Data-Driven State Observers for Measure-Preserving Systems

Abstract

The use of data-driven control strategies on systems with not fully measurable states induces the problem of learning-based state observation. Motivated by this need, the present work proposes a data-driven approach for the synthesis of state observers for discrete-time nonlinear systems with measure-preserving dynamics. To this end, Kazantzis--Kravaris/Luenburger (KKL) observers are shown to be well-defined, where the observer design boils down to determining a nonlinear injective mapping of states and its pseudo-inverse. For its learning-based construction, the KKL observer is related to the Koopman operator, well-defined on the square-integrable function space and restrictable to a Sobolev-type reproducing kernel Hilbert space (RKHS). Hence, observer synthesis algorithms, based on kernel interpolation/regression routines for the desired injective mapping in the observer and its pseudo-inverse, are proposed in various settings of the available dataset -- (i) many orbits, (ii) single long orbit, and (iii) snapshots. Theoretical error analyses are provided, and numerical studies on a chaotic Lorenz system are demonstrated.

30 pages, 11 figures, submitted to IEEE Transactions on Cybernetics on August 11, 2026. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible