分数阶系统变阶次状态空间建模及稳定理论

结合分数阶微积分算子的叠加原理,提出了变阶次状态空间建模方法.将分数阶系统推广到状态空间领域,实现了最小阶状态空间转换,并可根据实际需要通过增加状态变量来提取某一阶次的输出信号.对于各阶次均小于1的变阶次状态空间实现的分数阶系统,提出了变阶次分数阶系统的稳定性判定定理.最后通过实例仿真验证了所提出方法的有效性.     

State space approximation for general fractional order dynamic systems

Approximations for general fractional order dynamic systems are of much theoretical and practical interest. In this paper, a new approximate method for fractional order integrator is proposed. The poles of the approximate model are unrelated to the order of integrator. This feature shows benefits on extending the algorithm to the systems containing various fractional orders. Then a unified approximate method is derived for general fractional order linear or nonlinear dynamic systems via combining the proposed new method with the distributed frequency model approach. Numerical examples are given to show the wide applicability of our method and to illustrate the acceptable accuracy for approximations as well.     

质子交换膜燃料电池的分数阶非线性状态空间模型研究

针对质子交换膜燃料电池(PEMFC)发电过程中的分数阶和非线性特性,本文提出了一种分数阶子空间辨识方法建立了PEMFC非线性状态空间模型.首先,为了降低建模复杂度,采用典型相关分析法和相关分析法确定了模型输入变量;其次,将分数阶微分理论与Hammerstein模型子空间辨识方法相结合,采用Poisson矩函数对输入输出数据进行预处理,构造了子空间辨识方法的输入输出矩阵,并引入分数阶短时记忆法减少辨识算法计算量;最后,选取多项式作为Hammerstein模型前端静态非线性环节,采用模糊遗传算法优化系统分数阶阶次和系数矩阵.仿真结果验证了算法的有效性,改进的辨识算法可以明显减小计算时间,所得PEMFC辨识模型能够准确地描述PEMFC的动态过程.     

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.     

质子交换膜燃料电池模型的频域分数阶子空间辨识

针对质子交换膜燃料电池(PEMFC)系统发电过程中的分数阶特性, 本文提出了一种频域分数阶子空间辨 识方法建立PEMFC的分数阶状态空间(FOSS)模型. 考虑到时域分数阶的微分形式计算复杂度较大, 将时域中的分 数阶微分在频域中转化为乘积的形式. 首先, 采用随机多频正弦激励信号对时域采集的信号进行处理, 得到输入输 出的频率响应数据; 其次, 利用频率响应数据构造实、虚部矩阵; 接着, 通过RQ分解、SVD分解以及最小二乘法求取 系统系数矩阵A, B, C, D; 由于参数同元分数阶次α、辅助阶次q以及频域采样点数M未知, 本文提出了一种GA– PSO算法进行优化, 将PSO算法作为主线, 加入GA算法中的选择、交叉和变异操作, 以进一步提高个体的自适应调 整搜索方向、增强全局寻优的能力. 仿真结果验证了算法的有效性, 频域分数阶子空间辨识方法得到的输出能够较 好的跟随实测数据, 且优化后的辨识结果误差更小, 精确度更高, 能够更准确地描述PEMFC的电特性变化过程.     

Design of fractional order modeling based extended non-minimal state space MPC for temperature in an industrial electric heating furnace

In this paper, an improved approach of extended non-minimal state space (ENMSS) fractional order model predictive control (FMPC) is presented and tested on the temperature model of an industrial heating furnace. In the fractional order model predictive control algorithm, fractional order single-input single-output (SISO) system is discretized via fractional order Grünwald-Letnikov (GL) definition. The ENMSS fractional order model that contains the state variable and the fractional order output tracking error is formulated by choosing appropriate state variables. Meanwhile, the fractional order integral is introduced into the cost function and the GL definition is used to obtain the discrete form of the continuous cost function. Then the control signals are derived by minimizing the fractional order cost function. Lastly, the temperature process control of a heating furnace is illustrated to reflect the performance of the proposed FMPC method. Simulation results show the effectiveness of the proposed FMPC method.     

