On State Estimation Algorithm Of Discrete-time Nonlinear Fractional-order Systems

Due to the fractional calculus,a simpler and more accurate mathematical model can be obtained,and fractional systems play an increasingly important role in many scientific and engineering problems.In recent years,with the development of digital computers,the research on discrete fractional systems has also received much atten-tion.The control theory for discrete fractional state space systems emerges in an end-less stream.The analysis and control of fractional systems require the exactly known states.However,in practical systems,the state of systems is often contaminated by noises,and is also difficult to be obtained directly.Therefore,it is of great engineering practical significance to the design of the corresponding state estimation algorithms and framework for discrete time fractional systems.Firstly,a fractional central difference Kalman filter is proposed for nonlinear dis-crete time fractional systems.Different with the existing fractional extended Kalman filters with Taylor's formula,in this paper,the central difference formula is utilized to linearize nonlinear functions as no derivatives are needed.Without sacrificing compu-tational complexity and estimation accuracy,this algorithm can estimate system state unbiasedly.Then,a maximum a posteriori principle based adaptive fractional central difference Kalman filter is derived.This method can estimate noise statistics and sys-tem state simultaneously.Theoretical analysis and simulation experiments verify the accuracy and effectiveness of the proposed method.Secondly,a fractional particle filter algorithm for non-Gaussian fractional systems is presented.The Monte Carlo random sampling method is used to sample the non-Gaussian distribution,which aims to approximate the non-Gaussian posterior probabil-ity distribution function.In addition,the prior probability density function is selected as the importance density function to impro've the efficiency of this algorithm,and the resampling procedure is applied at each iteration to reduce the particle degradation.Furthermore,for the Non-Gaussian noise and Gaussian noise cases,the effectiveness of FPF is verified via several examples.Finally,based on the Bayesian formula,the recursive fractional Bayesian filtering framework is derived.Firstly,a recursive Bayesian filter is presented from probabili-ty density perspective,which can uniformly describe the nonlinear fractional filtering problem.Then for a nonlinear fractional Gaussian system,a general fractional Gaus-sian filtering framework is investigated.Under this framework,the performance of four suboptimal fractional filtering algorithms,including time complexity and estima-tion accuracy,is analyzed.