Control of fractional chaotic and hyperchaotic systems based on a fractional order controller

We present a new fractional-order controller based on the Lyapunov stability theory and propose a control method which can control fractional chaotic and hyperchaotic systems whether systems are commensurate or incommensurate.The proposed control method is universal, simple, and theoretically rigorous. Numerical simulations are given for several fractional chaotic and hyperchaotic systems to verify the effectiveness and the universality of the proposed control method.     

Control of fractional chaotic system based on fractional-order resistor—capacitor filter

We present a new fractional-order resistor-capacitor controller and a novel control method based on the fractional-order controller to control an arbitrary three-dimensional fractional chaotic system. The proposed control method is simple, robust, and theoretically rigorous, and its anti-noise performance is satisfactory. Numerical simulations are given for several fractional chaotic systems to verify the effectiveness and the universality of the proposed control method.     

Adaptive stabilization of an incommensurate fractional order chaotic system via a single state controller

In this paper, we investigate the stabilization of an incommensurate fractional order chaotic systems and propose a modified adaptive-feedback controller for the incommensurate fractional order chaos control based on the Lyapunov stability theory, the fractional order differential inequality and the adaptive control theory. The present controller, which only contains a single state variable, is simple both in design and in implementation. The simulation results for several fractional order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

基于自适应模糊控制的分数阶混沌系统同步

本文主要研究了带有未知外界扰动的分数阶混沌系统的同步问题.基于分数阶Lyapunov稳定性理论,构造了分数阶的参数自适应规则以及模糊自适应同步控制器.在稳定性分析中主要使用了平方Lyapunov函数.该控制方法可以实现两分数阶混沌系统的同步,使得同步误差渐近趋于0.最后,数值仿真结果验证了本文方法的有效性.     

间歇同步分数阶统一混沌系统

根据分数阶微分方程的性质,研究了间歇控制分数阶系统的稳定性,提出了间歇控制分数阶系统的一般理论并给出了数学证明. 根据该理论设计控制器实现了分数阶统一混沌系统的间歇同步, 数值仿真验证了该理论的正确性. 关键词： 分数阶 统一混沌系统 间歇同步 稳定性     

部分线性的分数阶混沌系统修正函数投影同步

针对一类部分线性的分数阶混沌系统修正函数投影同步问题, 可通过单变量耦合构造出这类系统的响应系统, 从而提出修正函数投影同步的设计方法. 根据Routh-Hurwitz条件, 给出耦合部分线性系统实现修正函数投影同步的方法, 并设计了控制器. 该方法中控制器和误差动态方程的选择都是确定的, 理论分析和数值仿真都说明了该方法在部分线性的分数阶混沌系统中应用具有可行性和有效性. 关键词： 分数阶混沌系统 部分线性 修正函数投影同步     

一种异结构分数阶混沌系统投影同步的新方法

基于Lyapunov稳定性理论和分数阶系统稳定理论以 及分数阶非线性系统性质,提出了一种用来判定分数阶混沌系统是 否稳定的新的判定定理,并把该理论运用于对分数阶混沌系统的控制与 同步,同时给出了数学证明过程,严格保证了该方法的正确性与一般适用性. 运用所提出的稳定性定理,实现了异结构分数阶混沌系统的投影同步. 对分数阶Lorenz混沌系统与分数阶Liu混沌系统实现了投影同步; 针对四维超混沌分数阶系统,也实现了异结构投影同步. 该稳定性定理避 免了求解分数阶平衡点以及Lyapunov指数的问题,从而可以方便地选 择出控制律,并且所得的控制器结构简单、适用范围广. 数值仿真的结果取得了预期的效果,进一步验证了这一稳定性定理的 正确性及普遍适用性.     

Designing synchronization schemes for fractional-order chaotic system via a single state fractional-order controller

In this paper the synchronization of fractional-order chaotic systems is studied and a new single state fractional-order chaotic controller for chaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can apply to an arbitrary three-dimensional fractional chaotic system whether the system is incommensurate or commensurate. This approach is universal, simple and theoretically rigorous. Numerical simulations of several fractional-order chaotic systems demonstrate the universality and the effectiveness of the proposed method.     

基于状态观测器的分数阶时滞混沌系统同步研究

研究分数阶时滞混沌系统同步问题,基于状态观测器方法和分数阶系统稳定性理论,设计分数阶时滞混沌系统同步控制器,使得分数阶时滞混沌系统达到同步,同时给出了数学证明过程.该同步控制器采用驱动系统和响应系统的输出变量进行设计,无需驱动系统和响应系统的状态变量,简化了控制器的设计,提高了控制器的实用性.利用Lyapunov稳定性理论和分数阶线性矩阵不等式,研究并给出了同步控制器参数的选择条件.以分数阶时滞Chen混沌系统为例,设计基于状态观测器的同步控制器,实现了分数阶时滞Chen混沌系统同步,并将其应用于保密通信系统中.仿真结果证明了该同步方法的有效性.     

一种新的分数阶系统稳定理论及在back-stepping方法同步分数阶混沌系统中的应用

对阶次小于1的分数阶系统的稳定问题,提出了一种新的判据.基于该理论,将back-stepping方法拓展到分数阶系统中并设计控制器实现了分数阶Newton-Leipnik混沌系统的同步.数值仿真验证了稳定性理论的准确性及控制器设计方法的有效性. 关键词： 分数阶 同步 back-stepping方法 Newton-Leipnik     

The stability control of fractional order unified chaotic system with sliding mode control theory

This paper studies the stability of the fractional order unified chaotic system with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for the fractional order unified chaotic system. By the existing proof of sliding manifold, the sliding mode controller is designed. To improve the convergence rate, the equivalent controller includes two parts: the continuous part and switching part. With Gronwall’s inequality and the boundness of chaotic attractor, the finite stabilization of the fractional order unified chaotic system is proved, and the controlling parameters can be obtained. Simulation results are made to verify the effectiveness of this method.     

