# Empirical Observability Gramian for Stochastic Observability of Nonlinear Systems

arXiv: https://arxiv.org/abs/2006.07451 · HTML: https://mlsanity.com/paper/2006.07451/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2020.

## Built on by (most cited first, 3 of 3)

- [Data-Driven Observability Analysis for Nonlinear Stochastic Systems](https://mlsanity.com/paper/2302.11979/) (2023)
- [Training Observable Control Policies to Expose Agent State Through Actions](https://mlsanity.com/paper/2606.27609/) (2026)
- [Information-Guided Safe Reinforcement Learning for Autonomous Gas Source Localization using sUAS](https://mlsanity.com/paper/2609.05569/) (2026)

## Abstract (arXiv)

We extend observability metrics based on the empirical observability Gramian from deterministic nonlinear systems to nonlinear stochastic systems in order to capture the impact of process noise on observability. We demonstrate that the empirical observability Gramian can be used to provide an equivalent condition for a definition of stochastic observability on linear systems, and that the Gramian can be used to extend stochastic observability to nonlinear stochastic systems. We further demonstrate through simulation that consideration of process noise can reveal observability in systems that would be considered unobservable using traditional deterministic tools.