Cyber-physical Systems as General Distributed Parameter Systems: Three Types of Fractional Order Models and Emerging Research Opportunities

Cyber-physical systems (CPSs) are man-made complex systems coupled with natural processes that, as a whole, should be described by distributed parameter systems (DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks.      

Novel techniques in parameter estimation for fractional dynamical models arising from biological systems

In recent years, both parameter estimation and fractional calculus have attracted a considerable interest. Parameter estimation of the fractional dynamical models is a new topic. In this paper, we consider novel techniques for parameter estimation of fractional nonlinear dynamical models in systems biology. First, a computationally effective fractional Predictor-Corrector method is proposed for simulating fractional complex dynamical models. Second, we convert the parameter estimation of fractional complex dynamical models into a minimization problem of the unknown parameters. Third, a modified hybrid simplex search (MHSS) and a particle swarm optimization (PSO) is proposed. Finally, these techniques are applied to a dynamical model of competence induction in a cell with measurement error and noisy data. Some numerical results are given that demonstrate the effectiveness of the theoretical analysis.     

Fractional partial differential equation denoising models for texture image

In this paper,a set of fractional partial differential equations based on fractional total variation and fractional steepest descent approach are proposed to address the problem of traditional drawbacks of PM and ROF multi-scale denoising for texture image.By extending Green,Gauss,Stokes and Euler-Lagrange formulas to fractional field,we can find that the integer formulas are just their special case of fractional ones.In order to improve the denoising capability,we proposed 4 fractional partial differential equation based multiscale denoising models,and then discussed their stabilities and convergence rate.Theoretic deduction and experimental evaluation demonstrate the stability and astringency of fractional steepest descent approach,and fractional nonlinearly multi-scale denoising capability and best value of parameters are discussed also.The experiments results prove that the ability for preserving high-frequency edge and complex texture information of the proposed denoising models are obviously superior to traditional integral based algorithms,especially for texture detail rich images.     

Implementation decision making for internetware driven by quality requirements

Internetware is an emerging software paradigm in the open,dynamic and ever-changing Internet environment.A successful internetware must demonstrate acceptable degree of quality when carrying out its functionality.Hence,when internetware is being dynamically constructed,making implementation decisions to satisfice the quality requirements becomes a critical issue.In the traditional software engineering,quality requirements are usually refined stepwise by sub-requirements utilizing goal modeling perspective,until some potential functional design alternatives are identified.The goal-oriented paradigms have adopted graphical goal models to reason about quality requirements and proposed qualitative or quantitative reasoning schemas.However,these techniques may become unviable due to the ever-changing operating environment and demands for run-time decision making.In this paper,we propose an approach for implementation decision making driven by quality requirements for internetware.It focuses on the symbolic formula representation of requirements goal models with the tree structure,which is of well-defined syntax and clear traceability.Furthermore,we explore some reasoning rules which effectively automate each reasoning action on the formulae.This supports multiple-factor decision making.A case study is also provided to illustrate our proposed approach.We have developed a supporting tool based on our theoretical approach that we also present in this paper.     

Nonlinear thermal system identification using fractional Volterra series

Linear fractional differentiation models have already proven their efficacy in modeling thermal diffusive phenomena for small temperature variations involving constant thermal parameters such as thermal diffusivity and thermal conductivity. However, for large temperature variations, encountered in plasma torch or in machining in severe conditions, the thermal parameters are no longer constant, but vary along with the temperature. In such a context, thermal diffusive phenomena can no longer be modeled by linear fractional models. In this paper, a new class of nonlinear fractional models based on the Volterra series is proposed for modeling such nonlinear diffusive phenomena. More specifically, Volterra series are extended to fractional derivatives, and fractional orthogonal generating functions are used as Volterra kernels. The linear coefficients are estimated along with nonlinear fractional parameters of the Volterra kernels by nonlinear programming techniques. The fractional Volterra series are first used to identify thermal diffusion in an iron sample with data generated using the finite element method and temperature variations up to 700 K. For that purpose, the thermal properties of the iron sample have been characterized. Then, the fractional Volterra series are used to identify the thermal diffusion with experimental data obtained by injecting a heat flux generated by a 200 W laser beam in the iron sample with temperature variations of 150 K. It is shown that the identified model is always more accurate than the finite element model because it allows, in a single experiment, to take into account system uncertainties.     

