离散时间分数阶多自主体系统的时延一致性

复杂工作环境中,许多自然现象的个体动力学特性用整数阶方程不能描述,只能用非整数阶（分数阶）动力学来描述个体的运动行为. 本文假设多自主体系统内部连接组成有向加权网络,个体的动态特性应用分数阶动力学方程描述,个体之间数据传输存在通信时延. 应用分数阶系统的Laplace变换和频域理论,研究了离散时间的分数阶多自主体系统的渐近一致性. 应用Hermit-Biehler 定理,研究了具有样本时延的分数阶多自主体系统的运动一致性,得到保证系统稳定的时延的上界阈值. 最后应用一个实例对结论进行了验证.     

分数阶时滞非线性多智能体系统的自适应控制

针对分数阶多智能体系统中存在时滞和非线性特性, 时滞往往会引起控制系统的性能下降甚至出现系统 不稳定等问题, 提出了一种含时滞非线性的分数阶多智能体系统自适应控制方法. 对于多智能体系统的控制协议, 设计了基于领导者和相邻智能体状态信息的自适应控制协议, 减小了过大常数控制增益带来的能源浪费. 对于一 致性, 利用图论基础、分数阶Halanay不等式稳定性定理、Kronecker积和Schur补引理, 获得了分数阶时滞非线性多 智能体系统的LMI一致性条件. 仿真结果验证了本文算法的正确性和有效性. 由于整数阶系统是分数阶系统的特殊 形式, 本文结论可以直接推广到整数阶多智能体系统中.     

分数阶奇异系统的Lie对称性与守恒量

对称性与守恒量可以简化动力学问题从而进一步求出力学系统的精确解,这样更加有利于研究动力学行为.分数阶模型相比于整数阶模型,能够描述复杂系统的动力学过程,因此在分数阶模型下研究对称性与守恒量是不可或缺的.首先介绍两个分数阶奇异系统,一个系统包含混合整数和Caputo分数阶导数,另一个系统仅含Caputo分数阶导数.由两个分数阶奇异系统分别给出两个分数阶固有约束,并给出对应的分数阶约束Hamilton方程.然后,基于微分方程在无限小变换下的不变性,给出了分数阶约束Hamilton方程Lie对称性的定义,导出了相应的确定方程,限制方程和附加限制方程.第三,建立并证明了两个分数阶约束Hamilton系统的Lie对称性定理,得到了相应的分数阶约束Hamilton系统的Lie守恒量.在特定条件下,本文所得结果可以退化为整数阶约束Hamilton系统的Lie守恒量.最后通过两个算例来说明此结果的应用.     

一类分数阶四维混沌系统及其投影同步

提出了一个新的四维自治类新混沌系统.首先在整数阶下分析了该系统的基本动力学特性.并利用数值仿真、功率谱分析了当参数固定时,分数阶新混沌系统随微分算子阶数变化时的动力学特性.研究表明:当微分算子阶数为0.85时,分数阶新系统随参数变化经短暂混沌和边界转折点分叉而进入混沌.针对一类结构部分未知分数阶混沌系统,基于Cheby...     

分数阶与整数阶混沌系统的同步与反同步

为了建立起整数阶与分数阶系统的桥梁,推进分数阶系统的应用,本文采用了滑模控制理论研究了一类整数阶与分数阶混沌系统的同步与反同步.文中,设计了一个新的滑模控制器,该控制器适用于一类系统,具有较好的鲁棒性,并且给出了严格的数学证明.本文实现了整数阶Sprott系统和分数阶Chen系统的同步和整数阶吕系统和分数阶Liu系统的反同步.这两个例子有效的证明了所提理论的可行性和正确性.同时,也将同步与反同步的概念统一在一起.     

基于分数阶滑模控制技术的永磁同步电机控制

针对传统整数阶滑模控制系统中存在的抖震问题,本文提出了分数阶滑模控制策略并应用到永磁同步电机的速度控制.传统滑模控制器中的开关函数由作用在切换流型或其整数阶导数面推广到其分数阶导数面,利用分数阶系统的特性,缓慢地传递系统的能量,有效地削减抖震.本文采用模糊逻辑推理算法,实现软开关切换增益的自整定.仿真和实验证明,本文提出的分数阶滑模控制系统不但能有效地削减抖震,而且能保持滑模控制器对系统参数变化和外部扰动的鲁棒性.     

