USING THE THEORY OF MARTINGALES TO PROVE THE SOUNDNESS OF THE ESTIMATES OF THE PARAMETERS OF LINEAR DYNAMIC SYSTEMS

V. I. Sushchuk-Slyusarenko, N. A. Rybachok, L. M. Oleshchenko

Abstract


The article is devoted to analytical methods of research of filtration algorithms in conditions of a priori uncertainty of information on statistical characteristics of state noise and measurement in linear dynamic systems.  For simple objects it is possible to apply simple evaluation algorithms.  In this case, the estimate of the matrix of the dynamics coincides with probability 1 to the true value, and the Kalman filter constructed on such an algorithm gives an estimate which also coincides with the probability of 1 to the estimation of the true Kalman filter.  To prove the validity of the estimates, the theory of martingales was applied.  Martingales and semimartingales form an important class of processes, which generalizes a class of processes with independent increments.  There is a special method for the study of random processes. But in practice, the condition that all components of the matrix of the dynamics are those that can be observed gives the limit to the use of this method.  The proposed technique will allow to extend the method of obtaining estimates of the parameters of linear dynamic systems in the case of an arbitrary dynamics matrix.

Keywords


Algorithm; filtration; matrix; probability; martingale; linear dynamical systems

References


W. Anderson, et al., “Consistent Estimates of the Parameters of a Liner System, Ann. Math. Statist., 40, 1969, pp. 2064–2075.

D. Grop, Methods for identifying systems, Moscow: Mir, 1979. (in Russian)

M. Zgurovsky, V. Podladchikov, Analytic methods of Kalman filtration for systems with a priori uncertainty, Kiev: Science opinion, 1995. (in Russian)

Р. Siminelakis, “Martingales and Stopping Times Use of martingales in obtaining bounds and analyzing al-gorithms”, http://www.corelab.ece.ntua.gr/ courses/rand-alg/slides/Martingales-topping_Times.pdf

“Martingale”, Encyclopedia of Mathematics. https://www.encyclo-pediaofmath.org /index.php /Martingale


Full Text: PDF

Refbacks

  • There are currently no refbacks.


Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.