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.
Highlights
Modal decomposition [1] for fluid dynamics has attracted attention from the viewpoints of data reduction, data analysis, and reduced-order modeling of complex dataset
This fact indicates that extended-Kalman-filter-based DMD (EKFDMD) can be used for noise reduction in the range we investigated for the case in which system noise is present, regardless of its strength
A dynamic mode decomposition method based on the extended Kalman filter (EKFDMD) was proposed for simultaneous parameter estimation and denoising
Summary
Modal decomposition [1] for fluid dynamics has attracted attention from the viewpoints of data reduction, data analysis, and reduced-order modeling of complex dataset. Galerkin projection method for instance, only a numerical approach can be used for reduced-order modeling in this way Another conventional method is global linear stability analysis (GLSA), [4,5,6] which shows that the eigenmodes of the system of linearized governing equations (i.e., the Navier-Stokes equations for most of the fluid problems) around the steady state of nonlinear dynamics. [8,9,10,11] Here, DMD has characteristics of both POD and GLSA, whereas DMD can be computed only by a time-series of snapshots of numerical and experimental data This method processes snapshots of sequential unsteady nonlinear flow fields and yields eigenvalues and corresponding eigenmodes for the case in which the dataset is assumed to be explained by a linear system xk+1 = Axk, where xk is the kth snapshot of sequential data and A is a system matrix. The proposed method is applied to various problems and its performance is illustrated
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