Robust fuzzy control for chaotic dynamics in Lorenz systems with uncertainties

This paper deals with the robust fuzzy control for chaotic systems in the presence of parametric uncertainties. An uncertain Takagi--Sugeno fuzzy model for a Lorenz chaotic system is first constructed. Then a robust fuzzy state feedback control scheme ensures the control for stable operations under bounded parametric uncertainties. For a chaotic system with known uncertainty bounds, a robust fuzzy regulator is designed by choosing the control parameters satisfying the linear matrix inequality. To verify the validity and effectiveness of the proposed controller design method, an analysis technique is suggested and applied to the control of an uncertain Lorenz chaotic system.     

Fault tolerant synchronization of chaotic systems based on T-S fuzzy model with fuzzy sampled-data controller

In this paper the fault tolerant synchronization of two chaotic systems based on fuzzy model and sample data is investigated. The problem of fault tolerant synchronization is formulated to study the global asymptotical stability of the error system with the fuzzy sampled-data controller which contains a state feedback controller and a fault compensator. The synchronization can be achieved no matter whether the fault occurs or not. To investigate the stability of the error system and facilitate the design of the fuzzy sampled-data controller, a Takagi--Sugeno (T--S) fuzzy model is employed to represent the chaotic system dynamics. To acquire the good performance and produce less conservative analysis result, a new parameter-dependent Lyapunov--Krasovksii functional and a relaxed stabilization technique are considered. The stability conditions based on linear matrix inequality are obtained to achieve the fault tolerant synchronization of the chaotic systems. Finally, a numerical simulation is shown to verify the results.     

Generating Li-Yorke chaos in a stable continuous-time T-S fuzzy model via time-delay feedback control

This paper investigates the chaotification problem of a stable continuous-time T-S fuzzy system.A simple nonlinear state time-delay feedback controller is designed by parallel distributed compensation technique.Then,the asymptotically approximate relationship between the controlled continuous-time T-S fuzzy system with time-delay and a discrete-time T-S fuzzy system is established.Based on the discrete-time T-S fuzzy system,it proves that the chaos in the discretetime T-S fuzzy system satisfies the Li-Yorke definition by choosing appropriate controller parameters via the revised Marotto theorem.Finally,the effectiveness of the proposed chaotic anticontrol method is verified by a practical example.     

Impulsive control of chaotic systems with exogenous perturbations

The impulsive control of chaotic systems, which are subjected to unbounded exogenous perturbations, is considered. By using the theory of impulsive differential equation together with the fuzzy control technique, the authors propose an impulsive robust chaos controlling criterion expressed as linear matrix inequalities (LMIs). Based on the proposed control criterion, the procedure for designing impulsive controllers of common (perturbed) chaotic systems is provided. Finally, a numerical example is given to demonstrate the obtained theoretical result.     

统一混沌系统的时延模糊控制

在将统一混沌系统表达为T-S模型的基础上, 利用泛函微分方程的Lyapunov-Krasovskii稳定性理论和线性矩阵不等式(LMIs)方法,设计出一个新的带时延状态反馈模糊控制器.仿真结果显示,所设计的控制器能有效地控制统一混沌系统的混沌时间轨迹到其平衡点,且控制简单可靠.     

基于线性矩阵不等式的一类新羽翼倍增混沌分析与控制

在对已有的混沌系统分析和研究的基础上,将一个二次混沌系统第三个方程关于x的线性项引入到第二个方程中,通过对该系统第二个等式中的线性项x作绝对值运算,提出了一类新的二次非线性系统.采用非线性动力学方法分析了系统参数变化时所经历的稳定、准周期、混沌的过渡过程,模拟电路实验结果与Matlab数值仿真结果相一致.分析发现混沌态时绝对值运算后的系统比原系统的Lyapunov指数更大,并可将原系统的混沌吸引子由两个翼的拓扑结构变为四翼的拓扑结构,从而实现羽翼倍增.针对该混沌特性更强的羽翼倍增混沌系统,基于Takagi-Sugeno(T-S)模糊模型和线性矩阵不等式(LMI),设计出使该羽翼倍增混沌系统渐近稳定的鲁棒模糊控制器.仿真结果证实了所提出定理和设计控制器的有效性.     

