Search

Search Results

• COMPUTER METHOD FOR ANALYZING THE STABILITY OF DIFFERENTIAL EQUATIONS SYSTEMS

  S.G. Bulanov

  2021-02-13

  Abstract ▼

  This article proposes approach to the stability analysis in the sense of Lyapunov for systems
  of ordinary differential equations. The approach is based on stability criteria in the form of necessary
  and sufficient conditions obtained on the basis of matrix multiplicative transformations of
  difference schemes of numerical integration. The matrix, multiplicative form of criteria implies the
  possibility of their cyclic program implementation in the form of a cycle by the number of multipliers.
  It is mathematically proved that the replacement of an infinite matrix product with a finite
  product, which is necessary in the programming process, preserves the certainty of the stability
  analysis according to the proposed criteria. The dependence of the certainty of computer stability
  analysis on the error of the difference solution of a system of ordinary differential equations is
  investigated. In order to improve the accuracy of difference approximations of the solution and
  linearization of the system, the method of variable piecewise polynomial approximation of the
  solution is used. The method gives continuous and continuously differentiable approximations of
  the desired solutions over the entire integration interval. The required approximations are obtained
  on the basis of a piecewise-polynomial approximation by Newtonian interpolation polynomials
  converted to the form of a polynomial with numerical coefficients. Computer approximation
  of integrands increases the accuracy of integral calculation. This increases the accuracy of calculating
  expressions in each multiplier of matrix products, and consequently increases the certainty
  of analysis using stability criteria. A program and numerical experiment was conducted to analyze
  the stability of the Lorentz system under given initial conditions and parameters changes. Based
  on the numerical data obtained during the experiment, the stability nature of the system under
  study is unambiguously established. In General, the proposed approach makes it possible to perform
  a stability analysis arbitrary systems of ordinary differential equations in real time mode
  without access to methods of the qualitative theory of differential equations and systems of computer
  mathematics.

• STABILITY ANALYSIS OF RIGID SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

  S. G. Bulanov

  2021-08-11

  Abstract ▼

  A method for analyzing stability in the sense of Lyapunov for systems of ordinary differential
  equations is proposed. The method is based on stability criteria in the form of necessary and sufficient
  conditions obtained on the basis of vector-matrix transformations of difference numerical
  integration schemes. The varieties of criteria in multiplicative, additive and matrix form are presented.
  The design of the criteria implies the possibility of their programmatic realization. To increase
  the reliability of the stability analysis, the approximations of the solution included in the
  construction of the criteria are based on piecewise interpolation approximation by Lagrange polynomials
  converted to a form with numerical coefficients. A programming and numerical experiment
  is carried out to analyze the stability of the Belousov-Jabotinsky periodic reaction model,
  which belongs to the class of rigid systems, under given initial conditions. The analysis is carried
  out on the basis of the presented criteria and the results of the program clearly determine the nature
  of the stability in real time. Based on the results of the experiment, it can be argued that replacing
  the difference approximations of the solution with piecewise interpolation approximations
  increases the reliability of the stability analysis, reduces the study time, and makes it possible to
  determine the asymptotic properties of the solution. In general, the proposed approach is an alternative
  to the methods of the qualitative theory of differential equations and makes it possible to
  reliably determine the stability of rigid systems of ordinary differential equations in real time.

• INVESTIGATION OF THE IMPACT OF COMPLEX FIRE CONDITIONS ON THE QUALITY OF SURVEILLANCE AND FLIGHT SAFETY OF UAVS

  M.I. Mokrova

  2021-04-04

  Abstract ▼

  Aviation monitoring of fires with the help of unmanned aerial vehicles (UAVs), in particular,
  forest ones, during which the search for various objects of interest is carried out: people, cars,
  etc., is one of the most effective measures to reduce the level of possible losses. In this paper, we
  consider approaches to the formation of algorithms for processing and improving images obtained
  in the process of monitoring the fire situation, based on the use of neural networks, as well as
  image filtering algorithms, in order to search for various objects of interest. Fire monitoring using
  an UAV is a two-criteria task: there is a need to protect the device from the thermal effects of the
  fire as much as possible, as well as to maximize the observability, which can be achieved by reducing
  the altitude of the flight. This paper presents the empirical models developed by the authors for
  the flight safety of an unmanned aerial vehicle and the observability of objects of interest in the
  process of monitoring the fire situation. The proposed models allow us to take into account the
  features of the monitoring conditions, such as the priority of detecting the object of interest to the
  security of the reconnaissance vehicle itself, air humidity, terrain and type of terrain, time of day,
  and so on. An example of the application of the contrast model is considered on the example of the
  search and detection of the "letter"label. On the basis of the conducted experiment on the recognition
  of the mark in the smoke, the analysis of the proposed models is carried out, the quantitative
  results are given. The paper describes the criteria for the optimal choice of the altitude of the
  flight of the device over the observed scene, which are formed on the basis of the base of expert
  assessments, as well as the proposed models of the observability and safety of the UAV flight. Depending
  on the target search task, the optimality criterion for choosing the UAV flight altitude
  over the observed scene may vary.