Generalized Total Least Squares Research Articles

Identification of parameters of induction motor with error of speed sensor

This paper proposes a new approach to parametric identification of induction motor with error of speed sensor. The ordinary least squares (OLS) estimates are shown to be biased. Generalized total least squares (GTLS) with a singular covariance matrix of the error are propose for estimation of K parameters of induction motor. Identification of parameters takes place in 2 steps. At the first step, two K parameters are estimated using GTLS with a singular error matrix. At the second step, the remaining three K parameters are estimated. An equivalent augmented system is used to solve a biased normal system. Based on the simulation results, a comparison of the accuracy of the estimates of the proposed algorithm and ordinary least squares estimates is given.

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

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.

Generalized total least squares for identification of electromagnetic parameters of an induction motor

This paper proposes an algorithm for electromagnetic parameters with errors in variables. The estimates obtained by the ordinary least squares (OLS) are biased due to errors in the variables. It is shown that even if there are errors in all variables with the same variance, the problem is reduced to generalized total least squares (GTLS). To find a solution to the generalized total least squares problem, an augmented system of equations is used that is equivalent to a biased normal system. This approach improves the conditionality of the system of equations being solved in comparison with the biased normal system and requires less memory compared to the solution based on the right singular vector. The simulation results show that the GTLS estimates are much more accurate than OLS estimates. Based on the proposed approach, recursive algorithms can be created for evaluating the parameters of an asynchronous electric motor online.

Structural-parametric identification of the sum of hyperbolic functions with additive noise

The paper proposes a two-step method for structural-parametric identification of hyperbolic functions with additive noise. At the first stage, a linear difference equation is associated with the sum of hyperbolic functions. For the estimation of the parameters of the difference equation, the method of generalized total least squares (GTLS). To solve the GTLS problem, an augmented system equivalent to a biased normal system is applied. In the second stage, the missing parameters are estimated. The simulation results showed that the use of GTLS in the first stage improves the accuracy of the estimate compared to Ordinary Least Squares (OLS).

Online estimation of the electrochemical impedance of polymer electrolyte membrane fuel cells using broad-band current excitation

In this work, an online algorithm to estimate the electrochemical impedance spectrum of polymer electrolyte membrane fuel cells (PEMFCs) is presented. The electrochemical impedance, constituting the linear dynamic relation between current and voltage, is described by a discrete time filter model. The unknown model parameters are estimated from streaming time domain data using a recursive algorithm based on the generalized total least squares (GTLS) technique. This allows for flexible noise assumptions, leading to unbiased parameter estimates in the presence of noisy current and voltage measurements. Therefore, the robustness of the proposed method is significantly increased. Broad-band current excitation signals are utilized as to sufficiently stimulate the fuel cell for the online parameter estimation. The model order is selected using Akaike's information criterion (AIC), leading to a parsimonious model. Equivalent circuits are automatically generated using Foster's method of analog circuit synthesis. The applicability and benefits of the proposed algorithm regarding fuel cell diagnostics are demonstrated using measurement data of a small scale single cell (5 cm2 active area) subjected to a broad-band excitation.

Iterative most-likely point registration (IMLP): a robust algorithm for computing optimal shape alignment.

We present a probabilistic registration algorithm that robustly solves the problem of rigid-body alignment between two shapes with high accuracy, by aptly modeling measurement noise in each shape, whether isotropic or anisotropic. For point-cloud shapes, the probabilistic framework additionally enables modeling locally-linear surface regions in the vicinity of each point to further improve registration accuracy. The proposed Iterative Most-Likely Point (IMLP) algorithm is formed as a variant of the popular Iterative Closest Point (ICP) algorithm, which iterates between point-correspondence and point-registration steps. IMLP’s probabilistic framework is used to incorporate a generalized noise model into both the correspondence and the registration phases of the algorithm, hence its name as a most-likely point method rather than a closest-point method. To efficiently compute the most-likely correspondences, we devise a novel search strategy based on a principal direction (PD)-tree search. We also propose a new approach to solve the generalized total-least-squares (GTLS) sub-problem of the registration phase, wherein the point correspondences are registered under a generalized noise model. Our GTLS approach has improved accuracy, efficiency, and stability compared to prior methods presented for this problem and offers a straightforward implementation using standard least squares. We evaluate the performance of IMLP relative to a large number of prior algorithms including ICP, a robust variant on ICP, Generalized ICP (GICP), and Coherent Point Drift (CPD), as well as drawing close comparison with the prior anisotropic registration methods of GTLS-ICP and A-ICP. The performance of IMLP is shown to be superior with respect to these algorithms over a wide range of noise conditions, outliers, and misalignments using both mesh and point-cloud representations of various shapes.

A Recursive Restricted Total Least-Squares Algorithm

We show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and restricted total least squares (RTLS) noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms.

Joint Time Synchronization and Localization of an Unknown Node in Wireless Sensor Networks

Time synchronization and localization are two important issues in wireless sensor networks. Although these two problems share many aspects in common, they are traditionally treated separately. In this paper, we present a unified framework to jointly solve time synchronization and localization problems at the same time. Furthermore, since the accuracy of synchronization and localization is very sensitive to the accuracy of anchor timings and locations, the joint time synchronization and localization problem with inaccurate anchors is also considered in this paper. For the case with accurate anchors, the joint maximum likelihood estimator and a more computationally efficient least squares (LS) estimator are proposed. When the anchor timings and locations are inaccurate, a generalized total least squares (GTLS) scheme is proposed. Crame¿r-Rao lower bounds (CRLBs) and the analytical mean square error (MSE) expressions of the LS based estimators are derived for both accurate and inaccurate anchor cases. Results show that the proposed joint estimators exhibit performances close to their respective CRLBs and outperform the separate time synchronization and localization approach. Furthermore, the derived analytical MSE expressions predict the performances of the proposed joint estimators very well.

