Koopman-based stability analysis

Abstract

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Abstract

In this chapter, the performance of a novel method to determine the stability of periodic solutions is analyzed. Using the Koopman framework, the linear time-periodic nonautonomous perturbed dynamics around a periodic solution can be approximated by a linear autonomous system of higher order, whose system matrix is the well-known Hill matrix. The evaluation of the closed-form solution reveals that the monodromy matrix can be approximated by the Hill matrix, using only a matrix exponential and a projection instead of solving a large eigenvalue problem or integrating numerically over one period. There does not exist a unique choice for the projection of the Hill matrix to the monodromy matrix. In this chapter, various ways to obtain a suitable projection are discussed. The computational efficiency of the novel method is analyzed for the vertically excited multiple pendulum, comparing the effect of the considered projection options.