Study On The Estimation Problem Of Linear Discrete Time Fractional-order System

Because fractional-order systems are more general than integer-order systems,research on fractional-order systems and related challenges has drawn increasing attention from control researchers in recent years.Although there have been notable advances in the study of Kalman filtering for linear discrete-time fractional-order systems,there is always room for improvement.For example,research into other related issues like white noise estimation is still needed.However,there hasn’t been a lot of research done on linear discrete-time fractional-order systems.In this essay,we first alleviate the white noise estimation problem for linear discrete-time fractional-order systems and then consider the system’s unknown disturbance.Based on the Krein spatial linear estimation theory,we solve the H_∞ state and disturbance joint one-step and multistep smoothing estimation problem for linear discrete-time fractional-order systems.Here is the main piece of work:Firstly,the estimation of process white noise and observation white noise for linear discrete-time fractional-order systems is discussed.First,the linear discrete-time fractional-order system is converted into a class of linear discrete delay systems by defining discrete fractional-order difference with the Grünwald-Letnikov operator.The obtained time delay system’s delay component,however,differs from the general time delay system in that it uses the cumulative sum from time 0 to the previous time.Then,a suboptimal one-step predictor for linear discrete-time fractional-order systems is created using the orthogonal projection theorem.Because of the accumulation and delay in the system,the obtained Riccati equation differs from the classical Kalman filter in that it includes an additional cumulative sum term for the state error variance.Next,utilizing the innovation analysis technology,the filter estimator and fixed-lag smoothing estimator of process white noise and observation white noise are provided.To demonstrate the usefulness of the process white noise estimator and the observation white noise estimator,two simulation examples are given in the last section.Secondly,the H_∞ state and disturbance joint one-step smoothing estimation problem for linear discrete-time fractional-order systems is discussed.The classic Kalman filter seeks to reduce the variance of the estimation error;however,due to the uncertainty of the system model and the statistical properties of noise,its application is restricted.The mentioned issues can be efficiently fixed using the H_∞ filter.The relationship between the Krein space Kalman filter and the H_∞ filter is established,and the necessary and sufficient conditions for the existence of the H_∞ state and disturbance joint one-step smoothing estimator are determined based on the definition of the H_∞performance index and state transition matrix of the linear discrete-time fractional-order system.The H_∞ state and disturbance joint one-step smoothing estimator based on the piecewise Riccati equation is provided using the innovation analysis technique.A simulation example is provided in order to confirm the efficiency of the suggested approach.Thirdly,the H_∞ state and disturbance joint multi-step smoothing estimation problem for linear discrete-time fractional-order systems is discussed.The research on the Kalman filtering algorithm for linear discrete-time fractional-order systems has made some progress,but the study of H_∞ estimation for these systems has not gotten significant attention.In this section,the corresponding relationship between the design problem of H_∞ state and disturbance joint multi-step smoothing estimator and Krein space Kalman filter is established by equating the H_∞ performance index to solve a class of problems of indeterminate quadratic form greater than zero and using the newly established state transition matrix of fractional-order system.The product uncertainty and stable point theory in Krein space are then used to derive the sufficient and necessary criteria for the existence of the H_∞ state and disturbance joint multi-step smoothing estimator.In addition,the innovation analysis technique is used to give a recursive methodology of the innovation analysis technique joint multi-step smoothing estimator based on the piecewise Riccati equation.A simulation example is provided in order to confirm the efficiency of the suggested approach.