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State Estimation with Protecting Exogenous Inputs via Cramér-Rao Lower Bound Approach

Liping Guo, Jimin Wang, Yanlong Zhao, Ji-Feng Zhang

TL;DR

This work tackles real-time state estimation with protection of exogenous inputs against honest-but-curious observers by enforcing a CRLB-based privacy constraint that bounds the adversary's MSE for ${f d}_{k-1}$. It introduces a perturbed-noise design, derives an explicit CRLB, and then develops a low-complexity approach using PCRLB surrogates and an SDP relaxation to compute the perturbation covariance. The method achieves $(\epsilon,oldsymbol{\delta})$-differential privacy while maintaining competitive state estimation accuracy, as demonstrated on building-occupancy and a 2D numerical example. The results show a clear privacy-utility trade-off and offer a scalable framework for privacy-preserving real-time estimation in dynamic systems.

Abstract

This paper addresses the real-time state estimation problem for dynamic systems while protecting exogenous inputs against adversaries, who may be honest-but-curious third parties or external eavesdroppers. The Cramér-Rao lower bound (CRLB) is employed to constrain the mean square error (MSE) of the adversary's estimate for the exogenous inputs above a specified threshold. By minimizing the MSE of the state estimate while ensuring a certain privacy level measured by CRLB, the problem is formulated as a constrained optimization. To solve the optimization problem, an explicit expression for CRLB is first provided. As the computational complexity of the CRLB increases with the time step, a low-complexity approach is proposed to make the complexity independent of time. Then, a relaxation approach is proposed to efficiently solve the optimization problem. Finally, a privacy-preserving state estimation algorithm with low complexity is developed, which also ensures $(ε, δ)$-differential privacy. Two illustrative examples, including a practical scenario for protecting building occupancy, demonstrate the effectiveness of the proposed algorithm.

State Estimation with Protecting Exogenous Inputs via Cramér-Rao Lower Bound Approach

TL;DR

This work tackles real-time state estimation with protection of exogenous inputs against honest-but-curious observers by enforcing a CRLB-based privacy constraint that bounds the adversary's MSE for . It introduces a perturbed-noise design, derives an explicit CRLB, and then develops a low-complexity approach using PCRLB surrogates and an SDP relaxation to compute the perturbation covariance. The method achieves -differential privacy while maintaining competitive state estimation accuracy, as demonstrated on building-occupancy and a 2D numerical example. The results show a clear privacy-utility trade-off and offer a scalable framework for privacy-preserving real-time estimation in dynamic systems.

Abstract

This paper addresses the real-time state estimation problem for dynamic systems while protecting exogenous inputs against adversaries, who may be honest-but-curious third parties or external eavesdroppers. The Cramér-Rao lower bound (CRLB) is employed to constrain the mean square error (MSE) of the adversary's estimate for the exogenous inputs above a specified threshold. By minimizing the MSE of the state estimate while ensuring a certain privacy level measured by CRLB, the problem is formulated as a constrained optimization. To solve the optimization problem, an explicit expression for CRLB is first provided. As the computational complexity of the CRLB increases with the time step, a low-complexity approach is proposed to make the complexity independent of time. Then, a relaxation approach is proposed to efficiently solve the optimization problem. Finally, a privacy-preserving state estimation algorithm with low complexity is developed, which also ensures -differential privacy. Two illustrative examples, including a practical scenario for protecting building occupancy, demonstrate the effectiveness of the proposed algorithm.

Paper Structure

This paper contains 15 sections, 13 theorems, 75 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Unbiased minimum-variance state estimate
  4. Privacy issue
  5. Design of the privacy-preserving state estimate via CRLB
  6. Perturbed noise approach
  7. Constrained optimization subject to privacy
  8. Explicit expression for CRLB
  9. Privacy-preserving state estimation with low complexity
  10. Privacy-preserving state estimation algorithm with low complexity
  11. Relation to differential privacy
  12. Examples
  13. Practical case: Building occupancy
  14. Numerical case: A two-dimensional model
  15. Conclusion

Key Result

Lemma 1

Let $X = \{{\bf x}_1, \dots, {\bf x}_{m}\}$ be a sample from $P \in {\mathcal{P}} = \{P_{\bm \theta}: \bm \theta \in \Theta\}$, where $\Theta$ is an open set in ${\mathbb R}^m$. Suppose that $T(X)$ is an estimator with $\mathbb E [T(X)] = g(\bm \theta)$ being a differential function of $\bm \theta$, where $\mathcal{I}(\bm \theta) = \mathbb E[\frac{\partial}{\partial \bm \theta} \log p_{\bm \theta}

Figures (6)

  • Figure 1: The privacy-preserving state estimation setup.
  • Figure 2: CO$_2$ level, occupancy and their estimates.
  • Figure 3: CO$_2$ level, occupancy and their estimates.
  • Figure 4: Comparison of unbiased minimum-variance and privacy-preserving state estimates.
  • Figure 5: MSEs of adversary's estimates for $d_k$ with unbiased minimum-variance and privacy-preserving state estimates.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Remark 1
  • Example 1: Building occupancy
  • Lemma 1: Cramér-Rao lower bound, Shao-Mathematical-2003
  • Theorem 1
  • Lemma 2: Fisher information matrix, Malago2015
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • Remark 2
  • Theorem 3
  • ...and 9 more