Optimal Structure of Recurrent Nonlinear Filters of Large Order for Diffusion Signals
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
OPTIMAL STRUCTURE OF RECURRENT NONLINEAR FILTERS OF LARGE ORDER FOR DIFFUSION SIGNALS E. A. Rudenko Moscow Aviation Institute (National Research University) 4, Volokokamskoe Shosse, Moscow 125993, Russia [email protected]
UDC 519.246.2: 681.518.22
We consider the optimal mean-square estimation problem for the state variables of a continuous nonlinear stochastic object by using results of time-discrete measurements. To obtain clock and inter-clock estimates on a computer of limited power in real time, we propose a procedure for the synthesis of a nonlinear structure of a discrete finitedimensional filter, the state vector of which is formed from the desired number of already obtained preceding clock estimates. We describe the synthesis algorithm for the filter and its suboptimal approximations. The advantage of the latter is shown in comparison with the corresponding generalizations of the Kalman filter. Bibliography: 8 titles.
To obtain estimates for the Markov state variables of a stochastic object of observation by an absolutely optimal filter [1]–[3], it is required to promptly find the posterior probability density of the estimated random process, which makes such a filter a distributed parameter system. Therefore, the state vector of such a filter is of infinite order, and it is difficult to implement an absolutely optimal filter in real time. Therefor, in practice, one has to use approximate finitedimensional filtering algorithms such as various generalizations of the Kalman filter, with a loss of accuracy, or create poly-Gaussian banks of such filters, which complicates the computer. The implementation of the particle filter [3] requires a very powerful computer due to the use of the cumbersome Monte Carlo method at each trajectory. A conditionally optimal filter is finite-dimensional and thereby can be easily realized [4, 5], but it is only parametric and its order is bounded by the order of the object of observation. Finite-dimensional filters of optimal structure of different orders, free from these restrictions, are synthesized in [6]–[8]. In the sense of potential accuracy, the optimal structure filters occupy an intermediate position between the absolutely and conditionally optimal filters ItAOF ItOSF ItCOF , where It is the mean-square estimation error at the time t. However, the accuracy of an optimal structure filter of small order [6, 7] is bounded exactly by its order, whereas an optimal finite memory filter [8] the order of which is a multiple of the dimension of the measurement vector, forgets the preceding measurements which could be more exact than the current ones.
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 121-128. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0134
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In this paper, we construct a rather simple recurrent optimal structure filter of a large order multiple to the dimension of the estimate vector. The memory of this filter is infinite, and a contradiction between accuracy and complexity
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