Extended-Kalman-filter-based dynamic mode decomposition for simultaneous system identification and denoising

Abstract

A new dynamic mode decomposition (DMD) method is introduced for simultaneous system identification and denoising in conjunction with the adoption of an extended Kalman filter algorithm. The present paper explains the extended-Kalman-filter-based DMD (EKFDMD) algorithm which is an online algorithm for dataset for a small number of degree of freedom (DoF). It also illustrates that EKFDMD requires significant numerical resources for many-degree-of-freedom (many-DoF) problems and that the combination with truncated proper orthogonal decomposition (trPOD) helps us to apply the EKFDMD algorithm to many-DoF problems, though it prevents the algorithm from being fully online. The numerical experiments of a noisy dataset with a small number of DoFs illustrate that EKFDMD can estimate eigenvalues better than or as well as the existing algorithms, whereas EKFDMD can also denoise the original dataset online. In particular, EKFDMD performs better than existing algorithms for the case in which system noise is present. The EKFDMD with trPOD, which unfortunately is not fully online, can be successfully applied to many-DoF problems, including a fluid-problem example, and the results reveal the superior performance of system identification and denoising.

Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother

“…Reduced-order Kalman filter-smoother. Considering the reduced-order process and measurement equations (16) and (19), the prediction equations of the Kalman filter (equations (11)) become…”

“…Here, a scheme is proposed to find an estimate to those matrices. Using the reduced-order process and measurement equations (16) and (19)…”

“…[15] proposed a novel DMD method based on a Kalman filter for parameter estimation [15]. They also imtroduced an online algorithm based on the extended Kalman filter and DMD for simultaneous system identification and denoising of datasets with a small number of degrees of freedom [16]. The Kalman filter is a real-time state predictor, i.e., the state is predicted based on observations until the current time.…”

Data-Driven Estimation of the Unsteady Flowfield Near an Actuated Airfoil

“…We refer to this filter as a DMD Kalman Filter (DMD-KF). The DMD-KF is different from [11] [12] in which the Kalman filter is used for obtaining the DMD modes more precisely in the presence of noise. To be useful for estimation, the DMD modes should not only represent the dataset in which DMD was performed, but also the ensemble of flow trajectories possible for the underlying dynamics.…”

Consistent Dynamic Mode Decomposition

“…To address this challenge, variants of DMD were proposed in the literature based on solving jointly for the basis and the evolution operator [36], formulating the problem as a total least squares minimization [17], and fitting an exponential model [1]. Other methods cope with noise by utilizing Kalman filters [26,27], adapting DMD to online data [18,16], and developing a Rayleigh-Ritz modal decomposition [8], among other approaches [7]. Under this classification, our method is applicable to non-sequential data and it performs extremely well when sensor noise corrupts the data, as we show in Section 5.…”