# Learning-based Design of Luenberger Observers for Autonomous Nonlinear Systems

arXiv: https://arxiv.org/abs/2210.01476 · HTML: https://mlsanity.com/paper/2210.01476/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 33 references that are in the map.
- First posted on arXiv 4 Oct 2022.

## Builds on (most cited first, 11 of 11)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [Generalization in Deep Learning](https://mlsanity.com/paper/1710.05468/) (2017)
- [Robust Large Margin Deep Neural Networks](https://mlsanity.com/paper/1605.08254/) (2016)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Generalization Error in Deep Learning](https://mlsanity.com/paper/1808.01174/) (2018)
- [Towards gain tuning for numerical KKL observers](https://mlsanity.com/paper/2204.00318/) (2022)

## Built on by (most cited first, 11 of 11)

- [Learning Robust State Observers using Neural ODEs (longer version)](https://mlsanity.com/paper/2212.00866/) (2022)
- [KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning](https://mlsanity.com/paper/2501.11655/) (2025)
- [Sensor Fault Detection and Isolation in Autonomous Nonlinear Systems Using Neural Network-Based Observers](https://mlsanity.com/paper/2304.08837/) (2023)
- [Learning Hybrid Dynamics Models With Simulator-Informed Latent States](https://mlsanity.com/paper/2309.02873/) (2023)
- [Adaptive Meta-Learning-Based KKL Observer Design for Nonlinear Dynamical Systems](https://mlsanity.com/paper/2310.19489/) (2023)
- [PINN-Obs: Physics-Informed Neural Network-Based Observer for Nonlinear Dynamical Systems](https://mlsanity.com/paper/2507.06712/) (2025)
- [Designing Robust Software Sensors for Nonlinear Systems via Neural Networks and Adaptive Sliding Mode Control](https://mlsanity.com/paper/2507.06817/) (2025)
- [HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning](https://mlsanity.com/paper/2602.22630/) (2026)
- [Verifiable Error Bounds for Physics-Informed Neural KKL Observers](https://mlsanity.com/paper/2603.20434/) (2026)
- [HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning](https://mlsanity.com/paper/2603.29744/) (2026)
- [Correct then Forecast: Observer State-Space Models for Time Series Forecasting](https://mlsanity.com/paper/2609.33566/) (2026)

## Abstract (arXiv)

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.