Design Of Kalman Filters For Fractional-order Systems Involving Colored Noises

The fractional-order Kalman filter is an effective robust state observer including input and output signal.The research results of the Kalman filter for integer-order systems are very rich,but the design of the Kalman filter for the fractional-order system is just beginning.Therefore,it is necessary to study the state estimation problem of fractional-order systems in theory and technology.Based on the existing research results,this paper further studies the state and parameter estimation of fractional-order systems with colored noise,and mainly accomplishes the following aspects.(1)The fractional-order Kalman filters for continuous-time linear fractional-order system are studied under the condition that the process noise and the measurement noise are colored noise,respectively.By the fractional-order average derivative,the linear fractional-order system equation with fractional-order colored noise can be discretized.We construct an augmented vector defined by the state and colored noise to deal with colored noises,establish the augmented equation of the fractional-order system,and propose an improved fractional-order Kalman filter for the fractional-order system with colored noise.In addition,the state and input information can be finitely truncated for the engineering implementation of the algorithm.This algorithm improves the accuracy of state estimation.(2)The fractional-order extended Kalman filters for continuous-time nonlinear fractionalorder systems are presented with fractional-order colored process and measurement noises,respectively.Firstly,the first-order Taylor expansion is used to linearize the nonlinear state equation.Then,based on Gr¨unwald-Letnikov(G-L)difference and fractional-order average derivative,nonlinear fractional-order systems with colored noise are discretized,and the fractionalorder extended Kalman filters are proposed.Three simulation examples show that the fractionalorder extended Kalman filter using fractional-order average derivative has better filtering effect on colored process or measurement noise.(3)The fractional-order adaptive extended Kalman filters for fractional-order systems involving colored process or measurement noises are studied under the condition of unknown parameters respectively.The continuous-time nonlinear fractional-order system equations are discretized by the method of G-L difference,and the nonlinear systems are linearized by the first-order Taylor expansion.By constructing the augmented vector defined by the state,the unknown parameter and the fractional-order colored noise,and establishing the related augmentation equations,the unknown parameter and the fractional-order colored noise existing in the system are performed.Simulation examples demonstrate that the adaptive extended Kalman filters using the G-L difference can effectively perform state and parameter estimation.