Accuracy improvement in Least-Squares estimation with harmonic regressor: New preconditioning and correction methods

Abstract

Abstract-Numerical aspects of least squares estimation have not been sufficiently studied in the literature. In particular, information matrix has a large condition number for systems with harmonic regressor in the initial steps of RLS (Recursive Least Squares) estimation. A large condition number indicates invertibility problems and necessitates the development of new algorithms with improved accuracy of estimation. Symmetric and positive definite information matrix is presented in a block diagonal form in this paper using transformation, which involves the Schur complement. Block diagonal sub-matrices have significantly smaller condition numbers and therefore can be easily inverted, forming a preconditioner for a large scale system. High order algorithms with controllable accuracy are used for solving least squares estimation problem. The second part of the paper is devoted to the performance improvement in classical RLS algorithm, which represents a feedforward estimation procedure with error accumulation. Two correction feedback terms originated from combined high order algorithms are introduced for performance improvement in classical RLS algorithms. Simulation results show significant performance improvement of modified algorithm compared to classical RLS algorithm in the presence of roundoff errors.

Towards accurate estimation of fast varying frequency in future electricity networks: The transition from model-free methods to model-based approach

“…where the matrix A i is an information matrix, and the parameter vector u i satisfies (7). The parameter vector is calculated with high accuracy using high-order algorithms, described in Stotsky, 18,19 see also Appendix 1 and section ''Accuracy, high-order algorithms, stepwise splitting and parallel computing'' for details. Finally, the variance V i of the measurement noise j k , associated with the multiple model is defined as follows…”

“…17 Accuracy of calculation of residual errors plays an important role in the frequency determination, using this method. Each model in the set represents the system with harmonic regressor, and the parameters of the models can be calculated using recently developed high-order algorithms, 18,19 which provide high accuracy of the parameter and residual error estimation. The family of high-order algorithms is extended in Appendix 1, where new stepwise splitting method is also developed.…”

Unified frameworks for high order Newton-Schulz and Richardson iterations: a computationally efficient toolkit for convergence rate improvement

“…For example, the matrix S can be chosen as a diagonal matrix, which contains the diagonal elements of SDD (Strictly Diagonally Dominant) and positive definite matrix A, [16]. For positive definite (not SDD) matrix A the simplest preconditioner can be chosen as S −1 = I /α with α = A ∞ /2 + ε, where • ∞ is the maximum row sum matrix norm, and ε > 0 is a small positive number, [9].…”

“…Notice that the initial mismatchθ 0 can be essentially reduced in recursive least squares estimation for example, using information available in the previous steps, [1,9] and by polynomial preconditioning, [18], described in the next Sect. 2.2.3.…”

“…Iterative methods for solving (1) are often preferable (especially for large-scale systems) due to simplicity, better accuracy and robustness, less processor time and memory space compared to direct methods, [8]. However, ill-conditioning of information matrix implies very slow convergence of the recursive procedures for matrix inversion, [9]. Moreover, limited computational capacity for on-board applications for example, results in significant loss of accuracy and error accumulation in finite-digit calculations, which in turn may result in numerical instability, [6].…”