Optimal state estimation: Turnpike analysis and performance results

TL;DR

This paper applies turnpike theory to optimal state estimation, demonstrating that solutions to truncated problems approximate the full solution with controllable error, thus providing performance guarantees and practical insights for moving horizon estimation.

Contribution

It introduces turnpike arguments into state estimation, formalizes the phenomenon, and derives conditions for approximation performance guarantees.

Findings

Turnpike property holds for optimal state estimation solutions.

Approximate solutions from truncated problems are nearly optimal.

Performance errors decrease with longer horizons.

Abstract

In this paper, we introduce turnpike arguments in the context of optimal state estimation. In particular, we show that the optimal solution of the state estimation problem involving all available past data serves as turnpike for the solutions of truncated problems involving only a subset of the data. We mathematically formalize this phenomenon and derive a sufficient condition that relies on a decaying sensitivity property of the underlying nonlinear program. As second contribution, we show how a specific turnpike property can be used to establish performance guarantees when approximating the optimal solution of the full problem by a sequence of truncated problems, and we show that the resulting performance (both averaged and non-averaged) is approximately optimal with error terms that can be made arbitrarily small by an appropriate choice of the horizon length. In addition, we discuss interesting implications of these results for the practically relevant case of moving horizon estimation and illustrate our results with a numerical example.