Obtaining Frequency-Domain Volterra Models From Port-Based Ordinary Differential Equations

Abstract

A frequency-domain Volterra model (FVM) is a nonlinear representation obtained when the multivariable Laplace transform is applied to a sum of multidimensional convolution integrals of increasing order. Two classes of FVMs can be identified. The first class of FVM is the Volterra transfer function (VTF) which has been recognized as a useful tool for nonlinear systems modeling and simulation. The second class of FVM is the Volterra dynamic model (VDM) which has been used in the modular assembly and condensation of port-based nonlinear models. Since physical nonlinear systems are frequently modeled using ordinary differential equations (ODEs), it is of significant value to derive their equivalent FVM representations from a corresponding ODE. Even though methods to obtain VTFs for multiple-input, multiple-output (MIMO) nonlinear ODEs are available, a general procedure to obtain the two classes of FVMs does not exist. In this work, a methodology to obtain the two classes of FVMs from port-based nonlinear ODEs is explained. Two cases are shown. In the first case, the ODEs do not include cross product nonlinearities. In the second case, cross products are included. An example is presented to clarify the idea, and the time response obtained from the nonlinear ODE model is compared to its corresponding third order VTF and its linearized model.