scholarly journals Symmetrical Augmented System of Equations for the Parameter Identification of Discrete Fractional Systems by Generalized Total Least Squares

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3250
Author(s):  
Dmitriy Ivanov ◽  
Aleksandr Zhdanov
Keyword(s):  
Parameter Identification ◽  
Least Squares ◽  
Singular Vector ◽  
Total Least Squares ◽  
System Of Equations ◽  
Errors In Variables ◽  
Fractional Systems ◽  
Augmented System ◽  
The Right ◽  
Complex Coefficients

This paper is devoted to the identification of the parameters of discrete fractional systems with errors in variables. Estimates of the parameters of such systems can be obtained using generalized total least squares (GTLS). A GTLS problem can be reduced to a total least squares (TLS) problem. A total least squares problem is often ill-conditioned. To solve a TLS problem, a classical algorithm based on finding the right singular vector or an algorithm based on an augmented system of equations with complex coefficients can be applied. In this paper, a new augmented system of equations with real coefficients is proposed to solve TLS problems. A symmetrical augmented system of equations was applied to the parameter identification of discrete fractional systems. The simulation results showed that the use of the proposed symmetrical augmented system of equations can shorten the time for solving such problems. It was also shown that the proposed system can have a smaller condition number.

scholarly journals Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformation

2012 ◽  
Vol 2 (2) ◽  
pp. 98-106 ◽  
Author(s):  
B. Schaffrin ◽  
F. Neitzel ◽  
S. Uzun ◽  
V. Mahboub
Keyword(s):  
Least Squares ◽  
Similarity Transformation ◽  
Optimal Solution ◽  
Total Least Squares ◽  
Errors In Variables ◽  
Parameter Adjustment ◽  
Structured Errors

Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformationIn 2005, Felus and Schaffrin discussed the problem of a Structured Errors-in-Variables (EIV) Model in the context of a parameter adjustment for a classical similarity transformation. Their proposal, however, to perform a Total Least-Squares (TLS) adjustment, followed by a Cadzow step to imprint the proper structure, would not always guarantee the identity of this solution with the optimal Structured TLS solution, particularly in view of the residuals. Here, an attempt will be made to modify the Cadzow step in order to generate the optimal solution with the desired structure as it would, for instance, also result from a traditional LS-adjustment within an iteratively linearized Gauss-Helmert Model (GHM). Incidentally, this solution coincides with the (properly) Weighted TLS solution which does not need a Cadzow step.

Inverse Problems With Errors in the Independent Variables Errors-in-Variables, Total Least Squares, and Bayesian Inference

2008 ◽  
Author(s):  
A. F. Emery
Keyword(s):  
Bayesian Inference ◽  
Maximum Likelihood ◽  
Inverse Problems ◽  
Least Squares ◽  
Total Least Squares ◽  
Errors In Variables ◽  
Independent Variables ◽  
Error In Variables ◽  
Best Estimates ◽  
Normally Distributed

Most practioners of inverse problems use least squares or maximum likelihood (MLE) to estimate parameters with the assumption that the errors are normally distributed. When there are errors both in the measured responses and in the independent variables, or in the model itself, more information is needed and these approaches may not lead to the best estimates. A review of the error-in-variables (EIV) models shows that other approaches are necessary and in some cases Bayesian inference is to be preferred.

scholarly journals Identification of linear dynamic systems of fractional order with errors in variables based on an augmented system of equations

2021 ◽  
Vol 25 (3) ◽  
pp. 508-518
Author(s):  
Dmitriy Vladimirovich Ivanov
Keyword(s):  
Fractional Order ◽  
Dynamic Systems ◽  
System Of Equations ◽  
Errors In Variables ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Augmented System

Уравнения с производными и разностями дробного порядка находят широкое применение для описания различных процессов и явлений. В настоящее время активно развиваются методы идентификации систем, описываемых уравнениями с разностями дробного порядка. Статья посвящена идентификации дискретных динамических систем, описываемых уравнениями с разностями дробного порядка с ошибками в переменных. Задачи идентификации систем с ошибками в переменных часто бывают плохо обусловленными. В статье предложен алгоритм, использующий представление нормальной смещенной системы в виде расширенной эквивалентной системы. Данное представление позволяет уменьшить число обусловленности решаемой задачи. Тестовые примеры показали, что предложенный алгоритм обладает более высокой точностью по сравнению с алгоритмами на основе разложения Холецкого и минимизации обобщенного отношения Релея.

scholarly journals Scaled weighted total least-squares adjustment for partial errors-in-variables model

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
J. Zhao
Keyword(s):  
Least Squares ◽  
Variance Component ◽  
Maximum Likelihood Method ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Likelihood Method ◽  
Errors In Variables ◽  
Weighted Total Least Squares ◽  
Least Squares Adjustment ◽  
Observation Vector

AbstractScaled total least-squares (STLS) unify LS, Data LS, and TLS with a different choice of scaled parameter. The function of the scaled parameter is to balance the effect of random error of coefficient matrix and observation vector for the estimate of unknown parameter. Unfortunately, there are no discussions about how to determine the scaled parameter. Consequently, the STLS solution cannot be obtained because the scaled parameter is unknown. In addition, the STLS method cannot be applied to the structured EIV casewhere the coefficient matrix contains the fixed element and the repeated random elements in different locations or both. To circumvent the shortcomings above, the study generalize it to a scaledweighted TLS (SWTLS) problem based on partial errors-in-variable (EIV) model. And the maximum likelihood method is employed to derive the variance component of observations and coefficient matrix. Then the ratio of variance component is proposed to get the scaled parameter. The existing STLS method and WTLS method is just a special example of the SWTLS method. The numerical results show that the proposed method proves to bemore effective in some aspects.

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