Robust fractional order differentiators using generalized modulating functions method

This paper aims at designing a fractional order differentiator for a class of signals satisfying a linear differential equation with unknown parameters. A generalized modulating functions method is proposed first to estimate the unknown parameters, then to derive accurate integral formulae for the left-sided Riemann-Liouville fractional derivatives of the studied signal. Unlike the improper integral in the definition of the left-sided Riemann-Liouville fractional derivative, the integrals in the proposed formulae can be proper and be considered as a low-pass filter by choosing appropriate modulating functions. Hence, digital fractional order differentiators applicable for on-line applications are deduced using a numerical integration method in discrete noisy case. Moreover, some error analysis are given for noise error contributions due to a class of stochastic processes. Finally, numerical examples are given to show the accuracy and robustness of the proposed fractional order differentiators. HighlightsThe proposed differentiators do not contain any sources of errors in continuous noise free case.The integrals in the proposed formulae can be proper and be considered as a low-pass filter.There are two sources of errors in discrete noise case: the numerical error and the noise error.They can be used for on-line applications to estimate a derivative with an arbitrary order.

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

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

[3]  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.

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

[5]  Sridhar Ungarala,et al.  Batch scheme recursive parameter estimation of continuous-time systems using the modulating functions method , 1997, Autom..

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

[7]  Giuseppe Fedele,et al.  A recursive scheme for frequency estimation using the modulating functions method , 2010, Appl. Math. Comput..

[8]  Vicente Feliú Batlle,et al.  An algebraic frequency estimator for a biased and noisy sinusoidal signal , 2007, Signal Process..

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

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

[11]  S. Diop,et al.  A numerical procedure for filtering and efficient high-order signal differentiation , 2004 .

[12]  Igor Podlubny,et al.  ADJOINT FRACTIONAL DIFFERENTIAL EXPRESSIONS AND OPERATORS , 2007 .

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

[14]  Olivier Gibaru,et al.  Non-asymptotic state estimation for a class of linear time-varying systems with unknown inputs , 2014 .

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

[16]  Richard L. Magin,et al.  On the fractional signals and systems , 2011, Signal Process..

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

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

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

[20]  Wilfrid Perruquetti,et al.  Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data , 2013, 2013 European Control Conference (ECC).

[21]  Joachim Rudolph,et al.  An Algebraic Approach for Identification of Linear Systems with Fractional Derivatives , 2013, 1302.4071.

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

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

[24]  B. Ross,et al.  The development of fractional calculus 1695–1900 , 1977 .

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

[26]  Massimo Ruzzene,et al.  . Proceedings of IDETC/CIE 2007, ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Las Vegas NV, USA , 2007 .

[27]  Wilfrid Perruquetti,et al.  An algebraic approach for human posture estimation in the sagittal plane using accelerometer noisy signal , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[28]  Yang Tian,et al.  Fast state estimation in linear time-invariant systems: an algebraic approach , 2008, 2008 16th Mediterranean Conference on Control and Automation.

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

[30]  Olivier Gibaru,et al.  Synthesis on a class of algebraic differentiators and application to nonlinear observation , 2014, Proceedings of the 33rd Chinese Control Conference.

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

[32]  Olivier Gibaru,et al.  Identification of fractional order systems using modulating functions method , 2013, 2013 American Control Conference.

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

[34]  Olivier Gibaru,et al.  Fractional Order Numerical Differentiation with B-Spline Functions , 2013 .

[35]  Heinz Unbehauen,et al.  A review of identification in continuous-time systems , 1998 .

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

[37]  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).

[38]  M. Fliess,et al.  Reconstructeurs d'état , 2004 .

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

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

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

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

[43]  José António Tenreiro Machado,et al.  Exploiting sensor redundancy for the calculation of fractional derivatives in the presence of noise , 2012, Signal Process..

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

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

[46]  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.

[47]  Heinz A. Preisig,et al.  Theory and application of the modulating function method—I. Review and theory of the method and theory of the spline-type modulating functions , 1993 .

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

[49]  Wilfrid Perruquetti,et al.  Fast state estimation in linear time-varying systems: An algebraic approach , 2008, 2008 47th IEEE Conference on Decision and Control.