A NEW APPROACH FOR DIRECT DISCRETIZATION OF FRACTIONAL ORDER OPERATOR IN DELTA DOMAIN

The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s±m, 0 < m < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s±mis expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain

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

Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas

This paper extends and illustrates a probabilistic interpretation of the fractional derivative operator proposed by Pr. José Tenreiro Machado. While his interpretation concerned the probability of finding samples of the derivate signal in the expression of the fractional derivative, the present paper proposes interpretations for other fractional models and more generally fractional behaviours (without using a model). It also proposes probabilistic interpretations in terms of time constants and time delay distributions. It shows that these probabilistic interpretations in terms of time delay distributions can be connected to the physical behaviour of real systems governed by adsorption or diffusion phenomena

Fractional order chaotic systems and their electronic design

"Con el desarrollo del cálculo fraccionario y la teoría del caos, los sistemas caóticos de orden fraccionario se han convertido en una forma útil de evaluar las características de los sistemas dinámicos. En esta dirección, esta tesis es principalmente relacionada, es decir, en el estudio de sistemas caóticos de orden fraccionario, basado en sistemas disipativos de inestables, un sistema disipativo de inestable de orden fraccionario es propuesto. Algunas propiedades dinámicas como puntos de equilibrio, exponentes de Lyapunov, diagramas de bifurcación y comportamientos dinámicos caóticos del sistema caótico de orden fraccionario son estudiados. Los resultados obtenidos muestran claramente que el sistema discutido presenta un comportamiento caótico. Por medio de considerar la teoría del cálculo fraccionario y simulaciones numéricas, se muestra que el comportamiento caótico existe en el sistema de tres ecuaciones diferenciales de orden fraccionario acopladas, con un orden menor a tres. Estos resultados son validados por la existencia de un exponente positivo de Lyapunov, además de algunos diagramas de fase. Por otra parte, la presencia de caos es también verificada obteniendo la herradura topológica. Dicha prueba topológica garantiza la generaci´n de caos en el sistema de orden fraccionario propuesto. En orden de verificar la efectividad del sistema propuesto, un circuito electrónico es diseñado con el fin de sintetizar el sistema caótico de orden fraccionario.""With the development of fractional order calculus and chaos theory, the fractional order chaotic systems have become a useful way to evaluate characteristics of dynamical systems and forecast the trend of complex systems. In this direction, this thesis is primarily concerned with the study of fractional order chaotic systems, based on an unstable dissipative system (UDS), a fractional order unstable dissipative system (FOUDS) is proposed. Dynamical properties, such as equilibrium points, Lyapunov exponents, bifurcation diagrams and phase diagrams of the fractional order chaotic system are studied. The obtained results shown that the fractional order unstable dissipative system has a chaotic behavior. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the fractional order three dimensional system with order less than three. The lowest order to yield chaos in this system is 2.4. The results are validated by the existence of one positive Lyapunov exponent, phase diagrams; Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order unstable dissipative system. In order to verify the effectiveness of the proposed system, an electronic circuit is designed with the purpose of synthesize the fractional order chaotic system, the fractional order integral is realized with electronic circuit utilizing the synthesis of a fractance circuit. The realization has been done via synthesis as passive RC circuits connected to an operational amplifier. The continuos fractional expansion have been utilized on fractional integration transfer function which has been approximated to integer order rational transfer function considering the Charef Method. The analogue electronics circuits have been simulated using HSPICE.