Synthesis of fractional Laguerre basis for system approximation

Fractional differentiation systems are characterized by the presence of non-exponential aperiodic multimodes. Although rational orthogonal bases can be used to model any L₂[0,∞[ system, they fail to quickly capture the aperiodic multimode behavior with a limited number of terms. Hence, fractional orthogonal bases are expected to better approximate fractional models with fewer parameters. Intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number. However, classical Laguerre, and by extension Kautz and generalized orthogonal basis functions, are divergent as soon as their differentiation order is non-integer. In this paper, the first fractional orthogonal basis is synthesized, extrapolating the definition of Laguerre functions to any fractional order derivative. Completeness of the new basis is demonstrated. Hence, a new class of fixed denominator models is provided for fractional system approximation and identification.     

Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation

The review was devoted to describing the dynamics of various systems and control processes in terms of the fractional integro-differential calculus. Consideration was given to particular types of the fractional differential equations and models of the fractional dynamic systems. Qualitative dynamics and the issues of stability and controllability of such systems were discussed. The analog and digital implementations of the fractional operations with the use of electrical and optical circuits were presented, as well as the models and methods of hardware implementation of the fractional controllers.     

Closed form tuning equations for model predictive control of first-order plus fractional dead time models

Many industrial processes can be effectively described with first-order plus fractional dead time models. In the case of plants with a large dead time relative to the time constant, approximations in discretizing the time delay can adversely affect the performance and if the sample time is enforced by system requirements, the fractional nature of the delay should be considered. In this paper, an analytical approach to model predictive control tuning for stable and unstable first-order plus dead time models with fractional delay is presented. The existing tuning methods are based on trial and error or numerical optimization approaches and the available closed form equations are limited to plants with integer delays. In this paper, an analytical approach is adopted and the issues of closed loop stability and achievable performance are addressed. Finally, simulation results are used to show the effectiveness of the proposed tuning strategy.     

变阶分数阶微分方程的数值解法综述

近年来,分数阶微积分作为一种工具已经被广泛应用于工程中的各个领域.较常阶分数阶微积分算子而言,变阶分数阶微积分算子能够更加准确地描述复杂系统的物理特性,变阶分数阶微积分建模作为一个强大的数学工具,为工程建模提供了便利.在前人优秀研究成果的基础上,结合近几年的国内外相关学者的研究成果对变分数阶微积分方程的研究作全面的综述.以变阶分数阶微分方程、变阶时间分数阶对流-扩散方程、变阶分数阶反应-扩散方程、变阶分数阶积分-微分方程和时滞变阶分数阶微分方程为主要研究对象,从变分数阶微积分算子的相关定义、模型、数值解及在工程中的相关应用等几个方面进行介绍.研究发现,近年来的算法多集中在多项式算法的基础上,通过构建不同的运算矩阵来实现变阶微分方程到代数方程组的转换.该综述可为相关领域的研究学者提供参考.     

分数阶动力学的分析力学方法及其应用

综述我们在分数阶动力学分析力学方法的研究进展,包括：分数阶动力学系统的分析力学表示,构造分数阶动力学模型的分析力学方法,构造分数阶动力学模型团簇的分析力学方法,三类分数阶Lie群无限小变换方法,分数阶动力学系统的对称性、对称性摄动和共形不变性的分析力学方法,分数阶动力学系统的代数结构与Poisson积分的分析力学方法,构造分数阶动力学系统积分不变量的分析力学方法,分数阶动力学系统梯度表示的分析力学方法,分数阶动力学系统稳定性的分析力学方法,分数阶微分方程的分析力学方法等,介绍了对于物理学、力学、生物学、非线性科学等领域的10多种分数阶动力学模型的应用,并指出了若干进一步研究的问题.     

Algorithms for the fractional calculus: A selection of numerical methods

Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.     

