Performance guarantees for optimization-based state estimation using turnpike properties

TL;DR

This paper introduces a novel delayed moving horizon estimation (MHE) approach for nonlinear systems, leveraging turnpike properties to provide performance guarantees and significantly improve estimation accuracy with minimal delay.

Contribution

The paper develops a new theoretical framework based on turnpike properties for performance guarantees in nonlinear state estimation, proposing a delayed MHE scheme with near-optimal performance.

Findings

Delayed MHE improves estimation accuracy by 20-25%

Performance bounds are established using turnpike properties

Small delays in MHE significantly enhance practical estimation results

Abstract

In this paper, we develop novel accuracy and performance guarantees for optimal state estimation of general nonlinear systems (in particular, moving horizon estimation, MHE). Our results rely on a turnpike property of the optimal state estimation problem, which essentially states that the omniscient infinite-horizon solution involving all past and future data serves as turnpike for the solutions of finite-horizon estimation problems involving a subset of the data. This leads to the surprising observation that MHE problems naturally exhibit a leaving arc, which may have a strong negative impact on the estimation accuracy. To address this, we propose a delayed MHE scheme, and we show that the resulting performance (both averaged and non-averaged) is approximately optimal and achieves bounded dynamic regret with respect to the infinite-horizon solution, with error terms that can be made arbitrarily small by an appropriate choice of the delay. In various simulation examples, we observe that already a very small delay in the MHE scheme is sufficient to significantly improve the overall estimation error by 20-25 % compared to standard MHE (without delay). This finding is of great importance for practical applications (especially for monitoring, fault detection, and parameter estimation) where a small delay in the estimation is rather irrelevant but may significantly improve the estimation results.