基于子空间辨识的状态空间模型预测控制

针对无法从工业过程中获得准确状态空间模型的问题,提出一种基于子空间辨识的状态空间模型预测控制方法。利用子空间辨识方法得到的状态空间模型作为系统模型,给出约束条件下的预测控制算法。以CD播放器机械臂系统为例,通过状态空间模型预测控制方法实现对系统输出的跟踪控制,仿真结果表明,该方法控制效果良好。     

Robust identification of MISO neuro‐fractional‐order Hammerstein systems

This paper introduces a multiple‐input–single‐output (MISO) neuro‐fractional‐order Hammerstein (NFH) model with a Lyapunov‐based identification method, which is robust in the presence of outliers. The proposed model is composed of a multiple‐input–multiple‐output radial basis function neural network in series with a MISO linear fractional‐order system. The state‐space matrices of the NFH are identified in the time domain via the Lyapunov stability theory using input‐output data acquired from the system. In this regard, the need for the system state variables is eliminated by introducing the auxiliary input‐output filtered signals into the identification laws. Moreover, since practical measurement data may contain outliers, which degrade performance of the identification methods (eg, least‐square–based methods), a Gaussian Lyapunov function is proposed, which is rather insensitive to outliers compared with commonly used quadratic Lyapunov function. In addition, stability and convergence analysis of the presented method is provided. Comparative example verifies superior performance of the proposed method as compared with the algorithm based on the quadratic Lyapunov function and a recently developed input‐output regression‐based robust identification algorithm.     

基于分数阶微分的超声斑点去噪

为了保留更多的纹理信息,构建了基于具有阻止扩散的梯度阈值k,和分数阶微分的阶数v平衡关系的分数偏微分方程的图像去噪模型,其有效结合了分数微积分理论和偏微分方程方法,并通过分数微分掩模算子实现了数值.超声体模信验和体内成像表明:基于分数阶微分的各向异性扩散方法可以提高组织的信噪比(SNR)和超声图像的质量.     

基于最优阶次分数阶神经网络的交通流预测

建立基于最优阶次的分数阶神经网络的动态预测模型,给出数据预处理、最优阶次优化和预测算法流程步骤,给定模型预测精确度的性能指标。分数阶神经网络是从时频两方面分析数据,比BP神经网络具有更灵活有效的函数逼近能力;针对短时数据分析,分数阶神经网络局部性与小波神经网络一致具有多分辨力,且有更强的自适应能力、更快的收敛速度和更高的预测精度。以短时交通流量数据为例进行仿真,与基于小波神经网络和BP神经网络模型的短时交通流量预测仿真比较,分析评价性能指标,结果表明分数阶神经网络最优阶次下可实现灵活快速有效的交通流量动态预测。     

基于分数阶神经网络的瓦斯涌出量预测

针对现有的煤与瓦斯涌出危险性区域预测模型存在收敛速度慢、极易陷入局部极值等问题,结合BP的局部搜索能力和分数阶算法快速的全局搜索能力,提出了一种基于分数阶神经网络的新预测模型,用于非线性瓦斯涌出量的动态预测。经训练和实验结果表明:该模型较其他模型具有更好的滤波效果、更强的抗干扰能力、更快的收敛速度、更高的收敛精度等特点,能够达到准确指导实践的要求。     

Subspace-based continuous-time identification of fractional order systems from non-uniformly sampled data

This work focuses on the identification of fractional commensurate order systems from non-uniformly sampled data. A novel scheme is proposed to solve such problem. In this scheme, the non-uniformly sampled data are first complemented by using fractional Laguerre generating functions. Then, the multivariable output error state space method is employed to identify the relevant system parameters. Moreover, an in-depth property analysis of the proposed scheme is provided. A numerical example is investigated to illustrate the effectiveness of the proposed method.     

Proportional stabilization and closed-loop identification of an unstable fractional order process

This paper deals with proportional stabilization and closed-loop step response identification of the fractional order counterparts of the unstable first order plus dead time (FOPDT) processes. At first, the necessary and sufficient condition for stabilizability of such processes by proportional controllers is found. Then, by assuming that a process of this kind has been stabilized by a proportional controller and the step response data of the closed-loop system is available, an algorithm is proposed for estimating the order and the parameters of an unstable fractional order model by using the mentioned data.     

