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

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Document pages: 6 pages

Abstract: We propose an analytical construction of observable functions in the extendeddynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method forapproximating the spectral properties of the Koopman operator. The choice ofobservable functions is fundamental for the application of EDMD to nonlinearproblems arising in systems and control. Existing methods either start from aset of dictionary functions and look for the subset that best fits theunderlying nonlinear dynamics or they rely on machine learning algorithms to "learn " observable functions. Conversely, in this paper, we start from thedynamical system model and lift it through the Lie derivatives, rendering itinto a polynomial form. This proposed transformation into a polynomial form isexact, and it provides an adequate set of observable functions. The strength ofthe proposed approach is its applicability to a broader class of nonlineardynamical systems, particularly those with nonpolynomial functions andcompositions thereof. Moreover, it retains the physical interpretability of theunderlying dynamical system and can be readily integrated into existingnumerical libraries. The proposed approach is illustrated with an applicationto electric power systems. The modeled system consists of a single generatorconnected to an infinite bus, where nonlinear terms include sine and cosinefunctions. The results demonstrate the effectiveness of the proposed procedurein off-attractor nonlinear dynamics for estimation and prediction; theobservable functions obtained from the proposed construction outperform methodsthat use dictionary functions comprising monomials or radial basis functions.

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