Research On Two Kinds Of Fractional Order Differentiators In Noisy Environment

In the vigorous development of the fractional calculus,an important research topic which is to estimate the fractional order derivatives of an unknown signal by designing a fractional order differentiator.However,in most cases,the fractional order derivatives of a signal cannot be analytically calculated.And when the signal is unknown,it is usually measured in noisy environment.Since in order to estimate the fractional order derivatives of an unknown signal from its discrete noisy observation,this paper mainly designs two kinds of fractional order differentiators to estimate the fractional derivatives of an unknown signal in noisy environment,which should be robust against the noises.And one is the generalized fractional order Jacobi differentiator,another is the modified fractional order Savitzky-Golay differentiator.The main contributions of this paper are outlined as follows:Firstly,this paper introduces the development history and research status of fractional calculus.Then,the research background of the fractional order differentiator is introduced.Moreover,the paper gives a brief introduction of the basic knowledge of fractional calculus,shifted Jacobi polynomials and the least squares algorithm.Secondly,based on shifted Jacobi polynomials,a new class of fractional order Jacobi functions is introduced,and the truncated fractional order Jacobi orthogonal series expansion is applied to filter the noisy signal.Hence,the generalized fractional order Jacobi differentiator is proposed by an integral formula to estimate the fractional order derivatives of the noisy signal,which depends on a set of design parameters.Then,the error analysis is provided of the noise error contribution and the truncated term error.The error bounds are also given,which permit to study the design parameters' influence.In particular,a digital fractional order Jacobi differentiator is deduced in discrete noise case.By comparing with two existing fractional order differentiators,numerical results are given to illustrate the accuracy and the robustness of the proposed fractional order differentiator.Finally,a fractional order Savitzky-Golay filter is introduced to filter the noisy signal,which is a power function obtained by the least squares algorithm.Then,by taking the fractional derivative of the power function,a modified fractional order Savitzky-Golay differentiator is obtained.By applying the generalized Taylor's formula,error analysis is provided,where the corresponding error bounds are also given.By comparing with two existing fractional order differentiators,the accuracy and the robustness of the modified fractional order Savitzky-Golay differentiator is illustrated in numerical simulations.