On Analytical Construction of Observable Functions in Extended Dynamic Mode Decomposition for Nonlinear Estimation and Prediction

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, in a certain sense, the underlying nonlinear dynamics; or they rely on machine learning algorithms, e.g., neural networks, to "learn" observable functions that are not explicitly available. Conversely, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This transformation into a polynomial form is exact, although not unique, 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, in which case 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 outperformed existing methods that use dictionary functions comprising monomials or radial basis functions.