Abstract
The study deals with the problem of identification of non-stationary parameters of a linear object which can be described by first-order Markovian model, with the help of the simplest in computational terms single-step adaptive identification algorithms – modified algorithms by Kaczmarz and Nagumo-Noda. These algorithms do not require knowledge of information on the degree of non-stationarity of the studied object. When building the model, they use the information only about one step of measurements. Modification involves the use of the regularizing addition in the algorithms to improve their computing properties and avoid division by zero. Using a Markovian model is quite effective because it makes it possible to obtain analytic estimates of the properties of algorithms. It was shown that the use of regularizing additions in identification algorithms, while improving stability of algorithms, leads to some slowdown of the process of model construction. The conditions for convergence of regularizing algorithms by Kaczmarz and Nagumo-Noda at the evaluation of stationary parameters in mean and root-mean-square and existing measurement interference were determined. The obtained estimates differ from the existing ones by higher accuracy. Despite this, they are quite general and depend both on the degree of non-stationarity of an object, and on statistical characteristics of interference. In addition, the expressions for the optimal values of the parameters of algorithms, ensuring their maximum rate of convergence under conditions of non-stationarity and the presence of Gaussian interferences, were determined. The obtained analytical expressions contain a series of unknown parameters (estimation error, degree of non-stationarity of an object, statistical characteristics of interferences). For their practical application, it is necessary to use any recurrent procedure for estimation of these unknown parameters and apply the obtained estimates to refine the parameters that are included in the algorithms
Highlights
Many problems of control, prediction, and pattern recognition, etc. relate to the construction of a model of the following type: yn+1 = θ∗T xn+1 + ξn+1, (1)where yn+1 is the observed output signal; x n+1 =( x1,n+1, x 2,n+1, .. x N )T,n+1 is the vector of input signals N ×1; θ∗ = (θ1∗,θ∗2,..θ∗N )T is the vector of sought-for parameters N ×1; ξn+1 is the interference and it is reduced to minimization of a certain functional of quality, chosen in advance
N+1 is the vector of input signals N ×1; θ∗ = (θ1∗,θ∗2,..θ∗N )T is the vector of sought-for parameters N ×1; ξn+1 is the interference and it is reduced to minimization of a certain functional of quality, chosen in advance
It should be noted that the effectiveness of application of this or that algorithm depends on availability of information about a drift
Summary
It should be noted that the effectiveness of application of this or that algorithm depends on availability of information about a drift. Mathematics and cybernetics – applied aspects available These algorithms include gradient algorithms, the algorithm by Kaczmarz and Nagumo-Noda. These algorithms, though they use only current information about x and y, are more responsive. This demonstrates the feasibility of their application for identification of non-stationary parameters. Existing estimates, characterizing the properties of these algorithms are quite rough This is caused by the difficulty of the theoretical studies of the properties of these algorithms under non-stationary conditions.
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