On analytical construction of observable functions in extended dynamic mode decomposition for nonlinear estimation and prediction

Marcos Netto, Yoshihiko Susuki, Venkat Krishnan, Yingchen Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Scopus citations

Abstract

We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for the application of EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to 'learn' observable functions. Conversely, in this paper, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact, and it provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. The proposed approach is illustrated with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.

Original languageEnglish (US)
Title of host publication2021 American Control Conference, ACC 2021
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages4190-4195
Number of pages6
ISBN (Electronic)9781665441971
DOIs
StatePublished - May 25 2021
Externally publishedYes
Event2021 American Control Conference, ACC 2021 - Virtual, New Orleans, United States
Duration: May 25 2021May 28 2021

Publication series

NameProceedings of the American Control Conference
Volume2021-May
ISSN (Print)0743-1619

Conference

Conference2021 American Control Conference, ACC 2021
Country/TerritoryUnited States
CityVirtual, New Orleans
Period5/25/215/28/21

All Science Journal Classification (ASJC) codes

  • Electrical and Electronic Engineering

Keywords

  • EDMD
  • Extended dynamic mode decomposition
  • Koopman spectral analysis
  • Lie derivative
  • nonlinear estimation and prediction
  • observable function
  • polynomialization

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