Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment

The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann-Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann-Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.

[1]  C. Lanczos.  Applied Analysis , 1961 .

[2]  M. Fliess,et al.  Compression différentielle de transitoires bruités , 2004 .

[3]  B. Onaral,et al.  Fractal system as represented by singularity function , 1992 .

[4]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[5]  Dayan Liu,et al.  An error analysis in the algebraic estimation of a noisy sinusoidal signal , 2008, 2008 16th Mediterranean Conference on Control and Automation.

[6]  Jian Li,et al.  A New FDTD Formulation for Wave Propagation in Biological Media With Cole–Cole Model , 2006, IEEE Microwave and Wireless Components Letters.

[7]  Olivier Gibaru,et al.  Error analysis for a class of numerical differentiator: application to state observation , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[8]  T. Hughes,et al.  Signals and systems , 2006, Genome Biology.

[9]  Olivier Gibaru,et al.  Fractional order differentiation by integration with Jacobi polynomials , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[10]  Chien-Cheng Tseng,et al.  Improved design of digital fractional-order differentiators using fractional sample delay , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[11]  Bijnan Bandyopadhyay,et al.  Finite-Time Stabilization of Fractional Order Uncertain Chain of Integrator: An Integral Sliding Mode Approach , 2013, IEEE Transactions on Automatic Control.

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

[13]  A. Charef,et al.  Digital fractional order operators for R-wave detection in electrocardiogram signal , 2009 .

[14]  Vahid Badri,et al.  On tuning fractional order [proportional-derivative] controllers for a class of fractional order systems , 2013, Autom..

[15]  Tom H. Koornwinder,et al.  Differentiation by integration using orthogonal polynomials, a survey , 2012, J. Approx. Theory.

[16]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

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

[18]  M. Mboup.  Parameter estimation for signals described by differential equations , 2009 .

[19]  Ronald W. Schafer,et al.  What Is a Savitzky-Golay Filter? [Lecture Notes] , 2011, IEEE Signal Processing Magazine.

[20]  M. Fliess,et al.  Questioning some paradigms of signal processing via concrete examples , 2003 .

[21]  Olivier Gibaru,et al.  Non-asymptotic fractional order differentiators via an algebraic parametric method , 2012, 2012 1st International Conference on Systems and Computer Science (ICSCS).

[22]  A. Ralston.  A first course in numerical analysis , 1965 .

[23]  Cédric Join,et al.  Numerical differentiation with annihilators in noisy environment , 2009, Numerical Algorithms.

[24]  M. Fliess,et al.  An algebraic framework for linear identification , 2003 .

[25]  Guido Maione,et al.  On the Laguerre Rational Approximation to Fractional Discrete Derivative and Integral Operators , 2013, IEEE Transactions on Automatic Control.

[26]  Shouming Zhong,et al.  Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems , 2014, Autom..

[27]  Jian Bai,et al.  Fractional-Order Anisotropic Diffusion for Image Denoising , 2007, IEEE Transactions on Image Processing.

[28]  Olivier Gibaru,et al.  Convergence Rate of the Causal Jacobi Derivative Estimator , 2010, Curves and Surfaces.

[29]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[30]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[31]  Michel Fliess,et al.  Analyse non standard du bruit , 2006, ArXiv.

[32]  J. Sabatier,et al.  Crone control of a nonlinear hydraulic actuator , 2002 .

[33]  Alain Oustaloup,et al.  From fractal robustness to the CRONE control , 1999 .

[34]  A. Savitzky,et al.  Smoothing and Differentiation of Data by Simplified Least Squares Procedures. , 1964 .

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

[36]  Fang Hu,et al.  Stabilization of Quasi Integrable Hamiltonian Systems With Fractional Derivative Damping by Using Fractional Optimal Control , 2013, IEEE Transactions on Automatic Control.

[37]  Per-Olof Persson,et al.  Smoothing by Savitzky-Golay and Legendre Filters , 2003, Mathematical Systems Theory in Biology, Communications, Computation, and Finance.

[38]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

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

[40]  J. Sabatier,et al.  From fractal robustness to the CRONE approach , 1998 .

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

[42]  G. Alexits,et al.  Convergence Problems of Orthogonal Series. , 1961 .

[43]  B. A. Shenoi,et al.  Introduction to Digital Signal Processing and Filter Design , 2005 .

[44]  E. H. Doha,et al.  A NEW JACOBI OPERATIONAL MATRIX: AN APPLICATION FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS , 2012 .

[45]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[46]  Olivier Gibaru,et al.  Error analysis of Jacobi derivative estimators for noisy signals , 2011, Numerical Algorithms.

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

[48]  Hugues Garnier,et al.  Parameter and differentiation order estimation in fractional models , 2013, Autom..

[49]  R. Schafer,et al.  What Is a Savitzky-Golay Filter? , 2022 .

[50]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

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

[52]  Indranil Pan,et al.  Fractional Order Signal Processing , 2011 .

[53]  Michel Fliess.  Critique du rapport signal à bruit en communications numériques -- Questioning the signal to noise ratio in digital communications , 2008, ArXiv.

[54]  M. Fliess,et al.  A revised look at numerical differentiation with an application to nonlinear feedback control , 2007, 2007 Mediterranean Conference on Control & Automation.

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

[56]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

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

[58]  Yangquan Chen,et al.  Digital Fractional Order Savitzky-Golay Differentiator , 2011, IEEE Transactions on Circuits and Systems II: Express Briefs.

[59]  Alain Oustaloup,et al.  The CRONE Control of Resonant Plants: Application to a Flexible Transmission , 1995, Eur. J. Control.

[60]  William H. Press,et al.  Numerical recipes in C , 2002 .

[61]  Olivier Gibaru,et al.  Differentiation by integration with Jacobi polynomials , 2011, J. Comput. Appl. Math..

[62]  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.

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

[64]  Olivier Gibaru,et al.  Parameters estimation of a noisy sinusoidal signal with time-varying amplitude , 2011, 2011 19th Mediterranean Conference on Control & Automation (MED).

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