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Privacy Preservation by Local Design in Cooperative Networked Control Systems

Chao Yang, Yuqing Ni, Wen Yang, Hongbo Shi

TL;DR

The paper tackles privacy in cooperative closed-loop networked control by enabling a user-local privacy design that distorts signals sent to a server computing LQG control. It introduces a noise-injection scheme $z_k=y_k+oldsymbol{ or delta_k}$ with covariance $oldsymbol{\Sigma_ ext{delta}}$, derives a Kalman-filtered recursion for the server-side estimation error, and proves that the resulting LQG performance loss can be bounded via $oldsymbol{Q}_{privacy}$ and $oldsymbol{Q}_{LQG}$. A privacy metric $oldsymbol{Q}_{privacy}$ is defined to quantify the deviation between the server’s private estimate and the true estimate, and the authors formulate a semidefinite programming problem to optimize privacy under a given performance loss constraint, including infinite-horizon steady-state analysis. The work demonstrates that privacy can be enhanced without destabilizing the system in many cases and provides a tractable optimization framework to balance privacy against control performance, with numerical examples supporting the theory. These results offer a practical approach to privacy-preserving cooperation in networked control with guaranteed performance bounds and scalable optimization tools.

Abstract

In this paper, we study the privacy preservation problem in a cooperative networked control system, which has closed-loop dynamics, working for the task of linear quadratic Guassian (LQG) control. The system consists of a user and a server: the user owns the plant to control, while the server provides computation capability, and the user employs the server to compute control inputs for it. To enable the server's computation, the user needs to provide the measurements of the plant states to the server, who then calculates estimates of the states, based on which the control inputs are computed. However, the user regards the states as privacy, and makes an interesting request: the user wants the server to have "incorrect" knowledge of the state estimates rather than the true values. Regarding that, we propose a novel design methodology for the privacy preservation, in which the privacy scheme is locally equipped at the user side not open to the server, which manages to create a deviation in the server's knowledge of the state estimates from the true values. However, this methodology also raises significant challenges: in a closed-loop dynamic system, when the server's seized knowledge is incorrect, the system's behavior becomes complex to analyze; even the stability of the system becomes questionable, as the incorrectness will accumulate through the closed loop as time evolves. In this paper, we succeed in showing that the performance loss in LQG control caused by the proposed privacy scheme is bounded by rigorous mathematical proofs, which convinces the availability of the proposed design methodology. We also propose an associated novel privacy metric and obtain the analytical result on evaluating the privacy performance. Finally, we study the performance trade-off between privacy and control, where the accordingly proposed optimization problems are solved by numerical methods efficiently.

Privacy Preservation by Local Design in Cooperative Networked Control Systems

TL;DR

The paper tackles privacy in cooperative closed-loop networked control by enabling a user-local privacy design that distorts signals sent to a server computing LQG control. It introduces a noise-injection scheme with covariance , derives a Kalman-filtered recursion for the server-side estimation error, and proves that the resulting LQG performance loss can be bounded via and . A privacy metric is defined to quantify the deviation between the server’s private estimate and the true estimate, and the authors formulate a semidefinite programming problem to optimize privacy under a given performance loss constraint, including infinite-horizon steady-state analysis. The work demonstrates that privacy can be enhanced without destabilizing the system in many cases and provides a tractable optimization framework to balance privacy against control performance, with numerical examples supporting the theory. These results offer a practical approach to privacy-preserving cooperation in networked control with guaranteed performance bounds and scalable optimization tools.

Abstract

In this paper, we study the privacy preservation problem in a cooperative networked control system, which has closed-loop dynamics, working for the task of linear quadratic Guassian (LQG) control. The system consists of a user and a server: the user owns the plant to control, while the server provides computation capability, and the user employs the server to compute control inputs for it. To enable the server's computation, the user needs to provide the measurements of the plant states to the server, who then calculates estimates of the states, based on which the control inputs are computed. However, the user regards the states as privacy, and makes an interesting request: the user wants the server to have "incorrect" knowledge of the state estimates rather than the true values. Regarding that, we propose a novel design methodology for the privacy preservation, in which the privacy scheme is locally equipped at the user side not open to the server, which manages to create a deviation in the server's knowledge of the state estimates from the true values. However, this methodology also raises significant challenges: in a closed-loop dynamic system, when the server's seized knowledge is incorrect, the system's behavior becomes complex to analyze; even the stability of the system becomes questionable, as the incorrectness will accumulate through the closed loop as time evolves. In this paper, we succeed in showing that the performance loss in LQG control caused by the proposed privacy scheme is bounded by rigorous mathematical proofs, which convinces the availability of the proposed design methodology. We also propose an associated novel privacy metric and obtain the analytical result on evaluating the privacy performance. Finally, we study the performance trade-off between privacy and control, where the accordingly proposed optimization problems are solved by numerical methods efficiently.
Paper Structure (21 sections, 10 theorems, 106 equations, 7 figures, 1 algorithm)

This paper contains 21 sections, 10 theorems, 106 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Studies
  3. Observation on the Related Studies
  4. The Study of This Paper
  5. Framework of the Privacy Preservation by Local Design
  6. Ordinary Cooperative System
  7. Local Design of Privacy Preservation
  8. Privacy Metric
  9. Problems to Study
  10. Performance Analysis
  11. Privacy Performance
  12. LQG Control Performance
  13. Optimization Problem
  14. Special Case Study: Infinite-Time Horizon
  15. Perfect Privacy Preservation without Performance Loss
  16. ...and 6 more sections

Key Result

Theorem 1

It holds that and where $K_k$ is the Kalman gain defined in eqn. (eqn:kalman-gain). The recursion starts with $\hat{x}^{svr}_{0|0}-\hat{x}_{0|0}=0$ and $\mathcal{Q}_{privacy}^0=0$.

Figures (7)

  • Figure 1: The user-server system.
  • Figure 2: The ordinary cooperative LQG control system.
  • Figure 3: A local privacy processor is used.
  • Figure 4: The plot of $\mathrm{tr}(\mathcal{Q}^\star_{privacy})$ when $\alpha$ varies from 0 to 50.
  • Figure 5: The plot of the first component of $\hat{x}_{k|k}$ and $\hat{x}^{svr}_{k|k}$. The presented time horizon is 100.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4: A Doubtful Server
  • ...and 18 more