Bifurcation Control Of A Fractional‐Order Van Der Pol Oscillator Based On The State Feedback

In this paper, a dynamic state feedback is applied to control Hopf bifurcations arising from a fractional‐order Van Der Pol oscillator. The degree parameter indicating the strength of the nonlinear damping is chosen as the bifurcation parameter. It is shown that in the absences of the dynamic state feedback controller, the fractional‐order Van Der Pol oscillator loses the stability via the Hopf bifurcation early, and can maintain the stability only in a certain domain of the degree parameter. When applying the state feedback controller to the fractional‐order Van Der Pol oscillator, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the system possesses the stability in a larger parameter range. Numerical simulations are given to justify the validity of the dynamic state feedback controller in bifurcation controls.     

一类分数阶非线性混沌系统的同步控制

在分数阶非线性系统同步控制的研究中,针对一类分数阶非线性混沌系统,研究了基于分数阶控制器的同步方法.利用状态反馈方法和分数阶微积分定义,设计了分数阶混沌系统同步控制器.进一步,根据分数阶非线性系统稳定性理论、Mittag-Leffler函数、Laplace变换以及Gronwall不等式,证明了同步控制器的有效性.最后,通过数值仿真,实现了初始值不同的两个分数阶非线性混沌系统同步.误差响应曲线表明研究的分数阶非线性系统同步响应速度快,控制精度高,验证了本文所设计的混沌同步控制方案的可行性.     

Fractional order [proportional derivative] controller for a class of fractional order systems

Recently, fractional order systems (FOS) have attracted more and more attention in various fields. But the control design techniques available for the FOS suffer from the lack of direct systematic approaches. In this paper, we focus on a given type of simple model of FOS. A fractional order [proportional derivative] (FO-[PD]) controller is proposed for this class of FOS, and a practical and systematic tuning procedure has been developed for the proposed FO-[PD] controller synthesis. The fairness issue in comparing with other controllers such as the traditional integer order PID (IO-PID) controller and the fractional order proportional derivative (FO-PD) controller has been addressed under the same number of design parameters and the same specifications. Fair comparisons of the three controllers (i.e., IO-PID, FO-PD and FO-[PD]) via the simulation tests illustrate that, the IO-PID controller designed may not always be stabilizing to achieve flat-phase specification while both FO-PD and FO-[PD] controllers designed are always stabilizing. Furthermore, the proposed FO-[PD] controller outperforms FO-PD controller for the class of fractional order systems.     

Convergence of Delayed Dynamical Systems

In this paper, we point out that the conditions given in [1] are sufficient but unnecessary for the global asymptotically stable equilibrium of a class of delay differential equations. Instead, we prove that under weaker conditions, it is still global asymptotically stable.     

Delay-induced bifurcation in a tri-neuron fractional neural network

This paper investigates the issue of stability and bifurcation for a delayed fractional neural network with three neurons by applying the sum of time delays as the bifurcation parameter. Based on fractional Laplace transform and the method of stability switches, some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. The analysis indicates that time delay can effectively enhance the stability of fractional neural networks. In addition, it is found that the stability interval can be varied by regulating the fractional order if all the parameters are fixed including time delay. Finally, numerical examples are presented to validate the derived theoretical results.     

时滞控制神经网络的稳定性和Turing斑图结构

本文讨论了一类时滞反应扩散神经网络模型.利用时滞来控制系统的稳定性、分岔和Turing斑图.研究结果表明,在一定条件下,时滞不仅能影响系统的稳定性和周期震荡性,还能影响系统的Turing不稳定性.数值模拟验证了理论分析的正确性,同时还说明了时滞能改变斑图的结构.     

一类推广的二元神经网络的稳定性分析

利用不动点定理和微分不等式的分析技巧,引入多个变时滞,去掉对激活函数光滑性与有界性的假设,研究了一类推广的二元神经网络的平衡点的存在性,得到了系统存在平衡点和全局指数稳定性的新的充分条件．     

Hopfield Neural Networks for Parametric Identification of Dynamical Systems

In this work, a novel method, based upon Hopfield neural networks, is proposed for parameter estimation, in the context of system identification. The equation of the neural estimator stems from the applicability of Hopfield networks to optimization problems, but the weights and the biases of the resulting network are time-varying, since the target function also varies with time. Hence the stability of the method cannot be taken for granted. In order to compare the novel technique and the classical gradient method, simulations have been carried out for a linearly parameterized system, and results show that the Hopfield network is more efficient than the gradient estimator, obtaining lower error and less oscillations. Thus the neural method is validated as an on-line estimator of the time-varying parameters appearing in the model of a nonlinear physical system.     

Partial differential equation modeling of malware propagation in social networks with mixed delays

With the wide applications of social networks, government and individuals increasingly emphasize information networks security. This paper is devoted to investigating a reaction–diffusion malware propagation model with mixed delays to describe the process of social networks. Applying matrix theory for characteristic values, we establish the local stability conditions of a positive equilibrium point. Based on the linear approximation method of nonlinear systems, the Hopf bifurcation at the positive equilibrium point is considered. Additionally, we identify some sensitive parameters in the process of malware propagation that are significant for control theory. Finally, numerical simulations are performed to illustrate the theoretical results.     

