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Optimal state estimation: Turnpike analysis and performance results

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

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.

Optimal state estimation: Turnpike analysis and performance results

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.
Paper Structure (12 sections, 5 theorems, 31 equations, 2 figures, 1 table)

This paper contains 12 sections, 5 theorems, 31 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. System description
  4. Optimal state estimation
  5. Turnpike in optimal state estimation
  6. Motivating example
  7. Turnpike under a decaying sensitivity condition
  8. Performance of optimal state estimation
  9. Approximation properties and performance guarantees
  10. Implications for moving horizon estimation
  11. Numerical example
  12. Conclusion

Key Result

Lemma 1

Any minimizer of ${P}(d_{0:T})$ is a minimizer of $\bar{P}(d_{\tau:\tau+N},x^*_\tau,x^*_{\tau+N})$ for any $N\in\mathbb{I}_{[0,T]}$, $\tau\in\mathbb{I}_{[0,T-N]}$.

Figures (2)

  • Figure 1: Difference between the optimal solution $x^*_j$ involving the full data batch with $T=70$ and the solution $\hat{x}^*_j$ of the truncated problem, plotted over $j\in\mathbb{I}_{[\tau,\tau+N]}$ for $N{=\,}5$ (cyan), $N{=\,}10$ (magenta), $N{=\,}15$ (blue), and $N{=\,}20$ (red) for $\tau{\;=\;}0$ (left), $\tau{\;=\;}\lfloor ({T{-\,}N})/2 \rfloor$ (middle), and $\tau{\;=\;}T{-\,}N$ (right).
  • Figure 2: Performance $J_T$ of the approximate estimator $\hat{z}^\mathrm{ae}_{0:T}$ (blue) and MHE $\hat{z}^\mathrm{mhe}_{0:T}$ (green) for different lengths $N$ of the truncated problems $P_N$ compared to the optimal performance of the FIE solution $z^*_{0:T}$ (red).

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Theorem 1: Decaying sensitivity implies turnpike
  • proof
  • Remark 1: Turnpike at the boundaries
  • Remark 2: Motivating example revisited
  • Remark 3: Turnpike via strict dissipativity
  • Proposition 1
  • proof
  • Theorem 2: Performance
  • ...and 3 more