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- Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomenaPublication . Barbosa, Ramiro S.; Machado, J. A. Tenreiro; Galhano, AlexandraThe 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.
- Fractional order dynamics in classical electromagnetic phenomenaPublication . Tenreiro Machado, J. A.; Jesus, Isabel S.; Galhano, Alexandra; Albert, W. Malpica; Silva, Fernando; Tar, József K.The Maxwell equations play a fundamental role in the well established formulation of the electromagnetic theory. These equations lead to the derivation of precise mathematical models useful in many applications in physics and engineering. The Maxwell equations involve only the integer-order calculus and, therefore, it is natural that the resulting classical models adopted in electrical engineering reflect this perspective. Recently, a closer look of some phenomena present in electrical systems, such as motors, transformers and lines, and the motivation towards the development of comprehensive models, seem to point out the requirement for a fractional calculus approach. Bearing these ideas in mind, in this study we shall address the well-known ‘skin effect’ and we reevaluate the results demonstrating its fractional-order dynamics.
- Fractional Dynamics: A Statistical PerspectivePublication . Tenreiro Machado, J. A.; Galhano, AlexandraFractional 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.
- Electric Fractional Order PotencialPublication . Tenreiro Machado, J. A.; Galhano, AlexandraIn this study we apply the concept of fractional calculus to electromagnetism and we develop a new fractional order approximation method to the electrical potential
- Application of fractional calculus in the system modelling and controlPublication . Tenreiro Machado, J. A.; Barbosa, Ramiro; Silva, Manuel; Figueiredo, Lino; Jesus, Isabel S.; Galhano, AlexandraFractional 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 a FC perspective in the study of the dynamics and control of several systems.
- Analysis of Fractional-Order Discrete Controllers in the Presence of NonlinearitiesPublication . Barbosa, Ramiro; Tenreiro Machado, J. A.; Galhano, AlexandraPresently, 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.
- A new method for approximating fractional derivatives: application in non-linear controlPublication . Tenreiro Machado, J. A.; Galhano, AlexandraThe 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 dynamics: a statistical perspectivePublication . Tenreiro Machado, J. A.; Galhano, AlexandraFractional 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 micro elements. In this line of thought, this article analyzed the statistical dynamics of a system composed of a large number of micro-mechanical 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.
- Approximating fractional derivatives through the generalized meanPublication . Machado, J. A. Tenreiro; Galhano, Alexandra; Oliveira, Anabela; Tar, József K.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.
- On the statistical and Fourier modelling of robot motionPublication . Tenreiro Machado, J. A.; Galhano, AlexandraA new method for the study and optimization of manipulator trajectories is developed. The novel feature resides on the modelling formulation. Standard system descriptions are based on a set of differential equations which, in general, require laborious computations and may be difficul to analyse. Moreover, the derived algorithms are suited to "deterministic" tasks, such as those appearing in a repetitive work, and are not well adapted to a "random" operation that occurs in intelligent systems interacting with a non-structured and changing environment. These facts motivate the development of alternative models based on distinct concepts. The proposed embedding of statistics and Fourier transform gives a new perspective towards the calculation and optimization of the robot trajectories in manipulating tasks.
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