with partially known uncertainties and completely unknown dead-zone nonlinearity

    J Franklin Inst

    (2013)
  • J.J. Yan et al.

    Adaptive variable structure control for uncertain chaotic systems containing dead-zone nonlinearity

    Chaos, Solitons Fractals

    (2005)
  • Y. Li et al.

    Adaptive fuzzy output feedback control of uncertain nonlinear systems with unknown backlash-like hysteresis

    Inf Sci

    (2012)
  • R. Shahnazi

    Output feedback adaptive fuzzy control of uncertain MIMO nonlinear systems with unknown input nonlinearities

    ISA Trans

    (2015)
  • Y.J. Liu et al.

    Observer-based adaptive fuzzy-neural control for a class of uncertain nonlinear systems with unknown dead-zone input

    ISA Trans

    (2010)
  • S. Labiod et al.

    Adaptive fuzzy control of a class of MIMO nonlinear systems

    Fuzzy Sets Syst

    (2005)
  • T. Kapitaniak

    Chaos for engineers: theory, applications, and control

    (2000)
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