On the Hankel singular values for a parametric system

Abstract

Given a system with a parameter k, its Hankel singular values, denoted by σ i (k) (i = 1, · · · , n), are naturally functions of k. In this paper, we show that σ i (k) can be expressed as a root of a bivariate polynomial f (x, k) with respect to x, and present an algorithm to compute the polynomial f (x, k). We then apply the expression of σ i (k) to examine the extrema and the asymptotic behaviors of σ i (k). We also show that the ratio σ i (k)/σ j (k) of two distinct Hankel singular values can also be expressed as a root of bivariate polynomial. This gives us a systematic method to examine various properties such as the extrema or the asymptotic behaviors of the ratio σ i (k)/σ j (k). Considering that the ratio σ i (k)/σ j (k) is quite important information for 'balanced model reduction', we can utilize the properties for a balanced model reduction of a parametric system.