Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case

In the first part of this work, the differentiation by integration method has been generalized from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. The estimation errors for the proposed fractional order Jacobi differentiators have been studied in continuous case. In this paper, the focus is on the study of these differentiators in discrete case. Firstly, the noise error contribution due to a large class of stochastic processes is studied in discrete case. In particular, it is shown that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, the mean value and variance functions of which are time-variant. Secondly, by using the obtained noise error bound and the error bound for the bias term error obtained in the first part, we analyze the design parameters' influence on the obtained fractional order differentiators. Thirdly, according to the knowledge of the design parameters' influence, the fractional order Jacobi differentiators are significantly improved by admitting a time-delay. In order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, numerical simulations show their accuracy and robustness with respect to corrupting noises

Fórmulas de Quadratura para o Cálculo do Integral Fracionário de Riemann-Liouville

The quadrature formulas for the fractional Riemann-Liouville integral are investigated in this article. A linear operator is introduced that associates ] , [ ) ( baCx  a polynomial n n P (; x)H satisfying the condition ( )( ) ( )( ) a n j a j I P x I  x   + + = , n j , 0 = , where j x – are Chebyshev points. The integrand is approximated by an algebraic polynomial. The formula of the remainder term for the quadrature formula is derived. The quadrature formulas obtained are verified using the Wolfram Mathematica computer algebra system.Las fórmulas en cuadratura para la integral fraccional de Riemann-Liouville se investigan en este artículo. Se introduce un operador lineal que asocia ] , [ ) ( baCx  un polinomio n n P (; x)H que satisface la condición ( )( ) ( )( ) a n j a j I P x I  x   + + =, n j , 0 = donde: j x son puntos de Chebyshev. El integrando es aproximado por un polinomio algebraico. Se deriva la fórmula del término restante para la fórmula de cuadratura. Las fórmulas en cuadratura obtenidas se verifican usando el sistema de álgebra computacional Wolfram Mathematica.As fórmulas de quadratura para a integral fracionária de Riemann-Liouville são investigadas neste artigo. Um operador linear é introduzido que associa (x)C[a,b] um polinômio n n P (; x)H satisfazendo a condição ( )( ) ( )( ) a n j a j I P x I  x   + + = , j = 0,n onde j x - são os pontos de Chebyshev.O integrando é aproximado por um polinômio algébrico. A fórmula do termo restante para a fórmula de quadratura é derivada. As fórmulas de quadratura obtidas são verificadas usando o sistema de álgebra computacional da Wolfram Mathematica.&nbsp

Lur’e Postnikov Lyapunov functional technique to global Mittag-Leffler stability of fractional-order neural networks with piecewise constant argument

International audienc

Fractional order differentiation by integration and error analysis in noisy environment

International audienceThe 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 contributiondue 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

Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case

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 are going to generalize this method from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. For sake of clarity, this work has been divided into two parts. The first part presented in this paper focuses on the continuous case while the second part that has been presented in another paper deals with the discrete case with on-line applications. In this paper, two kinds of 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 both for continuous-time and discrete-time models in on-line or off line applications. Then, some error bounds are provided for the corresponding estimation errors in continuous case. These bounds will be used to study the design parameters' influence on the obtained fractional order differentiators in the second part. Finally, numerical simulations are given to show the accuracy and the robustness with respect to corrupting noises of the proposed differentiators in off-line applications. The properties of our differentiators in on-line applications will be shown in the second part

Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case

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 are going to generalize this method from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. For sake of clarity, this work has been divided into two parts. The first part presented in this paper focuses on the continuous case while the second part that has been presented in another paper deals with the discrete case with on-line applications. In this paper, two kinds of 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 both for continuous-time and discrete-time models in on-line or off line applications. Then, some error bounds are provided for the corresponding estimation errors in continuous case. These bounds will be used to study the design parameters' influence on the obtained fractional order differentiators in the second part. Finally, numerical simulations are given to show the accuracy and the robustness with respect to corrupting noises of the proposed differentiators in off-line applications. The properties of our differentiators in on-line applications will be shown in the second part

Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case

In the first part of this work, the differentiation by integration method has been generalized from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. The estimation errors for the proposed fractional order Jacobi differentiators have been studied in continuous case. In this paper, the focus is on the study of these differentiators in discrete case. Firstly, the noise error contribution due to a large class of stochastic processes is studied in discrete case. In particular, it is shown that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, the mean value and variance functions of which are time-variant. Secondly, by using the obtained noise error bound and the error bound for the bias term error obtained in the first part, we analyze the design parameters' influence on the obtained fractional order differentiators. Thirdly, according to the knowledge of the design parameters' influence, the fractional order Jacobi differentiators are significantly improved by admitting a time-delay. In order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, numerical simulations show their accuracy and robustness with respect to corrupting noises