多自由度拟不可积哈密顿系统的随机分数阶最优控制

推广了适用于分数阶系统控制的随机分数阶最优控制策略,提出了高斯白噪声激励下多自由度拟不可积哈密顿系统以响应最小化为目标的随机分数阶最优控制策略.首先,应用拟不可积哈密顿系统随机平均法,将受控系统简化为关于能量的部分平均伊藤方程.然后,将控制性能指标中关于控制力的部分表示为分数阶形式,结合随机动态规划原理,建立并求解部分平均系统的无界遍历控制的随机动态规划方程,获得了随机分数阶最优控制律.最后,采用一个算例验证了随机分数阶控制策略的控制效果和控制效率.研究表明,随机分数阶最优控制策略对传统的整数阶随机动力学系统同样适用,能比传统的整数阶控制策略取得更好的控制效果.另外,随着激励强度增加,整数阶控制策略的控制效率显著降低;而分数阶控制策略的控制效率虽比整数阶控制策略的控制效率略低,但随着激励强度的增加,分数阶控制策略的控制效率缓慢上升并趋于平稳,可以有效地缓解控制效率与控制效果之间的矛盾.     

基于分数阶陈氏混沌系统的图像加密算法

由于分数阶混沌动力学系统比整数阶系统具有更复杂的动力学特性,且能为图像加密方案提供更多的自由度,基于分数阶陈氏混沌系统,提出了一种图像加密方法。在发送端,驱动系统产生混沌信号,利用混沌信号扰乱明文图像的像素位置,将扰乱后的图像掩盖在混沌信号中,得到传输的密文图像。在接收端,通过同步系统去掩盖,进行像素位置扰乱的逆操作,恢复明文图像。最后对提出的加密算法进行了安全性分析。实验结果表明,该加密算法安全性高,具有良好的研究价值和应用前景     

分数阶系统的分数阶PID控制器设计

对于一些复杂的实际系统,用分数阶微积分方程建模要比整数阶模型更简洁准确.分数阶微积分也为描述动态过程提供了一个很好的工具.对于分数阶模型需要提出相应的分数阶控制器来提高控制效果.本文针对分数阶受控对象,提出了一种分数阶PID控制器的设计方法.并用具体实例演示了对于分数阶系统模型,采用分数阶控制器比采用古典的PID控制器取得更好的效果.     

一类分数阶系统的分析及控制器设计

针对一类与传统一阶惯性环节传递函数结构类似的分数阶系统,推导出该类分数阶系统稳定的参数取值范围,并给出了不同时间响应与分数阶阶次的对应关系.然后基于该类分数阶系统同时设计了分数阶PIλ控制器和整数阶PI控制器,控制器参数采用粒子群优化算法得到.结果表明:在控制该类对象时两者均能取得很好的控制效果,证明了本文所提方法的有效性.但由于整数阶PI控制器比分数阶PIλ控制器简单且便于实现,因此在工程应用中针对该类分数阶对象选择PI控制器即可满足要求.     

分数阶非因果BP神经网络模型

由一阶因果、反因果微分的定义推导出Caputo分数阶因果、反因果微积分,并在此基础上定义Caputo分数阶非因果微积分。将它们分别应用于BP神经网络的反向传播过程中对权值进行处理,产生了Caputo分数阶因果、反因果和非因果BP神经网络模型。为了方便对比,将这些模型分别对波士顿房屋数据集和MNIST数据集进行处理。模拟结果表明：在整数阶因果、反因果和非因果的模型之间,整数阶非因果模型的结果最好;分数阶因果、反因果和非因果模型分别与其相应的整数阶模型进行比较,得出分数阶模型得到的准确率比整数阶的高;在分数阶因果、反因果和非因果的模型之间,非因果的准确性最高。总的来说,Caputo分数阶因果、反因果和非因果微积分都对传统BP神经网络有优化作用,尤其是分数阶非因果微积分的优化效果最好。     

Synchronization between a fractional-order system and an integer order system

This paper investigates chaotic synchronization between fractional-order chaotic systems and integer-order chaotic systems. Based on the idea of tracking control and the stability theory of the linear fractional-order system, we design the effective controller to realize the synchronization between fractional-order and integer-order chaotic systems. Theory analysis and numerical simulation results show that the method is effective and feasible.     

