A software for the probabilistic analysis of the systems with the piece-wise linear structure

Аuthors

Kan Y. S.^*, Travin A. A.^**

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

*e-mail: yu_kan@mail.ru
**e-mail: dron-mail2001@mail.ru

Abstract

The evaluation problems for the probabilistic and quantile criteria arise when estimating the terminal precision of the aircraft control under the random disturbances. The probabilistic criterion reflects usually the probability of attaining a control goal and the quantile one is a making decision result (the control precision) guaranteed with a given probability. The known evaluation techniques for these criteria are iterative numerical procedures having no stopping rules. For this reason they cannot allow us to obtain a result with the given precision. The paper presents more exact techniques for evaluating the cumulative distribution function and quantiles of the random variable which is piece-wise linear with respect to the Gaussian random vector.
The quantile evaluation problem is reduced to searching a root of equation for the cumulative distribution function. Such an equation can be solved with the given precision if the cumulative distribution function is approximated exactly by its two-sided bounds. The bounds are constructed by lower and upper approximations for the probability that the Gaussian vector belongs to the compact convex polyhedron.
The bounds on the probability are generated by special numerical integration procedures for the Gaussian probability density function over the compact convex polyhedron.
The efficiency of the presented techniques is demonstrated by a two-dimensional example.
The evaluation techniques are implemented as a software.

Keywords:

probability, the quantile, linear inequalities, convex polyhedrons

References

1. Malishev V.V., Kibzun A.I. Analys I sintez visokotochnogo upravleniya letatelnimi apparatami (Analysis and synthesis of precision aircraft control), Moscow, Mashinostroenie, 1987, 302 p.
2. Kuzmin V.P., Yaroshevskiy V.A. Ocenka predelnih otkloneniy fazovih koordinat dinamicheskoy systemi pri sluchaynih vozmusheniyah (Score limit deviations of phase coordinates of the dynamical system with random perturbations), Moscow, Fizmatlit, 1995, 296 p.
3. Kibzun A.I., Kan Yu.S. Zadachi stohasticheskogo programmirovaniya s veroyatnostnimi kriteriyami (Stochastic programming problem with probabilistic criteria), Moscow, Fizmatlit, 2009, 372 p.
4. Kan Yu.S., Travin A.A. Avtomatika I telemehanika, 2013, no. 6, pp. 57-65.
5. Naumov A.V., Ivanov C.V. Avtomatika I telemehanika, 2011, no. 2, pp. 142-158.
6. Ventcel E. S. Teoriya veroyatnostey (Probability theory), Moscow, Vishaya shkola, 1999, 576 p.

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