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The International Conference on Computational Mathematics


Statistic modeling and Monte Carlo methods

Algorithm of statistical simulation of dynamic systems with separated time

Averina T.A.

ICM MG SD RAS (Novosibirsk)

Many models of dynamic systems in various areas of science (including automatic control) are described by stochastic differential equations (SDEs). The term automatic systems with a sudden random changed structure (or random-structure system) appeared in the 60s. These are dynamic systems, which on random time intervals are described by different SDEs [1].

In conditions of the limited resources for processing the information, there is a necessity of their rational use in complex control systems. One of such ways is a consecutive use of these devices for alternate service of objects. Thus, we come to the separated time control. In some cases, systems with separated time control can be described as systems with distributed change of structure.

Approximate methods of the numerical analysis of systems with a separated time are:

1) integration of the generalized Focker - Planck Ц Kolmogorov equation - . a second order partial differential equation for the distribution density of the solution;

2) a method of two-moment parametric approximation of unknown densities of conditional distribution by the known functions, dependent on the first two moments;

3) a method of statistical simulation.

Difficulties, arising in solving problems by the first type method are associated with complicated computational procedures for solving partial differential equations. A serious disadvantage of methods of the second type is the complexity of obtaining approximation estimations and the fact that changing the original model causes essential changes in the equations for probabilistic characteristics.

In this paper, the algorithm of statistical simulation for the numerical analysis of the systems with separated time autonomous control is proposed. This algorithm is based on numerical methods for solution of the SDEs [2] and Monte Carlo methods [3]. The main features of the algorithm are universality and the possibility to calculate the estimates of different probabilistic characteristics of a solution (including the distribution density).

The differential equations are derived for unconditional and weighed moments in the case of linear autonomous control with additive noise, with exponential transitions and exact reconstruction. The results obtained are used as a test example for verifying work of the algorithm.

The problem of analysis of the influence of priority on the quality of control of an object was analytically solved and, numerically, in the case of two controlled objects. The results obtained enable us to draw a conclusion, that application of numerical methods of the SDEs has many advantages.


1. Kazakov I.YE., Artemiev V.M., Bukhalev V.A. Analysis of systems with random structure. Moscow, Nauka,, 1993 (272 p.) (in Russian).

2. Artemiev S.S., Averina T.A. Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations. VSP, Utrecht, The Netherlands, 1997 (176 p.).

3. Ermakov S.M., Mikhailov G.A. Statistical Modeling. Moscow, Nauka, 1982 (320 p.).(in Russian).

* Supported by scientific program "Russian Universities Ц basic research", Grant YP.04.01.34; program "Leading scientific schools", grant HIII 1271.2003.1; Russian Foundation for Basic Research, Grant 02-01-01178.

Note. Abstracts are published in author's edition

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