On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems
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On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems Subhrajit Sinha1
· Bowen Huang2 · Umesh Vaidya2
Received: 3 April 2018 / Accepted: 2 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In the paper, we consider the problem of robust approximation of transfer Koopman and Perron–Frobenius (P–F) operators from noisy time-series data. In most applications, the time-series data obtained from simulation or experiment are corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite-dimensional approximation of the deterministic system to a random uncertain case. However, these results hold only in asymptotic and under the assumption of infinite data set. In practice, the data set is finite, and hence it is important to develop algorithms that explicitly account for the presence of uncertainty in data set. We propose a robust optimization-based framework for the robust approximation of the transfer operators, where the uncertainty in data set is treated as deterministic norm bounded uncertainty. The robust optimization leads to a min–max type optimization problem for the approximation of transfer operators. This robust optimization problem is shown to be equivalent to regularized least-square problem. This equivalence between robust optimization problem and regularized leastsquare problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent trade-offs between the quality of approximation and complexity of approximation. These trade-offs are necessary to balance for the proposed application of transfer operators, for the design of optimal predictor. Simulation results demonstrate that our proposed robust approximation algorithm performs better than some of the existing algorithms like extended dynamic mode decomposition (EDMD), subspace DMD, noise-corrected DMD, and total DMD for systems with process and measurement noise. Keywords Data driven analysis · Koopman operator · Robust Koopman operator · Operator theoretic methods · Dynamical systems
Communicated by Dr. Alain Goriely. Financial support from the Department of Energy DOE Grant DE-OE0000876 is gratefully acknowledged. Extended author information available on the last page of the article
123
Journal of Nonlinear Science
Mathematics Subject Classification 37C30 · 37M10 · 37N35 · 37N40 · 70G60
1 Introduction In recent years, there has been an increased research trend toward the application of transfer operator theoretic methods involving transfer Perron–Frobenius and Koopman operators for the analysis and control of nonlinear systems (Dellnitz and Junge 1999; Mezic and Banaszuk 2000; Froyland 2001; Junge and Osinga 2004; Mezi´c and Banaszuk 2004; Dellnitz et al. 2005; Mezi´c 2005; Mehta and Vaidya 2005; Vaidya and Mehta 2008; Raghunathan and Vaidya 2014; Susuki and Mezic 2011; Budisi
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