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Recursive Binary Identification under Data Tampering and Non-Persistent Excitation with Application to Emission Control

Jian Guo, Lihong Pei, Wenchao Xue, Yanlong Zhao, Ji-Feng Zhang

TL;DR

This work tackles online parameter estimation for CPS with binary outputs in the presence of data tampering, addressing the lack of persistent excitation through gradient and second-order Newton schemes. It develops GRP-TB-KP and GRP-TB-UP (and their Newton counterparts NRPTB-KP/NRPTB-UP), handling known and unknown tampering, and proves almost-sure and mean-square convergence without requiring persistent excitation. An adaptive-control extension provides explicit tracking-error bounds under tampering, and a periodic-insertion scheme enables online estimation of tampering probabilities. The methods are validated via numerical simulations and a vehicle-emissions case study, showing robustness to tampering and improved detection of excess-emission events in real data.

Abstract

This paper studies the problem of online parameter estimation for cyber-physical systems with binary outputs that may be subject to adversarial data tampering. Existing methods are primarily offline and unsuitable for real-time learning. To address this issue, we first develop a first-order gradient-based algorithm that updates parameter estimates recursively using incoming data. Considering that persistent excitation (PE) conditions are difficult to satisfy in feedback control scenarios, a second-order quasi-Newton algorithm is proposed to achieve faster convergence without requiring the PE condition. For both algorithms, corresponding versions are developed to handle known and unknown tampering strategies, and their parameter estimates are proven to converge almost surely over time. In particular, the second-order algorithm ensures convergence under a signal condition that matches the minimal excitation required by classical least-squares estimation in stochastic regression models. The second-order algorithm is also extended to an adaptive control framework, providing an explicit upper bound on the tracking error for binary-output FIR systems under unknown tampering. Three numerical simulations verify the theoretical results and show that the proposed methods are robust against data tampering. Finally, the approach is validated via a vehicle emission control problem, where it effectively improves the detection accuracy of excess-emission events.

Recursive Binary Identification under Data Tampering and Non-Persistent Excitation with Application to Emission Control

TL;DR

This work tackles online parameter estimation for CPS with binary outputs in the presence of data tampering, addressing the lack of persistent excitation through gradient and second-order Newton schemes. It develops GRP-TB-KP and GRP-TB-UP (and their Newton counterparts NRPTB-KP/NRPTB-UP), handling known and unknown tampering, and proves almost-sure and mean-square convergence without requiring persistent excitation. An adaptive-control extension provides explicit tracking-error bounds under tampering, and a periodic-insertion scheme enables online estimation of tampering probabilities. The methods are validated via numerical simulations and a vehicle-emissions case study, showing robustness to tampering and improved detection of excess-emission events in real data.

Abstract

This paper studies the problem of online parameter estimation for cyber-physical systems with binary outputs that may be subject to adversarial data tampering. Existing methods are primarily offline and unsuitable for real-time learning. To address this issue, we first develop a first-order gradient-based algorithm that updates parameter estimates recursively using incoming data. Considering that persistent excitation (PE) conditions are difficult to satisfy in feedback control scenarios, a second-order quasi-Newton algorithm is proposed to achieve faster convergence without requiring the PE condition. For both algorithms, corresponding versions are developed to handle known and unknown tampering strategies, and their parameter estimates are proven to converge almost surely over time. In particular, the second-order algorithm ensures convergence under a signal condition that matches the minimal excitation required by classical least-squares estimation in stochastic regression models. The second-order algorithm is also extended to an adaptive control framework, providing an explicit upper bound on the tracking error for binary-output FIR systems under unknown tampering. Three numerical simulations verify the theoretical results and show that the proposed methods are robust against data tampering. Finally, the approach is validated via a vehicle emission control problem, where it effectively improves the detection accuracy of excess-emission events.

Paper Structure

This paper contains 20 sections, 14 theorems, 85 equations, 10 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Model Formulation and Identifiability
  3. Model Description Under Tampering Attack
  4. Identifiability Analysis
  5. First-order Gradient Identification Algorithm
  6. Known Attack Strategy
  7. Motivation and Algorithm
  8. Convergence Properties
  9. The Attack Strategy is Unknown
  10. Algorithm Design
  11. Convergence analysis
  12. Second-order Newton Identification Algorithm
  13. The Attack Strategy is Known
  14. The Attack Strategy is Unknown
  15. Numerical simulations and practical case study
  16. ...and 5 more sections

Key Result

Theorem 1

Consider system s1 with binary-valued observations s2 and tampering attacks s3. Under Assumptions A1–A5, the parameter estimate generated by the algorithm algo1–algo2 satisfies Moreover, if $\sum_{k=1}^{\infty} b_k^2 < \infty$, then the estimate $\hat{\theta}_k$ also converges almost surely to the true parameter:

Figures (10)

  • Figure 1: Closed‐loop control system flowchart with tampering attack and estimation/control centers
  • Figure 2: Convergence of the estimation shown by a trajectory of $\hat{\theta}_{n+1}$.
  • Figure 3: Convergence rate of the estimation shown by a trajectory of $k \tilde{\theta}_k^T \tilde{\theta}_k / \ln k$.
  • Figure 4: Convergence of parameter estimates $\hat{\theta}_k$ and attack probability estimates $(\hat{p}_k, \hat{q}_k)$ under unknown attack strategies with $(p, q) = (0.2, 0.3)$ and $(p, q) = (0.8, 0.9)$.
  • Figure 5: Simulation results of Algorithm \ref{['algorithm4']}. (a) Parameter estimates; (b) Estimation error and its theoretical bound; (c) Cumulative regret vs theoretical rate; (d) Online estimation of attack probabilities.
  • ...and 5 more figures

Theorems & Definitions (32)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1: Convergence
  • Theorem 2: Convergence Rate
  • proof
  • Remark 5
  • Remark 6
  • ...and 22 more