Inferring Dynamics of Discrete-time, Fractional-Order Control-affine Nonlinear Systems

In this paper, a system identification algorithm is proposed for control-affine, discrete-time fractional-order nonlinear systems. The proposed algorithm is a data-integrated framework that provides a mechanism for data generation and then uses this data to obtain the drift-vector, control-vector fields, and the fractional order of the system. The proposed algorithm includes two steps. In the first step, experiments are designed to generate the required data for dynamic inference. The second step utilizes the generated data in the first step to obtain the system dynamics. The memory-dependent property of fractional-order Grunwald-Letnikov difference operator is used to compute the fractional order of the system. Then, drift-vector and control-vector fields are reconstructed using orthonormal basis functions, and calculation of the coefficients is formulated as an optimization problem. Finally, simulation results are provided to illustrate the effectiveness of the proposed framework. Additionally, one of the methods for identifying integer-order systems is applied to the generated data by a fractional-order system, and the results are included to show the benefit of using a fractional-order model in long-range dependent processes.