Identification of Neurofuzzy Models Using GTLS Parameter Estimation

In this paper, nonlinear system identification utilizing generalized total least squares (GTLS) methodologies in neurofuzzy systems is addressed. The problem involved with the estimation of the local model parameters of neurofuzzy networks is the presence of noise in measured data. When some or all input channels are subject to noise, the GTLS algorithm yields consistent parameter estimates. In addition to the estimation of the parameters, the main challenge in the design of these local model networks is the determination of the region of validity for the local models. The method presented in this paper is based on an expectation-maximization algorithm that uses a residual from the GTLS parameter estimation for proper partitioning. The performance of the resulting nonlinear model with local parameters estimated by weighted GTLS is a product both of the parameter estimation itself and the associated residual used for the partitioning process. The applicability and benefits of the proposed algorithm are demonstrated by means of illustrative examples and an automotive application.

Geometric-Polar Tracking From Bearings-Only and Doppler-Bearing Measurements

Bearings-only tracking (BOT) and Doppler-bearing tracking (DBT) perform target motion analysis from measurements of bearings, Doppler and bearings, respectively. The state equations are simple for both the target and observer in the rectangular coordinates. Hence, BOT and DBT mostly work in that coordinate. By developing the trigonometric relations between target and observer postions over time, this paper first derives the pseudolinear equations for direct range and bearing estimation. It then provides a generalized total least squares (GTLS), and a maximum likelihood solution. Polar tracking has two advantages over rectangular. First, for applications that require range and bearing, polar output avoids a coordinate conversion at every output instant. Second, polar equations often have a smaller bias than rectangular. Consequently, polar estimates can give a closer (to the optimum) initialization for iterative algorithms. Simulation results have validated the theoretical development, confirming the near optimality of the GTLS and maximum likelihood solutions.

On the equivalence between total least squares and maximum likelihood PCA

The maximum likelihood PCA (MLPCA) method has been devised in chemometrics as a generalization of the well-known PCA method in order to derive consistent estimators in the presence of errors with known error distribution. For similar reasons, the total least squares (TLS) method has been generalized in the field of computational mathematics and engineering to maintain consistency of the parameter estimates in linear models with measurement errors of known distribution. The basic motivation for TLS is the following. Let a set of multidimensional data points (vectors) be given. How can one obtain a linear model that explains these data? The idea is to modify all data points in such a way that some norm of the modification is minimized subject to the constraint that the modified vectors satisfy a linear relation. Although the name “total least squares” appeared in the literature only 25 years ago, this method of fitting is certainly not new and has a long history in the statistical literature, where the method is known as “orthogonal regression”, “errors-in-variables regression” or “measurement error modeling”. The purpose of this paper is to explore the tight equivalences between MLPCA and element-wise weighted TLS (EW-TLS). Despite their seemingly different problem formulation, it is shown that both methods can be reduced to the same mathematical kernel problem, i.e. finding the closest (in a certain sense) weighted low rank matrix approximation where the weight is derived from the distribution of the errors in the data. Different solution approaches, as used in MLPCA and EW-TLS, are discussed. In particular, we will discuss the weighted low rank approximation (WLRA), the MLPCA, the EW-TLS and the generalized TLS (GTLS) problems. These four approaches tackle an equivalent weighted low rank approximation problem, but different algorithms are used to come up with the best approximation matrix. We will compare their computation times on chemical data and discuss their convergence behavior.

Frequency-domain generalized total least-squares identification for modal analysis

This contribution focuses on the area of modal analysis and studies the applicability of total least-squares (TLS) algorithms for the estimation of modal parameters in the frequency-domain from input–output Fourier data. These algorithms can be preferable to classical frequency response function based curve-fitting methods. This is certainly the case when periodic excitation is applied and an errors-in-variables noise model can be determined. The proposed generalized total least-squares (GTLS) algorithm provides an accurate modal parameter estimation by the integration of this noise model in the parametric identification process. Modal-based design and comfort improvement, damage assessment and structural health monitoring, and finite element model updating are important applications that strongly rely on a high accuracy of the modal model. In this paper it is shown how frequency-domain TLS and GTLS estimators can be numerically optimized to handle large amounts of modal data. In order to use an errors-in-variables noise model, a linear approximation is necessary in order to obtain a fast implementation of the GTLS algorithm. The validity of this approximation is a function of the signal-to-noise ratio of the input Fourier data and is evaluated by means of Monte Carlo simulations and experimental data.

A weighted total least squares estimator for multivariable systems with nearly maximum likelihood properties

The aim of the present paper is to develop a parametric estimator for linear time-invariant multivariable systems with nearly maximum likelihood properties. The estimator is based on the total least squares (TLS) method. It can be seen as an optimally weighted iterative generalized total least squares (GTLS) estimator, combining the nice asymptotic properties of the maximum likelihood (ML) method with the global minimization property of the GTLS estimator. The main difference with, for instance, the IQML method and the method of Sanathanan and Koerner is that it generates consistent estimates in each iteration step.