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arXiv · 2509.20596

Data-Driven State Observers for Measure-Preserving Systems

Wentao Tang
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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.

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Wentao Tang. 2025-09-24. Data-Driven State Observers for Measure-Preserving Systems. https://arxiv.org/abs/2509.20596

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