Globally Optimal SISO H2-Norm Model Reduction Using Walsh’s Theorem

We present a novel methodology for single-input single-output (SISO)
$\mathbf {H_{2}}$
-norm model reduction that guarantees global optimality of the obtained solution(s). By exploiting Walsh’s theorem, which is an elegant formulation of the first-order necessary conditions for optimality, we reformulate the model reduction problem as a multiparameter eigenvalue problem (MEVP), the real-valued eigentuples of which characterize the globally optimal solution(s) of the model reduction problem. While aiming for global optimality comes at the cost of a combinatorial growth of the problem complexity for increasing model orders, the novel methodology allows us to tackle larger problems compared to the few other globally optimal approaches in the literature. In particular, the degree of the obtained MEVP is independent of the order of the original higher order and obtained reduced-order model, a property that is favorable from a computational point of view. We perform three numerical experiments to illustrate the effectiveness of the methodology.