Quadratic and cubic Volterra polynomials: identification and application

Abstract

Volterra kernels identification is the main problem in constructing an input-output type mathematical model of nonlinear dynamical system by a Volterra polynomial of Nth order. Currently, various algorithms for solving this problem are proposed. Usually, it is assumed that the decomposition of the dynamical system response y(t) into components is preliminarily performed. Each of components is due to the influence of the concrete integral term. In general, the separation problem is invariant with respect to a particular family of test actions, and the choice of amplitudes of the test signals used to identify the Volterra kernels is related to the necessary conditions for the solvability of the corresponding multidimensional integral equations in special classes of functions. In the present paper, existence theorems for solutions of two-dimensional and three-dimensional Volterra integral equations of the first kind are given. This result is obtained in terms of the amplitudes of the test signals. This will allow us to remove the arbitrariness in the choice of amplitudes in construction of the quadratic and cubic Volterra polynomials in the case when external action x(t) = (x1(t), x2(t))T is a vector function of time. Illustrative calculations are given through the dynamic reference systems.