基于模型降阶的分数阶系统动态性能研究

由于大多数分数阶系统阶次过高,使得系统的控制器设计变得非常困难,会造成系统控制精度变差且动态性能降低等不利因素,而模型降阶技术是解决这一问题的有效工具.首先简要介绍了H2范数模型降阶的一般方法,并提出一种新的降阶模型结构,可以使降阶后的模型扩展到分数阶并且更加精确地逼近各种高阶系统.仿真结果证明,采用改进型H2范数模型降阶方法不仅保持了原有分数阶模型系统的动态性能而且还提升了原有整数阶模型系统的动态性能.     

基于径向基神经网络的分数阶混沌系统控制

针对分数阶混沌系统的控制问题,提出了一种基于径向基函数（RBF）神经网络的控制方法．利用RBF 神经网络对混沌系统的非线性进行补偿,并且神经网络的权值可以通过调整律在线调整．在有参数干扰和外部扰动 的情况下,所设计的控制器仍能使得控制误差渐近收敛到零．以分数阶Liu 混沌系统为例施加控制,仿真结果验证 了该方法的有效性和鲁棒性．     

几类动态反馈神经网络的稳定性分析

稳定性分析是动态反馈神经网络的重要研究内容 ,对于网络的设计和应用具有指导作用。本文针对几类典型的动态反馈神经网络模型 ,包括离散网络、非对称网络和时延网络 ,利用 L yapunov方法得到了网络的稳定条件     

分数阶PD控制器在时滞伺服系统中的研究

分数阶微积分理论的发展推动了分数阶微积分在各个领域的广泛应用,分数阶控制算法的研究也成为近几年的研究热点.目前分数阶控制理论的研究主要集中在理论分析,对于目前所研究的这类工程性较强的控制对象,成熟的研究成果较少.针对实际工程中含有时滞的伺服系统模型,将Flat phase法和系统稳定性裕度条件相结合,设计了分数阶PD控制器.此PD控制器结构简单,对参数变化稳定性好,控制精度高.为了对比仿真实验,文章还设计了整数阶PID控制器.仿真结果表明,分数阶控制器比传统整数阶PID控制器的控制精度更高,鲁棒性更好.     

Fault Diagnosis Based on the Fuzzy‐Recurrent Neural Network

A fuzzy‐recurrent neural network (FRNN) has been constructed by adding some feedback connections to a feedforward fuzzy neural network (FNN). The FRNN expands the modeling ability of a FNN in order to deal with temporal problems. A basic concept of the FRNN is first to use process or expert knowledge, including appropriate fuzzy logic rules and membership functions, to construct an initial structure and to then use parameter‐learning algorithms to fine‐tune the membership functions and other parameters. Its recurrent property makes it suitable for dealing with temporal problems, such as on‐line fault diagnosis. In addition, it also provides human‐understandable meaning to the normal feedforward multilayer neural network, in which the internal units are always opaque to users. In a word, the trained FRNN has good interpreting ability and one‐step‐ahead predicting ability. To demonstrate the performance of the FRNN in diagnosis, a comparison is made with a conventional feedforward network. The efficiency of the FRNN is verified by the results.     

Stabilization of Fractional‐Order Linear Systems with State and Input Delay

This paper describes a variable structure control for fractional‐order systems with delay in both the input and state variables. The proposed method includes a fractional‐order state predictor to eliminate the input delay. The resulting state‐delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional‐order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.     

Chaos control of a chemical chaotic system via time-delayed feedback control method

In this paper, the control of chemical chaotic dynamical system is investigated by time-delayed feedback control technique. The controllability and the stability of the equilibriums and local Hopf bifurcation of the system are verified. Some numerical simulations which show the effectiveness of the time-delayed feedback control method are provided.     

Fractional Control of a Humanoid Robot Reduced Model with Model Disturbances

There is an open discussion between those who defend mass-distributed models for humanoid robots and those in favor of simple concentrated models. Even though each of them has its advantages and disadvantages, little research has been conducted analyzing the control performance due to the mismatch between the model and the real robot, and how the simplifications affect the controller’s output. In this article we address this problem by combining a reduced model of the humanoid robot, which has an easier mathematical formulation and implementation, with a fractional order controller, which is robust to changes in the model parameters. This controller is a generalization of the well-known proportional–integral–derivative (PID) structure obtained from the application of Fractional Calculus for control, as will be discussed in this article. This control strategy guarantees the robustness of the system, minimizing the effects from the assumption that the robot has a simple mass distribution. The humanoid robot is modeled and identified as a triple inverted pendulum and, using a gain scheduling strategy, the performances of a classical PID controller and a fractional order PID controller are compared, tuning the controller parameters with a genetic algorithm.     

Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay

In this paper, a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations. A great variety of interesting dynamical behaviors of such a system including single-periodic, multiple-periodic, and chaotic motions are displayed. In particular, the effect of time delay on the chaotic behavior is investigated, it is found that an approximate time delay can enhance or suppress the emergence of chaos. Meanwhile, corresponding to different values of delay, the lowest orders for chaos to exist in the delayed fractional order financial systems are determined, respectively.     

Chaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback

In this paper, the problem of controlling chaos in a Sprott E system with distributed delay feedback is considered. By analyzing the associated characteristic transcendental equation, we focus on the local stability and Hopf bifurcation nature of the Sprott E system with distributed delay feedback. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form theory and center manifold theory. Numerical simulations for justifying the theoretical analysis are provided.