Design Of Kalman Filters For Continuous-time Fractional-order Systems Containing Correlated Colored Noises And Unknown Parameters

Because the computer is used for the practical industrial control system to achieve the information collection in general,the state estimation and parameters identification are the prerequisite and guarantee to analyze the safe operation of the system based on the input and output measurement information containing noises.Although the state information required in the sampling signals can be measured directly,it is still necessary to design the observer to obtain effective state estimation information due to the noises disturbance.Therefore,fractional-order Kalman filters are mainly explored to solve the problems on state estimation and parameters identification for the continuous-time linear and nonlinear fractional-order system involving correlated fractional-order colored noises and unknown parameters in this paper,and several aspects are accomplished as follows:(1)For the continuous-time linear fractional-order system disturbed by correlated fractionalorder colored noises,fractional-order Kalman filters are designed to achieve state estimation for the fractional-order system.Using the method of Tustin generating function,the differential equation of the fractional-order system is discretized a difference equation.Applying the augmented vector method,the augmented state equation can be established by the state equation and the noise equation contained in the fractional-order system to solve colored noises problem.Fractional-order Kalman filters using Tustin generating function are proposed to improve the accuracy of state estimation for the fractional-order system.(2)For the continuous-time nonlinear fractional-order system with correlated fractionalorder colored noises and unknown parameters,extended fractional-order Kalman filters are presented to achieve state estimation and parameters identification for the investigated systems.Using the first-order Taylor expansion formula,the nonlinear functions of the fractional-order system are transformed into the linear functions.Applying the augmented vector method,the augmented state equation can be established by the state equation and the parameters equation contained in the fractional-order system to deal with the problem on unknown parameters.Extended fractional-order Kalman filters based on Tustin generating function and Gr ¨unwaldLetnikov difference are designed,which illustrates that the extended fractional-order Kalman filters based on Tustin generation function can obtain a satisfactory effect of state estimation and parameters identification.(3)For the continuous-time linear fractional-order system including unknown fractionalorder,adaptive extended fractional-order Kalman filters are proposed to achieve state estimation and order identification for the linear fractional-order system.Because the fractional-order of the system is unknown,the linear fractional-order system can be viewed as a nonlinear fractional-order system.According to the augmented vector method,the augmented state equation can be got by the state equation and the order equation for the fractional-order system to estimate effectively unknown order.The fractional-order system is discretized by Gr ¨unwaldLetnikov difference,and the nonlinear system is linearized by the first-order Taylor expansion formula.The adaptive extended fractional-order Kalman filter with initial value compensation and the adaptive extended fractional-order Kalman filter without initial value compensation based on Gr ¨unwald-Letnikov difference are provided to achieve the state and order estimation of the fractional-order system,and illustrate that the estimation error is smaller using the Kalman filter algorithm with initial value compensation.