H基于模糊观测器参数摄动的耦合时空混沌H跟踪控制

由于子系统的时空耦合作用及参数的摄动性，实现参数摄动的耦合时空混沌的跟踪控制非常困难。然而模型未知的耦合时空混沌的每个子系统可由一系列模糊逻辑模型逼近，每个模糊逻辑模型代表子系统在特定运行点的局部线性化模型，同时考虑子系统状态的不可测性，采用模糊观测器来估计子系统的状态。基于模糊模型及状态观测器，计及混沌参数的摄动性，提出一种模糊跟踪控制方案，实现了参数摄动的耦合时空混沌的鲁棒跟踪控制，并将模糊跟踪控制表征为线性矩阵不等式问题，用线性矩阵不等式的凸优化方法求解控制器参数，确保系统的全局渐近稳定性。仿真验证了方案的有效性。     

Synchronization of chaotic Lur'e systems with delayed feedback control using deadzone nonlinearity

In this paper we present a synchronization method for chaotic Lur'e systems by constructing a new piecewise Lyapunov function. Using a delayed feedback control scheme, a delay-dependent stability criterion is derived for the synchronization of chaotic systems that are represented by Lur'e systems with deadzone nonlinearity. Based on the Lyapunov--Krasovskii functional and by using some properties of the nonlinearity, a new delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The criterion is less conservative than existing ones, and it will be verified through a numerical example.     

含不确定性混沌系统的模糊自适应同步

研究了含不确定性混沌系统的同步问题.基于Takagi-Sugeno(T-S)模糊动态模型，给出了一个新的自适应模糊同步控制设计方法.该方法同时适用于相同结构混沌系统的同步以及异构混沌系统的同步.为说明问题，给出了Lorenz混沌系统和Rossler混沌系统的同步控制设计和仿真结果. 关键词： 混沌系统 模糊控制 同步     

Synchronization of chaotic systems based on adaptive observer design

The synchronizing problem of a chaotic system is investigated based on the observer design. The nonlinear section is assumed to satisfy the Lipschitz condition. Firstly, the normal observer is designed based on the known Lipschitz constant and the results are given in linear matrix inequality (LMI) form. Then a fairly simple adaptive observer is designed with the Lipschitz constant unknown. Simulations on synchronizing the Lorenz system are investigated and the results show the validity and feasibility of our main results.     

受扰统一混沌系统的主动自适应模糊积分滑模控制

针对受扰统一混沌系统,提出一种主动白适应模糊积分滑模控制方法．在保持原非线性积分滑模控制暂态性能的同时,设计了一种改进非线性积分滑模控制方法,保证了系统的稳定性．所提出控制方法减小了统一混沌系统的不可控状态和自适应模糊补偿项的逼近精度对系统状态误差的影响．基于Lyapunov稳定性理论分析了系统的稳定性．数值仿真结果表明,在存在参数摄动和外部扰动时,能将统一混沌系统控制到目标点上．数值仿真验证了所提出控制方法的有效性．     

H∞ synchronization of chaotic systems using output feedback control design

This article investigates the H_∞ synchronization problem for a general class of chaotic systems. Based on Lyapunov theory, linear matrix inequality (LMI) and linear matrix equality (LME) formulation, the output feedback controller is established to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance to an H_∞-norm constraint. Two illustrative examples are provided to demonstrate the effectiveness of the developed theoretical results.     

Hybrid TS fuzzy modelling and simulation for chaotic Lorenz system

The projection of the chaotic attractor observed from the Lorenz system in the $X$--$Z$ plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined.In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi--Sugeno (TS) fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex (two-wing) butterfly attractors observed from this system in the three dimensional (3D) $X$--$Y$--$Z$ space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.     

