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Modeling and Model Reduction by Analytic Interpolation and Optimization
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of six papers. The main topic of all these papers is modeling a class of linear time-invariant systems. The system class is parameterized in the context of interpolation theory with a degree constraint. In the papers included in the thesis, this parameterization is the key tool for the design of dynamical system models in fields such as spectral estimation and model reduction. A problem in spectral estimation amounts to estimating a spectral density function that captures characteristics of the stochastic process, such as covariance, cepstrum, Markov parameters and the frequency response of the process. A  model reduction problem consists in finding a small order system which replaces the original one so that the behavior of both systems is similar in an appropriately defined sense.  In Paper A a new spectral estimation technique based on the rational covariance extension theory is proposed. The novelty of this approach is in the design of a spectral density that optimally matches covariances and approximates the frequency response of a given process simultaneously.In Paper B  a model reduction problem is considered. In the literature there are several methods to perform model reduction. Our attention is focused on methods which preserve, in the model reduction phase, the stability and the positive real properties of the original system. A reduced-order model is computed employing the analytic interpolation theory with a degree constraint. We observe that in this theory there is a freedom in the placement of the spectral zeros and interpolation points. This freedom can be utilized for the computation of a rational positive real function of low degree which approximates the best a given system. A problem left open in Paper B is how to select spectral zeros and interpolation points in a systematic way in order to obtain the best approximation of a given system. This problem is the main topic in Paper C. Here, the problem is investigated in the analytic interpolation context and spectral zeros and interpolation points are obtained as solution of a optimization problem.In Paper D, the problem of modeling a floating body by a positive real function is investigated. The main focus is  on modeling the radiation forces and moment. The radiation forces are described as the forces that make a floating body oscillate in calm water. These forces are passive and usually they are modeled with system of high degree. Thus, for efficient computer simulation it is necessary to obtain a low order system which approximates the original one. In this paper, the procedure developed in Paper C is employed. Thus, this paper demonstrates the usefulness of the methodology described in Paper C for a real world application.In Paper E, an algorithm to compute the steady-state solution of a discrete-type Riccati equation, the Covariance Extension Equation, is considered. The algorithm is based on a homotopy continuation method with predictor-corrector steps. Although this approach does not seem to offer particular advantage to previous solvers, it provides insights into issues such as positive degree and model reduction, since the rank of the solution of the covariance extension problem coincides with the degree of the shaping filter. In Paper F a new algorithm for the computation of the analytic interpolant of a bounded degree is proposed. It applies to the class of non-strictly positive real interpolants and it is capable of treating the case with boundary spectral zeros. Thus, in Paper~F, we deal with a class of interpolation problems which could not be treated by the optimization-based algorithm proposed by Byrnes, Georgiou and Lindquist. The new procedure computes interpolants by solving a system of nonlinear equations. The solution of the system of nonlinear equations is obtained by a homotopy continuation method.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , xiii, 29 p.
Series
Trita-MAT. OS, ISSN 1401-2294 ; 08:08
Keyword [en]
Analytic Interpolation theory with a degree constraint, rational covariance extension problem, spectral estimation, model reduction, optimization, passive system
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-9125ISBN: 978-91-7415-111-4 (print)OAI: oai:DiVA.org:kth-9125DiVA: diva2:24306
Public defence
2008-10-10, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20100721Available from: 2008-09-26 Created: 2008-09-22 Last updated: 2010-07-21Bibliographically approved
List of papers
1. Spectral estimation by least-squares optimization based on rational covariance extension
Open this publication in new window or tab >>Spectral estimation by least-squares optimization based on rational covariance extension
2007 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 43, no 2, 362-370 p.Article in journal (Refereed) Published
Abstract [en]

This paper proposes a new spectral estimation technique based on rational covariance extension with degree constraint. The technique finds a rational spectral density function that approximates given spectral density data under constraint on a covariance sequence. Spectral density approximation problems are formulated as nonconvex optimization problems with respect to a Schur polynomial. To formulate the approximation problems, the least-squares sum is considered as a distance. Properties of optimization problems and numerical algorithms to solve them are explained. Numerical examples illustrate how the methods discussed in this paper are useful in stochastic model reduction and stochastic process modeling.

Keyword
spectral estimation, optimization, rational covariance extension, least-squares sum, Schur polynomial
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-14156 (URN)10.1016/j.automatica.2006.09.003 (DOI)000243670600018 ()2-s2.0-33845931345 (Scopus ID)
Note
QC 20100720Available from: 2010-07-20 Created: 2010-07-20 Last updated: 2017-12-12Bibliographically approved
2. Passivity-preserving model reduction by analytic interpolation
Open this publication in new window or tab >>Passivity-preserving model reduction by analytic interpolation
2007 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 425, no 2-3, 608-633 p.Article in journal (Refereed) Published
Abstract [en]

Antoulas and Sorensen have recently proposed a passivity-preserving model-reduction method of linear systems based on Krylov projections. The idea is to approximate a positive-real rational transfer function with one of lower degree. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models. We also show that, in the multi-input/multi-output case, Sorensen's algorithm actually amounts to tangential Nevanlinna-Pick interpolation.

