# MAP moving horizon estimation for threshold measurements with application to field monitoring

## TL;DR

This paper introduces a convex optimization-based MAP moving horizon estimator for spatially distributed systems with threshold sensors, demonstrating its effectiveness and computational efficiency through PDE-based simulations.

## Contribution

It develops a convex optimization approach for MAP moving horizon estimation applicable to large-scale spatial systems and PDEs, with a noise-assisted accuracy phenomenon.

## Key findings

- Estimator solves convex optimization problems at each step.
- Feasible for large-scale PDE-based spatial systems.
- Estimation accuracy improves with measurement noise.

## Abstract

The paper deals with state estimation of a spatially distributed system given noisy measurements from pointwise-in-time-and-space threshold sensors spread over the spatial domain of interest. A Maximum A posteriori Probability (MAP) approach is undertaken and a Moving Horizon (MH) approximation of the MAP cost-function is adopted. It is proved that, under system linearity and log-concavity of the noise probability density functions, the proposed MH-MAP state estimator amounts to the solution, at each sampling interval, of a convex optimization problem. Moreover, a suitable centralized solution for large-scale systems is proposed with a substantial decrease of the computational complexity. The latter algorithm is shown to be feasible for the state estimation of spatially-dependent dynamic fields described by Partial Differential Equations (PDE) via the use of the Finite Element (FE) spatial discretization method. A simulation case-study concerning estimation of a diffusion field is presented in order to demonstrate the effectiveness of the proposed approach. Quite remarkably, the numerical tests exhibit a noise-assisted behavior of the proposed approach in that the estimation accuracy results optimal in the presence of measurement noise with non-null variance.