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Robust ${H}_{\mathrm{\infty}}$ synchronization of chaotic systems with input saturation and timevarying delay
Advances in Difference Equations volume 2014, Article number: 124 (2014)
Abstract
This paper investigates driveresponse robust synchronization of chaotic systems with disturbance, timevarying delay and input saturation via state feedback control. Sufficient conditions for achieving the synchronization of two chaotic systems are derived on the basis of the Lyapunov theory and the linear matrix inequality (LMI) technique, which is not only to guarantee the asymptotic synchronization but also to attenuate the effects of the perturbation on the overall error system to a prescribed level. Finally, an illustrative numerical simulation is also given to demonstrate the effectiveness of the proposed scheme.
1 Introduction
In 1963, Lorenz found the first chaotic attractor in a threedimensional autonomous system when he studied the atmosphere convection [1]. Since then, more chaotic systems have been constructed, such as Chua’s circuit, logistic map, Chen system, and generalized Lorenz system (see [2–6]), and their complex behaviors have also been widely studied. Nowadays, there has been considerable interest in the control of chaos in nonlinear dynamical systems, and many different techniques, such as OGY method [7], PC technique [8], backstepping approach [9], adaptive control [10], fuzzy control [11], digital control [12], state feedback control [13], timedelay feedback control [14], sampled driving signals [15], and observerbased approach [16], have been proposed to control chaos. Since the pioneering work by Pecora and Carroll [17] who originally proposed the driveresponse concept for achieving the synchronization of coupled chaotic systems, chaotic synchronization has received considerable attention due to its potential applications in physics, biology, and engineering and has become an important topic in control theory [18, 19].
However, all of these works and many others in the literature have focused on the study of chaotic synchronization between two chaotic systems without model uncertainties and external disturbance. In real physical systems, some noise or disturbance always exists, which may cause instability and poor performance. Therefore, how to reduce the effect of the noise or disturbance in synchronization process for chaotic systems has become an important issue, see [20–22]. On the other hand, there has been increasing interest in timedelay chaotic systems since the chaos phenomenon in timedelay systems was first found by Mackey and Glass [23]. For chaotic systems with timedelay and disturbance, several works have proposed the problem for various chaotic systems in the literature [24–26]. In [24], an adaptive control law was derived and applied to achieve the state lagsynchronization of two nonidentical timedelayed chaotic systems with unknown parameters. In [25], an output coupling and feedback scheme were proposed to achieve the robust synchronization of noiseperturbed chaotic systems with multiple timedelays. An impulse control was proposed by Qian and Cao [26] to synchronize two nonidentical chaotic systems with timevarying delay. Most of them are based on the fact that the timedelay is a constant, while, in real world applications, the timedelay is also varying over time. Hence the study of chaotic synchronization with timevarying delay is an important topic.
Besides, in a practical chaos system, there exist not only disturbance and varyingtime delay but also the input saturation. Many literature works are based on the assumption that the actuator will not be saturated during the control process, but actuator will saturate due to its physical limitations in practice. Due to its high sensitivity to system parameters, the presence of saturation of control input may cause serious influence on system stability and performance. Hence, the derivation of controller with input saturation is an important problem. In [27], an adaptive sliding mode control scheme for Lorenz chaos subject saturating input was presented. Rehan studied the synchronization and antisynchronization of chaotic oscillators under input saturation via simple state feedback control in [28], and the design of dynamic controller and static antiwindup compensator for Lipschitz nonlinear systems under input saturation was described in [29] and [30]. However, most of them studied the normal chaotic system without timedelay and the inner uncertainty and the external disturbance. Motivated by the above discussion, in this paper we investigate the synchronization of chaotic systems with disturbance and varying timedelay under input saturation. Based on the Lyapunov stability theory, a robust controller is designed and its robustness and stability are analytically proved. Finally, we present a numerical simulation to demonstrate the feasibility and usefulness.
This paper is organized as follows. Section 2 provides the system description. In Section 3, LMIbased conditions for chaotic synchronization are developed. In Section 4, a numerical example is given to illustrate the main result. Finally, conclusion is made in Section 5.
Standard notation is used in this paper. For a matrix M, the i th row is denoted by ${M}_{(i)}$. For a vector $u\in {R}^{m}$, $sat(u)=sign({u}_{(i)})min({\overline{u}}_{(i)},{u}_{(i)})$ represents the classical nonlinear saturation function, where ${\overline{u}}_{(i)}>0$ denotes the i th bound on the saturation.
