Modeling of nonlinear dynamic and stationary systems based on Volterra integro-functional series and various classes of quadrature formulas
The article deals with the analysis of nonlinear dynamic and stationary systems based on Volterra integro-functional series and various classes of quadrature formulas. A mathematical model of the input-output type is used, which does not take into account the specific physical nature of the dynamic process, which is commonly called a black box. The methods of the article are applicable to the main variants of the Volterra integral-functional decomposition, including for the case of stationary dynamical systems, a vector input signal. An example of an optimization problem based on the considered integrative series is given. It is noted that when analyzing and optimizing nonlinear dynamical systems by the method of integro-functional series, the problem of calculating multidimensional integrals may arise. The article considers the application of the combined method based on the Volterra integrative series and grid methods for solving the corresponding one -- and multidimensional integral equations for the analysis of nonlinear dynamic and stationary systems. This article considers the case when a certain set of implementations of input and output signals is known, which can be in principle random processes. According to these data, the kernels are found in the decomposition based on the solution of the corresponding linear multidimensional Fredholm integral equation of the first kind. The corresponding problem belongs to the incorrectly posed ones and the regularization method according to A.N. Tikhonov is used to solve it. The article proposes to apply the quasi Monte-Carlo method, characterized by satisfactory convergence, in this problem in the case of large dimensions. The computational qualities in the considered problem of a semi-statistical method for solving integral equations of large dimension, the quasi Monte-Carlo method, the method of central rectangles (cells) and the quadrature formulas of Gauss-Legendre are studied. The approaches under consideration allow us to expand the range of problems to be solved in the theory of analysis and optimization of systems, since methods are proposed that are practically acceptable for large dimensions of integral equations in conditions of limited information about the system.