Parameter Identification of Nonlinear Systems Using Generalized Associated Linear Equations (GALEs)

For a weakly nonlinear system described by Volterra series, if the nonlinear differential equation (NDE) model of the system is unique, the nonlinear auto-regressive with exogenous input (NARX) model identified based on the input and output data of that system is not unique. Fortunately, the NDE model and the NARX model of the same nonlinear system have unique frequency domain representations, e.g., generalized frequency response functions (GFRFs), so these two models also have the same nonlinear output response spectrum. And the generalized associated linear equations (GALEs) theory can be used to directly calculate the arbitrary order nonlinear output response of the system according to the parameters of the NDE model or the NARX model. Therefore, this paper proposes a parameter identification method of the NDE model for the nonlinear system based on GALEs. Firstly, the NARX model of the system is identified according to the input and output data generated by the NDE model of the system with unknown parameters. Secondly, the GALEs of the NDE model containing the unknown parameters are determined. Finally, the unknown parameters in the NDE model are identified based on the principle that the NDE model and the NARX model of the same system have the same nonlinear output response spectrum. The method is convenient and effective, and provides a new idea for parameter identification of NDE models of nonlinear systems.