Robust synchronization of FitzHugh–Nagumo network with parameter disturbances by sliding mode control

In this paper, based on the sliding mode control method, the robust synchronization for a coupled FitzHugh–Nagumo (FHN) neurobiological network with parameter disturbances is investigated. Some theoretical criteria are derived to realize the robust synchronization of the FHN network with disturbed parameters, and the synchronization occurs without dependence on the type and magnitude of the noise, which greatly extend some existing results for two or three coupled FHN neurons. Finally, a numerical example is given to illustrate the effectiveness of the proposed theoretical results.     

Chaos synchronization of coupled neurons via adaptive sliding mode control

In this paper, an adaptive neural network (NN) sliding mode controller (SMC) is proposed to realize the chaos synchronization of two gap junction coupled FitzHugh–Nagumo (FHN) neurons under external electrical stimulation. The controller consists of a radial basis function (RBF) NN and an SMC. After the RBFNN approximating the uncertain nonlinear part of the error dynamical system, the SMC realizes the desired control property regardless of the existence of the approximation errors and external disturbances. The weights of the NN are tuned online based on the sliding mode reaching law. According to the Lyapunov stability theory, the stability of the closed error system is guaranteed. The control scheme is robust to the uncertainties such as approximate error, ionic channel noise and external disturbances. Chaos synchronization is obtained by the proper choice of the control parameters. The simulation results demonstrate the effectiveness of the proposed control method.     

基于自适应模糊控制方法的四翼混沌系统有限时间同步

研究了一类带有未知外部摄动的四翼混沌主从系统的有限时间同步控制问题.首先,基于自适应模糊控制方法,对四翼混沌系统的不确定项进行了处理.其次,基于Lyapunov有限时间稳定性准则,设计了一种有限时间同步控制器,使得主系统与从系统能在有限时间内实现状态同步.最后,通过数值仿真,检验了该方法的有效性和鲁棒性.     

Synchronization of fractional‐order chaotic systems with disturbances via novel fractional‐integer integral sliding mode control and application to neuron models

In this paper, a novel fractional‐integer integral type sliding mode technique for control and generalized function projective synchronization of different fractional‐order chaotic systems with different dimensions in the presence of disturbances is presented. When the upper bounds of the disturbances are known, a sliding mode control rule is proposed to insure the existence of the sliding motion in finite time. Furthermore, an adaptive sliding mode control is designed when the upper bounds of the disturbances are unknown. The stability analysis of sliding mode surface is given using the Lyapunov stability theory. Finally, the results performed for synchronization of three‐dimensional fractional‐order chaotic Hindmarsh‐Rose (HR) neuron model and two‐dimensional fractional‐order chaotic FitzHugh‐Nagumo (FHN) neuron model.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

Adaptive backstepping sliding mode control for chaos synchronization of two coupled neurons in the external electrical stimulation

In this paper, a robust control system combining backstepping and sliding mode control techniques is used to realize the synchronization of two gap junction coupled chaotic FitzHugh-Nagumo (FHN) neurons in the external electrical stimulation. A backstepping sliding mode approach is applied firstly to compensate the uncertainty which occur in the control system. However, the bound of uncertainty is necessary in the design of the backstepping sliding mode controller. To relax the requirement for the bound of uncertainty, an adaptive backstepping sliding mode controller with a simple adaptive law to adapt the uncertainty in real time is designed. The adaptive backstepping sliding mode control system is robust for time-varying external disturbances. The simulation results demonstrate the effectiveness of the control scheme.     

Introducing internal model to robust output synchronization of FitzHugh–Nagumo neurons in external electrical stimulation

A novel internal model control method is proposed for the robust output synchronization of FizHugh–Nagumo (FHN) neurons under external electrical stimulation. The output synchronization problem can be converted into a robust stabilization problem of an augmented system consisting of the original given plant and an internal model. Hence, the design procedures include the design of a satisfied internal model and a proper state-feedback stabilizer for this augmented system. The stability analysis of the resulting closed-loop system leads to semiglobal Lyapunov asymptotic stability of the robust output synchronization achieved for some appointed initial condition in the state space and for all possible values of the uncertain parameter vector. Finally, the simulation results demonstrate the validity of the proposed method.     

Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller

In this paper, a robust adaptive sliding mode controller (RASMC) is proposed to realize chaos synchronization between two different chaotic systems with uncertainties, external disturbances and fully unknown parameters. It is assumed that both master and slave chaotic systems are perturbed by uncertainties, external disturbances and unknown parameters. The bounds of the uncertainties and external disturbances are assumed to be unknown in advance. Suitable update laws are designed to tackle the uncertainties, external disturbances and unknown parameters. For constructing the RASMC a simple sliding surface is first designed. Then, the RASMC is derived to guarantee the occurrence of the sliding motion. The robustness and stability of the proposed RASMC is proved using Lyapunov stability theory. Finally, the introduced RASMC is applied to achieve chaos synchronization between three different pairs of the chaotic systems (Lorenz–Chen, Chen–Lorenz, and Liu–Lorenz) in the presence of the uncertainties, external disturbances and unknown parameters. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed RASMC.     

Motion synchronization in unmanned aircrafts formation control with communication delays

This paper proposes a formation control strategy for unmanned aircrafts using a virtual structure. Cross coupled sliding mode controllers are introduced to cope with uncertainties in the attitude measurement systems of the unmanned aircrafts and unmeasurable bounded external disturbances such as wind effects, and also to provide motion synchronization in the multi-agent system. This motion synchronization strategy improves the agents convergence to their desired positions, and this is useful for a multi-agent system with faulty agents.Moreover, the proposed motion synchronization strategy is not restricted to specific communication topologies, and sufficient conditions are provided to guarantee the multi-agent system stability in the presence of communication delays. Numerical simulations are presented for a team of five unmanned aircrafts to make a pentagon formation and confirm the accepted performance of the proposed control strategy.     

