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

This work employs a distributional, unifying framework for the analysis and design of moving-horizon estimation filters with stability and differential-privacy properties. 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 distributional 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 data generated by the estimator, based on an entropic 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.