Abstract
Let G(s) = C(sI - A)-1B + D be a given system where entries of A, B, C, D are polynomials in a parameter k. Then H∞ norm ||G(s)||∞ of G(s) is a function of k, and [9] presents an algorithm to express 1/(||G(s)||∞)2 as a root of a bivariate polynomial, assuming feedthrough term D to be zero. This paper extends the algorithm in two ways: The first extension is the form of the function to be expressed. The extended algorithm can treat, not only H∞ norm, but also functions that appear in the celebrated KYP Lemma. The other extension is the range of the frequency. While H∞ norm considers the supremum of the maximum singular value of G(iω) for the infinite range 0 ≤ ω ≤ ∞ of ω, the extended algorithm treats the norm for the finite frequency range ω ≤ ω ≤ \overline{\omega}(ω, \overline{\omega} ∈ R ∪ ∞). Those two extensions allow the algorithm to be applied to wider area of control problems. We give illustrative numerical examples where we apply the extended algorithm to the computation of the frequency-restricted norm, i.e., the supremum of the maximum singular value of G(iω) (ω ≤ ω ≤ \overline{\omega}).
