1,720,998 research outputs found
Consistent least squares fitting of ellipsoids
No full textA parameter estimation problem for ellipsoid fitting in the presence of measurement errors is considered. The ordinary least squares estimator is inconsistent, and due to the nonlinearity of the model, the orthogonal regression estimator is inconsistent as well, \ie, these estimators do not converge to the true value of the parameters, as the sample size tends to infinity. A consistent estimator is proposed, based on a proper correction of the ordinary least squares estimator. The correction is explicitly given in terms of the true value of the noise variance
Consistency of the structured total least squares estimator in a multivariate errors-in-variables model
No full textThe structured total least squares estimator, defined via a constrained optimization problem, is a generalization of the total least squares estimator when the data matrix and the applied correction satisfy given structural constraints. In the paper, an affine structure with additional assumptions is considered. In particular, Toeplitz and Hankel structured, noise free and unstructured blocks are allowed simultaneously in the augmented data matrix. An equivalent optimization problem is derived that has as decision variables only the estimated parameters. The cost function of the equivalent problem is used to prove consistency of the structured total least squares estimator. The results for the general affine structured multivariate model are illustrated by examples of special models. Modification of the results for block-Hankel/Toeplitz structures is also given. As a by-product of the analysis of the cost function, an iterative algorithm for the computation of the structured total least squares estimator is proposed
On the computation of the structured total least squares estimator
No full textA class of structured total least squares problems is considered, in which the extended data matrix is partitioned into blocks and each of the blocks is (block) Toeplitz/Hankel structured, unstructured, or noise free. We describe the implementation of two types of numerical solution methods for this problem: i) standard local optimization methods in combination with efficient evaluation of the cost function and its gradient, and ii) an iterative procedure proposed originally for the element-wise weighted total least squares problem. The computational efficiency of the proposed methods is compared with this of alternative methods. Application of the structured total least squares problem for system identification and model reduction is described and illustrated with numerical examples
Consistent estimation in the bilinear multivariate errors-in-variables model
No full textA bilinear multivariate errors-in-variables model is considered. It corresponds to an overdetermined set of linear equations AXB=C, A?Rm×n, B?Rp×q, in which the data A, B, C are perturbed by errors. The total least squares estimator is inconsistent in this case. An adjusted least squares estimator hat X is constructed, which converges to the true value X, as m -> infty, q -> infty. A small sample modification of the estimator is presented, which is more stable for small m and q and is asymptotically equivalent to the adjusted least squares estimator. The theoretical results are confirmed by a simulation study
The element-wise weighted total least squares problem
No full textA new technique for parameter estimation is considered in a linear measurement error model AX approx B, A = A0 + tilde A, B = B0 + tilde B, with row-wise independent and non-identically distributed measurement errors tilde A, tilde B. The total least squares method yields an inconsistent estimate of the parameter X in this case. We formulate a modified total least squares problem, called element-wise weighted total least squares, which provides a consistent estimator, and propose two iterative algorithms for its solution. A local convergence and the rate of convergence of the algorithms is discussed. As a computationally cheap initial approximation we use the generalized total least squares estimate
A comparison between structured low-rank approximation and correlation approach for data-driven output tracking
No full textData-driven control is an alternative to the classical model-based control paradigm. The main idea is that a model of the plant is not explicitly identified prior to designing the control signal. Two recently proposed methods for data-driven control a method based on correlation analysis and a method based on structured matrix low-rank approximation and completion solve identical control problems. The aim of this paper is to compare the methods, both theoretically and via a numerical case study. The main conclusion of the comparison is that there is no universally best method: the two approaches have complementary advantages and disadvantages. Future work will aim to combine the two methods into a more effective unified approach for data-driven output tracking. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved
Consistent estimation in an implicit quadratic measurement error model
No full textAn adjusted least squares estimator is derived that yields a consistent estimate of the parameters of an implicit quadratic measurement error model. In addition, a consistent estimator for the measurement error noise variance is proposed. Important assumptions are: (1) all errors are uncorrelated identically distributed and (2) the error distribution is normal. The estimators for the quadratic measurement error model are used to estimate consistently conic sections and ellipsoids. Simulation examples, comparing the adjusted least squares estimator with the ordinary least squares method and the orthogonal regression method, are shown for the ellipsoid fitting problem
Linear dynamic filtering with noisy input and output
No full textEstimation problems for linear time-invariant systems with noisy input and output are considered. The smoothing problem is a least norm problem. An efficient algorithm using a Riccati-type recursion is derived. The equivalence between the optimal filter and an appropriately modified Kalman filter is established. The optimal estimate of the input signal is derived from the optimal state estimate. The result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem
Block-Toeplitz/Hankel structured total least squares
No full textA multivariate structured total least squares problem is considered, in which the extended data matrix is partitioned into blocks and each of the blocks is block-Toeplitz/Hankel structured, unstructured, or noise free. An equivalent optimization problem is derived and its properties are established. The special structure of the equivalent problem enables to improve the computational efficiency of the numerical solution via local optimization methods. By exploiting the structure, the computational complexity of the algorithms per iteration is linear in the sample size. Application of the method for system identification and for model reduction is illustrated by simulation examples
Addition and intersection of linear time-invariant behaviors
Full text linkWe define and analyze the operations of addition and intersection of linear time-invariant systems in the behavioral setting, where systems are viewed as sets of trajectories rather than input–output maps. The classical definition of addition of input–output systems is addition of the outputs with the inputs being equal. In the behavioral setting, addition of systems is defined as addition of all variables. Intersection of linear time-invariant systems was considered before only for the autonomous case in the context of “common dynamics” estimation. We generalize the notion of common dynamics to open systems (systems with inputs) as intersection of behaviors. This is done by proposing trajectory-based definitions. The main results of the paper are (1) characterization of the link between the complexities (number of inputs and order) of the sum and intersection systems, (2) algorithms for computing their kernel and image representations and (3) a duality property of the two operations. Our approach combines polynomial and numerical linear algebra computations
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