一种分数阶混沌系统同步的自适应滑模控制器设计

针对分数阶混沌系统的同步问题, 基于滑模控制理论和自适应控制理论, 设计了一个具有较强鲁棒性的分数阶积分滑模面, 并提出了一种自适应滑模控制器在不消除非线性项的情况下实现一类三维分数阶混沌系统同步的方法. 利用所设计的自适应滑膜控制器实现了分数阶Chen系统、分数阶Liu系统以及分数阶Arneodo系统的混沌同步. 数值模拟仿真结果验证了所设计的控制器的有效性和可行性.     

基于自适应滑模控制的不同维分数阶混沌系统的同步

针对异结构不同维分数阶混沌系统的广义同步问题进行研究, 设计了一种将滑模变结构理论和自适应控制理论相结合的方法.通过设计一种对外界干扰具有强鲁棒性的分数阶滑模面, 以及构造合适的自适应滑模控制器, 该控制器将系统的运动控制到滑模面上, 使系统轨道沿滑动模运动到所需的控制状态, 最终实现了两个不同维异结构混沌系统之间的广义同步.以四维超混沌Chen系统和三维Chen混沌系统为例, 对这两个系统分别进行升维和降维的同步仿真. 仿真模拟结果表明, 运用本文设计的控制器, 经过短暂的时间, 两系统的广义误差变量始终平稳地趋于零, 即证明了这种控制器的有效性. 关键词： 分数阶混沌系统 异结构 自适应滑模控制 混沌同步     

基于最小二乘支持向量机的分数阶混沌系统控制

提出了基于最小二乘支持向量机(LS-SVM)的分数阶混沌系统控制方法.基于分数阶线性系统稳定理论,通过线性分离的方法将系统分解为稳定的线性部分和相应的非线性部分,再利用支持向量机良好的非线性函数逼近和泛化能力设计了主动控制器,对非线性部分进行补偿,从而将分数阶混沌系统控制到平衡点.分别以分数阶Liu系统和分数阶Chen系统为例进行了仿真研究,表明该方法是有效和可行的.     

基于滑模控制的分数阶混沌系统的自适应同步

基于滑模控制理论和自适应控制理论,研究了分数阶混沌系统的同步问题.设计了新的分数阶积分滑模面,并提出了用单一自适应控制器实现一类三维分数阶混沌系统同步的方法.数值模拟证实了所提方法的有效性. 关键词： 滑模控制 分数阶混沌系统 单一控制器 自适应同步     

Comparison between two different sliding mode controllers for a fractional-order unified chaotic system

Two different sliding mode controllers for a fractional order unified chaotic system are presented. The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system, and the fractional-order system can be made asymptotically stable by this controller. By proving the existence of a sliding manifold containing fractional integral, the controller for a fractional-order system is obtained, which can stabilize it. A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.     

Synchronization between a novel class of fractional-order and integer-order chaotic systems via a sliding mode controller

<正>In order to figure out the dynamical behaviour of a fractional-order chaotic system and its relation to an integerorder chaotic system,in this paper we investigate the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems via sliding mode control method.Stability analysis is performed for the proposed method based on stability theorems in the fractional calculus.Moreover,three typical examples are carried out to show that the synchronization between fractional-order chaotic systems and integer-orders chaotic systems can be achieved. Our theoretical findings are supported by numerical simulation results.Finally,results from numerical computations and theoretical analysis are demonstrated to be a perfect bridge between fractional-order chaotic systems and integer-order chaotic systems.     

Hybrid projective synchronization of chaotic fractional order systems with different dimensions

The hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper. It is shown that the slave system can be synchronized with the projection of the master system generated through state transformation. Based on the stability theorem of linear fractional order systems, a suitable controller for achieving the synchronization is given. The hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order. The corresponding numerical results verify the effectiveness of the proposed method.     

Synchronization of chaotic fractional-order systems via active sliding mode controller

In this paper, we propose a controller based on active sliding mode theory to synchronize chaotic fractional-order systems in master-slave structure. Master and slave systems may be identical or different. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Finally, three numerical simulations (synchronizing fractional-order Lü-Lü systems, synchronizing fractional order Chen-Chen systems and synchronizing fractional-order Lü-Chen systems) are presented to show the effectiveness of the proposed controller. The simulations are implemented using two different numerical methods to solve the fractional differential equations.     

Chaos in a fractional-order micro-electro-mechanical resonator and its suppression

The present paper investigates the existence of chaos in a non-autonomous fractional-order micro-electromechanical resonator system(FOMEMRS).Using the maximal Lyapunov exponent criterion,we show that the FOMEMRS exhibits chaos.Strange attractors of the system are plotted to validate its chaotic behavior.Afterward,a novel fractional finite-time controller is introduced to suppress the chaos of the FOMEMRS with model uncertainties and external disturbances in a given finite time.Using the latest version of the fractional Lyapunov theory,the finite time stability and robustness of the proposed scheme are proved.Finally,we present some computer simulations to illustrate the usefulness and applicability of the proposed method.