# Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systems

arXiv: https://arxiv.org/abs/2208.13006 · HTML: https://mlsanity.com/paper/2208.13006/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 50 references that are in the map.
- First posted on arXiv 27 Aug 2022.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Towards Robust Neural Networks via Close-loop Control](https://mlsanity.com/paper/2102.01862/) (2021)
- [Offset-free setpoint tracking using neural network controllers](https://mlsanity.com/paper/2011.14006/) (2020)
- [The Impact of Data on the Stability of Learning-Based Control- Extended Version](https://mlsanity.com/paper/2011.10596/) (2020)
- [A Control-Theoretic Perspective on Optimal High-Order Optimization](https://mlsanity.com/paper/1912.07168/) (2019)
- [How Training Data Impacts Performance in Learning-based Control](https://mlsanity.com/paper/2005.12062/) (2020)
- [Safe Learning-based Observers for Unknown Nonlinear Systems using Bayesian Optimization](https://mlsanity.com/paper/2005.05888/) (2020)
- [Solving Optimal Control Problems of Rigid-Body Dynamics with Collisions Using the Hybrid Minimum Principle](https://mlsanity.com/paper/2205.08622/) (2022)
- [Neural Network Based Nonlinear Observers](https://mlsanity.com/paper/2003.07269/) (2020)

## Built on by (most cited first, 2 of 2)

- [The Novel Adaptive Fractional Order Gradient Decent Algorithms Design via Robust Control](https://mlsanity.com/paper/2303.04328/) (2023)
- [Learning Provable Neural Network Observer for Uncertain Dynamical Systems](https://mlsanity.com/paper/2609.30819/) (2026)

## Abstract (arXiv)

In this paper, we propose a novel nonlinear observer based on neural networks, called neural observer, for observation tasks of linear time-invariant (LTI) systems and uncertain nonlinear systems. In particular, the neural observer designed for uncertain systems is inspired by the active disturbance rejection control, which can measure the uncertainty in real-time. The stability analysis (e.g., exponential convergence rate) of LTI and uncertain nonlinear systems (involving neural observers) are presented and guaranteed, where it is shown that the observation problems can be solved only using the linear matrix inequalities (LMIs). Also, it is revealed that the observability and controllability of the system matrices are required to demonstrate the existence of solutions of LMIs. Finally, the effectiveness of neural observers is verified on three simulation cases, including the X-29A aircraft model, the nonlinear pendulum, and the four-wheel steering vehicle.