Formal Characterization and Optimization of Algorithm for the Modelling of Strongly Nonlinear Dependencies Using the Method 'Cut-Glue' Approximation of Experimental Data

摘要

Mathematical modeling of technical objects is most frequently connected with mathematical processing of experimental data. The obtained pointlike dependencies of output variables on input ones are often strongly nonlinear, piecewise, and sometimes discontinuous. Approximation of these dependencies using polynomial resolution and spline-functions is problematic and may cause low accuracy. A radically new solution to this problem was suggested in a number of previous works. The method is based on partitioning of experimental dependencies into patches, approximation of each patch by analytic functions, multiplicative cutting of fragments from each function along the patch border and additive gluing of the fragments into a single function -- namely the model of approximated dependence. The analytic properties of this approximating glued function appear to be the major distinguishing feature and advantage of the method. This property allows for analytical research of the model and application to vehicle dynamics modeling. In relation to the described technology of data processing (cutting of locally approximated fragments and their subsequent additive compound) this method is called Cut-glue approximation (CGA). Published scientific papers already demonstrated that it can be used in descriptions of either one-dimensional or two-dimensional experimental dependencies. The research showed that the method CGA can also be applied for dependencies of higher order. However, the efficiency of the proposed solution has been proved only in theory and confirmed practically for one-dimensional and two-dimensional dependencies. For widespread introduction of CGA into research and experimental modeling practice, it is necessary to develop, justify and explore some local techniques for implementing its main stages. The present paper dwells on investigation of possible approaches to the implementation of these stages and the development of appropriate software. Some examples illustrated how the encouraging results were obtained, while using heuristic methods. It is shown, that the algorithms of ant colonies (ACA) effectively solve the problem of fragmentation. Evolutionary genetic algorithms (EGA), modified by the task, allow to improve the effectiveness of the selected fragments approximation. Algorithms of swarming particles (ASP) significantly reduce inaccuracy of the cut-glue basic procedure. The actual efficiency of the researched heuristic algorithms is illustrated by the solution of specific objectives of data processing and by numerical results of their application.