Electrostatics in Fractal Geometry: Fractional Calculus Approach

The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume $V_d\sim r^d$ with the fractal dimension $d$ and a complementary host volume $V_h=V_3-V_d$. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation

Geometrical enhancement of the electric field: Application of fractional calculus in nanoplasmonics

We developed an analytical approach, for a wave propagation in metal-dielectric nanostructures in the quasi-static limit. This consideration establishes a link between fractional geometry of the nanostructure and fractional integro-differentiation. The method is based on fractional calculus and permits to obtain analytical expressions for the electric field enhancement.Comment: Published in EP

Impact of fractional filter in PI control loop applied to induction motor speed drive

Introduction. One of the main problems of electrical machine control systems is to obtain a satisfactory performance in the rejection of load disturbances, as well as in the set-point tracking tasks. Generally, the development of control algorithms does not take into account the presence of noise. Appropriate filtering is, therefore, essential to reduce the impact of noise on the output of the controller, in addition to the machine output. Recently, there has been a great tendency toward using fractional calculus to solve engineering problems. The filtering is one of the fields in which fractional calculus has received great attention. The importance of filters in signal processing and other engineering areas is unquestionable Novelty. The proposed work is intended to be a contribution in the recent works conducted on the influence of the fractional filtering on the control robustness of induction machines control. Purpose. The main contribution of this research is the application of fractional filtering to the standard PI control loop for an induction motor speed drive. Methods. In order to assess its impact and benefit, different structures for introducing the filters are investigated, A first order filter is considered in different positions, whether before or after the controller or even in both positions at the same time, with a noise source. A review of the index performance evolution (the Integral Square Error, Integral Absolute Error and Integral Time Absolute Error) has allowed a configuration design of the filter. Results. Intensive simulations were performed with a control setup using integer and fractional order filters, which permitted to conclude that the fractional filters give better performance indices compared to the integer one and thus improve the dynamic characteristics of the system.Вступ. Однією з основних проблем систем керування електричними машинами є отримання задовільних характеристик при придушенні збурень навантаження, а також завдання відстеження уставок. Зазвичай, при розробці алгоритмів керування наявність шуму не враховується. Тому потрібна відповідна фільтрація для зниження впливу шуму на вихідний сигнал контролера на додаток до вихідного сигналу машини. Останнім часом спостерігається чітка тенденція до використання дробового обчислення для вирішення інженерних завдань. Фільтрація – це одна з областей, в якій дрібному обчисленню приділяється велика увага. Важливість фільтрів у обробці сигналів та інших галузях техніки незаперечна. Новизна. Запропонована робота покликана стати внеском у недавні роботи, присвячені впливу дробової фільтрації на надійність керування асинхронними машинами. Мета. Основним внеском цього дослідження є застосування дробової фільтрації до стандартного контуру ПІ-регулювання для приводу швидкості асинхронного двигуна. Методи. Щоб оцінити його вплив та користь, досліджуються різні конструкції для введення фільтрів. Фільтр першого порядку розглядається в різних положеннях до або після контролера або навіть в обох положеннях одночасно з джерелом шуму. Огляд розвитку показників ефективності (інтегральна квадратична помилка, інтегральна абсолютна помилка та інтегральна абсолютна помилка за часом) дозволив розробити конфігурацію фільтра. Результати. Значний обсяг моделювання був проведений з налаштуванням керування з використанням фільтрів цілочисельного та дробового порядку, що дозволило зробити висновок, що дробові фільтри дають кращі показники ефективності порівняно з цілочисельним і таким чином покращують динамічні характеристики системи

Fractional Calculus as a Macroscopic Manifestation of Randomness

We generalize the method of Van Hove so as to deal with the case of non-ordinary statistical mechanics, that being phenomena with no time-scale separation. We show that in the case of ordinary statistical mechanics, even if the adoption of the Van Hove method imposes randomness upon Hamiltonian dynamics, the resulting statistical process is described using normal calculus techniques. On the other hand, in the case where there is no time-scale separation, this generalized version of Van Hove's method not only imposes randomness upon the microscopic dynamics, but it also transmits randomness to the macroscopic level. As a result, the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus.Comment: 20 pages, 1 figur

Control of a novel chaotic fractional order system using a state feedback technique

We consider a new fractional order chaotic system displaying an interesting behavior. A necessary condition for the system to remain chaotic is derived. It is found that chaos exists in the system with order less than three. Using the Routh-Hurwitz and the Matignon stability criteria, we analyze the novel chaotic fractional order system and propose a control methodology that is better than the nonlinear counterparts available in the literature, in the sense of simplicity of implementation and analysis. A scalar control input that excites only one of the states is proposed, and sufficient conditions for the controller gain to stabilize the unstable equilibrium points derived. Numerical simulations confirm the theoretical analysis. © 2013 Elsevier Ltd. All rights reserved

Design of digital differentiators

A digital differentiator simply involves the derivation of an input signal. This work includes the presentation of first-degree and second-degree differentiators, which are designed as both infinite-impulse-response (IIR) filters and finite-impulse-response (FIR) filters. The proposed differentiators have low-pass magnitude response characteristics, thereby rejecting noise frequencies higher than the cut-off frequency. Both steady-state frequency-domain characteristics and Time-domain analyses are given for the proposed differentiators. It is shown that the proposed differentiators perform well when compared to previously proposed filters. When considering the time-domain characteristics of the differentiators, the processing of quantized signals proved especially enlightening, in terms of the filtering effects of the proposed differentiators. The coefficients of the proposed differentiators are obtained using an optimization algorithm, while the optimization objectives include magnitude and phase response. The low-pass characteristic of the proposed differentiators is achieved by minimizing the filter variance. The low-pass differentiators designed show the steep roll-off, as well as having highly accurate magnitude response in the pass-band. While having a history of over three hundred years, the design of fractional differentiator has become a ‘hot topic’ in recent decades. One challenging problem in this area is that there are many different definitions to describe the fractional model, such as the Riemann-Liouville and Caputo definitions. Through use of a feedback structure, based on the Riemann-Liouville definition. It is shown that the performance of the fractional differentiator can be improved in both the frequency-domain and time-domain. Two applications based on the proposed differentiators are described in the thesis. Specifically, the first of these involves the application of second degree differentiators in the estimation of the frequency components of a power system. The second example concerns for an image processing, edge detection application

Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions