MINIMIZING THE EUCLIDEAN CONDITION NUMBER

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