EDMD-Based Robust Observer Synthesis for Nonlinear Systems

Abstract

This paper presents a data-driven approach for designing state observers for continuous-time nonlinear systems, where an extended dynamic mode decomposition (EDMD) procedure is used to identify an approximate linear lifted model. Since such a model on a finite-dimensional space spanned by the dictionary functions has an inevitable mismatch, we first establish, based on our theory of reproducing kernel Hilbert space with a linear--radial kernel, that the nonlinear error magnitude in the approximate linear model is sectorially bounded by the lifted state. The sector bound comprises a deterministic part due to the finite dictionary and a stochastic part due to the random data samples, and the observer design needs to account for both of these errors in a robust formulation. Hence, the observer synthesis is performed using linear matrix inequalities (LMIs), specified by the desired exponential decay rate of the observation error (when the system is asymptotically stable) or the L2 gain from the modeling error to the observation error. Numerical studies demonstrate the effectiveness and flexibility of the proposed method. As such, this work entails an explicit elementary use of linear systems theory for nonlinear state observation in a Koopman operator-theoretic framework.