Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control

This paper is concerned with the global exponential synchronization problem of two identical nonlinear time-delay Lur’e systems via delayed impulsive control. Some novel impulsive synchronization criteria are obtained by introducing a discontinuous Lyapunov function and by using the Lyapunov–Razumikhin technique, which are expressed in forms of linear matrix inequalities. The derived criteria reveal the effects of impulsive input delays and impulsive intervals on the stability of synchronization error systems. Then, sufficient conditions on the existence of a delayed impulsive controller are derived by employing these newly-obtained synchronization criteria. Additionally, some synchronization criteria for two identical time-delay Lur’e systems with impulsive effects are presented by using delayed continuous feedback control. The synchronization criteria via delayed continuous feedback control can deal with the case when the impulsive control strategy fails to synchronize two identical impulsive time-delay Lur’e systems. Three numerical examples are provided to illustrate the efficiency of the obtained results.     

Exponential H∞ synchronization for discrete‐time chaotic neural networks with time delays and stochastic perturbations via output feedback control

This paper investigates the problem of exponential H_∞ synchronization of discrete‐time chaotic neural networks with time delays and stochastic perturbations. First, by using the Lyapunov‐Krasovskii (Lyapunov) functional and output feedback controller, we establish the H_∞ performance of exponential synchronization in the mean square of master‐slave systems, which is analyzed using a matrix inequality approach. Second, the parameters of a desired output feedback controller can be achieved by solving a linear matrix inequality. Finally, 2 simulated examples are presented to show the effectiveness of the theoretical results.     

Exponential synchronization of stochastic coupled oscillators networks with delays

This paper investigates the exponential synchronization problem of coupled oscillators networks with disturbances and time-varying delays. On basis of graph theory and stochastic analysis theory, a feedback control law is designed to achieve exponential synchronization. By constructing a global Lyapunov function for error network, both pth moment exponential synchronization and almost sure exponential synchronization of drive-response networks are obtained. Finally, a numerical example is given to show the effectiveness of the proposed criteria.     

Dupire It\^{o}’s formula for the exponential synchronization of stochastic semi-Markov jump systems with mixed delay under impulsive control

This paper emphasizes the exponential synchronization for a class of stochastic semi-Markov jump systems with mixed delay via stochastic hybrid impulsive control. The impulsive sequence includes synchronous and asynchronous impulses with the impulsive gains being a sequence of stochastic variables. Inspired by the idea of average, a concept of ``average stochastic impulsive gain" is used to qualify the impulse intensity. Our approach expands Dupire functional It\^{o}$"s formula to the stochastic semi-Markov jump systems with mixed delay for the first time. Moreover, in view of the established Lyapunov functional, graph theory, and stochastic analysis theory, some exponential synchronization criteria for the systems are derived. The theoretical results are applied to a class of Chua"s circuit systems with semi-Markov jump and mixed delay. Some synchronization criteria for the circuit systems are provided. The simulation results verify the effectiveness of the theoretical results.     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

Function vector synchronization based on fuzzy control for uncertain chaotic systems with dead‐zone nonlinearities

In this article, a fuzzy adaptive control scheme is designed to achieve a function vector synchronization behavior between two identical or different chaotic (or hyperchaotic) systems in the presence of unknown dynamic disturbances and input nonlinearities (dead‐zone and sector nonlinearities). This proposed synchronization scheme can be considered as a generalization of many existing projective synchronization schemes (namely the function projective synchronization, the modified projective synchronization, generalized projective synchronization, and so forth) in the sense that the master and slave outputs are assumed to be some general function vectors. To practically deal with the input nonlinearities, the adaptive fuzzy control system is designed in a variable‐structure framework. The fuzzy systems are used to appropriately approximate the uncertain nonlinear functions. A Lyapunov approach is used to prove the boundedness of all signals of the closed‐loop control system as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two identical systems (chaotic satellite systems) and two different systems (chaotic Chen and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 234–249, 2016     

New estimations for ultimate boundary and synchronization control for a disk dynamo system

We consider globally exponentially attractive sets and synchronization control for a disk dynamo system. First, based on generalized Lyapunov function theory and the extremum principle of function, we derive some new 4D ellipsoid estimations and a polydisk domain estimation of the globally exponentially attractive set of a 4D disk dynamo system without existence assumptions. Our results improve existing results on the globally exponentially attractive set as special cases and can lead to a series of new estimations. Second, we propose linear feedback control with a single input or two inputs to realize globally exponential synchronization of two 4D disk dynamo systems using inequality techniques. Some new sufficient algebraic criteria for the globally exponential synchronization of two 4D disk dynamo systems are obtained analytically. The controllers designed here have a simple structure and less conservation. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

