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Statistical Linear Regression Approach to Kalman Filtering and Smoothing under Cyber-Attacks

Kundan Kumar, Muhammad Iqbal, Simo Särkkä

TL;DR

This work addresses robust remote state estimation for cyber-physical systems under DoS and false data injection attacks. It introduces a generalized statistical linear regression (GSLR) framework to approximate the faulty measurement model, yielding a linearized measurement form $y_k \approx H_k^{+} x_k + b_k^{+} + \tilde{\nu}_k$, which enables a Kalman filter and Rauch--Tung--Striebel (RTS) smoother to operate under attack. The approach unifies DoS and additive/multiplicative FDI attack models, derives the necessary conditional moments, and provides a complete forward-backward estimation algorithm. Numerical experiments on an aircraft tracking scenario demonstrate improved position and velocity RMSE over standard KF/RTS methods when cyber-attacks are present, illustrating the method's practical potential for resilient CPS state estimation.

Abstract

Remote state estimation in cyber-physical systems is often vulnerable to cyber-attacks due to wireless connections between sensors and computing units. In such scenarios, adversaries compromise the system by injecting false data or blocking measurement transmissions via denial-of-service attacks, distorting sensor readings. This paper develops a Kalman filter and Rauch--Tung--Striebel (RTS) smoother for linear stochastic state-space models subject to cyber-attacked measurements. We approximate the faulty measurement model via generalized statistical linear regression (GSLR). The GSLR-based approximated measurement model is then used to develop a Kalman filter and RTS smoother for the problem. The effectiveness of the proposed algorithms under cyber-attacks is demonstrated through a simulated aircraft tracking experiment.

Statistical Linear Regression Approach to Kalman Filtering and Smoothing under Cyber-Attacks

TL;DR

This work addresses robust remote state estimation for cyber-physical systems under DoS and false data injection attacks. It introduces a generalized statistical linear regression (GSLR) framework to approximate the faulty measurement model, yielding a linearized measurement form , which enables a Kalman filter and Rauch--Tung--Striebel (RTS) smoother to operate under attack. The approach unifies DoS and additive/multiplicative FDI attack models, derives the necessary conditional moments, and provides a complete forward-backward estimation algorithm. Numerical experiments on an aircraft tracking scenario demonstrate improved position and velocity RMSE over standard KF/RTS methods when cyber-attacks are present, illustrating the method's practical potential for resilient CPS state estimation.

Abstract

Remote state estimation in cyber-physical systems is often vulnerable to cyber-attacks due to wireless connections between sensors and computing units. In such scenarios, adversaries compromise the system by injecting false data or blocking measurement transmissions via denial-of-service attacks, distorting sensor readings. This paper develops a Kalman filter and Rauch--Tung--Striebel (RTS) smoother for linear stochastic state-space models subject to cyber-attacked measurements. We approximate the faulty measurement model via generalized statistical linear regression (GSLR). The GSLR-based approximated measurement model is then used to develop a Kalman filter and RTS smoother for the problem. The effectiveness of the proposed algorithms under cyber-attacks is demonstrated through a simulated aircraft tracking experiment.

Paper Structure

This paper contains 8 sections, 1 theorem, 28 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Measurement model under DoS and FDI attacks
  4. Approximating the Faulty Measurement using GSLR Method
  5. Conditional moment computation
  6. Kalman Filter and RTS Smoother under Cyber-Attacks
  7. Simulation Results
  8. Conclusion

Key Result

Lemma 1

Consider the probability distribution of attack parameters, $a_k \sim \mathcal{N}(\mu_a, \Sigma_a)$, $m_k \sim \mathcal{N}(\mu_m, \sigma_m^2)$ and BRVs $\xi_{a,k}, \, \xi_{b, k}, \, \xi_{c, k}, \, \xi_{m,k}$. Then, we have the following: and

Figures (3)

  • Figure 1: Schematic diagram of the proposed estimation algorithm under DoS and FDI attacks. The filtering (forward pass) and smoothing algorithms are developed based on the GSLR-based approximated faulty measurement model.
  • Figure 2: The true trajectory, faulty measurement, the proposed KF and RTSS for the aircraft tracking problem in the presence of cyber-attacks in a single representative run.
  • Figure 3: The position and velocity RMSE of different estimators for the aircraft tracking problem under DoS and FDI attacks, obtained from 100 MC runs.

Theorems & Definitions (2)

  • Lemma 1
  • proof