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.

Optimal controllers with complex order derivatives

This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complexorder control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy.

Fractional order modelling of fractional-order holds

Discrete time control systems require sample- and-hold circuits to perform the conversion from digital to analog. Fractional-Order Holds (FROHs) are an interpolation between the classical zero and first order holds and can be tuned to produce better system performance. However, the model of the FROH is somewhat hermetic and the design of the system becomes unnecessarily complicated. This paper addresses the modelling of the FROHs using the concepts of Fractional Calculus (FC). For this purpose, two simple fractional-order approximations are proposed whose parameters are estimated by a genetic algorithm. The results are simple to interpret, demonstrating that FC is a useful tool for the analysis of these devices.

Optimal tuning of fractional controllers using genetic algorithms

This study addresses the optimization of fractional algorithms for the discrete-time control of linear and non-linear systems. The paper starts by analyzing the fundamentals of fractional control systems and genetic algorithms. In a second phase the paper evaluates the problem in an optimization perspective. The results demonstrate the feasibility of the evolutionary strategy and the adaptability to distinct types of systems.

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.

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.

An optimal boundary fair scheduling

Nowadays, many real-time operating systems discretize the time relying on a system time unit. To take this behavior into account, real-time scheduling algorithms must adopt a discrete-time model in which both timing requirements of tasks and their time allocations have to be integer multiples of the system time unit. That is, tasks cannot be executed for less than one time unit, which implies that they always have to achieve a minimum amount of work before they can be preempted. Assuming such a discrete-time model, the authors of Zhu et al. (Proceedings of the 24th IEEE international real-time systems symposium (RTSS 2003), 2003, J Parallel Distrib Comput 71(10):1411–1425, 2011) proposed an efficient “boundary fair” algorithm (named BF) and proved its optimality for the scheduling of periodic tasks while achieving full system utilization. However, BF cannot handle sporadic tasks due to their inherent irregular and unpredictable job release patterns. In this paper, we propose an optimal boundary-fair scheduling algorithm for sporadic tasks (named BF TeX ), which follows the same principle as BF by making scheduling decisions only at the job arrival times and (expected) task deadlines. This new algorithm was implemented in Linux and we show through experiments conducted upon a multicore machine that BF TeX outperforms the state-of-the-art discrete-time optimal scheduler (PD TeX ), benefiting from much less scheduling overheads. Furthermore, it appears from these experimental results that BF TeX is barely dependent on the length of the system time unit while PD TeX —the only other existing solution for the scheduling of sporadic tasks in discrete-time systems—sees its number of preemptions, migrations and the time spent to take scheduling decisions increasing linearly when improving the time resolution of the system.

Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Real time manufacturing systems simulator

A área da simulação computacional teve um rápido crescimento desde o seu apareciment, sendo actualmente uma das ciências de gestão e de investigação operacional mais utilizadas. O seu princípio baseia-se na replicação da operação de processos ou sistemas ao longo de períodos de tempo, tornando-se assim uma metodologia indispensável para a resolução de variados problemas do mundo real, independentemente da sua complexidade. Das inúmeras áreas de aplicação, nos mais diversos campos, a que mais se destaca é a utilização em sistemas de produção, onde o leque de aplicações disponível é muito vasto. A sua aplicação tem vindo a ser utilizada para solucionar problemas em sistemas de produção, uma vez que permite às empresas ajustar e planear de uma maneira rápida, eficaz e ponderada as suas operações e os seus sistemas, permitindo assim uma rápida adaptação das mesmas às constantes mudanças das necessidades da economia global. As aplicações e packages de simulação têm seguindo as tendências tecnológicas pelo que é notório o recurso a tecnologias orientadas a objectos para o desenvolvimento das mesmas. Este estudo baseou-se, numa primeira fase, na recolha de informação de suporte aos conceitos de modelação e simulação, bem como a respectiva aplicação a sistemas de produção em tempo real. Posteriormente centralizou-se no desenvolvimento de um protótipo de uma aplicação de simulação de ambientes de fabrico em tempo real. O desenvolvimento desta ferramenta teve em vista eventuais fins pedagógicos e uma utilização a nível académico, sendo esta capaz de simular um modelo de um sistema de produção, estando também dotada de animação. Sem deixar de parte a possibilidade de integração de outros módulos ou, até mesmo, em outras plataformas, houve ainda a preocupação acrescida de que a sua implementação recorresse a metodologias de desenvolvimento orientadas a objectos.

Discrete fractional order system vibrations

A theory of free vibrations of discrete fractional order (FO) systems with a finite number of degrees of freedom (dof) is developed. A FO system with a finite number of dof is defined by means of three matrices: mass inertia, system rigidity and FO elements. By adopting a matrix formulation, a mathematical description of FO discrete system free vibrations is determined in the form of coupled fractional order differential equations (FODE). The corresponding solutions in analytical form, for the special case of the matrix of FO properties elements, are determined and expressed as a polynomial series along time. For the eigen characteristic numbers, the system eigen main coordinates and the independent eigen FO modes are determined. A generalized function of visoelastic creep FO dissipation of energy and generalized forces of system with no ideal visoelastic creep FO dissipation of energy for generalized coordinates are formulated. Extended Lagrange FODE of second kind, for FO system dynamics, are also introduced. Two examples of FO chain systems are analyzed and the corresponding eigen characteristic numbers determined. It is shown that the oscillatory phenomena of a FO mechanical chain have analogies to electrical FO circuits. A FO electrical resistor is introduced and its constitutive voltage–current is formulated. Also a function of thermal energy FO dissipation of a FO electrical relation is discussed.