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Robust Data-Driven Kalman Filtering for Unknown Linear Systems using Maximum Likelihood Optimization

Peihu Duan, Tao Liu, Yu Xing, Karl Henrik Johansson

TL;DR

This work addresses state estimation for unknown linear systems with both process and measurement noise by exploiting a pre-collected high-frequency input-output trajectory and a lower-frequency state trajectory. It introduces the Robust Data-Driven Kalman Filter (RDKF), built from a maximum-likelihood formulation and a recursive, tractable modification that yields state estimates without knowing $A$, $B$, or $C$; it also derives a data-driven sample-complexity bound that shows the estimation gap to the model-based Kalman filter vanishes as the amount of pre-collected data grows. The analysis covers data informativity, matrix-identification error bounds, and conditions under which the learned filter remains observable and stable, with extensions to cases where only input-output data are available, yielding a balanced realization suitable for LQG control. Numerical experiments on a chemical reactor (CSTR) demonstrate improved filtering performance with more pre-collected data and highlight robustness to data noise, validating the theoretical guarantees and practical applicability in scenarios where state sampling is slower than input-output measurements.

Abstract

This paper investigates the state estimation problem for unknown linear systems subject to both process and measurement noise. Based on a prior input-output trajectory sampled at a higher frequency and a prior state trajectory sampled at a lower frequency, we propose a novel robust data-driven Kalman filter (RDKF) that integrates model identification with state estimation for the unknown system. Specifically, the state estimation problem is formulated as a non-convex maximum likelihood optimization problem. Then, we slightly modify the optimization problem to get a problem solvable with a recursive algorithm. Based on the optimal solution to this new problem, the RDKF is designed, which can estimate the state of a given but unknown state-space model. The performance gap between the RDKF and the optimal Kalman filter based on known system matrices is quantified through a sample complexity bound. In particular, when the number of the pre-collected states tends to infinity, this gap converges to zero. Finally, the effectiveness of the theoretical results is illustrated by numerical simulations.

Robust Data-Driven Kalman Filtering for Unknown Linear Systems using Maximum Likelihood Optimization

TL;DR

This work addresses state estimation for unknown linear systems with both process and measurement noise by exploiting a pre-collected high-frequency input-output trajectory and a lower-frequency state trajectory. It introduces the Robust Data-Driven Kalman Filter (RDKF), built from a maximum-likelihood formulation and a recursive, tractable modification that yields state estimates without knowing , , or ; it also derives a data-driven sample-complexity bound that shows the estimation gap to the model-based Kalman filter vanishes as the amount of pre-collected data grows. The analysis covers data informativity, matrix-identification error bounds, and conditions under which the learned filter remains observable and stable, with extensions to cases where only input-output data are available, yielding a balanced realization suitable for LQG control. Numerical experiments on a chemical reactor (CSTR) demonstrate improved filtering performance with more pre-collected data and highlight robustness to data noise, validating the theoretical guarantees and practical applicability in scenarios where state sampling is slower than input-output measurements.

Abstract

This paper investigates the state estimation problem for unknown linear systems subject to both process and measurement noise. Based on a prior input-output trajectory sampled at a higher frequency and a prior state trajectory sampled at a lower frequency, we propose a novel robust data-driven Kalman filter (RDKF) that integrates model identification with state estimation for the unknown system. Specifically, the state estimation problem is formulated as a non-convex maximum likelihood optimization problem. Then, we slightly modify the optimization problem to get a problem solvable with a recursive algorithm. Based on the optimal solution to this new problem, the RDKF is designed, which can estimate the state of a given but unknown state-space model. The performance gap between the RDKF and the optimal Kalman filter based on known system matrices is quantified through a sample complexity bound. In particular, when the number of the pre-collected states tends to infinity, this gap converges to zero. Finally, the effectiveness of the theoretical results is illustrated by numerical simulations.
Paper Structure (16 sections, 7 theorems, 70 equations, 3 figures, 1 algorithm)

This paper contains 16 sections, 7 theorems, 70 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Model
  4. Data Collection
  5. Problem Statement
  6. RDKF Design
  7. Data Informativity Analysis
  8. RDKF Performance Evaluation
  9. Simulation
  10. Conclusion
  11. Appendix
  12. Proof of Lemma \ref{['lemmaf']}
  13. Proof of Proposition \ref{['thm3']}
  14. Proof of Theorem \ref{['thm4']}
  15. Proof of Theorem \ref{['thm5']}
  16. ...and 1 more sections

Key Result

Lemma 1

For system equ:systemstate, the joint probability density function $f_{{\bf{x}}, {\bf{y}}, {\bf{y^p}}} (\hat{x}_{[0,k+1]}, y_{[1,k+1]}, y^{\textup{p}})$ is equivalent to where and $\hat{\omega}_{[0,k]} \triangleq [\hat{\omega}_0^T$, $\ldots$, $\hat{\omega}_{k}^T]^T$ and $\hat{\nu}_{[1,k+1]} \triangleq [\hat{\nu}_1^T$, $\ldots$, $\hat{\nu}_{k+1}^T]^T$ are variables to approximate th

Figures (3)

  • Figure 1: The pre-collected and online system trajectories, where red, yellow, and blue solid dots denote the state, input, and output, respectively. In Fig. (a), the states are sampled at time instants $k_1$, $k_2$, $\ldots$, $k_{N+1}$, and the inputs and outputs are sampled at every time instant from $k_1$ to $k_{N+1}$. This paper aims to estimate the online state at every time instant, using the pre-collected input-state-output trajectory and the online input-output trajectory.
  • Figure 2: The values of $\textup{AMSE}$ under different numbers of samples and magnitudes of noise using the proposed RDKF, where the number $N$ is set as $10, 20, \ldots, 190$, respectively.
  • Figure 3: The values of MSE of 200 Monte Carlo trials using different filtering methods, where the red solid line and the blue dotted line denote the median and mean, respectively; the tops and bottoms of each box represent the 25th and 75th percentiles, respectively; and black circles denote outliers beyond 1.5 times the interquartile range.

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Proposition 1
  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • ...and 1 more