On computing the H2 norm using the polynomial Diophantine equation

Abstract

An explicit algorithm will be presented for computing the H2 norm of a single-input single-output system from the coe fficients in its transfer function. The algorithm follows directly from Cauchy's residue theorem, and the most computationally intensive step involves solving a polynomial Diophantine equation. This can be e fficiently solved using subresultant sequences in a fraction-free variant of the extended Euclidean algorithm. The coe fficients in these subresultant sequences correspond to the Hurwitz determinants, whereby a stability test can be obtained alongside computing the H2 norm with little additional computational effort. Implementations of the algorithm symbolically, in exact arithmetic, and in floating-point arithmetic will be presented. These will be applied to examples of passive train suspension systems that optimise passenger comfort. The examples will demonstrate the algorithm's greater robustness and computational e fficiency relative to H2 norm algorithms requiring the computation of the controllability or observability Gramians. Finally, applications to the realisation of optimal lumped-parameter systems will be discussed.