Characterization and Computation of $\mathcal{H}_{\infty}$ Norms for Time-Delay Systems

Abstract

We consider the characterization and computation of H-infinity norms for a class of time-delay systems. It is well known that in the finite dimensional case the H-infinity norm of a transfer function can be computed using the connections between the corresponding singular value curves and the imaginary axis eigenvalues of a Hamiltonian matrix, leading to the established level set methods. We show a similar connection between the transfer function of a time-delay system and the imaginary axis eigenvalues of an infinite dimensional linear operator $\mathcal{L}_{\xi}^N$. Based on this result, we propose a predictor-corrector algorithm for the computation of the H-infinity norm. In the prediction step, a finite-dimensional approximation of the problem, induced by a spectral discretization of the operator $\mathcal{L}_{\xi}$, and an adaptation of the algorithms for finite-dimensional systems, allow to obtain an approximation of the H-infinity norm of the transfer function of the time-delay system. In the next step the approximate results are corrected to the desired accuracy by solving a set of nonlinear equations which are obtained from the reformulation of the eigenvalue problem for the linear infinite-dimensional operator $\mathcal{L}_{\xi}^N$ as a finite dimensional nonlinear eigenvalue problem. These equations can be interpreted as characterizations of peak values in the singular value plot. The effects of the discretization in the predictor step are fully characterized and the choice of the number of discretization points is discussed. The paper concludes with a numerical example and the presentation of the results of extensive benchmarking.