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This paper focuses on the model-free synthesis of state observers fornonlinear autonomous systems without knowing the governing equations.Specifically, the Kazantzis-Kravaris/Luenberger (KKL) observer structure isleveraged, where the outputs are fed into a linear time-invariant (LTI) systemto obtain the observer states, which can be viewed as the states nonlinearlytransformed by an immersion mapping, and a neural network is used toapproximate the inverse of the nonlinear immersion and estimate the states. Inview of the possible existence of noises in output measurements, this workproposes to impose an upper bound on the Lipschitz constant of the neuralnetwork for robust and safe observation. A relation that bounds thegeneralization loss of state observation according to the Lipschitz constant,as well as the $H_2$-norm of the LTI part in the KKL observer, is established,thus reducing the model-free observer synthesis problem to that ofLipschitz-bounded neural network training, for which a direct parameterizationtechnique is used. The proposed approach is demonstrated on a chaotic Lorenzsystem.

Synthesis of Data-Driven Nonlinear State Observers using Lipschitz-Bounded Neural Networks