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Designing Luenberger observers for nonlinear systems involves the challenging
task of transforming the state to an alternate coordinate system, possibly of
higher dimensions, where the system is asymptotically stable and linear up to
output injection. The observer then estimates the system's state in the
original coordinates by inverting the transformation map. However, finding a
suitable injective transformation whose inverse can be derived remains a
primary challenge for general nonlinear systems. We propose a novel approach
that uses supervised physics-informed neural networks to approximate both the
transformation and its inverse. Our method exhibits superior generalization
capabilities to contemporary methods and demonstrates robustness to both neural
network's approximation errors and system uncertainties.
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