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Minimizing the Euclidean Condition Number

Braatz, Richard D. and Morari, Manfred (1994) Minimizing the Euclidean Condition Number. SIAM Journal on Control and Optimization, 32 (6). pp. 1763-1768. ISSN 0363-0129. doi:10.1137/S0363012992238680.

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This paper considers the problem of determining the row and/or column scaling of a matrix A that minimizes the condition number of the scaled matrix. This problem has been studied by many authors. For the cases of the ∞-norm and the 1-norm, the scaling problem was completely solved in the 1960s. It is the Euclidean norm case that has widespread application in robust control analyses. For example, it is used for integral controllability tests based on steady-state information, for the selection of sensors and actuators based on dynamic information, and for studying the sensitivity of stability to uncertainty in control systems. Minimizing the scaled Euclidean condition number has been an open question—researchers proposed approaches to solving the problem numerically, but none of the proposed numerical approaches guaranteed convergence to the true minimum. This paper provides a convex optimization procedure to determine the scalings that minimize the Euclidean condition number. This optimization can be solved in polynomial-time with off-the-shelf software.

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Additional Information:© 1994 Society for Industrial and Applied Mathematics. Received October 13, 1992. Accepted June 22, 1993. This author was supported by the Fannie and John Hertz Foundation.
Funding AgencyGrant Number
Fannie and John Hertz FoundationUNSPECIFIED
Subject Keywords:scaling, conditioning, condition number
Issue or Number:6
Classification Code:AMS subject classifications: 65F35, 93B35, 93D21
Record Number:CaltechAUTHORS:20120229-153642155
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Official Citation:Minimizing the Euclidean Condition Number Richard D. Braatz and Manfred Morari SIAM J. Control Optim. 32, pp. 1763-1768
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29534
Deposited By: Ruth Sustaita
Deposited On:29 Feb 2012 23:54
Last Modified:09 Nov 2021 17:07

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