On the Completeness Problem for Fractional Rationals with Incommensurable Differentiation Orders

In this paper, completeness problem for a class of fractional rational basis functions with incommensurable differentiation orders is studied. Using the Muntz-Szasz theory, it is established that the completeness problem for this class of the basis functions is equivalent to another completeness problem for a particular set of uncountably many basis functions. This equivalence allows one to draw fairly general conclusions on the nature of the completeness problem for fractional rational basis functions with incommensurable differentiation orders.

[1]  A. Oustaloup,et al.  On Fractional Systems H-Norm Computation , 2005 .

[2]  S. Rodrigues,et al.  A review of state-of-charge indication of batteries by means of a.c. impedance measurements , 2000 .

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

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

[5]  A. Oustaloup,et al.  La commande crone : du scalaire au multivariable , 1999 .

[6]  I Podlubny.  Fractional-order systems and (PID mu)-D-lambda-controllers , 1999 .

[7]  Hüseyin Akçay.  SYNTHESIS OF COMPLETE ORTHONORMAL FRACTIONAL BASIS FUNCTIONS WITH PRESCRIBED POLES , 2007 .

[8]  A. Oustaloup,et al.  Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes , 2004, First International Symposium on Control, Communications and Signal Processing, 2004..

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

[10]  N. Heymans,et al.  Fractal rheological models and fractional differential equations for viscoelastic behavior , 1994 .

[11]  J. Battaglia,et al.  Solving an inverse heat conduction problem using a non-integer identified model , 2001 .

[12]  J. Sabatier,et al.  On Fractional Systems H∞, -Norm Computation , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[13]  J. Partington,et al.  Coprime factorizations and stability of fractional differential systems , 2000 .

[14]  V. Vorperian,et al.  A fractal model of anomalous losses in ferromagnetic materials , 1992, PESC '92 Record. 23rd Annual IEEE Power Electronics Specialists Conference.

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

[16]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[17]  Bo Wahlberg,et al.  On approximation of stable linear dynamical systems using Laguerre and Kautz functions , 1996, Autom..

[18]  Jerome Spanier,et al.  A general solution of the diffusion equation for semiinfinite geometries , 1972 .

[19]  H. Hjalmarsson,et al.  Generalized Fourier and Toeplitz Results for Rational Orthonormal Bases , 1999 .

[20]  D. Matignon,et al.  Some Results on Controllability and Observability of Finite-dimensional Fractional Differential Systems , 1996 .

[21]  K. B. Oldham.  Diffusive transport to planar, cylindrical and spherical electrodes , 1973 .

[22]  Jerry M. Mendel,et al.  A Unified Approach to the Synthesis of Orthonormal Exponential Functions Useful in Systems Analysis , 1966, IEEE Trans. Syst. Sci. Cybern..

[23]  J. Sabatier,et al.  TUTORIAL ON SYSTEM IDENTIFICATION USING FRACTIONAL DIFFERENTIATION MODELS , 2006 .

[24]  K. B. Oldham,et al.  The replacement of Fick's laws by a formulation involving semidifferentiation , 1970 .

[25]  W. Rudin.  Real and complex analysis , 1968 .

[26]  P. V. D. Hof,et al.  A generalized orthonormal basis for linear dynamical systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[27]  Brett Ninness,et al.  Orthonormal basis functions for modelling continuous-time systems , 1999, Signal Process..

[28]  Brett Ninness,et al.  Orthonormal Basis Functions for Continuous-Time Systems and Lp Convergence , 1999, Math. Control. Signals Syst..

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

[30]  H. W. Bode,et al.  Network analysis and feedback amplifier design , 1945 .

[31]  D. Matignon.  Stability properties for generalized fractional differential systems , 1998 .

[32]  B. Epstein.  Orthogonal families of analytic functions , 1965 .

[33]  Alain Oustaloup,et al.  Synthesis of fractional Laguerre basis for system approximation , 2007, Autom..