Dynamics and Control of Initialized Fractional-Order Systems

Due to the importance of historical effects in fractional-order systems,this paper presents a general fractional-order system and control theorythat includes the time-varying initialization response. Previous studieshave not properly accounted for these historical effects. Theinitialization response, along with the forced response, forfractional-order systems is determined. The scalar fractional-orderimpulse response is determined, and is a generalization of theexponential function. Stability properties of fractional-order systemsare presented in the complex w-plane, which is a transformation of thes-plane. Time responses are discussed with respect to pole positions inthe complex w-plane and frequency response behavior is included. Afractional-order vector space representation, which is a generalizationof the state space concept, is presented including the initializationresponse. Control methods for vector representations of initializedfractional-order systems are shown. Finally, the fractional-orderdifferintegral is generalized to continuous order-distributions whichhave the possibility of including all fractional orders in a transferfunction.     

A Frequency-Domain Approach to Optimal Fractional-Order Damping

In this paper, we will consider the single term optimal fractional-order damper for an otherwise undamped oscillator. First, we will find the single term damper that minimizes the time domain integral of the squared step error (2-norm) and the integral of the time-weighted squared error (Hilbert–Schmidt–Hankel norm). Next we will consider a more intuitive frequency domain approach that insures the maximally flat magnitude response. Time and frequency domain plots are given for comparison with the integer-order solutions. Further improvements in performance are shown to be possible using multiple active fractional-order dampers.     

Chaotic Phenomena and Fractional-Order Dynamics in the Trajectory Control of Redundant Manipulators

Redundant manipulators have some advantages when compared with classicalarms because they allow the trajectory optimization, both on the freespace and on the presence of obstacles, and the resolution ofsingularities. For this type of arms the proposed kinematic controlalgorithms adopt generalized inverse matrices but, in general, thecorresponding trajectory planning schemes show important limitations.Motivated by these problems this paper studies the chaos revealed by thepseudoinverse-based trajectory planning algorithms, using the theory offractional calculus.     

Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems

Delayed Linear Time-Invariant (LTI) fractional-order dynamic systems areconsidered. The analytical stability bound is obtained by using Lambertfunction. Two examples are presented to illustrate the obtainedanalytical results.     

The Time-Scaled Trapezoidal Integration Rule for Discrete Fractional Order Controllers

In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/s^k control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.     

A Suggestion of Fractional-Order Controller for Flexible Spacecraft Attitude Control

A controller design method for flexible spacecraft attitude control isproposed. The system is first described by partial differential equationwith internal damping. Then the frequency response is analyzed, and thethree basic characteristics of the flexible system, namely, averagefunction, lower bound and upper bound are defined. On this basis, afractional-order controller is proposed, which functions as phasestabilization control for lower frequency and smoothly enters toamplitude stabilization at higher frequency by proper amplitudeattenuation. It is shown that the equivalent damping ratio increases inproportion to the square of frequency.     

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphists equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.     

Fractional Calculus via Functional Calculus: Theory and Applications

This paper demonstrates the power of the functional-calculus definition oflinear fractional (pseudo-)differential operators via generalised Fouriertransforms.Firstly, we describe in detail how to get global causal solutions of linearfractional differential equations via this calculus. The solutions arerepresented as convolutions of the input functions with the related impulseresponses. The suggested method via residue calculus separates an impulseresponse automatically into an exponentially damped (possibly oscillatory)part and a `slow' relaxation. If an impulse response is stable it becomesautomatically causal, otherwise one has to add a homogeneous solution to getcausality.Secondly, we present examples and, moreover, verify the approach alongexperiments on viscolelastic rods. The quality of the method as an effectivefew-parameter model is impressively demonstrated: the chosen referenceexample PTFE (Teflon) shows that in contrast to standard classical modelsour model describes the behaviour in a wide frequency range within theaccuracy of the measurement. Even dispersion effects are included.Thirdly, we conclude the paper with a survey of the required theory. Therethe attention is directed to the extension from the L ₂-approachon the space of distributions .     

Impulse Responses of Fractional Damped Systems

Considered are systems of single-mass oscillators with different fractional damping behaviour. Similar to the classical model, where the damping terms are represented by first derivatives, the eigensystem can be used to decompose the fractional system in frequency domain, if mass, stiffness and damping matrices are linearly dependent. The solution appears as a linear combination of single-mass oscillators. This is true even in the general case such that stability and causality are insured by the same argumentation as used in the linear dependent case.     

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. The flow near a wall suddenly set in motion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model. Exact solutions of velocity and stress are obtained by using the discrete inverse Laplace transform of the sequential fractional derivatives. It is found that the effect of the fractional orders in the constitutive relationship on the flow field is significant. The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate, for large times the viscoelastic effects become weak. The project supported by the National Natural Science Foundation of China (10002003), Foundation for University Key Teacher by the Ministry of Education, Research Fund for the Doctoral Program of Higher Education     

Exact solutions in generalized Oldroyd-B fluid

This investigation deals with the influence of slip condition on the magnetohydrodynamic (MHD) and rotating flow of a generalized Oldroyd-B (G.Oldroyd-B)fluid occupying a porous space.Fractional calcul...     

粘弹性流体的分数元模型及圆管起动流

分数元模型所描述的非牛顿流体属于复杂粘弹性流体,其应力与应变的分数阶时间导数成正比.本文提出一种用弹簧和油壶连接组成的分形网络结构来比拟分数元模型的应力-应变特性,利用Heaviside运算微积,证明了该分形网络结构对应的粘弹性流体为1/2阶导数的分数元.并证明了构成其他分数阶导数分数元模型需要引入弹簧和油壶的多重分形网络结构.本文还导出了分数元模型的圆管起动流的解析解,研究了分数元模型起动过程振荡特征与该模型导数阶β之间的关系;发现在β≠1的情况下,随时间的进程,圆管内分数元模型的运动最终均将趋于静止,只有β=1的情况是一个例外.     

The solution for axisymmetrical shells with abrupt curvature change (corrugated shells) by the finite element method

It has been noted in the present paper that the finite element method using linear elements for solving axisymmetrical shells cannot be applied to the analysis of axisymmetrical shells with abrupt curvature change, owing to the fact that the influence of the curvature upon the angular displacements has been neglected. The present paper provides a finite element method using linear elements in which the influence of curvature is considered and the angular displacements are treated as continuous parameters. This method has been applied to the calculation of corrugated shells of the type C, and compared with the experimental results obtained by Turner-Ford as well as with the analytical solution given by Prof. Chien Wei-zang. The comparisons have been proved that this theory is correct.     

FE formulation for the viscoelastic body modeled by fractional constitutive law

This paper presents finite element (FE) formulation of the viscoelastic materials described by fractional constitutive law. The time-domain three-dimensional constitutive equation is constructed. The FE equations are set up by equations are solved by numerical integration method. The numerical algorithm developed by the authors for Liouville-Riemann's fractional derivative was adopted to formulate FE procedures and extended to solve the more general case of the hereditary integration. The numerical examples were given to show the correctness and effectiveness of the integration algorithm. The project supported by the Ministry of Education of China for the returned overseas Chinese scholars     

Using Fractional Order Adjustment Rules and Fractional Order Reference Models in Model-Reference Adaptive Control

This paper investigates the use of Fractional Order Calculus (FOC) inconventional Model Reference Adaptive Control (MRAC) systems. Twomodifications to the conventional MRAC are presented, i.e., the use offractional order parameter adjustment rule and the employment offractional order reference model. Through examples, benefits from theuse of FOC are illustrated together with some remarks for furtherresearch.     

A numerical method for fractional integral with applications

IntroductionThefractionalcalculushasalonghistoryandthereareamassofworkstodiscussthefractionalderivativesandfractionalintegralswitharbitrary (realorcomplex)order[1- 3 ].Thefractionalcalculushasawideapplicationbackground ,especiallyinthefieldsofchemistry ,electromagnetics,materialscienceandmechanics.Forexample,Gement[4 ]proposedthefractionalderivativeconstitutivemodelsofaviscoelasticmaterialatfirst.Themodelshavereceivedincreasingattention[5 - 7].Onlyafewparametersarecontainedinthemodelsandthemo…