Design On Cubature Kalman Filters For Continuous-time Nonlinear Fractional-order Systems

In the actual industrial control system,information collection is usually realized by computer.Therefore,it is the premise and guarantee to analyze the safe operation of the system to use the input and output information containing noise for state estimation and parameter identification,it is still necessary to design observers to obtain effective state estimation information.Cubature Kalman filter is a state observer for state estimation in the case of noise and disturbance existing.It is widely used not only for state estimation of integer-order system,but also for state information estimation of fractional-order system.Cubature Kalman filter can effectively solve the disturbance problem of nonlinear function and the state estimation problem of nonlinear fractional-order system with noise.The main contents of this paper are as follows:For the continuous-time nonlinear fractional-order system with uncorrelated and correlated noises,a hybrid extended-cubature Kalman filter is designed to estimate the state of the fractional-order system.The fractional-order average derivative method is used to discretize the fractional-order system and a difference equation is obtained.The nonlinear functions in the system description are processed by extended Kalman filter and cubature Kalman filter.The first-order Taylor expansion is used to estimate the nonlinear functions at the current time.Based on the third-order spherical-radial rule,cubature points are used to represent the nonlinear functions in the state equation and output equation.Hybrid extended-cubature Kalman filters with uncorrelated and correlated noises are proposed.The results show that the proposed algorithm has higher state estimation accuracy than the fractional-order extended Kalman filter and cubature Kalman filter.Adaptive extended Kalman filter and adaptive cubature Kalman filter are proposed for continuous-time nonlinear fractional-order systems with unknown parameters and fractional-order by using the Gr¨unwald-letnikov difference.The first-order Taylor expansion method is used to deal with nonlinear functions,and the adaptive extended Kalman filter is designed.Based on the third-order spherical-radial rule,the adaptive cubature Kalman filter is proposed by using cubature points to represent nonlinear functions.An augmented vector composed of state vector,unknown fractional-order and unknown parameters is constructed,and the corresponding augmented equation is established to solve the problems of state estimation,fractional-order and parameter identification.It is shown that the proposed adaptive cubature Kalman filter is better than the adaptive extended Kalman filter in aspects of state estimation,fractional-order and parameter identification,but the calculation time of the adaptive extended Kalman filter is shorter.For continuous-time nonlinear fractional-order system,the cubature Kalman filter with initial value compensation and the hybrid extended-cubature Kalman filter to reduce the impact of initial value are designed to realize the state estimation of fractionalorder system.Through the augmented vector method,the augmented state equation constructed by fractional-order system state equation and initial value state equation is established.By using the Gr¨unwald-letnikov difference method,the fractional-order system is discretized,and the cubature Kalman filter with initial value compensation is designed to effectively estimate the state information of the fractional-order system.By using the model transformation and combining the first-order Taylor expansion and the third-order spherical-radial rule,the fractional-order hybrid extended-cubature Kalman filter is designed to reduce the impact of initial value.It is shown that the estimation error of the cubature Kalman filter and the hybrid extended-cubature Kalman filter considering the impact of initial value is smaller than that of the cubature Kalman filter without initial value compensation.