2 System description and preliminaries
Consider a class of uncertain chaotic systems with timevarying delay which is described by
where $x\in {R}^{n}$ is the state vector. The vector $f(\cdot )\in {R}^{n}$ is a continuous nonlinear vector function satisfying the Lipschitz condition $\parallel f({x}_{1})f({x}_{2})\parallel \le \rho \parallel {x}_{1}{x}_{2}\parallel $ (1a), where ρ is a positive constant. $\tau (t)$ denotes the varying timedelay. A and B are known real constant matrices with suitable dimensions. ΔA and ΔB are perturbation matrices representing parametric uncertainties and are assumed to be of the following form:
where ${H}_{1}$, ${H}_{2}$, ${E}_{1}$, and ${E}_{2}$ are known real constant matrices with appropriate dimensions, $F(t)\in {R}^{n\times n}$ is an unknown real and possibly timevarying matrix satisfying
The uncertainties ΔA and ΔB are said to be admissible if both (2) and (3) hold. Eq. (1) is considered as the drive system and the controlled response system is given by the following differential Eq. (4):
where $y(t)\in {R}^{n}$, $u\in {R}^{m}$ and $w(t)\in {R}^{n}$ are the state, the input, and the external disturbance vectors for the response system, respectively, and $sat(u)\in {R}^{m}$ represents the saturated input. $C\in {R}^{n\times m}$ represents a constant matrix.
Define the synchronization error as $e(t)=(x(t)y(t))\in {R}^{n}$. Subtracting the drive system (1) from the response system (4) yields the dynamical system
This paper aims at designing the controller to not only asymptotically synchronize between the drive and the response systems but also to guarantee a prescribed performance of the external perturbation attenuation γ.
Before presenting the main result, we introduce the following definition.
Definition [25]
For the synchronization error system (5), it is said to have the ${H}_{\mathrm{\infty}}$ synchronization with the external perturbation attenuation γ if the following conditions are satisfied:

(i)
With $w(t)=0$, the dynamics error system (5) is asymptotically stable.

(ii)
Given a desired positive scalar γ and under the zeroinitial condition, the following performance index is satisfied:
$$J={\int}_{0}^{\mathrm{\infty}}({e}^{T}(t)e(t){\gamma}^{2}{w}^{T}(t)w(t))\phantom{\rule{0.2em}{0ex}}dt\le 0.$$(6)
For a positive definite diagonal matrix $W\in {R}^{m\times m}$, the saturation nonlinearity satisfies the classical global sector condition [31] given by
where $\varphi (u)=usat(u)$ represents the dead zone nonlinearity. This sector condition can be used to design a global controller for synchronization of nonlinear systems under input saturation. However, if global results cannot be achieved, a more general sector condition can be utilized to design a local synchronization controller. Define the following associated set:
where $\overline{u}\in {R}^{m}$ represents the bound on saturation. If (8) holds, the local sector condition
is satisfied.
In dealing with this study, the following assumptions and lemmas are necessary for the sake of convenience.
Assumption 1 The timedelay $\tau (t)$ is a bounded and continuously differentiable function such that $0\le \tau (t)\le {\mu}_{1}$ and $0<\dot{\tau}(t)\le \mu <1$.
Lemma 1 [32]
Given any vector x, y of appropriate dimensions and a positive number ε, the following inequality holds:
Lemma 2 [33]
Let ${S}_{11}$ be a regular $n\times n$ matrix, ${S}_{12}$ can be an $n\times q$ matrix, and ${S}_{22}$ is a regular matrix. Let a Hermitian matrix S be represented as $S=\left(\begin{array}{cc}{S}_{11}& {S}_{12}\\ {S}_{12}^{T}& {S}_{22}\end{array}\right)$.
Then the matrix S is positive definite if and only if the matrices ${S}_{11}$ and ${S}_{22}{S}_{12}^{T}{S}_{11}^{1}{S}_{12}$ are positive definite.
3 Chaotic synchronization
To synchronize the driveresponse systems (1) and (4), the following state feedback control law is considered:
where $F\in {R}^{m\times n}$. By using $\varphi (u)=usat(u)$, (5) and (12), the overall closedloop system becomes
Theorem 1 Consider the driveresponse systems (1) and (2) satisfying Assumption 1 and condition (1a). Given a scalar $\gamma >0$ and a matrix $Q={Q}^{T}>0$, if there exist a matrix ${X}_{1}={X}_{1}^{T}>0\in {R}^{n\times n}$, a matrix $R={R}^{T}>0\in {R}^{n\times n}$, a diagonal matrix $U\in {R}^{m\times m}$, a matrix ${X}_{2}\in {R}^{m\times n}$, a matrix ${X}_{3}\in {R}^{m\times n}$, and scalars ${\epsilon}_{i}>0$ ($i=1,2,3$) satisfying the following linear matrix inequalities (LMIS):
with $\mathrm{\Xi}={X}_{1}{A}^{T}+A{X}_{1}C{X}_{2}{{X}_{2}}^{T}{C}^{T}+\frac{1}{{\epsilon}_{3}}I$, then the overall closedloop system with Eq. (5) is ${H}_{\mathrm{\infty}}$ synchronized with the disturbance attenuation level γ.
Proof Choose the following Lyapunov functional candidate:
where ${V}_{1}(t)={e}^{T}(t)Pe(t)$, ${V}_{2}(t)=\frac{1}{1\mu}{\int}_{t\tau (t)}^{t}{e}^{T}(s)Re(s)\phantom{\rule{0.2em}{0ex}}ds$, $P={P}^{T}$, $R={R}^{T}$.
First, evaluating the time derivative of ${V}_{1}(t)$ along the trajectory given in Eq. (13) gives
By using Lemma 1 and the Lipschitz condition, we have
Substituting Eqs. (18), (19), and (20) into Eq. (17) results in
By using $u=Fe$, we take $v=Ge$, then the local sector conditions (8) and (9) can be rewritten as
Consider the set $\epsilon (P,\delta )=\{e(t)\in {R}^{n};{e}^{T}(t)Pe(t)\le \delta \}$, then LMI (15) is obtained by including the region ${e}^{T}(t)Pe(t)\le \delta $ into $S(\overline{u})$. Hence the region $S(\overline{u})$ in (22) remains valid, which further implies that the sector condition (23) is satisfied. By using (23), we have
By using Assumption 1, we have
In order to obtain LMI (16), we give a matrix $Q={Q}^{T}$ such that the following inequality holds:
Then we have
Define a functional $J(e(t),w(t))$ as follows:
Substituting (27) into (28) yields
where
From the above, if the following inequality holds:
where
Applying the Schur complement and congruence transform by using diag $({P}^{1},I,{W}^{1},I,I,I,I,I,I,I)$, and further, substituting ${P}^{1}={X}_{1}$, ${W}^{1}=U$, ${X}_{2}=F{X}_{1}$, ${X}_{3}=G{X}_{1}$, LMI (16) is obtained. Then integrating the function in (26) yields
With the zeroinitial condition, we have
which completes the proof of Theorem 1. □
4 Examples and simulation results
To demonstrate the validity of the proposed synchronization approach with input saturation and timedelays, we consider the Lorenz chaotic system with:
The Lipschitz constant is chosen as $\rho =1$, and the parameter δ is given as $\delta =1$. The disturbance is selected as $w(t)={(\begin{array}{ccc}0.1sin10t& 0.2sin20t& 0.1sin30t\end{array})}^{T}$. For convenience, we choose $\tau (t)=1$.
The chaotic behavior of Lorenz system with timedelays is shown in Figures 1 and 2.
By applying the conditions in Theorem 1 with ${\epsilon}_{1}=\frac{1}{4}$, ${\epsilon}_{2}=\frac{1}{3}$, ${\epsilon}_{3}=\frac{1}{5}$, and the disturbance attenuation $\gamma =0.5$, we can obtain the following matrices:
Applying the controller $u=Fe$ without the disturbance signal, the synchronization error between the drive system and the response system with the initial conditions ${x}_{0}={(\begin{array}{ccc}1& 2& 3\end{array})}^{T}$ and ${y}_{0}={(\begin{array}{ccc}1.2& 1.8& 3.6\end{array})}^{T}$, respectively, is shown in Figure 3, which implies that the synchronization error converges to zero. Figure 4 shows that the effect of the disturbance $w(t)$ on the dynamic error system has been reduced within a prescribed level to $\gamma =0.5$ by the control gain F.
5 Conclusions
The problem of robust ${H}_{\mathrm{\infty}}$ synchronization for an uncertain chaotic system with timevarying delay and input saturation has been presented. Based on the Lyapunov theory and the LMI technique, the sufficient condition has been derived not only to guarantee the asymptotic synchronization but also to ensure a prescribed perturbation attenuation performance. Finally, a simulation example is presented to verify the validity of the proposed method.
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Acknowledgements
This paper is supported by the National Natural Science Foundation of China (No. 61273004). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
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Ma, Y., Jing, Y. Robust ${H}_{\mathrm{\infty}}$ synchronization of chaotic systems with input saturation and timevarying delay. Adv Differ Equ 2014, 124 (2014). https://doi.org/10.1186/168718472014124
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Keywords
 chaotic system
 robust synchronization
 saturation
 timevarying delay