Abstract
This paper addresses the task of identifying the parameters of a linear object in the presence of non-Gaussian interference. The identification algorithm is a gradient procedure for minimizing the combined functional. The combined functional, in turn, consists of the fourth-degree functional and a modular functional, whose weights are set using a mixing parameter. Such a combination of functionals makes it possible to obtain estimates that demonstrate robust properties. We have determined the conditions for the convergence of the applied procedure in the mean and root-mean-square measurements in the presence of non-Gaussian interference. In addition, expressions have been obtained to determine the optimal values of the algorithm's parameters, which ensure its maximum convergence rate. Based on the estimates obtained, the asymptomatic and non-asymptotic values of errors in estimating the parameters and identification errors. Because the resulting expressions contain a series of unknown parameters (the values of signal and interference variances), their practical application requires that the estimates of these parameters should be used. We have investigated the issue of stability of the steady identification process and determined the conditions for this stability. It has been shown that determining these conditions necessitates solving the third-degree equations, whose coefficients depend on the specificity of the problem to be solved. The resulting ratios are rather cumbersome but their simplification allows for a qualitative analysis of stability issues. It should be noted that all the estimates reported in this work depend on the choice of a mixing parameter, the task of determining which remains to be explored. The estimates obtained in this paper allow the researcher to pre-evaluate the capabilities of the identification algorithm and the effectiveness of its use in solving practical problems.
Highlights
Underlying many of the tasks related to processing information is the task of building a model of the following form: y(k) = θ∗T x (k) + ξ(k), (1)where y(k) is the observed output signal;x (k) = (x1 (k), x2 (k),..xN (k))T ( ) is the vector of input signals N×1; θ∗ = θ1∗,θ∗2,..θ∗N T is the vector of the desired parameters N×1; ξ(k) is the interference that implies minimizing some of the predefined quality functional
A quadratic functional, the most widely used in practice, leads to various identification algorithms, making it possible to obtain the estimates of the desired vector θ* at the normal interference distributions, ( ) that is, ξ(k) N 0,σ2ξ
Available papers do not include the results of studying the features of the robust algorithms for evaluating a model’s parameters built by using the combined criterion. All this allows us to argue that it is appropriate to conduct a study on the analysis of the properties of the robust identification algorithm, which minimizes a combined functional, allowing for the combination of the LSM and LMM benefits
Summary
A quadratic functional, the most widely used in practice, leads to various identification algorithms, making it possible to obtain the estimates of the desired vector θ* at the normal interference distributions,. The LSM-solution is asymptotically optimal with minimal variance in the class of non-displaced grades This assumption does not generally hold in real-world conditions as almost always the a priori information about distributions is typically inaccessible, or interference is clogged with non-Gaussian noise. (M-assessment) assessment by minimizing the optimal criterion, which is the reverse logarithm of the interference distribution function If such information is not available, a non-quadratic criterion must be applied to assess the vector of θ* parameters. This ensures that the estimate obtained is robust. One of these criteria is a modular one, whose minimization leads to a symbolic algorithm
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