Fractional-order Kalman Filters For Continuous-time Fractional-order Systems Involving Uncorrelated And Correlated Noises

Fractional-order calculus is widely used in control theory systems.Compared with the traditional integral-order calculus,fractional-order systems,which is an effective tool for describing nonlinear dynamics,have better memory properties.The modeling meth-ods and theory of fractional-order calculus have been successfully used in many areas,such as high energy physics,abnormal diffusion,system control,rheology,geophysics,biomedical engineering,economics.The introduction of fractional-order operators brings the flexibility of the controller designs.Considering the memory property of fractional-order differential operators,a lot of historical data of input and measurement signals need to be provided during the process of state estimation of continuous time fractional-order systems.The processing of historical information,which is different from integer-order systems,is the key problem need to be solved.In addition,due to the non-localized nature of fractional-order calculus,the computational and storage quantities of the nu-merical simulation of the fractional-order derivative increase with the increase of problem size.Some numerical mothods that are effective for calculating the integer-order equation are also completely invalid for the fractional-order equations.Therefore,the purpose of this paper is to design a new method for state estimation in fractional-order systems.Due to the process and measurement noises existing in the measurements,the mea-surement signals must be filtered to obtain the effective state estimation.Kalman filter is an effective robust state observer containing input and output signals and widely used in algorithm estimation.Kalman filter is an optimized autoregressive data processing algo-rithm,and its filtering standard is the minimum mean square error.In the time domain,the algorithm uses the state space method to design the filter,which can estimate the stochastic process of multidimensional systems and non-stationary systems.The Kalman filter has a wide range of applications due to its own advantages,such as recursive oper-ation,calculation simple,adaptive,capable of optimal estimation of nonlinear fractional dynamic systems under random disturbances.This paper mainly focuses on the design of fractional-order Kalman filter involving correlated and uncorrelated process noise and measurement noise for linear and nonlinear fractional-order systems.Traditional state estimation method was discretized fractional-order system by us-ing Grunwald-Letnikov(G-L)difference and Tustin generating function,which can design the fractional-order Kalman filter.The G-L difference can effectively estimate the state information when the sampling period is shorter.However,the state observation can be invalid if the sampling period increase.Compared with G-L difference,Tustin generating function can improve the estimate accuracy but takes a long computation time.Hence,the purpose of this paper is to design a new fractional-order Kalman filter which can improve the state estimation accuracy,increase system stability and save calculation time for fractional-order systems.This paper introduces the concept of fractional-order av-erage derivative operator and fractional-order systems can be discreted according to the fractional-order average derivative operator.Based on the linear and nonlinear fractional-order system involving uncorrelated and correlated noises,this paper design fractional-order Kalman filters for different situation and give five recursive formulas for the Kalman filters.This paper design simulation experiments for fractional-order system involving un-correlated and correlated noises.Give a comparison of state estimation accuracy based on the method of G-L difference and fractional-order average derivative.Compare simulation effects based on G-L difference and fractional-order average derivative method when the system sampling period increases.Give a comparison of running times based on Tustin generating function and fractional-order average derivative method.Simulation exper-iments verify the effectiveness of this method for linear and nonlinear fractional-order system.The state estimation method proposed in this paper can improve the state esti-mation accuracy,increase system stability and save calculation time,which can verify the feasibility and effectiveness of the fractional-order average derivative method proposed in this paper.