Tensor products and matrix differential calculus

Pollock, D.S.G.

doi:10.1016/0024-3795(85)90194-6

Cited by 31 publications

(20 citation statements)

References 14 publications

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“…We shall begin, as the author has done in Pollock (1984Pollock ( ), (1985, by using, as the criterion function, an expression of the likelihood function which is in terms of the reduced-form parameters:…”

Section: The π Methodsmentioning

confidence: 99%

“…The fundamental difficulty of determining the two functions from observations on prices and quantities sold was highlighted by the brothers Holbrook Working, 1895-1985and Elmer Joseph Working, 1900-1968. Whereas the elder brother was the first to grapple with the perplexities of the problem (in H. Woking, 1925), it was the younger brother who discovered the true nature of the so-called identification problem in econometrics and who suggested the solution to it (in E. Woking, 1927).…”

Section: Introductionmentioning

confidence: 99%

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The classical econometric model

Pollock¹

1987

International Studies in Economics and Econometrics

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A compendium is presented of the various approaches that may be taken in deriving the estimators of the simultaneous-equations econometric model according to the principle of maximum likelihood. The structural equations of the model have the character both of a regression equation and of an errors-in-variables equation. This partly accounts for way in which the various approaches that have been followed appear to differ widely. In the process of achieving a synthesis of the methods of estimation, some elements that have been missing from the theory are supplied.

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Section: The π Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The classical econometric model

Pollock¹

1987

International Studies in Economics and Econometrics

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“…This makes that the optimization problem in this form is not directly useful for optimization. It can however be proved that (20) can be locally approximated by a set of Linear Matrix Inequalities (LMI's). Based upon these LMI's, methods from interior points LMI solvers can be used to compute the gradient of the direction that reduces the maximum singular value.…”

Section: Norm Minimizationmentioning

confidence: 99%

“…To compute this expansion, matrix differential calculus is exploited. We use the notation introduced by [20] and further described in [15].…”

Section: Lemmamentioning

confidence: 99%

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Model-free norm-based fixed structure controller synthesis

Hamer

Weiland

Steinbuch

2009

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference

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Model-free norm-based fixed structure controller synthesisCitation for published version (APA): Hamer, den, A. J., Weiland, S., & Steinbuch, M. (2009). Model-free norm-based fixed structure controller synthesis. In Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference (CDC / CCC), 16-18 december 2009, Shanghai, China (pp. 4030-4035 Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at:openaccess@tue.nl providing details and we will investigate your claim. Abstract-This paper presents a method to perform modelfree fixed structure controller synthesis. Based on frequency response data of the plant, the parameters of a predefined controller structure are optimized directly with respect to closedloop performance specifications. As a result, no parametric plant model is required such that time consuming iterative identification-synthesis procedures can be omitted.A framework is presented to both assure stability and optimize closed-loop performance based on frequency response data. Furthermore, based upon these results, a cost-function is formulated that can be exploited to converge from a destabilizing-to the stabilizing controller parameter region. Both the stability guarantee and performance optimization procedures are combined in one optimization algorithm that is illustrated by means of an example. I. INTRODUCTIONApplication of model-based optimal control synthesis methodologies require a low order parametric model of the plant. When only input-output data of a system is available, the most appr...

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Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

Kinghorn

1996

Int. J. Quantum Chem.

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The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theorctically and computationally insightful. ( International Journal of Quantum Chemistry, Vol 57, 141 -155 (1996) c 1996 John Wiley & Sons, Inc tedious to verify, noninsightful, and not optimally designed for high-performance computing. On the other hand, if derivatives are written in matrix form, all the tools of matrix algebra can be applied and the advantages of matrix notation can be exploited for new theoretical and computational insight. This article addresses these issues and illustrates them for a long-standing quantum chemical basis set.An elegant and very useful form of matrix differential calculus was presented by the mathematical economists J. R. Magnus and H. Neudecker [l].This powerful mathematical tool is not well known outside of the disciplines of mathematical statistics CCC 0020-7608 I 9 6 I0201 41 -1 5

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Tensor products and matrix differential calculus

Cited by 31 publications

References 14 publications

The classical econometric model

The classical econometric model

Model-free norm-based fixed structure controller synthesis

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

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