Problems of Information Transmission, Vol. 37, No. 4, 2001, pp. 365–379. Translated from Problemy Peredachi Informatsii, No. 4, 2001, pp. 97–111.
Original Russian Text Copyright
2001 by Tertychnyi − Dauri.
METHODS OF SIGNAL PROCESSING
Parametric Filtering: Optimal Synthesis
on a Nonlinear Dynamic Variety
V. Yu. Tertychnyi-Dauri
Received January 10, 2000; in ﬁnal form, June 19, 2001
Abstract—We analyze the optimal parametric ﬁltering problem for a wide class of nonlinear
stochastic systems. Much attention is paid to validation of formed models of observation and
estimation since derivation of equations that determine the mean-square optimal ﬁlter directly
depends on a rational choice of the models. We ﬁnd conditions under which the original optimal
ﬁltering problem is equivalent to the dual optimal control problem. Synthesis of the optimal
ﬁlter is accompanied by the analysis of properties of solutions of ﬁltering equations obtained.
In the problem of synthesizing optimal algorithms for parametric nonlinear ﬁltering considered
below, solutions that we suggest are based on the well-known method of reducing an optimal
ﬁltering problem to a dual optimal control problem [1–4]. We believe that this way is the most
convenient and natural with regard to the complicated form of the problem of estimating a state of a
nonlinear stochastic object from observations of output against noise background. In some optimal
parametric ﬁltering problems, it is apparently possible to apply other methods, say, ﬁltering for
the extended state vector, decomposition method, etc. (see, in particular, [5–11]).
The present paper is devoted to the solution of the optimal dynamic nonlinear ﬁltering problem
in a parametric setting. The main questions that arise here and should be answered explicitly are
1. What are methods for algorithmization of optimal estimation in nonlinear dynamic systems?
2. If the Kalman–Busy ﬁltering scheme admits extension to the nonlinear case, then under what
assumptions and by employing what additional resources?
Below, answers to these questions are mainly sought for by means of using the apparatus of
optimal parameter correction .
If a number of parameters of a stochastic system vary in time, the problem of constructing the
optimal ﬁlter becomes much more complicated, even provided that statistical properties of the noise
are known. In this situation, it is useful to perform the process of ﬁltering and estimation of the
state of the object under analysis simultaneously with adaptation of the parameter feedback. In
fact, if there is nonlinearity in the system, realization of this program is most probably a problem
that is hard to solve. Therefore, we suggest to change the optimal parametric ﬁltering problem
to the equivalent dual deterministic optimal control problem and solve it by the above-mentioned
method of parameter correction.
Classical results of linear optimal stochastic Kalman–Busy ﬁltering allow one to obtain consistent
estimators, which possess the roughness property with respect to white noise, where “properties
of the estimators do not change signiﬁcantly under small deviations of properties of real processes
from assumptions a priori made” [1, p. 190]. It is known that, in the class of nonlinear models, the
Kalman–Busy ﬁlter (the general solution was obtained in [11,13,14]) does not have this roughness;
i.e., the Kalman–Busy ﬁlter is divergent. In the parametric case in question, it was possible to
synthesize a stable ﬁlter due to, ﬁrst, “symbolic linearization” of equations of the object and
2001 MAIK “Nauka/Interperiodica”