On stochastic system position correction by quantile criterion

Аuthors

Kibzun A. I.^1*, Khromova O. M.^2**

1. ,
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: kibzun@mail.ru
**e-mail: khromova-om@mail.ru

Abstract

The article is devoted to solving problem of stochastic system position correction by quantile criterion. The single-pulse correction case is considered in the article. It is required to determine the correction pulse which will provide system movement from one position to another in minimal time. This problem can be written as a two-stage stochastic programming problem. The strategy is determined on the first stage, and different random factors (interference) appear with the realization of the first stage strategy. The impact of random factors is partially compensated by the choice of the new strategy (the second stage strategy). Traditionally, the expectation value (average loss) is considered as the problem criterion in the two-stage stochastic programming problems. But the control accuracy which is guaranteed with a given probability is more appropriate criterion in the aircraft control problems. Mathematically, this criterion is described with the quantile function.
The two-stage stochastic programming problem with the normal distribution of the random factors is considered in the article. The feature of the problem statement is the kind of the loss function, namely its linearity both random factors and strategies. Thus the loss function is the bilinear function. It is assumed that the random factors have the normal distribution. But the class of the random factors distribution can be expanded. For example, the random factors may have the spherically symmetrical distribution. In this case only size of the confidence sphere and the kernel of probability measure are changes.
An algorithm to obtain a guarantees solutions is proposed in this article. This algorithm based on the confidence method and reducing the initial problem to the convex programming problem which is parameterized by a scalar parameter. This parameter is determined in the algorithm by dichotomy. The counting procedure of the polyhedral set probability measure is the main complexity of the proposed algorithm. The discretization of the normal distribution and a process of reducing the search of points are offered for the time decreased.

Keywords:

stochastic programming, two-stage problem, quantile criterion, normal distribution, convex programming

References

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