Full adaptive Kalman filters for nonlinear fractional-order systems containing unknown parameters and fractional-orders

In this study, full adaptive Kalman filters are designed for continuous-time nonlinear fractional-order systems (FOSs) containing unknown parameters and fractional-orders. First, the estimated FOS is discretized using the Grünwald–Letnikov difference method to transform the fractional-order differential equation into a difference equation. Then, in terms of the nonlinear function contained in the investigated system, the Taylor expansion formula is adopted to linearize the discretized equation. Based on the method of augmented vector, an augmented state equation is established by state equations, unknown parameter equations, and fractional-order equation to achieve the state estimation. Besides, the sigmoid function is brought to ensure that the estimation of the fractional-order is performed in a suitable range. Because the covariance matrices of noises are difficult to be measured in physical systems, we also concern the problems on the state estimation, parameter estimation, and fractional-order estimation under the cases that the covariance matrix of process noise is unknown or the covariance matrix of measurement noise is unknown. Considering that the initial value can produce an error of state estimation and parameter identification for the FOS defined under the Caputo sense, the augmented vector method is used to achieve the initial value compensation. Finally, the effectiveness of the proposed algorithms is validated by four examples.