Parameterization of kernels of the Volterra series for systems given by nonlinear differential equations

The presented article is devoted on an issue regarding to the transformation of nonlinear models of a certain class to the Volterra functional series. The new identification method based on analytical input and output of a system was developed. The key task of representing kernels was achieved by Frechet functional derivative. Explicit calculation of the functional derivative for the output was solved using mean-value theorem, while the sifting property of the Dirac function was used in order to solve derivative of input. Attention is given to the calculation of the second-order functional derivative. The procedure for adding linear kernels to the composition of a quadratic kernel is described. All techniques of the method to other components of the model are described in detail. The method's outcome is differential equation, which allows representation of kernels. An illustrative example of the transformation of the nonlinear differential Riccati equation is considered. The form of a subsystem consisting of linear and quadratic kernels for adding to a complex system is shown. Linear and quadratic kernels were parameterized using operational calculus within the example. The agreement of the obtained analytical results, with the frequency characteristics, which was obtained by the test signals method, is shown.