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

A new method for approximating fractional derivatives: application in non-linear 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.N/

Numerical calculation of fractional derivatives of non-smooth data

This paper addresses the calculation of fractional derivatives of fractional order for non-smooth data. The noise is avoided by adopting an optimization formulation using genetic algorithms (GA). Given the flexibility of the evolutionary schemes it is established a hierarchical GA composed by a series of two GAs, each one with having a distinct fitness function.N/

Analysis of Fractional-Order Discrete Controllers in the Presence of Nonlinearities

Presently, the development of fractional-order controllers is one of the most promising fields of research. However, most of the work in this area addresses the case of linear systems. In this paper we consider the analysis of fractional-order control of nonlinear systems. The performance of discrete fractional-order controllers in the presence of several nonlinearities is discussed. Some results are provided that assesses the superior robustness of such algorithms.N/

A Clustering Perspective of the Collatz Conjecture

This manuscript focuses on one of the most famous open problems in mathematics, namely the Collatz conjecture. The first part of the paper is devoted to describe the problem, providing a historical introduction to it, as well as giving some intuitive arguments of why is it hard from the mathematical point of view. The second part is dedicated to the visualization of behaviors of the Collatz iteration function and the analysis of the resultsThe work of D. Cao Labora was partially supported by grant number MTM2016-75140-P (AEI/FEDER, UE)S

Fractional dynamics and MDS visualization of earthquake phenomena

This paper analyses earthquake data in the perspective of dynamical systems and fractional calculus (FC). This new standpoint uses Multidimensional Scaling (MDS) as a powerful clustering and visualization tool. FC extends the concepts of integrals and derivatives to non-integer and complex orders. MDS is a technique that produces spatial or geometric representations of complex objects, such that those objects that are perceived to be similar in some sense are placed on the MDS maps forming clusters. In this study, over three million seismic occurrences, covering the period from January 1, 1904 up to March 14, 2012 are analysed. The events are characterized by their magnitude and spatiotemporal distributions and are divided into fifty groups, according to the Flinn–Engdahl (F–E) seismic regions of Earth. Several correlation indices are proposed to quantify the similarities among regions. MDS maps are proven as an intuitive and useful visual representation of the complex relationships that are present among seismic events, which may not be perceived on traditional geographic maps. Therefore, MDS constitutes a valid alternative to classic visualization tools for understanding the global behaviour of earthquakes

Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms

This study addresses the optimization of rational fraction approximations for the discrete-time calculation of fractional derivatives. The article starts by analyzing the standard techniques based on Taylor series and Padé expansions. In a second phase the paper re-evaluates the problem in an optimization perspective by tacking advantage of the flexibility of the genetic algorithms

Approximating fractional derivatives through the generalized mean

This paper addresses the calculation of fractional order expressions through rational fractions. The article starts by analyzing the techniques adopted in the continuous to discrete time conversion. The problem is re-evaluated in an optimization perspective by tacking advantage of the degree of freedom provided by the generalized mean formula. The results demonstrate the superior performance of the new algorithm

A Computational Approach for Obtaining the Root Locus of Fractional Systems

In this paper a new method for the calculation of the root locus of fractional systems is presented. The proposed algorithm takes advantage of present day computational resources and processes directly the characteristic equation. The results demonstrate the feasibility of the method for different types of expressions.N/

Statistical modelling of dual-arm robotic systems

The article presents a new approach to the analysis and design of two-arm robotic systems. Usually, system descriptions are based on a set of differential equations which require laborious computations. This motivates the need of alternative models based on other mathematical concepts. The proposed statistical method gives clear guidelines towards the robotic systems analysis and development.N/