Total Least Squares Algorithm for Errors-in-Variables Systems: Iterative Algorithm or Two-Step Algorithm

• Published in: IEEE transactions on automation science and engineering Vol. 22; pp. 10753 - 10763
• Main Authors: Chen, Jing, Na, Jing, Li, Junhong, Zhu, Quanmin
• Format: Journal Article
• Language: English
• Published: IEEE 2025
• Subjects: Accuracy; Automation; Computational modeling; Convergence; Cost function; eigenvalue calculation; Eigenvalues and eigenfunctions; Errors-in-variables system; iterative algorithm; Iterative algorithms; Large-scale systems; Mathematical models; Noise; total least squares; two-step algorithm
• ISSN: 1545-5955; 1558-3783
• DOI: 10.1109/TASE.2025.3528532

The total least squares (TLS) algorithm is a superior identification tool for low-order errors-in-variables (EIV) systems, where the estimate can be obtained by solving an eigenvector of the minimum eigenvalue of an augmented matrix. However, the TLS algorithm demonstrates inefficiency when applied to high-order EIV systems. This study introduces two innovative TLS algorithms: an iterative TLS algorithm, offering superior performance for low-order EIV models, and a two-step TLS algorithm, designed to effectively handle high-order EIV models. In comparison to the conventional TLS algorithm, these proposed methodologies present noteworthy advantages, including: 1) reduced computational costs, 2) the utilization of an iterative technique to calculate the inverse, and 3) the diversification of EIV identification methods. Simulation bench test examples are selected to show the efficacy of the proposed algorithms and transparent procedure for applications. Note to Practitioners-This paper was motivated by the problem of identifying network systems which are contaminated by noises. For network systems, the input and output data are usually contaminated by noises. Existing approaches to estimating such systems have the assumption that the noises are in little level scenarios or only the output data are contaminated by noises. This paper suggests two new total least squares approaches which can deal with systems contaminated by noises in medium level scenarios or whose input and output are both contaminated by noises. These two algorithms, using iterative technique and two-step technique, can: 1) avoid the matrix inverse calculation; 2) reduce the computational efforts; 3) increase the convergence rates. The proposed algorithms can also be extended to various fields such as inverse scattering, pattern recognition, image restoration, and computer vision.