Synchronization and adaptive synchronization of nuclear spin generator system

This study addresses the synchronization and adaptive synchronization problems of nuclear spin generator (NSG) systems with unknown system parameters. We show that the NSG system can be synchronized by using drive-response systems. Adaptive control law is applied to achieve the state synchronization of two identical NSG systems. Lyapunov direct method of stability is used to prove the asymptotic stability of solutions for the error dynamical system. Numerical simulation is used to show the effectiveness of the proposed control schemes.     

Impulsive generalized synchronization for a class of nonlinear discrete chaotic systems

The problem of impulsive generalized synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly the response system is constructed based on the impulsive control theory. Then by the asymptotic stability criteria of discrete systems with impulsive effects, some sufficient conditions for asymptotic H-synchronization between the drive system and response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Finite‐time synchronization of a class of N‐coupled complex partial differential systems with time‐varying delay

This study examines finite‐time synchronization for a class of N‐coupled complex partial differential systems (PDSs) with time‐varying delay. The problem of finite‐time synchronization for coupled drive‐response PDSs with time‐varying delay is similarly considered. The synchronization error dynamic of the PDSs is defined in the q‐dimensional spatial domain. We construct a feedback controller to achieve finite‐time synchronization. Sufficient conditions are derived by using the Lyapunov‐Krasoviskii stability approach and inequalities technology to ensure that the proposed networks achieve synchronization in finite time. The proposed systems demonstrate extensive application. Finally, an example is used to verify the theoretical results.     

Universal chaos synchronization control laws for general quadratic discrete systems

This paper addresses the reliable universal synchronization problem between two coupled chaotic quadratic discrete systems. A general nonlinear control method of synchronization for coupled 2D and 3D quadratic dynamical systems in discrete-time is proposed. The proposed synchronization method is based on universal controllers. The synchronization results are derived theoretically using active control method and Lyapunov stability theory. Numerical simulations are performed to assess the performance of the presented analysis.     

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.     

Synchronization of mechanical horizontal platform systems in finite time

The horizontal platform system (HPS) is a mechanical device that exhibits rich and chaotic dynamics. In this paper, the problem of finite-time synchronization of two non-autonomous chaotic HPSs is investigated. It is assumed that both drive and response systems are disturbed by model uncertainties, external disturbances and fully unknown parameters. Appropriate update laws are proposed to undertake the unknown parameters. Using the update laws and finite-time control theory, a robust adaptive controller is derived to synchronize the two uncertain HPSs in a given finite time. Subsequently, the effects of input nonlinearities are taken into account and a robust adaptive controller is introduced to synchronize the two uncertain HPSs within a finite time. The finite-time stability and convergence of the proposed schemes are analytically proved. Two illustrative examples are presented to show the robustness and applicability of the proposed adaptive finite-time control techniques.     

Synchronization of the unified chaotic systems using a sliding mode controller

The unified chaotic system incorporates the behaviors of the Lorenz, the Chen and the Lü chaotic systems. This paper deals with the synchronization of two identical unified chaotic systems where the slave system is assumed to have a single input. A sliding mode controller is proposed to synchronize the two systems. The asymptotic convergence to zero of the errors between the states of the master and the slave systems is shown. Simulations results are presented to illustrate the proposed controller; they indicate that the designed controller is able to synchronize the unified chaotic systems. Also, simulation results show that the proposed control scheme is robust to random bounded disturbances acting on the master system. Moreover, the proposed scheme is applied to the secure communications field, where simulation results indicate that the proposed scheme is effective.     

Quadratic optimal synchronization of uncertain chaotic systems

This paper investigates the quadratic optimal synchronization of uncertain chaotic systems with parameter mismatch, parametric perturbations and external disturbances on both master and slave systems. A robust control scheme based on Lyapunov stability theory and quadratic optimal control approach is derived to realize chaotic synchronization. The sufficient criterion for stability condition is formulated in a linear matrix inequality (LMI) form. The effect of uncertain parameters and external disturbance is suppressed to an H_∞ norm constraint. An adaptive algorithm is proposed to adjust the uncertain bound in the robust controller avoiding the chattering phenomena. The simulation results for synchronization of the Chua’s circuit system and the Lorenz system demonstrate the effectiveness of the proposed scheme.     

A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models

The paper is concerned with development of a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. The first step in the derivation of the new method is made by adding an equation for the chemoattractant concentration gradient to the original system. We then show that the convective part of the resulting system is typically of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable. The proposed method is based on the application of the second-order central-upwind scheme, originally developed for hyperbolic systems of conservation laws in Kurganov et al. (SIAM J Sci Comput 21:707–740, 2001), to the extended system of PDEs. We show that the proposed second-order scheme is positivity preserving, which is a very important stability property of the method. The scheme is applied to a number of two-dimensional problems including the most commonly used Keller–Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue. Our numerical results demonstrate high accuracy, stability, and robustness of the proposed scheme.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

一类非线性时滞混沌系统的自适应脉冲同步

针对一类非线性时滞混沌系统,提出了一种新的自适应脉冲同步方案.首先基于Lyapunov稳定性理论、自适应控制理论及脉冲控制理论设计了自适应控制器、脉冲控制器及参数自适应律,然后利用推广的Barbalat引理,理论证明响应系统与驱动系统全局渐近同步,并给出了相应的充分条件.方案利用参数逼近Lipschitz常数,从而取消了Lipschitz常数已知的假设.两个数值仿真例子表明本方法的有效性.