Extended Kalman filters for fractional‐order nonlinear continuous‐time systems containing unknown parameters with correlated colored noises

By using the Grunwald-Letnikov (G-L) difference method and the Tustin generating function method, this study presents extended Kalman filters to achieve satisfactory state estimation for fractional-order nonlinear continuous-time systems that containing some unknown parameters with the correlated fractional-order colored noises. Based on the G-L difference method and the Tustin generating function method, the difference equations corresponding to fractional-order nonlinear continuous-time systems are constructed respectively. The first-order Taylor expansion is used to linearize the nonlinear functions in the estimated system, which provides the system model for extended Kalman filters. Using the augmented vector method, the unknown parameters are regarded as new state vectors, and the augmented difference equation is constructed. Based on the augmented difference equation, extended Kalman filters are designed to estimate the state of fractional-order nonlinear systems with process noise as fractional-order colored noise or measurement noise as fractional-order colored noise. Meanwhile, the extended Kalman filters proposed in this paper can also estimate the unknown parameters effectively. Finally, the effectiveness of the proposed extended Kalman filters is validated in simulation with two examples.