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Robust Data-Driven Moving Horizon Estimation for Linear Discrete-Time Systems

Tobias M. Wolff, Victor G. Lopez, Matthias A. Müller

TL;DR

The paper tackles robust state estimation for linear time-invariant discrete-time systems without identifying a model, using a data-driven moving horizon estimation (MHE) approach grounded in Willems' fundamental lemma. It builds a robust optimization framework that combines offline noisy state/output data with online measurements, employing Hankel representations and slack variables to handle data imperfections, and proves practical robust exponential stability ($pRES$) under incremental detectability (e-UOSS). Key contributions include the data-driven MHE formulation with prior and regularization terms that bound the impact of offline noise, a derivation of sufficient conditions for $pRES$, and a comprehensive numerical study on a four-tank system comparing data-driven MHE to model-based MHE variants (SID/PEM). The results demonstrate that the data-driven method can be competitive with, and sometimes superior to, traditional model-based schemes, especially when the offline data are sufficiently informative, highlighting the practical value of direct data-driven estimation in settings with limited online sensing or evolving models.

Abstract

In this paper, a robust data-driven moving horizon estimation (MHE) scheme for linear time-invariant discrete-time systems is introduced. The scheme solely relies on offline collected data without employing any system identification step. We prove practical robust exponential stability for the setting where both the online measurements and the offline collected data are corrupted by non-vanishing and bounded noise. The behavior of the novel robust data-driven MHE scheme is illustrated by means of simulation examples and compared to a standard model-based MHE scheme, where the model is identified using the same offline data as for the data-driven MHE scheme.

Robust Data-Driven Moving Horizon Estimation for Linear Discrete-Time Systems

TL;DR

The paper tackles robust state estimation for linear time-invariant discrete-time systems without identifying a model, using a data-driven moving horizon estimation (MHE) approach grounded in Willems' fundamental lemma. It builds a robust optimization framework that combines offline noisy state/output data with online measurements, employing Hankel representations and slack variables to handle data imperfections, and proves practical robust exponential stability () under incremental detectability (e-UOSS). Key contributions include the data-driven MHE formulation with prior and regularization terms that bound the impact of offline noise, a derivation of sufficient conditions for , and a comprehensive numerical study on a four-tank system comparing data-driven MHE to model-based MHE variants (SID/PEM). The results demonstrate that the data-driven method can be competitive with, and sometimes superior to, traditional model-based schemes, especially when the offline data are sufficiently informative, highlighting the practical value of direct data-driven estimation in settings with limited online sensing or evolving models.

Abstract

In this paper, a robust data-driven moving horizon estimation (MHE) scheme for linear time-invariant discrete-time systems is introduced. The scheme solely relies on offline collected data without employing any system identification step. We prove practical robust exponential stability for the setting where both the online measurements and the offline collected data are corrupted by non-vanishing and bounded noise. The behavior of the novel robust data-driven MHE scheme is illustrated by means of simulation examples and compared to a standard model-based MHE scheme, where the model is identified using the same offline data as for the data-driven MHE scheme.
Paper Structure (9 sections, 3 theorems, 71 equations, 1 figure, 1 table)

This paper contains 9 sections, 3 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. PRELIMINARIES & SETUP
  3. PRACTICAL RES FOR NOISY OFFLINE STATE & OUTPUT MEASUREMENTS
  4. APPLICATION TO FOUR-TANK SYSTEM
  5. Linear System
  6. Nonlinear System
  7. CONCLUSION
  8. Proof of Theorem \ref{['thm:pRGES']}
  9. Data-Driven Computation of e-UOSS Constants

Key Result

Lemma 1

Suppose $\{x(k)\}_{k=0}^{N-1}$, $\{u(k)\}_{k=0}^{N-1}$ is an input/state trajectory of the controllable LTI system (system) where $\{u(k)\}_{k=0}^{N-1}$ is persistently exciting of order $L + n$, then the matrix has full row rank.

Figures (1)

  • Figure 1: The simulation results illustrate the performance of the robust data-driven MHE scheme (where the estimated states are denoted by $\hat{x}^{\mathrm{DB}}$) defined in (\ref{['MHE_noisy']}) compared to a model-based MHE scheme (where the model is identified by SID techniques using the same data used in the data-driven MHE scheme and where the estimated states are denoted by $\hat{x}^{\mathrm{MB, SID}}$) for system (\ref{['example_system']}). The offline input is drawn from a uniform random distribution $\mathcal{U}(0,10)$ and $R$ is chosen as $R = 100$.

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Theorem 2
  • Remark 3
  • Remark 4