Mathematics PhD Thesis Defense by Nicat Aliyev

KOÇ UNIVERSITY

GRADUATE SCHOOL OF SCIENCES & ENGINEERING

MATHEMATICS

PhD THESIS DEFENSE BY NİCAT ALIYEV

Title: Approximation and Minimization of the H-infinity Norms of Large-Scale Control Systems

Speaker: Nicat Aliyev

Time: January 25, 2018, 10:30

Place: ENG 127

Koç University

Rumeli Feneri Yolu

Sariyer, Istanbul

Thesis Committee Members:

Assoc. Prof.Dr. Emre Mengi (Advisor, Koc University)

Prof. Dr. Attila Aşkar (Koc University)

Prof. Dr. Alper Demir (Koc University)

Asst. Prof. Dr. Fatih Ecevit (Boğaziçi University)

Asst. Prof. Dr. Hamdullah Yücel (Middle East Technical University)

Abstract:

The H-infinity norm of the transfer function of a control system is more than a norm on the operator that maps the input of the system to its output; it is a metric that is often associated with the robust stability of the system. For instance, for a standard time-invariant linear system, it is the reciprocal of a structured stability radius, and such interpretations carry over to many other control systems including delay systems.

The first part of this thesis work concerns the computation of the H-infinity norm for a large-scale system. We derive a subspace framework that is based on tools from model order reduction. In particular, we restrict the state space to a small subspace, and impose the Petrov-Galerkin conditions with respect to another small subspace on the differential part of the system. The resulting small-scale problem is solved by existing methods, then the two subspaces are expanded with the inclusion of certain singular vectors of the transfer function evaluated on the imaginary axis. We prove rigorously that our expansion strategy leads to a superlinear rate-of-convergence with respect to the dimensions of the subspaces. This results in significant run-time  speed-ups compared to best existing methods, often ten times or more.

The second part translates the ideas in the first part to the setting ofH-infinity norm minimization for a large-scale parametrized descriptor system over a set of admissible parameter values. As in the first part, state space is restricted to a small subspace and Petrov-Galerkin conditions are imposed. Now the small-scale problems involve H-infinity norm minimization over the same admissible values of the parameters. We again devise a subspace expansion strategy that leads to superlinear convergence, which we prove in theory and observe in practice. The H-infinity norm minimization problem comes with additional challenges, since this is a minmax problem; for instance the proof of superlinear convergence treats the maximizers of the inner problem implicitly as a function of the parameters.

The last part turns attention to a special type of descriptor systems with a particular structure, namely port-Hamiltonian (PH) systems, which arise from various applications  such as circuit simulation models and the brake squeal problem. A particular notion of a stability radius of a PH system is linked to the H-infinity norm of a descriptor system, so we adapt the subspace framework from the first part for the computation of this stability  radius for a large-scale PH system. We employ structure-preserving model reduction techniques, meaning the subspace are designed in a way to ensure that small-scale problems also have PH structure. All of the approaches are tested extensively on synthetic examples, as well as several examples that originate from real applications.