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Designing Luenberger observers for nonlinear systems involves the challengingtask of transforming the state to an alternate coordinate system, possibly ofhigher dimensions, where the system is asymptotically stable and linear up tooutput injection. The observer then estimates the system's state in theoriginal coordinates by inverting the transformation map. However, finding asuitable injective transformation whose inverse can be derived remains aprimary challenge for general nonlinear systems. We propose a novel approachthat uses supervised physics-informed neural networks to approximate both thetransformation and its inverse. Our method exhibits superior generalizationcapabilities to contemporary methods and demonstrates robustness to both neuralnetwork's approximation errors and system uncertainties.

Learning-based Design of Luenberger Observers for Autonomous Nonlinear Systems