On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems

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 is corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite dimensional approximation of deterministic system to a random uncertain case. However, these results hold true 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 least square problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent tradeoffs between the quality of approximation and complexity of approximation. These tradeoffs 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 the Extended Dynamic Mode Decomposition (EDMD) and DMD algorithms for a system with process and measurement noise.

arXiv admin note: text overlap with arXiv:1803.08558

References in corpus (7)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
• Applied Koopmanism
• De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis
• Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition
• Subspace dynamic mode decomposition for stochastic Koopman analysis
• Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition

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