A novel approach to synchronization conditions for delayed chaotic Lur’e systems with state sampled-data quantized controller

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Abstract

This paper, by designing a state quantized sampled-data controller (SQSDC), studies the synchronization issue for delayed chaotic Lur’e systems (CLSS). A novel method, the auxiliary function-based-inequality Lyapunov–Krasovskii functional (LKF), is for the first time to be introduced. This method can make the best of more the obtainable information about the practical sampling pattern in comparison with some existing approaches. A new zero-valued equation, which increases the combinations of some resulting vectors, is firstly formulated. According to the novel method and zero-valued equation, some new master-slave synchronization (MSS) conditions are established. Meanwhile, the corresponding SQSDC gain matrix is obtained with the bigger sampling interval (SI) than those in the existing works. Finally, the effectiveness and advantages of the proposed method are shown by two numerical examples.

Introduction

Chaotic synchronization, which was firstly proposed by Pecora et al. [1], has aroused more and more interests since it can be extensively applied to many engineering control fields like chemical reaction, secure communication, chaos generator design and signal processing [2], [3], [4], [5], [6], [7]. In fact, it has been displayed that lots of nonlinear systems, for example, network systems, Chua’s circuit, can be described exactly in terms of CLSS. Therefore, the MSS of CLSS has received considerable attention in many practical application and research areas, and a lot of significant and interesting results on this topic have been presented [4], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Moreover, it should be noticed that, due to the finite switching speed of amplifiers and the inherent signal transmission time among the neurons in the realization of electronic circuits, inevitably time delays exist in the CLSS. Generally speaking, time delays that present in the CLSS usually can lead to many poor performances, such as oscillation, divergence, and even instability. As a consequence, it is very important to consider the time delay for the MSS of CLSS.

From the viewpoint of control tactics, the control is a very efficient MSS approach for the CLSS. As a result, a great number of interesting control schemes have been developed to achieve the MSS of CLSS, for instance, impulsive control [12], sampled-data control (SDC) [9], [15], [16], [17], [18], [19], [20], [21], time-delay feedback control [7], [8], [10], PD control [13], adaptive control [22], [23]. Among these control methods, the SDC enjoys great popularity owing to its low installation cost, better reliability and easy maintenance [24], [25], [26], [27], [28], [37], [38]. In general, in using the SDC to implement the MSS of CLSS, only the sampling information at discrete time instants is sent to the controller, which can greatly decrease the amount of transmitted information and increase the efficiency of bandwidth usage. Thus, the research on the MSS of CLSS with time-delay by SDC is of great significance both in theory and practice.

As stated in [28], the particular formed discontinuous LKF in [29], which is very valid to decrease the conservation of MSS criteria for SDC systems, was constructed via Wirtinger’s inequality. After this work, many scholars only have utilized it to enhance their results, but no one has attempted to propose novel discontinuous LKF terms on the basis of the auxiliary function-based integral inequalities [30]. The key difficulties lie in the fact that one is that the above mentioned single integral inequality in [30] include terms, 1ttktktx(s)ds and 1(ttk)2tktθtx(s)dsdθ, the other is that if the sawtooth structure term, (tk+1t)tktθtx˙T(s)Rx˙(s)dsdθ, is introduced into the time-dependent discontinuous LKF (T-DDLKF) candidate, the quadratic term (tk+1t)(ttk)x˙T(t)Rx˙(t) would appear in the derivative of the T-DDLKF. Therefore, how to introduce the auxiliary function-based single integral inequality (AFBSII) T-DDLKF term and handle the quadratic term (tk+1t)(ttk)x˙T(t)Rx˙(t) are still challenging.

On the other hand, the existing works [14], [15], [16], [17], [18], [19], [20], [21], [25], [26], [28], [37], [38] about MSS of CLSS were only supposed that all sampling signals are executed with infinite precision, resulting in the neglect for the influence of data quantization. However, we notice that in a networked system, since the restriction of the network’s transmission capacity, sampling signals are generally quantized before being transmitted. In other words, the quantization is an absolutely necessary step to economize the finite capacity and the system’s energy consumption. Following this, many researchers have drawn their interests to SDC systems with state quantization [42], [43], [44], [45], [46]. In [42], [43], the issue of the stability for linear and switched SDC systems with state quantization was investigated, respectively. In [44], [45], the problems of stability and stabilization for fuzzy chaotic SDC systems with state quantization were studied. Very recently, the MSS issue for a class of switched nonlinear systems based on SQSDC was addressed in [46]. By noting the significance of issue discussed above, it is very important to explore the MSS of CLSS with SQSDC. In [31], by using input delay approach and Lyapunov functional method, the MSS problem for the CLSS with SQSDC was solved. Inspired by Xiao et al. [31], new quantized sampled-data MSS conditions for delayed CLSS were proposed by novel extended Wirtinger-inequality-based LKF approach [32], and simultaneously modified a mistake in the Theorem 1 of [31]. Unfortunately, until now, to the best of our knowledge, only a few literatures [31], [32] investigate the MSS problem for delayed CLSS with SQSDC. Therefore, there is still room to further study such subject.

