Outer set polynomial modeling in constructing an optimal type aircraft system

Аuthors

Akimov E. N.^*, Balyk V. M.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: K608@mai.ru
**e-mail: balikv@gmail.com

Abstract

The article considers a method of statistical synthesis of a distribution function of targets between various types of aircraft. It presents statistical samples, describing an optimal type aircraft system in large, as well as samples describing the function of target distribution.

The process of building of design and functional bindings that model aircraft systems at large starts from forming a statistical sample, which input data represents the characteristics of a goal set, while elementary functions of targets distribution by aircraft types are taken as intermediate characteristics. We accept the values of criterion of optimality as output data. This work assumes as criterion of optimality the cost of aircraft system. Intermediate characteristics have special meaningfulness when modeling aircraft systems.

According to the principle of mathematical model self-organization, at successive complexity increase of a model (in the course of the transition from a linear model to a square one and further to higher degree models) all external criteria pass through their minimum. It gives the possibility to obtain a model of optimal complexity, unique for each criterion. In self-organizing theory of complex systems models all basic algorithms for building models of optimal complexity are based on grouped account of arguments. Combinatorial algorithms appear effective for problem order no more than the specified sample size. For problems of higher complexity, such as designing systems for an optimal type aircraft, the multi-row algorithms become more effective. The perspective of multi-row algorithms should be noted, since, in principle, they are the prototypes of genetic algorithms, which allow solving simulation problems of with dimension of several thousand variables. However, whichever the algorithm is, it does not allow going beyond the framework of the specified class of basic functions. These algorithms increase only the complexity of the model within a specified basis.

With statistical synthesis employed in the paper the modeling algorithms are built in the way that provides the possibility in principle to obtain the models involving various basic functions.

This corresponds to the basic statistical synthesis concept, according to which the output data of the initial statistical sample is modified in a certain manner in the process of building of a mathematical model. This adjustment is carried out according to the conditions and requirements that the formed model should meet. Thus, at each stage of the mathematical model building we form the statistical sample inherent to it, and the inherent optimal system of basic functions corresponds to each sample.

The article presents the analytical models of the targets distribution function, which represent an approximation of statistics reported in terms of trigonometric polynomials.

The article considers the operation of statistical sampling reduction, which reduces the initial n — dimensional sample to n one-dimensional samples, and operation of inversion allowing obtaining the inverse sampling required for the formation of inverse functions. Based on these operations, build one-dimensional functional relationships between characteristic functions and outer target set characteristics presented in the form of trigonometric polynomials. The paper presents a simple, but at the same time effective way to meet the statistical functional limitations. This method is based on statistical sampling regularizing. It consists in replacing or eliminating fragments of statistical sampling, which do not agree with the specified limitations.

Keywords:

statistical synthesis, distribution function, statistical sampling, basic functions, reduction, aircraft system, optimal type

References

1. Piyavskii S.A., Brusov V.S., Khvilon E.A. Optimizatsiya parametrov mnogotselevykh letatel’nykh apparatov (Multipurpose aircraft parameters optimization), Moscow, Mashinostroenie, 1974, 168 p.

2. Balyk V.M., Vedenkov K.V., Kulakova R.D. Vestnik Moskovskogo aviatsionnogo instituta, 2014, vol. 21, no. 4, pp. 49-58.

3. Balyk V.M. Statisticheskii sintez proektnykh reshenii prirazrabotke slozhnykh system (Staistical synthesis of design decisions when developing complex systems), Moscow, MAI, 2011, 280 p.

4. Ivakhnenko A.G. Induktivnyi metod samoorganizatsii modelei slozhnykh system (Inductive method of selforganization of complex systems models), Kiev, Naukova dumka, 1981, 296 p.

5. Balyk V.M., Kulakova R.D., Khesin L.B. Polet, 2010, no. 9, pp. 32-41.

6. Balyk V.M., Vedenkov K.V., Kulakova R.D. Vestnik Moskovskogo aviatsionnogo instituta, 2014, vol. 21, no. 4, pp. 49-59.

7. Balyk V.M., Severtsev N.A. Voprosy teorii bezopasnosti I ustoichivosti sistem, Moscow, VTs RAN, 2015, no. 17, pp. 128-141.

8. Konstantinov M.S., Petukhov V.G. Vestnik Moskovskogo aviatsionnogo instituta, 2008, vol. 15, no. 1, pp. 8-16.