IIR approximations to the fractional differentiator/integrator using Chebyshev polynomials theory.

This paper deals with the use of Chebyshev polynomials theory to achieve accurate discrete-time approximations to the fractional-order differentiator/integrator in terms of IIR filters. These filters are obtained using the Chebyshev-Padé and the Rational Chebyshev approximations, two highly accurate numerical methods that can be computed with ease using available software. They are compared against other highly accurate approximations proposed in the literature. It is also shown how the frequency response of the fractional-order integrator approximations can be easily improved at low frequencies.

[1]  Chien-Cheng Tseng,et al.  Digital IIR integrator design using recursive Romberg integration rule and fractional sample delay , 2008, Signal Process..

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

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

[4]  G. Maione.  A Rational Discrete Approximation , 2006 .

[5]  J. Terpak,et al.  Comparison of the methods for discrete approximation of the fractional-order operator , 2003 .

[6]  Piotr Ostalczyk.  Fundamental properties of the fractional-order discrete-time integrator , 2003, Signal Process..

[7]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[8]  Miguel Romero,et al.  A survey of Fractional-Order Generalized Predictive Control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[9]  Anthony Ralston,et al.  Mathematical Methods for Digital Computers , 1960 .

[10]  Mohammad Saleh Tavazoei,et al.  Identifiability of fractional order systems using input output frequency contents. , 2010, ISA transactions.

[11]  Igor Podlubny,et al.  A New Discretization Method for Fractional Order Differentiators via Continued Fraction Expansion , 2003 .

[12]  Alain Oustaloup,et al.  Fractional differentiation for edge detection , 2003, Signal Process..

[13]  E. Ya. Remez,et al.  General computational methods of Chebyshev approximation : the problems with linear real parameters , 1967 .

[14]  Y. Chen,et al.  Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review , 2004 .

[15]  J. Shanks.  RECURSION FILTERS FOR DIGITAL PROCESSING , 1967 .

[16]  Raoul R. Nigmatullin,et al.  Signal processing and recognition of true kinetic equations containing non-integer derivatives from raw dielectric data , 2003, Signal Process..

[17]  Y. Chen,et al.  A variable-order fractional operator based synthesis method for multifractional Gaussian noise , 2010, Proceedings of 2010 IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications.

[18]  A. J. Calderón,et al.  Fractional PID Controllers for Industry Application. A Brief Introduction , 2007 .

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

[20]  Chien-Cheng Tseng,et al.  Design of Fractional Order Digital Differentiator Using Radial Basis Function , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

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

[22]  W. J. Cody,et al.  A Survey of Practical Rational and Polynomial Approximation of Functions , 1970 .

[23]  H. V. Rossum,et al.  Padé Approximation and its Applications Amsterdam 1980 , 1981 .

[24]  New direct discretization of the fractional-order differentiator/integrator by the Chebyshev-Padé a , 2006 .

[25]  Guido Maione,et al.  Conditions for a Class of Rational Approximants of Fractional Differentiators/Integrators to Enjoy the Interlacing Property , 2011 .

[26]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[27]  M. Omair Ahmad,et al.  Exact fractional-order differentiators for polynomial signals , 2004, IEEE Signal Processing Letters.

[28]  Alain Oustaloup,et al.  CRONE control of continuous linear time periodic systems: application to a testing bench. , 2003, ISA transactions.

[29]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

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

[31]  Celaleddin Yeroglu,et al.  Classical controller design techniques for fractional order case. , 2011, ISA transactions.

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

[33]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[34]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .