Computation of Extremum Singular Values and the Strong H-infinity Norm of SISO Time-Delay Systems

• Published in: arXiv.org
• Main Authors: Gumussoy, Suat, Michiels, Wim
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.03.2020
• Subjects: Algebra; Algorithms; Computation; Computer Science - Systems and Control; Differential equations; Eigenvalues; Feedback control; H-infinity control; Iterative methods; Matrix; Norms; Numerical analysis; Numerical methods; Predictor-corrector methods; Time delay systems; Transcendental functions; Transfer functions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2003.10048

We consider the computation of H-infinity norms for Single-Input-Single-Output (SISO) time-delay systems, which are described by delay differential algebraic equations. Unlike the iterative level set methods in the literature, we present a novel numerical method to compute the H-infinity norm. This method requires solving one eigenvalue problem of at most twice the size of the eigenvalue problem in every iteration of a level set method, but in practice often considerably lower. We first show that the computation of extrema of the transfer function can be turned into the computation of the imaginary axis zeros of a transcendental function. We compute these zeros by a predictor-corrector type algorithm. It is known that the H-infinity norm of delay differential algebraic systems, which can model both retarded and neutral type systems, might be sensitive with respect to arbitrarily small delay perturbations. This recently led to the concept of strong H-infinity norms, which explicitly take into account such small delay perturbations. We present a direct numerical method to compute the strong H-infinity norm of SISO time-delay systems. Our algorithm is applicable to the closed-loop system of interconnections (series, parallel, feedback, junctions) of time-delay systems and/or controllers.