分数阶微积分在图像处理中的研究综述*

综述了关于分数阶微积分理论在数字图像底层处理中的应用研究,具体包括:分数阶微积分和分数阶偏微分方程的基本理论及分数阶傅里叶变换的基本性质;分数阶微分滤波器的构造及在图像增强中的应用研究;分数阶积分滤波器的构造及在图像去噪中的应用研究;分数阶偏微分方程在图像去噪中的应用研究。最后,总结了分数阶微积分理论在图像处理中已取得的研究成果,并结合已有的基于分数阶微积分理论的图像底层处理模型,展望了分数阶微积分理论在图像处理中的应用前景。     

Robust analysis and synthesis for a class of fractional order systems with coupling uncertainties

A class of uncertain fractional order systems is concerned where intervals of system matrices parameters are affected by the fractional order. Moreover, the order is also uncertain and belongs to an interval containing the integer one. This kind of system models are reasonable results of fractional order system identifications. In this note, we derive the robust stability analysis and synthesis of such uncertain fractional order systems. Two numerical examples are presented to illustrate the effectiveness and potential of the developed theoretical results.

     

A new operator splitting method for American options under fractional Black–Scholes models

A new operator splitting method is proposed for American options under time-fractional Black–Scholes models. The fractional linear complementarity problem is split into two easy sub-problems, with the leading coefficients separated from the convolution sum and matched through a general correction step. The method is implementation friendly in the sense that one can easily modify a fractional European solver to obtain the proposed method, since the correction step is decoupled and is trivial to solve. The method is validated through numerical experiments and demonstrated to be superior to the traditional approach. The paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models.     

基于分式规划的区间直觉梯形模糊数多属性决策方法

针对属性值为区间梯形直觉模糊且属性权重为区间数的多属性决策问题,提出一种基于分式规划的决策方法.定义了区间梯形直觉模糊数的Hamming距离和Euclidean距离,采用优劣解距离法构建了相对贴近度的非线性分式规划模型,并通过Charnes and Cooper变换转化为线性规划模型求解,得到各方案相对贴近度的区间数,进而提出了决策方法.数值算例分析验证了所提出方法的有效性.     

A tau approach for solution of the space fractional diffusion equation

Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.     

Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems

In this article we investigate possible scenarios of pattern formations in reaction-diffusion systems with time fractional derivatives. Linear stability analysis is performed for different values of derivative orders. Results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations. Most attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer–van der Pol kinetics and to the Brusselator model.     

Analytical and numerical solutions of the unsteady 2D flow of MHD fractional Maxwell fluid induced by variable pressure gradient

The nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations. This paper studies the analytical and numerical methods of unsteady 2D flow of Magnetohydrodynamic (MHD) fractional Maxwell fluid in a rectangular pipe driven by variable pressure gradient. The governing equation is formulated with Caputo time dependent fractional derivatives whose orders are distributed in interval (0, 2). A challenge is to firstly obtain the exact solution by combining modified separation of variables method with Mikusiński-type operational calculus. Meanwhile, the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed. Different to the most classical works, both the stability and convergence analysis of two-dimensional multi-term time fractional momentum equation are derived. Based on numerical analysis, the results show that the velocity increases with the rise of the fractional parameter and relaxation time. While an increase in the values of Hartmann number leads to a slower velocity in the rectangular pipe.     

Exponential integrator methods for systems of non-linear space-fractional models with super-diffusion processes in pattern formation

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this paper, we propose an exponential integrator method for space fractional models as an attractive and easy-to-code alternative for other existing second-order exponential integrator methods. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. One of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The scheme is established to be second-order convergent; and proven to be robust for nonlinear space fractional reaction–diffusion problems involving non-smooth initial data. Our approach is exhibited by solving a system of two-dimensional problems which exhibits pattern formation and has applications in cell-division. Empirically, super-diffusion processes are displayed by investigating the effect of the fractional power of the underlying Laplacian operator on the pattern formation found in these models. Furthermore, the superiority of our method over competing second order ETD schemes, BDF2 scheme, and IMEX schemes is demonstrated.