Electrical skin phenomena: a fractional calculus analysis

The internal impedance of a wire is the function of the frequency. In a conductor, where the conductivity is sufficiently high, the displacement current density can be neglected. In this case, the conduction current density is given by the product of the electric field and the conductance. One of the aspects of the high-frequency effects is the skin effect (SE). The fundamental problem with SE is it attenuates the higher frequency components of a signal.

A fractional calculus perspective in the evolutionary design of combinational circuits

This paper analyses the performance of a genetic algorithm (GA) in the synthesis of digital circuits using two novel approaches. The first concept consists in improving the static fitness function by including a discontinuity evaluation. The measure of variability in the error of the Boolean table has similarities with the function continuity issue in classical calculus. The second concept extends the static fitness by introducing a fractional-order dynamical evaluation.

Integer vs. fractional order control of a hexapod robot

This paper studies the performance of integer and fractional order controllers in a hexapod robot with joints at the legs having viscous friction and flexibility. For that objective the robot prescribed motion is characterized in terms of several locomotion variables. The controller performance is analised through the Nyquist stability criterion. A set of model-based experiments reveals the influence of the different controller implementations upon the proposed metrics.

Approximating fractional derivatives in the perspective of system control

The theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received attention only recently. In the area of automatic control some work was developed, but the proposed algorithms are still in a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear system control.

Fractional describing function of systems with Coulomb friction

This paper studies the describing function (DF) of systems constituted by a mass subjected to nonlinear friction. The friction force is decomposed into two components, namely, the viscous and the Coulomb friction. The system dynamics is analyzed in the DF perspective revealing a fractional-order behavior. The reliability of the DF method is evaluated through the signal harmonic contents.

Development of fractional order capacitors based on electrolyte processes

In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.

Fractional control of heat diffusion systems

The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared. Simulations are presented assessing the performance of the proposed fractional-order algorithms.

Fractional electrical impedances in botanical elements

Fractional calculus (FC) is no longer considered solely from a mathematical viewpoint, and is now applied in many emerging scientific areas, such as electricity, magnetism, mechanics, fluid dynamics, and medicine. In the field of dynamical systems, significant work has been carried out proving the importance of fractional order mathematical models. This article studies the electrical impedance of vegetables and fruits from a FC perspective. From this line of thought, several experiments are developed for measuring the impedance of botanical elements. The results are analyzed using Bode and polar diagrams, which lead to electrical circuit models revealing fractional-order behaviour.

Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.

Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomena

The development of fractional-order controllers is currently one of the most promising fields of research. However, most of the work in this area addresses the case of linear systems. This paper reports on the analysis of fractional-order control of nonlinear systems. The performance of discrete fractional-order PID controllers in the presence of several nonlinearities is discussed. Some results are provided that indicate the superior robustness of such algorithms.

Analysis of the Van der Pol oscillator containing derivatives of fractional order

In this paper a modified version of the classical Van der Pol oscillator is proposed, introducing fractional-order time derivatives into the state-space model. The resulting fractional-order Van der Pol oscillator is analyzed in the time and frequency domains, using phase portraits, spectral analysis and bifurcation diagrams. The fractional-order dynamics is illustrated through numerical simulations of the proposed schemes using approximations to fractional-order operators. Finally, the analysis is extended to the forced Van der Pol oscillator.

Fractional order electromagnetics

The Maxwell equations constitute a formalism for the development of models describing electromagnetic phenomena. The four Maxwell laws have been adopted successfully in many applications and involve only the integer order differential calculus. Recently, a closer look for the cases of transmission lines, electrical motors and transformers, that reveal the so-called skin effect, motivated a new perspective towards the replacement of classical models by fractional-order mathematical descriptions. Bearing these facts in mind this paper addresses the concept of static fractional electric potential. The fractional potential was suggested some years ago. However, the idea was not fully explored and practical methods of implementation were not proposed. In this line of thought, this paper develops a new approximation algorithm for establishing the fractional order electrical potential and analyzes its characteristics.

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

A fractional perspective to financial indices

The application of mathematical methods and computer algorithms in the analysis of economic and financial data series aims to give empirical descriptions of the hidden relations between many complex or unknown variables and systems. This strategy overcomes the requirement for building models based on a set of ‘fundamental laws’, which is the paradigm for studying phenomena usual in physics and engineering. In spite of this shortcut, the fact is that financial series demonstrate to be hard to tackle, involving complex memory effects and a apparently chaotic behaviour. Several measures for describing these objects were adopted by market agents, but, due to their simplicity, they are not capable to cope with the diversity and complexity embedded in the data. Therefore, it is important to propose new measures that, on one hand, are highly interpretable by standard personal but, on the other hand, are capable of capturing a significant part of the dynamical effects.

Numerical analysis of the initial conditions in fractional systems

Fractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of adequate values for the initial conditions in numerical simulations.