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Performance guarantees for optimization-based state estimation using turnpike properties

Julian D. Schiller, Lars Grüne, and Matthias A. Müller

TL;DR

The paper tackles the challenge of accuracy and performance in nonlinear state estimation when using optimization-based methods like moving horizon estimation (MHE). It identifies a turnpike property, whereby the acausal infinite-horizon solution serves as a reference for finite-horizon problems, which creates a leaving arc that can degrade estimation quality. To counteract this, the authors propose a delayed MHE scheme ($\delta$MHE) and a turnpike prior for MHE with prior weighting, proving that the delayed scheme achieves bounded dynamic regret and approximately matches the infinite-horizon benchmark as the delay increases, with exponentially fast convergence under an exponential turnpike assumption. They also provide offline and online results, complemented by extensive simulations (batch reactor, CSTR, quadrotor, and large-scale LTI problems) showing that even small delays yield substantial improvements (20–25%) over standard MHE. Overall, the work offers principled guarantees and practical strategies to leverage turnpike structure for more accurate and efficient state estimation in nonlinear systems.

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.

Performance guarantees for optimization-based state estimation using turnpike properties

TL;DR

The paper tackles the challenge of accuracy and performance in nonlinear state estimation when using optimization-based methods like moving horizon estimation (MHE). It identifies a turnpike property, whereby the acausal infinite-horizon solution serves as a reference for finite-horizon problems, which creates a leaving arc that can degrade estimation quality. To counteract this, the authors propose a delayed MHE scheme (MHE) and a turnpike prior for MHE with prior weighting, proving that the delayed scheme achieves bounded dynamic regret and approximately matches the infinite-horizon benchmark as the delay increases, with exponentially fast convergence under an exponential turnpike assumption. They also provide offline and online results, complemented by extensive simulations (batch reactor, CSTR, quadrotor, and large-scale LTI problems) showing that even small delays yield substantial improvements (20–25%) over standard MHE. Overall, the work offers principled guarantees and practical strategies to leverage turnpike structure for more accurate and efficient state estimation in nonlinear systems.

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.

Paper Structure

This paper contains 22 sections, 7 theorems, 56 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. System description
  4. Optimal state estimation
  5. Benchmark solution
  6. Turnpike in optimal state estimation problems
  7. Motivating example
  8. Turnpike characterization
  9. Optimization-based state estimation
  10. A delay improves the estimation results
  11. Performance estimates
  12. MHE with prior weighting
  13. Offline state estimation
  14. Good performance implies accurate state estimates
  15. Numerical examples
  16. ...and 7 more sections

Key Result

Proposition 1

Let Assumption ass:turnpike_inf hold and $\mathcal{X}$ be compact. Then, there exists $\sigma\in\mathcal{L}$ such that the estimated state sequence of $\delta$MHE in eq:MHE_delay_x satisfies for all $t\in\mathbb{I}_{\geq\delta}$, $\delta\in[0,N/2]$, $N\in\mathbb{I}_{\geq0}^\mathrm{e}$, and all $d_{-\infty:\infty}$.

Figures (8)

  • Figure 1: Difference between the solution of the infinite-horizon problem $P_\infty$ and solutions of the finite-horizon problem $P_N$ for different values of $N$.
  • Figure 2: Sketch of the infinite-horizon solution $x^\infty$ (blue), the current FIE solution $\zeta_t(j,d_{0:t})$ (green), and solutions of the finite-horizon estimation problem $\zeta_N(j,d_{\tau-N:\tau})$ for different values of $\tau$ (red).
  • Figure 3: Performance $J_{[0,T]}$ of the AE $\hat{x}^\mathrm{ae}_{0:T}$ (cyan) and MHE $\hat{x}^\mathrm{mhe}_{0:T}$ (red) for different lengths $N$ of the problem $P_N$ compared to the performance achieved by full solution $\hat{x}^*_{0:T}$ (green).
  • Figure 4: CSTR. Finite-horizon solutions $\tilde{x}^*_{j|t}$ compared to $x^\infty_j$, $j\in\mathbb{I}_{[t-N,t]}$ for $t\in\mathbb{I}_{[130,180]}$ using the filtering prior (blue), smoothing prior (black), and turnpike prior (red); highlighted are particular solutions obtained at $t=149$ and $t=169$.
  • Figure 5: CSTR. Distance between state estimates using different MHE schemes and the IHE. Dots indicate values at time $t$, lines indicate the moving average over a sliding window of size $N+1$.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Remark 1: Existence of solutions to $P_N$
  • Definition 1: Turnpike for optimal state estimation
  • Remark 2: Approaching and leaving arcs
  • Remark 3: $\delta$FIE
  • Remark 4: Smoothing form of MHE
  • Proposition 1
  • proof
  • Theorem 1: Performance of $\delta$MHE
  • Remark 5: Performance of $\delta$MHE
  • Lemma 1
  • ...and 11 more