Computing H-infinity Norms of Time-Delay Systems

In this paper we consider the computation of H-infinity norm of retarded time-delay systems with discrete pointwise state delays. It is well known that in the finite dimensional case H-infinity norm of a system is computed using the connection between the singular values of the transfer function and the imaginary axis eigenvalues of an Hamiltonian matrix. We show a similar connection between the singular values of a transfer function of a time-delay system and the imaginary axis eigenvalues of an infinite dimensional operator $\mathcal{L}_{\xi}$. Using spectral methods, this linear operator is approximated with a matrix. The approximate H-infinity norm of the time-delay system is calculated using the connection between the imaginary eigenvalues of this matrix and the singular values of a finite dimensional approximation of the time-delay system. Finally the approximate results are corrected by solving a set of equations which are obtained from the reformulation of the eigenvalue problem for $\mathcal{L}_{\xi}$ as a finite dimensional nonlinear eigenvalue problem.