Iterative Method for Online Fractional Order and Parameter Identification

This paper deals with fractional order and theparameter identification of a non-commensurable fractionalorder system using an iterative method consisting of two steps.The novelty is that the system needs not to be at rest and that animplementable algorithm is given using the Gr ̈unwald-Letnikovderivative. In the first step of the algorithm, the instrumentalvariable least-squares method identifies the parameters bymeans of given values of the fractional orders. To apply thesecond step, the system is interpreted as a non-linear equationwith respect to fractional orders. After setting up a system ofequations, one step of Newton’s method is performed to improvethe estimate of the fractional orders. The required Jacobian aswell as a convergence proof of the applied Newton’s methodis given considering fractional order dependent parameters.Executing both steps iteratively yields the online identificationof fractional orders and parameters. A numerical exampledemonstrating the efficacy of the proposed algorithm concludesthe paper.