Liu混沌系统的线性反馈同步控制及电路实验的研究

研究了新型混沌系统——Liu混沌系统的同步控制问题，基于Liu混沌系统的混沌特性，采用线性反馈控制方法，给出了实现Liu混沌系统同步的控制参数取值范围，数值仿真和电路实验证实了该方法的有效性. 关键词： Liu混沌系统 线性反馈控制 混沌同步 电路     

Exponential stability of Takagi-Sugeno fuzzy systems with impulsive effects and small delays

This paper deals with the exponential stability of impulsive Takagi-Sugeno fuzzy systems with delay. Impulsive control and delayed fuzzy control are applied to the system, and the criterion on exponential stability expressed in terms of linear matrix inequalities （LMIs） is presented.     

Generalized projective synchronization of the fractional-order chaotic system using adaptive fuzzy sliding mode control

An adaptive fuzzy sliding mode strategy is developed for the generalized projective synchronization of a fractionalorder chaotic system, where the slave system is not necessarily known in advance. Based on the designed adaptive update laws and the linear feedback method, the adaptive fuzzy sliding controllers are proposed via the fuzzy design, and the strength of the designed controllers can be adaptively adjusted according to the external disturbances. Based on the Lyapunov stability theorem, the stability and the robustness of the controlled system are proved theoretically. Numerical simulations further support the theoretical results of the paper and demonstrate the efficiency of the proposed method. Moreover,it is revealed that the proposed method allows us to manipulate arbitrarily the response dynamics of the slave system by adjusting the desired scaling factor λi and the desired translating factor ηi, which may be used in a channel-independent chaotic secure communication.     

基于模糊模型的耦合时空混沌H_\infty$跟踪控制

由于子系统的时空耦合作用，实现耦合时空混沌的跟踪控制比较困难。然而模型未知的耦合时空混沌的子系统可由一系列模糊逻辑模型逼近，每个模糊逻辑模型代表子系统在特定运行点的局部线性化模型。基于该系列模糊模型，采用模糊跟踪控制方法实现了耦合时空混沌的模型参考跟踪控制，并用线性矩阵不等式的凸优化方法求解控制器参数，确保系统的全局渐近稳定性。仿真验证了方案的有效性。     

Linear matrix inequality approach for synchronization control of fuzzy cellular neural networks with mixed time delays

Fuzzy cellular neural networks (FCNNs) are special kinds of cellular neural networks (CNNs). Each cell in an FCNN contains fuzzy operating abilities. The entire network is governed by cellular computing laws. The design of FCNNs is based on fuzzy local rules. In this paper, a linear matrix inequality (LMI) approach for synchronization control of FCNNs with mixed delays is investigated. Mixed delays include discrete time-varying delays and unbounded distributed delays. A dynamic control scheme is proposed to achieve the synchronization between a drive network and a response network. By constructing the Lyapunov–Krasovskii functional which contains a triple-integral term and the free-weighting matrices method an improved delay-dependent stability criterion is derived in terms of LMIs. The controller can be easily obtained by solving the derived LMIs. A numerical example and its simulations are presented to illustrate the effectiveness of the proposed method.     

基于积分观测器实现一类受扰混沌系统的同步

在系统参数和系统输出同时受扰的情形下,基于积分观测器的方法,实现了一类不确定混沌系统的同步.首先利用Lyapunov 稳定性理论,给出了混沌系统同步所需的充分条件,此条件可以快速地实现驱动系统与响应系统的同步,然后结合Schur补定理,将此条件转化为求解线性矩阵不等式组.最后,借助Matlab软件包中的LMI工具箱,通过对受扰MLC电路系统的同步数值仿真,证实了该方法的有效性.     

Impulsive control for Takagi--Sugeno fuzzy model with time-delay and its application to chaotic system

A control approach where the fuzzy logic methodology is combined with impulsive control is developed for controlling some time-delay chaotic systems in this paper. We first introduce impulses into each subsystem with delay of the Takagi--Sugeno (TS) fuzzy IF--THEN rules and then present a unified TS impulsive fuzzy model with delay for chaos control. Based on the new model, a simple and unified set of conditions for controlling chaotic systems is derived by the Lyapunov--Razumikhin method, and a design procedure for estimating bounds on control matrices is also given. Several numerical examples are presented to illustrate the effectiveness of this method.