Observer-based Differentially Private Consensus for Linear Multi-agent Systems

TL;DR

The paper tackles observer-based, privacy-preserving consensus for general linear multi-agent systems by injecting Laplace noise into inter-agent communications to achieve $\epsilon$-differential privacy. It develops full-order and reduced-order observer frameworks, derives mean-square and almost-sure consensus conditions via backstepping and almost-supermartingale theory, and proves a separation principle under decaying privacy noise. A unified co-design approach jointly selects the state-estimation gain, control gain, and noise parameters to guarantee $\epsilon$-DP at every time step, with explicit $\epsilon^*$-DP conditions and per-agent design guidelines. Simulations validate the theory, demonstrating consensus and privacy preservation across both observer architectures and illustrating the scalability to large networks.

Abstract

This paper investigates the differentially private consensus problem for general linear multi-agent systems (MASs) based on output feedback protocols. To protect the output information, which is considered private data and may be at high risk of exposure, Laplace noise is added to the information exchange. The conditions for achieving mean square and almost sure consensus in observer-based MASs are established using the backstepping method and the convergence theory for nonnegative almost supermartingales. It is shown that the separation principle remains valid for the consensus problem of linear MASs with decaying Laplace noise. Furthermore, the convergence rate is provided. Then, a joint design framework is developed for state estimation gain, feedback control gain, and noise to ensure the preservation of ε-differential privacy. The output information of each agent is shown to be protected at every time step. Finally, sufficient conditions are established for simultaneously achieving consensus and preserving differential privacy for linear MASs utilizing both full-order and reduced-order observers. Meanwhile, an ε*-differentially private consensus is achieved to meet the desired privacy level. Two simulation examples are provided to validate the theoretical results.

Figures (7)

• Figure 1: The communication topology of MAS (1).
• Figure 2: The trajectory of observation errors (Full-order Observer).
• Figure 3: The trajectory of consensus errors (Full-order Observer).
• Figure 4: The histograms of $\theta_1^a(k)$ and $(\theta_1^a(k))^\prime$ at $k = 2$ (Full-order Observer).
• Figure 5: The trajectory of observation errors (Reduced-order Observer).

Theorems & Definitions (62)

• Remark 1
• Definition 1: Mean square consensus
• Definition 2: Almost sure consensus
• Definition 3: Adjacent
• Remark 2
• Remark 3
• Definition 4: $\epsilon$-differential privacy
• Remark 4
• Definition 5: Mean square convergence rate
• Remark 5