# Towards gain tuning for numerical KKL observers

arXiv: https://arxiv.org/abs/2204.00318 · HTML: https://mlsanity.com/paper/2204.00318/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2022.

## Built on by (most cited first, 14 of 14)

- [Learning-based Design of Luenberger Observers for Autonomous Nonlinear Systems](https://mlsanity.com/paper/2210.01476/) (2022)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [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)
- [Learning Reduced Nonlinear State-Space Models: an Output-Error Based Canonical Approach](https://mlsanity.com/paper/2206.04791/) (2022)
- [Data-Driven Observability Analysis for Nonlinear Stochastic Systems](https://mlsanity.com/paper/2302.11979/) (2023)
- [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)

## Abstract (arXiv)

This paper presents a first step towards tuning observers for general nonlinear systems. Relying on recent results around Kazantzis-Kravaris/Luenberger (KKL) observers, we propose an empirical criterion to guide the calibration of the observer, by trading off transient performance and sensitivity to measurement noise. We parametrize the gain matrix and evaluate this criterion over a family of observers for different parameter values. We then use neural networks to learn the mapping between the observer and the nonlinear system, and present a novel method to sample the state-space efficiently for nonlinear regression. We illustrate the merits of this approach in numerical simulations.