Digital Filters

The new technology advances provide that a great number of system signals can be easily measured with a low cost. The main problem is that usually only a fraction of the signal is useful for different purposes, for example maintenance, DVD-recorders, computers, electric/electronic circuits, econometric, optimization, etc. Digital filters are the most versatile, practical and effective methods for extracting the information necessary from the signal. They can be dynamic, so they can be automatically or manually adjusted to the external and internal conditions. Presented in this book are the most advanced digital filters including different case studies and the most relevant literature

Engineering Education and Research Using MATLAB

MATLAB is a software package used primarily in the field of engineering for signal processing, numerical data analysis, modeling, programming, simulation, and computer graphic visualization. In the last few years, it has become widely accepted as an efficient tool, and, therefore, its use has significantly increased in scientific communities and academic institutions. This book consists of 20 chapters presenting research works using MATLAB tools. Chapters include techniques for programming and developing Graphical User Interfaces (GUIs), dynamic systems, electric machines, signal and image processing, power electronics, mixed signal circuits, genetic programming, digital watermarking, control systems, time-series regression modeling, and artificial neural networks

Analog Implementation of Fractional-Order Elements and Their Applications

With advancements in the theory of fractional calculus and also with widespread engineering application of fractional-order systems, analog implementation of fractional-order integrators and differentiators have received considerable attention. This is due to the fact that this powerful mathematical tool allows us to describe and model a real-world phenomenon more accurately than via classical “integer” methods. Moreover, their additional degree of freedom allows researchers to design accurate and more robust systems that would be impractical or impossible to implement with conventional capacitors. Throughout this thesis, a wide range of problems associated with analog circuit design of fractional-order systems are covered: passive component optimization of resistive-capacitive and resistive-inductive type fractional-order elements, realization of active fractional-order capacitors (FOCs), analog implementation of fractional-order integrators, robust fractional-order proportional-integral control design, investigation of different materials for FOC fabrication having ultra-wide frequency band, low phase error, possible low- and high-frequency realization of fractional-order oscillators in analog domain, mathematical and experimental study of solid-state FOCs in series-, parallel- and interconnected circuit networks. Consequently, the proposed approaches in this thesis are important considerations in beyond the future studies of fractional dynamic systems

Dissipation in the superfluid helium film

Experimental apparatus to study dissipation in the saturated superfluid helium film has been developed. The low temperature parts comprise a sealed cell containing liquid helium, to which are affixed two parallel plate capacitors, functioning both as liquid reservoirs and as a way of measuring the liquid level. A small hole in a thin plastic film located in the flow path between the two capacitors forms the flow-limiting constriction. This arrangement introduces large velocity gradients in the vicinity of the hole. Film flow is initiated and sustained by an electric field in one capacitor, generated by a purpose-built Film Drive Unit (FDU) and a high-voltage amplifier. Detailed study of the helium film under steady flow conditions was not possible, but those results which were obtained indicate that the transfer rate is about 30% higher than was anticipated. By applying positive feedback to the film through the FDU, the inertial oscillations can be studied over many cycles. This new method has revealed some unexpected results, and a variety of types of oscillation behaviour have been observed. A theoretical model of dissipation has been developed, based on the premise that vortices in the film are oriented perpendicular to the film plane and are free to move and cross streamlines. According to this model, the large steady film transfer rates are due to the separation of the region of dissipation and the region of maximum velocity, an effect caused by the radial-flow geometry. Numerical simulation of the inertial oscillations using the model reproduces some of the behaviour observed experimentally, provided that the rate of vortex creation is taken to be a step function of the velocity. The shape of the liquid helium surface tension meniscus has been calculated numerically. The calculation is valid for the moving and static film in the absence of dissipation

An Updated Vision of Continuous-Time Fractional Models

A few days before the end of the revision procedure, my friend J. Tenreiro Machado had a sudden cardio-respiratory arrest and died. Here I want to express my gratitude and tribute to a great man and scientist. He was a very friendly and helpful person, with an unusual work capacity that allowed him to publish interesting articles on a wide range of topics.This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregressive-moving average type and the tempered linear system. For both classes, the computations of the impulse response, transfer function, and frequency response are discussed. It is shown that such systems can have integer and fractional components. From the integer component we deduce the stability. The fractional order component is always stable. The initial-condition problem is analyzed and it is verified that it depends on the structure of the system. For a correct definition and backward compatibility with classic systems, suitable fractional derivatives are also introduced. The Grünwald-Letnikov and Liouville derivatives, as well as the corresponding tempered versions, are formulated.authorsversionpublishe

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding