On the Hankel singular values for a parametric system

Given a system with a parameter k, its Hankel singular values, denoted by sigma_i(k) (i = 1,ldrldrldr, n), are naturally functions of k. In this paper, we show that sigma_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 sigma_i(k) to examine the extrema and the asymptotic behaviors of sigma_i(k). We also show that the ratio sigma_i(k)/sigma_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 sigma_i(k)/sigma_j(k). Considering that the ratio sigma_i(k)/sigma_j(k) is quite important information for dasiabalanced model reductionpsila, we can utilize the properties for a balanced model reduction of a parametric system.