分数阶时滞传染病模型的传播动力学

本文研究了一个具有时滞的分数阶SEIR传染病模型,并且着重研究了时滞的引入对模型的动力学行为的影响.首先,建立了分数阶SEIR传染病模型并给出了无时滞情况下地方病平衡点稳定的充分条件,以此来确保时滞的引入具有实际意义.其次,结合分岔理论求得了Hopf分岔发生的条件以及分岔阈值的表达式.研究发现,系统的动力学行为依赖于分...     

分数阶多机器人的领航–跟随型环形编队控制

针对多机器人系统的环形编队控制复杂问题,提出一种基于分数阶多机器人的环形编队控制方法,应用领航–跟随编队方法来控制多机器人系统的环形编队和目标包围,通过设计状态估测器,实现对多机器人的状态估计.由领航者获取系统中目标状态的信息,跟随者监测到领航者的状态信息并完成包围环绕编队控制,使多机器人系统形成对动态目标的目标跟踪....     

自适应分数阶微分的复合双边滤波算法

分数阶微分的图像滤波和增强方法多数通过尝试不同的分数阶得到结果,并以固定分数阶进行纹理细节提取,这种方法对于复杂环境难以鲁棒的增强整幅图像中的纹理细节。为此,我们提出了一种自适应的分数阶微分的复合双边滤波方法。通过分析纹理特性,建立幅值频率非线性联合指数模型自适应选择分数阶微分阶数检测图像纹理细节,有效克服图像中纹理细节的变化;在双边滤波的框架下,引入自适应分数阶微分构建的引导图像,借助细节转移方法,确保在图像去噪的同时保持/增强纹理图像细节。实验结果表明,自适应分数阶微分的复合双边滤波算法在图像滤波、去雾、细节增强等计算机视觉应用方面具有良好的效果。     

基于分数阶统一混沌系统的图像加密算法

为了解决数字图像信息传输所面临的安全性问题,基于分数阶统一混沌系统,提出了一种新的图像加密算法.采用经典的置乱-扩散机制,整个加密策略分为图像像素位置置乱和像素值替代两个过程.在像素置乱的过程中,采用排序的方式分别对图像的行和列进行置乱.在像素值替代的过程中,通过与密钥序列进行异或运算来实现加密.而混沌系统则作为伪随机序列发生器,并作用于加密的各个阶段.安全性和时间复杂度分析表明:该算法具有高的安全性和低的时间复杂度,且能够抵御几种常见的攻击方式.     

基于互逆分数阶算子的GM(1,1) 阶数优化模型

在互逆的分数阶累加生成算子和分数阶累减生成算子的基础上, 建立分数阶算子GM(1,1) 模型, 均值GM(1,1) 模型是当?? = 1 时的特例. 给出分数阶算子GM(1,1) 模型最小平均相对误差下最优阶数的粒子群优化算法.多个验证实例表明, 通过对阶数进行优化, 分数阶算子GM(1,1) 模型可具有比GM(1,1)、DGM(1,1) 等模型更高的拟合精度.

     

State feedback with fractional integral control design based on the Bode’s ideal transfer function

State feedback technique through a gain matrix has been a well-known method for pole assignment of a linear system. The technique could encounter a difficulty in eliminating the steady-state errors in some states. Introducing an integral element can effectively eliminate these errors. State feedback with fractional integral control is proposed, in this work, for pole placement of a linear time invariant system. The proposed method yields simple gain formulae. The paper presents the derivation of the design formulae. The method is applied to stabilise an inherently unstable inverted pendulum-cart system. Simulation and experimental results show the effectiveness of the proposed method for set-point tracking, disturbance rejection and stabilising the inverted pendulum. Comparison with the results obtained from applying Achermann’s formula is also presented.     

On fractional order composite model reference adaptive control

This paper presents a novel composite model reference adaptive control approach for a class of fractional order linear systems with unknown constant parameters. The method is extended from the model reference adaptive control. The parameter estimation error of our method depends on both the tracking error and the prediction error, whereas the existing method only depends on the tracking error, which makes our method has better transient performance in the sense of generating smooth system output. By the aid of the continuous frequency distributed model, stability of the proposed approach is established in the Lyapunov sense. Furthermore, the convergence property of the model parameters estimation is presented, on the premise that the closed-loop control system is stable. Finally, numerical simulation examples are given to demonstrate the effectiveness of the proposed schemes.