# A Probabilistic Framework for Moving-Horizon Estimation: Stability and Privacy Guarantees

## TL;DR

This paper introduces a probabilistic framework for stable moving-horizon estimation of nonlinear systems, incorporating differential privacy, and compares two variants based on Wasserstein distance and KL-divergence.

## Contribution

It unifies stability analysis and privacy guarantees in moving-horizon estimators using probabilistic metrics and introduces two novel estimator variants.

## Key findings

- W2-MHE provides a gradient-based estimation approach.
- KL-MHE functions as a particle filter with stability properties.
- Differential privacy can be achieved through entropy regularization.

## Abstract

This work proposes a unifying probabilistic framework for the design of robustly asymptotically stable moving-horizon estimators (MHE) for discrete-time nonlinear systems, and a mechanism to incorporate differential privacy in moving-horizon estimation. We begin with an investigation of the classical notion of strong local observability of nonlinear systems and its relationship to optimization-based state estimation. We then present a general moving-horizon estimation framework for strongly locally observable systems, as an iterative minimization scheme in the space of probability measures. This framework allows for the minimization of the estimation cost with respect to different metrics. In particular, we consider two variants, which we name $W_2$-MHE and KL-MHE, where the minimization scheme uses the 2-Wasserstein distance and the KL-divergence, respectively. The $W_2$-MHE yields a gradient-based estimator whereas the KL-MHE yields a particle filter, for which we investigate asymptotic stability and robustness properties. Stability results for these moving-horizon estimators are derived in the probabilistic setting, against the backdrop of the classical notion of strong local observability which, to the best of our knowledge, differentiates it from other previous works. We then propose a mechanism to encode differential privacy of the measurements used by the estimator via an entropy regularization of the MHE objective functional. In particular, we find sufficient bounds on the regularization parameter to achieve the desired level of differential privacy. Numerical simulations demonstrate the performance of these estimators.