On distributed order integrator/differentiator

In this paper, we derive the impulse response of the distributed order integrator/differentiator and its asymptotic property by using the complex path integral. Based on the derived analytical impulse response, we present a technique to perform the discretization of the above distributed order integrator/differentiator. The derived asymptotic property can be applied to verify the feasibility of this method. Some new features of the distributed order integrator/differentiator are shown in time and frequency domains. A number of illustrated figures are presented to validate the concepts.

[1]  Teodor M. Atanackovic,et al.  Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  Kai Diethelm,et al.  Numerical analysis for distributed-order differential equations , 2009 .

[3]  C. Lubich.  Discretized fractional calculus , 1986 .

[4]  Hu Sheng,et al.  On mean square displacement behaviors of anomalous diffusions with variable and random orders , 2010 .

[5]  Carl F. Lorenzo,et al.  Identification of Complex Order-distributions , 2008 .

[6]  F. Mainardi,et al.  The role of the Fox-Wright functions in fractional sub-diffusion of distributed order , 2007, 0711.3779.

[7]  Ahmed S. Elwakil,et al.  Fractional-order sinusoidal oscillators: Design procedure and practical examples , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Yangquan Chen,et al.  Analytical impulse response of a fractional second order filter and its impulse response invariant discretization , 2011, Signal Process..

[9]  Tom T. Hartley,et al.  Fractional-order system identification based on continuous order-distributions , 2003, Signal Process..

[10]  L. Mcbride,et al.  A technique for the identification of linear systems , 1965 .

[11]  T. Hartley,et al.  Initialization, conceptualization, and application in the generalized (fractional) calculus. , 2007, Critical reviews in biomedical engineering.

[12]  Wenchang Tan,et al.  Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics , 2006 .

[13]  Stevan Pilipović,et al.  On a nonlinear distributed order fractional differential equation , 2007 .

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

[15]  Stanly Steinberg,et al.  Random walk models associated with distributed fractional order differential equations , 2006 .

[16]  C. Halijak,et al.  Approximation of Fractional Capacitors (1/s)^(1/n) by a Regular Newton Process , 1964 .

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

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

[19]  R. Gorenflo,et al.  Time-fractional Diffusion of Distributed Order , 2007, cond-mat/0701132.

[20]  The Semi-Empirical and Empirical Models for Predicting Sound Absorption Coefficients for a Novel Porous Laminated Composite Material , 2003 .

[21]  Tomasz Srokowski.  Lévy flights in nonhomogeneous media: distributed-order fractional equation approach. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  J. Machado,et al.  Implementation of Discrete-Time Fractional- Order Controllers based on LS Approximations , 2006 .

[23]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

[24]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

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

[26]  Teodor M. Atanackovic,et al.  Time distributed-order diffusion-wave equation. I. Volterra-type equation , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[28]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[29]  Stevan Pilipović,et al.  On a fractional distributed-order oscillator , 2005 .

[30]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[31]  Alain Oustaloup,et al.  Fractional order sinusoidal oscillators: Optimization and their use in highly linear FM modulation , 1981 .

[32]  G. Bohannan.  Application of fractional calculus to polarization dynamics in solid dielectric materials , 2000 .

[33]  Hongguang Sun,et al.  Anomalous diffusion modeling by fractal and fractional derivatives , 2010, Comput. Math. Appl..

[34]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[35]  K. Moore,et al.  Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[36]  Francesco Mainardi,et al.  The Two Forms of Fractional Relaxation of Distributed Order , 2007 .

[37]  Ahmed S. Elwakil,et al.  Design equations for fractional-order sinusoidal oscillators: Practical circuit examples , 2007 .

[38]  Stevan Pilipović,et al.  Existence and calculation of the solution to the time distributed order diffusion equation , 2009 .

[39]  Joseph Arthur Connolly.  The numerical solution of fractional and distributed order differential equations , 2004 .

[40]  Carl F. Lorenzo,et al.  Fractional System Identification: An Approach Using Continuous Order-Distributions , 1999 .

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

[42]  Youcef Ferdi,et al.  Impulse invariance-based method for the computation of fractional integral of order 0alpha , 2009, Comput. Electr. Eng..

[43]  Wenchang Tan,et al.  Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics , 2006 .

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

[45]  I. Podlubny,et al.  Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives , 2005, math-ph/0512028.

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

[47]  Michele Caputo,et al.  Mean fractional-order-derivatives differential equations and filters , 1995, ANNALI DELL UNIVERSITA DI FERRARA.

[48]  Smith,et al.  Physical audio signal processing : for virtual musical instruments and audio effects , 2010 .

[49]  Anatoly N. Kochubei,et al.  Distributed order calculus and equations of ultraslow diffusion , 2008 .

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

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

[52]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

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

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