Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 934094, 12 pages doi:10.1155/2011/934094 Research Article Time-Delay and Fractional Derivatives J. A. Tenreiro Machado Department of Electrical Engineering, Institute of Engineering of Porto, Rua Dr. Ant´ onio Bernardino de Almeida, 431, 4200-072 Porto, Portugal Correspondence should be addressed to J. A. Tenreiro Machado, jtm@isep.ipp.pt Received 7 January 2011; Accepted 4 February 2011 Academic Editor: J. J. Trujillo Copyright q 2011 J. A. Tenreiro Machado. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper proposes the calculation of fractional algorithms based on time-delay systems. The study starts by analyzing the memory properties of fractional operators and their relation with time delay. Based on the Fourier analysis an approximation of fractional derivatives through time- delayed samples is developed. Furthermore, the parameters of the proposed approximation are estimated by means of genetic algorithms. The results demonstrate the feasibility of the new perspective. 1. Introduction Fractional calculus FC deals with the generalization of integrals and derivatives to a noninteger order 1–7. In the last decades the application of FC verified a large development in the areas of physics and engineering and considerable research about a multitude of applications emerged such as, viscoelasticity, signal processing, diffusion, modeling, and control 8–17. The area of dynamical systems and control has received a considerable attention, and recently several papers addressing evolutionary concepts and fractional algorithms can be mentioned 18, 19. Nevertheless, the algorithms involved in the calculation of fractional derivatives require the adoption of numerical approximations 20– 26, and new research directions are clearly needed. Bearing these ideas in mind, this paper addresses the optimal system control using fractional order algorithms and is organized as follows. Section 2 introduces the calculation of fractional derivatives and formulates the problem of optimization through genetic algorithms GAs. Section 3 presents a set of experiments that demonstrate the effectiveness of the proposed optimization strategy. Finally, Section 4 outlines the main conclusions.