Keynote IUTAM Symposium Tsukuba: Koopman as a gate to Hill, a quest for a stability criterion

17. April 2023

Remco Leine and Fabia Bayer, IUTAM Symposium on Nonlinear dynamics for design of mechanical systems across different length/time scales July 31 - August 4, 2023, Tsukuba, Japan

Remco Leine & Fabia Bayer

The Koopman framework has gained immense popularity in recent years as a versatile tool for various engineering applications, such as system identification, model order reduction and feedback control. This is due to an auspicious promise: A linear (but infinite-dimensional) operator to globally represent a nonlinear system. Classically, the Koopman framework has been developed for time-autonomous systems, whereas nonautonomous systems can generally only be represented with considerable compromises. The application of the Koopman framework to time-nonautonomous systems therefore remains a major challenge.

The Hill stability method is a well-known frequency-based method to obtain stability information of linear time-periodic systems, e.g. systems with parametric excitation such as the Mathieu equation or, more generally, the linearization of systems around a periodic solution. The Hill method is closely related to the Harmonic Balance Method and uses the frequency content of the system to form an infinite dimensional matrix. Its eigenvalues, called Floquet exponents, determine the stability of the linear time-periodic system or, correspondingly, of the periodic solution. An approximation is obtained by truncating the size of the infinite dimensional Hill matrix to a finite dimension. The truncation, however, comes at a price as it may compromise an accurate stability analysis.

This talk is a journey with a demanding mission: Where can we find structural similarities between these two concepts and how can we apply the Koopman framework to nonautonomous time-periodic systems – all with the goal to understand (or even improve) the Hill stability method? 

Link to the IUTAM Symposium

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