Alexandra M. S. F. Galhano

Some Applications of Fractional Calculus in Engineering

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the progress in the area of chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses FC in the study of system dynamics and control. In this perspective, this paper investigates the use of FC in the fields of controller tuning, legged robots, redundant robots, heat diffusion, and digital circuit synthesis.

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.

On development of fractional calculus during the last fifty years

Fractional calculus generalizes integer order derivatives and integrals. During the last half century a considerable progress took place in this scientific area. This paper addresses the evolution and establishes an assertive measure of the research development.

Fractional Dynamics : A Statistical Perspective

Fractional calculus is a mathematical paradigm that has been increasingly adopted to describe the dynamics of systems with hereditary characteristics, or that reflect an average of a large population of microelements. In this line of thought, this article analyzes the statistical dynamics of a system composed of a large number of micromechanical masses with backlash and impacts. We conclude that, while individual dynamics of each element has an integer-order nature, the global dynamics reveal the existence of both integer and fractional dynamics.

Science metrics on fractional calculus development since 1966

During the last fifty years the area of Fractional Calculus verified a considerable progress. This paper analyzes and measures the evolution that occurred since 1966.

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 robot dynamics and compensation using classical and computed torque techniques

A classical analysis of the dynamics of robot manipulators is presented. It is shown that these systems have configuration-dependent properties and can be open-loop unstable. Due to this fact, present day linear controllers are inefficient. On the other hand, nonlinear hardware and software compensation methods also are shown to have some limitations. Controllers based on the direct design algorithm and the computed torque method have superior performances. These algorithms have nonlinear loops, yet it is shown that a linear analysis is still feasible. Therefore, classical design tools can be adopted in order to develop practical implementations. >

Multidimensional Scaling Visualization Using Parametric Similarity Indices

In this paper, we apply multidimensional scaling (MDS) and parametric similarity indices (PSI) in the analysis of complex systems (CS). Each CS is viewed as a dynamical system, exhibiting an output time-series to be interpreted as a manifestation of its behavior. We start by adopting a sliding window to sample the original data into several consecutive time periods. Second, we define a given PSI for tracking pieces of data. We then compare the windows for different values of the parameter, and we generate the corresponding MDS maps of ‘points’. Third, we use Procrustes analysis to linearly transform the MDS charts for maximum superposition and to build a globalMDS map of “shapes”. This final plot captures the time evolution of the phenomena and is sensitive to the PSI adopted. The generalized correlation, theMinkowski distance and four entropy-based indices are tested. The proposed approach is applied to the Dow Jones Industrial Average stock market index and the Europe Brent Spot Price FOB time-series.

A statistical and harmonic model for robot manipulators

A new approach to the analysis and design of robot manipulators is presented. The novel feature resides on a non-standard formulation to the modelling problem. Usually, system descriptions are based on a set of differential equations which, in general, require laborious computations and may be difficult to analyse. These facts motivate the need of alternative models based on different mathematical concepts. The proposed statistical approach to the Fourier modelling gives clear guidelines towards the optimisation of the robot kinematics and points out structural characteristics of the trajectory planning algorithms.

On the Analysis and Design of Robot Manipulators: A Statistical Approach

Abstract A new approach to the analysis and design of robot manipulators is presented. The novel feature resides on a non-standard approach to the modelling problem. Usually, system descriptions are based on a set of differential equations which, due to their nature lead to very precise results and strategies but in general lead to laborious computations. This motivates the need of alternative models based on other mathemati c al concepts . The proposed statistical method is a step in this direction which gives clear guidelines towards the robot kinematic and dynami c optimization . Furthermore, the use of histograms allows not only fast design procedures but above all, the use of experimental data; consequently, complex dynamic modelling exercises can be avoided.