In light of the aforementioned discussions, the main contributions are listed as follows:

✦ New AFBSII T-DDLKF approach is proposed, which can make the most of more information on the sawtooth structure of sampling pattern.

✦ A novel T-DDLKF candidate, (dkd(t))tktθtx˙T(s)Rx˙(s)dsdθ, is introduced and a bound of its time derivative is estimated by a maximum value of a function and using the auxiliary function-based double integral inequality.

✦ Two new T-DDLKFs, V5(t) and V6(t), are firstly proposed according to a nonlinear function condition to improve the feasible region of quantized sampled-data synchronization criteria.

✦ A novel zero-valued equation is established to enhance the combinations of some resulting vectors.

The structure of this paper is as follows. The main problem statement and preliminaries are described in Section 2. In Section 3, some less conservative MSS criteria, which were established by novel AFBSII T-DDLKF, are provided to ensure the MSS of delayed CLSS. In Section 4, two numerical examples are provided to illustrate the feasibility and effectiveness of our method proposed. A conclusion is given in Section 5.

Notations: Some significant symbols used throughout this paper are considerably standard. The superscript “T” stands for the transpose of a matrix or a vector. Rn denotes the n-dimensional Euclidean space. Rn×m is the set of all n × m real matrices. P > 0 means that the matrix P is symmetric and positive-definite. Symmetric terms in a symmetric matrix are denoted by *. In and 0n × m represent n × n identity matrix and n × m zero matrices, respectively. diag{⋅⋅⋅} denotes a block-diagonal matrix. sym{X} stands for X+XT. col{⋅⋅⋅} represents a column vector. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

Section snippets

Problem statement and preliminaries

Consider a fundamental MSS scheme of CLSS with SQSDC as follows:U:{p˙(t)=Ap(t)+Bp(tτ)+Wϕ(Dp(t)),φ1(t)=Cp(t),V:{q˙(t)=Aq(t)+Bq(tτ)+Wϕ(Dq(t))+ϝ(t),φ2(t)=Cq(t),W:ϝ(t)=Kf(φ1(tk)φ2(tk)),t[tk,tk+1),where U, V, and W are the master system (MS), the controlled slave system (CSS), and the SQSDC, respectively. p(t)Rn, q(t)Rn are state vectors. τ > 0 is the time delay. φ1(t)Rl and φ2(t)Rl are the output vector functions. ARn×n, BRn×n, WRn×m, DRm×n, and CRl×n are known constant matrices. KRn×l

Main result

According to the new AFBSII T-DDLKF approach and the novel zero-valued equation, we will establish a new MSS criterion for delayed CLSS (1) with SQSDC in this section. For convenience, one needs to introduce block entry matrices as Ii(i=1,2,,13)R13n×n (e.g., I3=[0,0,I,0,0,0,0,0,0,0,0,0,0]T) and define some notations asy1(t)={x(tk),t=tk1ttktktx(s)ds,ttk,y2(t)={x(tk)2,t=tk1(ttk)2tktθtx(s)dsdθ,ttk,y3(t)=d(t)x(t),y4(t)=d(t)x˙(t),ξ(t)=col{x(t),x(tk),x˙(t),x(tτ),y1(t),y2(t),y3(t),y4(t),tktx

Illustrative examples and simulation

Two interesting examples are given to show the superiority and validity of the presented design approach in this section.

Example 1

Consider the CLSS (1) with the following [15], [18], [26], [32]:A=[am1a01110b0],B=[c00c002c0c],W=[a(m0m1)00],C=D=[100]T,where m0=17, m1=27, a=9, b=14.28, the nonlinear function Φ(x1(t))=12(|x1(t)+1||x1(t)1|) belongs to sector [0,1] with γ=0 and γ+=1. Taking the incipient values as p(0)=[0.3,0.2,0.4]T and q(0)=[0.2,0.3,0.3]T.

Now, one will verify the effectiveness

Conclusion

In this paper, the MSS issue for delayed CLSS has been studied via designing a SQSDC. Compared with the existing MSS conditions on delayed CLSS, our results have two merits. First, AFBSII T-DDLKF approach is developed, which can fully capture the features of system and the available information for the practical sampling mode. Second, a novel double integral T-DDLKF and zero-valued equality are constructed, less conservative MSS criteria are established to ensure that the error system is

Declaration of Competing Interest

We declare that we have no conflicts of interest in our work submitted. We claim that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, which mean that no professional or other personal interest of any nature or kind in any product, service and/or company.

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    This work was supported by National Natural Science Foundation under Grant 11461082, 11601474.

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