Iterative Method for Online Fractional Order and Parameter Identification

This paper deals with fractional order and the parameter identification of a non-commensurable fractional order system using an iterative method consisting of two steps. The novelty is that the system needs not to be at rest and that an implementable algorithm is given using the Grünwald-Letnikov derivative. In the first step of the algorithm, the instrumental variable least-squares method identifies the parameters by means of given values of the fractional orders. To apply the second step, the system is interpreted as a non-linear equation with respect to fractional orders. After setting up a system of equations, one step of Newton's method is performed to improve the estimate of the fractional orders. The required Jacobian as well as a convergence proof of the applied Newton's method is given considering fractional order dependent parameters. Executing both steps iteratively yields the online identification of fractional orders and parameters. A numerical example demonstrating the efficacy of the proposed algorithm concludes the paper.