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

Abstract 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 Pade 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.

[1]  J. Machado.  Analysis and design of fractional-order digital control systems , 1997 .

[2]  Mohamad Adnan Al-Alaoui,et al.  Novel digital integrator and differentiator , 1993 .

[3]  I. Podlubny.  Fractional differential equations , 1998 .

[4]  Yangquan Chen,et al.  A new IIR-type digital fractional order differentiator , 2003, Signal Process..

[5]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[6]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[7]  Chien-Cheng Tseng,et al.  Design of fractional order digital FIR differentiators , 2001, IEEE Signal Processing Letters.

[8]  K. Moore,et al.  Discretization schemes for fractional-order differentiators and integrators , 2002 .

[9]  Yangquan Chen,et al.  Two direct Tustin discretization methods for fractional-order differentiator/integrator , 2003, J. Frankl. Inst..

[10]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[11]  Dumitru Baleanu,et al.  About fractional quantization and fractional variational principles , 2009 .

[12]  José António Tenreiro Machado,et al.  Time domain design of fractional differintegrators using least-squares , 2006, Signal Process..

[13]  Katica R. Hedrih.  Vibration Modes of an Axially Moving Double Belt System with Creep Layer , 2008 .

[14]  Peter J. Torvik,et al.  Fractional calculus in the transient analysis of viscoelastically damped structures , 1983 .

[15]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[16]  G. Zaslavsky,et al.  Fractional dynamics of systems with long-range interaction , 2006, 1107.5436.

[17]  Raoul R. Nigmatullin,et al.  The statistics of the fractional moments: Is there any chance to "read quantitatively" any randomness? , 2006, Signal Process..

[18]  Thomas J. Anastasio,et al.  The fractional-order dynamics of brainstem vestibulo-oculomotor neurons , 1994, Biological Cybernetics.

[19]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[20]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[21]  J. A. Tenreiro Machado,et al.  Discrete-time fractional-order controllers , 2001 .

[22]  J. Machado.  Calculation of fractional derivatives of noisy data with genetic algorithms , 2009 .

[23]  I. Podlubny.  Fractional-Order Systems and -Controllers , 1999 .

[24]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[25]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[26]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .