Optimizing the EV of the data matrix: a case study in non-smooth analysis

We study some properties of the extrema of the eigenvalues (EVs) associated to the data covariance matrix arising in the robust identification problem for discrete time, finite dimensional linear systems. It is well known that the EVs are continuous but, in general, non-differentiable functions of the parameters of the matrix. As a consequence, the maximization of a pre-specified EV cannot be performed using tools from smooth analysis and is typically performed numerically using LMI methods. Nevertheless, we show that nondifferentiable extrema have a simple interpretation enabling their detection. Finally, the convexity properties of the EVs of the data covariance matrix are studied