Mean square exponential synchronization for impulsive coupled neural networks with time‐varying delays and stochastic disturbances

In this article, the mean square exponential synchronization of a class of impulsive coupled neural networks with time‐varying delays and stochastic disturbances is investigated. The information transmission among the systems can be directed and lagged, that is, the coupling matrices are not needed to be symmetrical and there exist interconnection delays. The dynamical behaviors of the networks can be both continuous and discrete. Specially, the time‐varying delays are taken into consideration to describe the impulsive effects of the system. The control objective is that the trajectories of the salve system by designing suitable control schemes track the trajectories of the master system with impulsive effects. Consequently, sufficient criteria for guaranteeing the mean square exponential convergence of the two systems are obtained in view of Lyapunov stability theory, comparison principle, and mathematical induction. Finally, a numerical simulation is presented to show the verification of the main results in this article. © 2015 Wiley Periodicals, Inc. Complexity 21: 190–202, 2016     

A graph‐theoretic approach to stability of neutral stochastic coupled oscillators network with time‐varying delayed coupling

In this paper, a graph‐theoretic approach for checking exponential stability of the system described by neutral stochastic coupled oscillators network with time‐varying delayed coupling is obtained. Based on graph theory and Lyapunov stability theory, delay‐dependent criteria are deduced to ensure moment exponential stability and almost sure exponential stability of the addressed system, respectively. These criteria can show how coupling topology, time delays, and stochastic perturbations affect exponential stability of such oscillators network. This method may also be applied to other neutral stochastic coupled systems with time delays. Finally, numerical simulations are presented to show the effectiveness of theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization

In this article, we investigate globally exponentially attractive sets and chaos synchronization for a hyperchaotic system, namely, Lorenz–Stenflo system. For this system, two ellipsoidal globally exponentially attractive sets are derived based on generalized Lyapunov function theory and the extremum principle of function. Furthermore, we propose linear feedback control with a one, two, three, and four inputs to realize globally exponential synchronization of two four‐dimesional hyperchaotic systems using inequality techniques. Numerical simulations are presented to show the effectiveness of the proposed synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 20: 30–44, 2015     

Intermittent control for synchronization of hybrid multi-weighted complex networks with reaction-diffusion effects

In this paper, exponential synchronization for hybrid multi-weighted complex networks is studied via aperiodically intermittent control. Different from previous work, both Markov jump and reaction-diffusion effects are simultaneously considered into multi-weighted complex networks. By employing network split technique, graph theory, and Lyapunov method, several synchronization criteria are derived. These criteria show the effects of multiple weights, Markov jump, and reaction-diffusion on exponential synchronization. Furthermore, an application to Cohen–Grossberg neural networks is conducted, and the corresponding synchronization criterion is given. Finally, some numerical simulations are presented to show the effectiveness of the obtained theoretical results.     

Finite‐time synchronization of memristive neural networks with time‐varying delays via two control methods

In this paper, on the basis of the Lyapunov stability theory and finite‐time stability lemma, the finite‐time synchronization problem for memristive neural networks with time‐varying delays is studied by two control methods. First, the discontinuous state‐feedback control rule containing integral part for square sum of the synchronization error and the discontinuous adaptive control rule are designed for realizing synchronization of drive‐response memristive neural networks in finite time, respectively. Then, by using some important inequalities and defining suitable Lyapunov functions, some algebraic sufficient criteria guaranteeing finite‐time synchronization are deduced for drive‐response memristive neural networks in finite time. Furthermore, we give the estimation of the upper bounds of the settling time of finite‐time synchronization. Lastly, the effectiveness of the obtained sufficient criteria guaranteeing finite‐time synchronization is validated by simulation.     

Robust global exponential synchronization of general Lur’e chaotic systems subject to impulsive disturbances and time delays

In this paper, we aim to study the robust global exponential synchronization problem for a general class of Lur’e chaotic systems subject to time delays and impulsive disturbances. Furthermore, we also provide an estimation of the maximum Lyapunov exponent. By using the Lyapunov function method and linear matrix inequality (LMI) technique, sufficient conditions for the robust global exponential synchronization and estimation of its maximum Lyapunov exponent are obtained for the class of Lur’e chaotic systems with and without time delays, respectively. Furthermore, by applying the M-matrix theory, some of these sufficient conditions are shown to be expressible in forms of fairly simple algebraic conditions. For illustration, several examples are solved by using the sufficient conditions obtained.     

Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control

In this paper, the exponential generalized synchronization for a class of coupled systems with uncertainties is defined. A novel and powerful method is proposed to investigate the generalized synchronization based on the adaptive control technique. According to the Lyapunov stability theory, rigorous proof is given for the exponential stability of error system. In comparison with previous schemes, the presented method shortens the synchronization time and is more applicable in practice. Besides, it is shown that the synchronization effect is robust against the uncertain factors. Some typical chaotic and hyper-chaotic systems are taken as examples to illustrate above approach. The corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.     

Dynamical behaviors of a new hyperchaotic system

Currently, chaotic systems and chaos‐based applications are commonly used in the engineering fields. One of the main structures used in these applications is chaotic control and synchronization. In this paper, the dynamical behaviors of a new hyperchaotic system are considered. Based on Lyapunov Theorem with differential and integral inequalities, the global exponential attractive sets and positively invariant sets are obtained. Furthermore, the rate of the trajectories is also obtained. The global exponential attractive sets of the system obtained in this paper also offer theoretical support to study chaotic control, chaotic synchronization for this system. Computer simulation results show that the proposed method is effective. Copyright © 2014 John Wiley & Sons, Ltd.     

参数不确定非自治混沌系统的自适应指数同步

针对参数不确定非自治混沌系统,研究了指数同步问题。给出了自适应控制器的构造方法,并运用Lyapunov稳定性定理证明了在该控制器下的误差系统是指数稳定的,且可以通过调整控制参数控制同步时间。最后,利用MATLAB软件对两个含有不确定参数的非自治混沌系统进行了数值仿真,验证了所提出方法的有效性和正确性.     

Synchronization of quaternion-valued coupled systems with time-varying coupling via event-triggered impulsive control

In this paper, the problem of exponential synchronization of quaternion-valued coupled systems based on event-triggered impulsive control is investigated for the first time. It should be pointed out that the coupling strength is quaternion-valued and time-varying, which makes our model more in line with practical models. First, we prove that event-triggered impulsive control can exclude Zeno behavior. Then, based on the Lyapunov method and the graph theory, some sufficient conditions are derived to ensure that quaternion-valued coupled systems reach synchronization. Furthermore, as an application of our theoretical results, exponential synchronization of quaternion-valued Kuramoto oscillators is studied in detail and a synchronization criterion is presented. Finally, some numerical simulations are given to show the effectiveness of our theoretical results.     

Synchronization for the coupled stochastic strict-feedback nonlinear systems with delays under pinning control

In this paper, a class of new coupled stochastic strict-feedback nonlinear systems with delays (CSFND) on networks without strong connectedness (NWSC) is considered, and the issue pertaining to the synchronization of the systems is discussed by pinning control. Towards CSFND, the controllers are approached by combining the back-stepping method and the design of virtual controllers. A key novel design ingredient is that the global Lyapunov function is obtained based on each Lyapunov function of stochastic strict-feedback nonlinear systems with delays (SFND). Moreover, a sufficient criterion is presented to realize the exponential synchronization by employing the graph theory and Lyapunov method. As a subsequent result, we apply the obtained theoretical results to the second-order oscillator systems and robotic arm systems. Meanwhile, numerical simulations are provided to demonstrate the validity and feasibility of our theoretical results.     

Exponential stabilization and L2‐gain for uncertain switched nonlinear systems with interval time‐varying delay

This paper is concerned with the exponential stabilization and L₂‐gain for a class of uncertain switched nonlinear systems with interval time‐varying delay. Based on Lyapunov–Krasovskii functional method, novel delay‐dependent sufficient conditions of exponential stabilization for a class of uncertain switched nonlinear delay systems are developed under an average dwell time scheme. Then, novel criteria to ensure the exponential stabilization with weighted L₂‐gain performance for a class of uncertain switched nonlinear delay systems are established. Furthermore, an effective method is proposed for the designing of a stabilizing feedback controller with L₂‐gain performance. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

An exponential polynomial observer for synchronization of chaotic systems

In this paper, we consider the synchronization problem via nonlinear observer design. A new exponential polynomial observer for a class of nonlinear oscillators is proposed, which is robust against output noises. A sufficient condition for synchronization is derived analytically with the help of Lyapunov stability theory. The proposed technique has been applied to synchronize chaotic systems (Rikitake and Rössler systems) by means of numerical simulation.