Technology of Cut-Glue Approximation Method for Modeling Strongly Nonlinear Multivariable Objects. Theoretical Bases and Prospects of Practical Application

摘要

The main difficulties of the mathematical models vehicles creation are defined by strongly nonlinearity of dependences which connect various variables their states and conditions of the movement environment. Most it belongs to aircrafts as aerodynamic interactions are characterized by essential nonlinearity up to discontinuity of variables and their derivatives. Creation process of these models is complicated by high-dimensionality, characteristic for the mechanical movement laws. Experimental creation of the mathematical models (MM) of such dependences is carried out by various mathematical methods of approximation of data. Universal remedies of the solution of the formulated task don't exist. Each of it possesses both benefits, and considerable shortcomings. In this regard the possibilities of a method creation of high-precision analytical approximations of the strongly nonlinear dependences using the analytical functions have been investigated. The Cut-Glue approximation (CGA) method for one-dimensional dependences is justified, and then this method is developed for approximation by functions of two arguments. In this method approximation is carried out by splitting of data into fragments similarly as in the method of piecewise approximation. Each data fragment is approximated by suitable function from which along its borders is cut out a fragment of this function. Cutting out of the function fragment is carried by multiplication its on the other nonlinear function with special characteristics. Properties of a nonlinear multiplier are such that in borders of a fragment of its value coincide with the approximating function, but are almost equal to zero in other range of definition. The received fragments are glued together in the united function which is model of the overall approximated dependence. The advantage of the Cut-Glue approximation method is differentiability of the mathematical models. It gives the chance to investigate functions analytically and to use their in MM of dynamics. In the offered article the possibility of application of a method for creation of nonlinear models of any dimension is proved and the ideology of accomplishment of all stages of "Cut-Glue" approximation is formulated. Tasks of optimum splitting experimental data into fragments, approximations of these fragments by analytical functions and their effective pasting in single in MM are formalized. The received result significantly expands a application area of a method and its opportunity in problems of mathematical modeling of dynamics of aircrafts and vehicles in general. Possibilities of optimal multidimensional "Cut-Glue" of approximation are illustrated by an example.