A Spectral Expression for the Frequency-Limited H2-norm

In this paper, a new simple but yet efficient spectral expression of the frequency-limited H2-norm, denoted H2,ω-norm, is introduced. The proposed new formulation requires the computation of the system eigenvalues and eigenvectors only, and provides thus an alternative to the well established Gramian-based approach. The interest of this new formulation is in three-folds: (i) it provides a new theoretical framework for the H2,ω-norm-based optimization approach, such as controller synthesis, filter design and model approximation, (ii) it improves the H2,ω-norm computation velocity and it applicability to models of higher dimension, and (iii) under some conditions, it allows to handle systems with poles on the imaginary axis. Both mathematical proofs and numerical illustrations are provided to assess this new H2,ω-norm expression.