Fractional discrete-time consensus models for single- and double-summator dynamics

The leader–following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the systems, interactions between opinions are defined like in Krause and Cucker–Smale models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear systems. In this paper, we investigate fractional-order models of opinions for the single- and double-summator dynamics of discrete-time by analytical methods as well as by computer simulations. The necessary and sufficient conditions for the leader–following consensus are formulated by proposing a consensus control law for tracking the virtual leader.     

A note on time-domain simulation of feedback fractional-ordersystems

The study of feedback fractional-order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional-order models, and that fractional-order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer-order controllers. Owing to the lack of effective analytic methods for the time-domain analysis and simulation of linear feedback fractional-order systems, we suggest in this paper two reliable and accurate numerical methods for inverting fractional-order Laplace transforms. One is based on computing Bromwich's integral with a numerical integration scheme capable of accuracy control, and the other is based on expanding the time response function in a B-spline series. In order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the Grunwald-Letniknov approximation method and Podlubny's analytic formulas, which are in a form of double infinite series, the time-domain simulations of the feedback control of a fractional-order process with a PD^μ-controller and a fractional-order band-limited lead compensator are worked out. The simulation results indicate that a convergence problem indeed occurs in using Podlubny's infinite series expressions, and that the problem could not be overcome by a series acceleration scheme     

Dynamics of a neuron exposed to integer- and fractional-order discontinuous external magnetic flux

We propose a modified Fitzhugh-Nagumo neuron (MFNN) model. Based on this model, an integer-order MFNN system (case A) and a fractional-order MFNN system (case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractional-order magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity.

     

Stability and non-standard finite difference method of the generalized Chua’s circuit

In this paper, we develop a framework to obtain approximate numerical solutions of the fractional-order Chua’s circuit with Memristor using a non-standard finite difference method. Chaotic response is obtained with fractional-order elements as well as integer-order elements. Stability analysis and the condition of oscillation for the integer-order system are discussed. In addition, the stability analyses for different fractional-order cases are investigated showing a great sensitivity to small order changes indicating the poles’ locations inside the physical s-plane. The Grünwald-Letnikov method is used to approximate the fractional derivatives. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is an effective and convenient method to solve fractional-order chaotic systems, and to validate their stability.     

基于改进Oustaloup滤波器的分数阶PID控制器简化设计

针对分数阶微积分算子的直接和间接近似化方法所表现出来的形式复杂、运算量大的问题,对Oustaloup滤波器的结构进行改进,对常规分数阶PID控制器进行了简化设计,并采用自适应遗传算法对控制器的参数进行整定.选取两种代表性分数阶系统,在模型处于两种典型状态下,对简化型分数阶PID控制器、常规分数阶PID控制器和整数阶PID控制器的控制性能进行仿真实验对比.结果表明,在控制器性能基本相同的情况下,通过该方法设计的简化型分数阶PID控制器性具有结构简单,耗时量小的优点,提高了工程可实现性.     

Fractional-order adaptive backstepping control of a class of uncertain systems with external disturbances

Fractional control schemes are powerful tools for fulfilling robust tracking performance of different systems. This paper is the pioneering one in developing a fractional-order adaptive backstepping controller (FOABC) for a general class of integer-order and fractional-order (FO) systems. Model uncertainties and external disturbances can perturb system response and the controller is designed such that it can suppress the performance degradation caused by these factors. Moreover, rigorous mathematical analyses are carried out based on fractional Lyapunov theorems to ensure stability of the controlled systems. To justify the claims, worked-out examples including integer-order and FO systems are simulated. Good tracking performance of the proposed controller as well as robustness against uncertainties and insensitivity to external disturbances make it a good candidate for a broad range of systems. The results of implementing the proposed controller on different systems are compared with some newly proposed control approaches which highlight the outperformance of the FOABC.     

Leader-following consensus of nonlinear fractional-order multi-agent systems via event-triggered control

This paper investigates the consensus problem of leader-following multi-agent systems with fractional-order nonlinear dynamics. A typical event is defined as some error signals exceeding a given threshold. By applying Lyapunov functional approach and skills of computing function limit, consensus of the controlled multi-agent systems can be reached asymptotically. Meanwhile, the event-triggered algorithm can exclude Zeno behaviours. Finally, a numerical simulation is exploited to verify the effectiveness of the theoretical result.