Software for weighted structured low-rank approximation

Markovsky, Ivan; Usevich, Konstantin

doi:10.1016/j.cam.2013.07.048

Cited by 61 publications

(91 citation statements)

References 34 publications

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“…These problems can be solved by existing methods, e.g., the prediction error (Söderström and Stoica 1989;Ljung 1999) and the low-rank approximation methods (Markovsky 2008;Markovsky and Usevich 2014). Deriving the the maximum likelihood estimator for the dynamic measurement problem with unknown dynamics is our second contribution.…”

Section: Novelty and Contributionsmentioning

confidence: 99%

An application of system identification in metrology

Markovsky

2015

Control Engineering Practice

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Metrology is advancing by development of new measurement techniques and corresponding hardware. A given measurement technique, however, has fundamental speed and precision limitations. In order to overcome the hardware limitations, we develop signal processing methods based on the prior knowledge that the measurement process dynamics is linear time-invariant.Our approach is to model the measurement process as a step response of a dynamical system, where the input step level is the quantity of interest. The solution proposed is an algorithm that does real-time processing of the sensor's measurements. It is shown that when the measurement process dynamics is known, the input estimation problem is equivalent to state estimation. Otherwise, the input estimation problem can be solved as a system identification problem. The main underlying assumption is that the measured quantity is constant and the measurement process is a low-order linear time-invariant system. The methods are validated and compared on applications of temperature and weight measurement.

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Section: Novelty and Contributionsmentioning

confidence: 99%

An application of system identification in metrology

Markovsky

2015

Control Engineering Practice

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“…The method, presented in this paper (general affine structure) and the efficient methods of [26] are implemented in Matlab (using Optimization Toolbox) and in C++ (using the Levenberg-Marquardt algorithm [20] from the GNU Scientific Library [7]), respectively. Description of the software and overview of its applications is given in [16].…”

Section: Note 8 (Choice Of γ) γ = Max R∈r F M(r) Always Satisfies Comentioning

confidence: 99%

“…In contrast, the kernel representation (KER) has m(m − r) variables, so that it is suitable for problems with small co-rank m − r. The implementation [17] of the methods in the paper is applicable for relatively small size problems (say, m < n < 100, and m − r < 3). For larger problems, the efficient C implementation [16] (denoted by slra-c below), of the variable projection approach without missing values, can be used by setting the missing values to zeros and the corresponding weights to a small number (10 −6 in the simulation examples). The In the reported results, slra-m corresponds to (SLRA ′ R ) and slra-r corresponds to (SLRA ′′ R ) with γ = p G 2 2 (see Note 8).…”

Section: Approximate Matrix Completionmentioning

confidence: 99%

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Structured Low-Rank Approximation with Missing Data

Markovsky¹,

Usevich²

2013

SIAM J. Matrix Anal. & Appl.

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We consider low-rank approximation of affinely structured matrices with missing elements. The method proposed is based on reformulation of the problem as inner and outer optimization. The inner minimization is a singular linear least-norm problem and admits an analytic solution. The outer problem is a nonlinear least squares problem and is solved by local optimization methods: minimization subject to quadratic equality constraints and unconstrained minimization with regularized cost function. The method is generalized to weighted low-rank approximation with missing values and is illustrated on approximate low-rank matrix completion, system identification, and data-driven simulation problems. An extended version of the paper is a literate program, implementing the method and reproducing the presented results.

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“…Specifically, the SLRA software package [46] covers the block Toeplitz structure of the matrix W k and can hence be used to compute k . We have applied this software to all examples from [15].…”

Section: Distance To Singularity and Structured Low-rank Approximationmentioning

confidence: 99%

Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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This chapter is concerned with distance problems for linear timeinvariant differential and differential-algebraic equations. Such problems can be formulated as distance problems for matrices and pencils. In the first part, we discuss characterizations of the distance of a regular matrix pencil to the set of singular matrix pencils. The second part focuses on the distance of a stable matrix or pencil to the set of unstable matrices or pencils. We present a survey of numerical procedures to compute or estimate these distances by taking into account some of the historical developments as well as the state of the art. IntroductionConsider a linear time-invariant differential-algebraic equation (DAE)with coefficient matrices E; A 2 R n n , a sufficiently smooth inhomogeneity f W OE0; 1/ ! R n , and an initial state vector x 0 2 R n . When E is nonsingular, (20.1) is turned into a system of ordinary differential equations by simply multiplying with E 1 on both sides. The more interesting case of singular E arises, for example, when imposing algebraic constraints on the state vector x.t/ 2 R n . Equations of this type play an important role in a variety of applications, including electrical circuits [54,55] and multi-body systems [48]. The analysis and numerical solution of more general DAEs (which also take into account time-variant coefficients, nonlinearities and time delays) is a central part of Volker Mehrmann's work, as witnessed by the monograph [43] and by the several other chapters of this book

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Software for weighted structured low-rank approximation

Cited by 61 publications

References 34 publications

An application of system identification in metrology

An application of system identification in metrology

Structured Low-Rank Approximation with Missing Data

Distance Problems for Linear Dynamical Systems

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