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

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Abstract

This paper is concerned with the problem of synchronization controller design for Genesio–Tesi chaotic systems with plant uncertainties and dead-zone input. The upper bounds of plant uncertainties are partially known, while dead-zone nonlinearity is completely unknown. The prior knowledge on the parameters in the uncertainties and dead-zone is not required to be known in advance. A novel control law with adaptive methodology is proposed to compensate for input nonlinearity. An adaptive law with exponent function is designed to estimate the unknown lumped parameters. It is shown that complete synchronization between two identical Genesio–Tesi chaotic systems is achieved and the synchronization errors converge to zero exponentially. Numerical studies are provided to illustrate the effectiveness of the presented scheme.

Introduction

In the past more than two decades, the control and synchronization problems of chaotic systems have attracted much attention. The increasing interest in this topic lies in the potential applications of chaos theory in many engineering areas such as secure communication, biological systems, signal generator, chemical reaction, etc. Chaos synchronization was firstly proposed in [1]. From then on, many control techniques such as OGY method, active control, adaptive control, backstepping approach, impulse control, have been presented to implement synchronization (see e.g. [2], [3], [4], [5], [6], [7], [8], [9] and the references therein).

The Genesio–Tesi system was firstly reported in [10]. Chaos control and synchronization of such system have been investigated in several papers. In [11], the Genesio–Tesi chaotic system with unknown plant parameters, system uncertainties and external disturbance was considered. Adaptive sliding mode approach was employed to design tracking controller. In [12], a dynamic output-feedback controller was developed to stabilize the Genesio–Tesi system with known parameters, where linear matrix inequality (LMI) technique was utilized and a LMI condition was provided. In [13], a systematic design procedure, called as backstepping, was proposed for the synchronization of two identical or nearly identical Genesio–Tesi systems. In [14], the control idea of [12] was extended to design adaptive synchronization controller. In [15], the stability theory of general nonlinear systems was introduced to modify the existing results for control and synchronization of the Genesio–Tesi system. However, there are two main drawbacks in the aforementioned papers for the Genesio–Tesi system. One is that the used control signal was free of nonlinear input, and input nonlinearity was not considered in the control scheme. Another drawback lies in that most of the synchronization schemes were based on asymptotic stability, and exponential stability has not been achieved.

As we know, dead zone, backlash, hysteresis are typical nonlinearity characteristics, which are common in hydraulic actuators, electric servomotors, etc. In the literature, it has been shown that the presence of these nonlinearities in the control input severely limits system performance. Hence it is important that input nonlinearity is taken into account in control design. Recently, chaos synchronization with nonlinearity constraints has received much attention (see e.g. [16], [17], [18], [19], [20]). Dead zone, as one of the most important nonsmooth nonlinearities, has been considered in the problem of chaos synchronization (see e.g. [21], [22]). Particularly, in the recent paper [22], the synchronization principle was presented for the Genesio–Tesi system with dead-zone input. However, it is noted that the dead-zone parameters in [22] were used in the control law, which implied that the proposed synchronization control was based on the exact knowing of these parameters. Nevertheless, in real situations, some or all of the parameters in the dead-zone nonlinearity may be not available to the designer.

On the other hand, compared with asymptotic stability, exponential stability is a more strengthened one [23]. If the exponential convergence of synchronization errors can be ensured, the master–salve synchronization can be rapidly accomplished within a short time. In recent years, the exponential master–slave synchronization problem of several class of chaotic systems has been addressed in the literature (see e.g. [22], [24], [25], [26]). When input nonlinearity is involved in the design of synchronization control, the exponential synchronization problem becomes more difficult and more complicated. In the aforementioned paper [22], the so-called practically exponential synchronization was developed for the Genesio–Tesi system with both plant uncertainties and dead-zone input. However, it is worth pointing out that asymptotic convergence with zero synchronization errors was not guaranteed. Accurately speaking, in [22] the synchronization errors with exponential performance were established and the bounded-error synchronization was derived.

Motivated by the above discussions, in this paper we further investigate the problem of exponential master–slave synchronization between two identical Genesio–Tesi systems, which are subject to both plant uncertainties and dead-zone input nonlinearity. A novel adaptive synchronization scheme is proposed, where a decreasing exponent function is incorporated into control law. Different from the existing literature, the restrictive assumption on the known parameters in the uncertain functions and dead-zone model is removed by using adaptive technique. The parameter estimate is updated according to a novel adaptive rule with an increasing exponent term. It is proven that the synchronization errors converge to zero exponentially. Finally, a numerical example is provided to demonstrate the validity of the presented scheme.

The rest of the paper is organized as follows. In Section 2, some preliminaries and problem formulation are given. Then, an adaptive synchronization controller is designed and the exponential convergence is proved in Section 3. In Section 4, simulation studies are performed. In Section 5, some conclusions are drawn.

Section snippets

Preliminaries and problem formulation

Before presenting the problem formulation, we introduce several useful lemmas and the form of dead-zone nonlinearity.

Lemma 1

If a continuous differential real function s(t) satisfies the inequalitys˙(t)g(t)2αs(t)t0,where α>0 and g(t) is a real function, thens(t)e2αts(0)+e2αt0te2ατg(τ)dτt0.

Proof

Multiplying (1) by e2αt, we havee2αts˙(t)+e2αt·2αs(t)e2αtg(t)t0,that is,ddt[e2αts(t)]e2αtg(t)t0.Integrating (4) on [0, t] yieldse2αts(t)s(0)0te2ατg(τ)dτ.Thus, (2) immediately follows from (5). 

Lemma 2

If g(

Synchronization controller design

We first describe Eq. (17) as a more compact forme˙(t)=Ae+b[x12z12Δf1+Δf2+Δϕ(u)],whereA=010001a1b1c1,b=001.It is easily verified that (A,b) is a controllable pair. So for a given α>0, (A+αI,b) is also controllable. Thus, we can choose a vector k such that (A+αI)+bk is stable. Hence, for any given Q>0, there exists a P>0 satisfying[(A+αI)+bk]TP+P[(A+αI)+bk]=Q,which further implies that(A+bk)TP+P(A+bk)=Q2αP<2αP.Then, we defineθ=max1m,θ1m,θ2m,d¯,f=|x12z12ke|+f1+f2+1.Now we present the

Simulation

In this section, simulation results are provided to verify the effectiveness and correctness of the above theoretical results. We consider the same numerical example as in [22]:

Master system:x˙1=x2,x˙2=x3,x˙3=6x12.92x21.2x3+x12Δa·x22.Slave system:z˙1=z2,z˙2=z3,z˙3=6z12.92z21.2z3+z12Δb·sin(t)Δϕ(u),where0.1Δa0.1,0.1Δb0.1,Δϕ(u)D(u,m,r¯,r̲).Clearlya1=6,b1=2.92,c1=1.2,Δf1=Δa·x22,Δf2=Δbsin(t).Thus, Assumption 1 is satisfied if we choosef1=x22,f2=|sin(t)|.Then, let us choose α=0.01,k=[5

Conclusion

In this paper, we address the problem of adaptive synchronization for two identical Genesio–Tesi chaotic systems with partially known uncertainties and completely unknown dead-zone nonlinearity. A simple synchronization controller is developed and an adaptive law for parameter estimation is presented. Exponential convergence with zero synchronization errors is guaranteed. The effectiveness of the proposed synchronization scheme is verified by numerical simulation.

Acknowledgements

This work was supported by the National Natural Science Foundation of Peoples Republic of China under Grants 61104007, 61074040 and 61273091, China Postdoctoral Science Foundation funded project under Grant 2012M511465, Shandong Postdoctoral Science Foundation funded project under Grant 201203031, the Young and Middle-Aged Scientists Research Foundation of Shandong Province under Grant BS2011DX013, Dr. start-up fund research of Qufu Normal University and Youth Foundation of Qufu Normal

References (27)

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