Keyword
model reduction, passivity, interpolation, spectral zeros, positive-real functions, rational approximation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-14157 (URN)10.1016/j.laa.2007.03.014 (DOI)000249065400021 ()2-s2.0-34447551585 (Scopus ID)
Note
QC 20100721Available from: 2010-07-20 Created: 2010-07-20 Last updated: 2017-12-12Bibliographically approved
3. Passive system with degree bound designed by analytic interpolation
Open this publication in new window or tab >>Passive system with degree bound designed by analytic interpolation
2008 (English)Article in journal (Other academic) Submitted
Keyword
Analytic Interpolation with a degree constraint; passive system; interpolation point; spectral zero; quasi-convex optimization; least-square optimization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-14158 (URN)
Note
QS 20120314. Baserad på "Low order passive system designed by Analytic Interpolation", Proceeding of the 18th International Symposium on Mathematical Theory and Systems, Blacksburg, Virginia, USA, August 2008.Available from: 2010-07-21 Created: 2010-07-21 Last updated: 2012-03-14Bibliographically approved
4. Low order radiation forces by analytic interpolation with degree constraint
Open this publication in new window or tab >>Low order radiation forces by analytic interpolation with degree constraint
2007 (English)In: Proceedings of the 46th IEEE conference on decision and control, 2007, 2405-2410 p.Conference paper, Published paper (Refereed)
Abstract [en]

The positive real modeling of a floating body is considered, whereas the main focus is on the radiation forces and moments. The radiation forces and moments describe the interaction of a floating body with the surrounding fluid. This type of mathematical model is of interest in among control and simulation of dynamical positioned vessels (i.e. ships and offshore platforms) and wave power plants. It has been proven that the radiation forces are passive, but very little attention have been drawn towards low order passive identification of these forces. Traditionally high order models have been obtained, and subsequently model order reduction have been applied to obtain low order models. Here, a direct approach for obtaining low order passive models using analytic interpolation with a degree constraint is applied. A case study involving a 3 degrees of freedom surface vessel is shown to illustrate the features of the proposed approach.

Series
IEEE conference on decision and control - proceedings, ISSN 0191-2216
Keyword
Constraint theory; Degrees of freedom (mechanics); Fluid structure interaction; Interpolation; Mathematical models; Degree constraint; Radiation forces; Surface vessels
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-14159 (URN)10.1109/CDC.2007.4434656 (DOI)000255181702048 ()2-s2.0-62749102896 (Scopus ID)978-1-4244-1497-0 (ISBN)
Conference
46th IEEE Conference on Decision and Control, New Orleans, LA, DEC 12-14, 2007
Note
QC 20100721Available from: 2010-07-21 Created: 2010-07-21 Last updated: 2010-07-21Bibliographically approved
5. A homotopy continuation solution of the covariance extension equation
Open this publication in new window or tab >>A homotopy continuation solution of the covariance extension equation
2005 (English)In: Lecture notes in control and information sciences, ISSN 0170-8643, E-ISSN 1610-7411, Vol. 321, 27-42 p.Article in journal (Refereed) Published
Keyword
rational covariance extension problem; rational covariance extension equation; continuation method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-14160 (URN)10.1007/10984413_3 (DOI)000233074300003 ()2-s2.0-33947511777 (Scopus ID)
Note
QC 20100721Available from: 2010-07-21 Created: 2010-07-21 Last updated: 2017-12-12Bibliographically approved
6. Computation of bounded degree Nevanlinna-Pick interpolants by solving nonlinear equations
Open this publication in new window or tab >>Computation of bounded degree Nevanlinna-Pick interpolants by solving nonlinear equations
2003 (English)In: 42nd IEEE Conference on Decision and Control: Maui, HI, DEC 09-12, 2003, 2003, 4511-4516 p.Conference paper, Published paper (Refereed)
Abstract [en]

This paper provides a procedure for computing scalar real rational Nevanlinna-Pick interpolants of a bounded degree. It applies to a wider class of interpolation problems and seems numerically more reliable than previous, optimization-based, procedures. It is based on the existence and the uniqueness of the solution guaranteed by Georgiou's proof of bijectivity of a map between a class of nonnegative trigonometric polynomials and a class of numerator/denominator polynomial pairs of interpolants. Further analysis of this map suggests a numerical continuation method for determining the interpolant from a system of nonlinear equations. A numerical example illustrates the reliability of the proposed procedure.

Keyword
Nevanlinna-Pick interpolation, positive realness, rationality, system of nonlinear equations, continuation method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-14162 (URN)10.1109/CDC.2003.1272255 (DOI)000189434100774 ()0-7803-7924-1 (ISBN)
Note
QC 20100721Available from: 2010-07-21 Created: 2010-07-21 Last updated: 2010-07-21Bibliographically approved

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