An Autocovariance Least-Squares-Based Data-Driven Kalman Filter for Unknown Systems

TL;DR

This work addresses data-driven state estimation for unknown linear systems by jointly identifying system dynamics and noise covariances. It introduces the Autocovariance Least-Squares-based Data-driven Kalman Filter (ADKF), which uses pre-collected trajectories and a stabilizing feedback controller to estimate $\hat{A},\hat{B},\hat{C},\hat{Q},\hat{R}$ via ALS and a semidefinite program. The authors establish probabilistic and sample-complexity guarantees (Theorems 1–3, Propositions 1–2) for the covariance estimates and the resulting filter performance, showing convergence to the optimal Kalman filter as data length and trajectory count grow. Numerical experiments on a DC motor and a wheel-legged robot demonstrate ADKF’s robustness to mismatched nominal covariances and its competitive performance relative to i-KF and EKF, highlighting its potential for data-driven estimation in unknown systems.

Abstract

This article investigates the problem of data-driven state estimation for linear systems with both unknown system dynamics and noise covariances. We propose an Autocovariance Least-squares-based Data-driven Kalman Filter (ADKF), which provides a unified framework for simultaneous system identification and state estimation by utilizing pre-collected input-output trajectories and estimated initial states. Specifically, we design a SDP-based algorithm for estimating the noise covariances. We quantify the impact of model inaccuracy on noise covariances estimation using this identification algorithm, and introduce a feedback control mechanism for data collection to enhance the accuracy and stability of noise covariance estimation. The estimated noise covariances account for model inaccuracy, which are shown to be more suitable for state estimation. We also quantify the performance gap between the ADKF and the traditional Kalman filter with known system dynamics and noise covariances, showing that this gap decreases as the number and length of pre-collected trajectories increase. Finally, numerical simulations validate the robustness and effectiveness of the proposed ADKF.

Figures (4)

• Figure 1: DC Motor: (a) The estimation error of noise covariance matrices as $\tau_1$ increases, with $Q_u = 10Q, R_u = 5R$, $N = 5000$, $L=15$. (b) The estimation error of noise covariance matrices as $L$ increases, with $\tau_1 = 150$,$Q_u = 10Q, R_u = 5R$ and $N = 5000$. (c) MSE with $Q_u = \gamma Q$, $R_u = 5R$, $\gamma = [10.0, 20.0, 50.0, 100.0], L=15, \tau_1=200$.
• Figure 2: Wheel-legged Robot: (a) $\theta$ is the tilt angle. (b) $\dot{\theta}$ is the angular velocity. (c) The gap between velocity and determinate velocity.
• Figure 3: Template model of the wheel-legged robot.
• Figure 4: MSE for three filters, with $R_u = \gamma R$, $Q_u = 5Q$, where $\gamma = [1.0, 10.0, 50.0, 100.0 ]$.

Theorems & Definitions (7)

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