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

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.

💡 Research Summary

This paper addresses the problem of achieving consensus in general linear multi‑agent systems (MASs) while guaranteeing differential privacy of each agent’s output information. Unlike most existing works that protect only the initial states and rely on exponentially decaying Laplace noise, the authors propose an observer‑based framework that protects the entire output trajectory throughout the execution of the algorithm. The agents’ dynamics are discrete‑time linear systems (x_i(k+1)=Ax_i(k)+Bu_i(k),; y_i(k)=Cx_i(k)). Because the full state is generally not directly measurable, a state observer is constructed from the output (y_i). Both full‑order and reduced‑order observers are considered, allowing a trade‑off between estimation accuracy and computational/communication load.

The core technical contribution lies in the rigorous convergence analysis. Using a backstepping design, the observer gain (L) and the feedback gain (K) are synthesized step‑by‑step. The closed‑loop error dynamics, together with the injected Laplace noise (\gamma_i(k)\sim\text{Lap}(0,b(k))), are modeled as a non‑negative almost supermartingale. This enables the derivation of sufficient conditions for both mean‑square consensus and almost‑sure consensus. Moreover, the authors obtain an explicit convergence rate for the scale parameter (b(k)) when it decays exponentially, and they show that the classical separation principle still holds even in the presence of decaying stochastic noise.

A unified design framework is then introduced to jointly select the observer gain (L), the control gain (K), and the noise parameters (b(k)) so that the system satisfies (\varepsilon)-differential privacy at every time step. The paper generalizes the adjacency relation between neighboring data sets, allowing larger deviations than the standard definition, and relaxes the requirement that the noise variance must decay exponentially; slower (e.g., polynomial) decay rates are admissible provided the derived LMI conditions are met.

For a prescribed privacy budget (\varepsilon^\star), the authors give explicit inequalities that guarantee (\varepsilon^\star)-differential privacy (denoted (\varepsilon^\star)-DP) while still achieving dynamic average consensus. These inequalities link (|K|), (|L|), and the noise scale (b(k)) in a transparent way, facilitating practical parameter tuning.

The theoretical results are extended from full‑order observers to reduced‑order observers. The reduced‑order design significantly reduces the dimension of the observer state, leading to lower computational complexity and communication overhead, yet the same privacy‑consensus guarantees are retained under the derived conditions.

Two simulation studies illustrate the effectiveness of the proposed approach. In the first example, a five‑agent linear MAS reaches consensus with mean‑square error decreasing exponentially, while each agent’s transmitted output is protected by Laplace noise satisfying the prescribed (\varepsilon) and (\varepsilon^\star) privacy levels. The second example, involving a ten‑agent higher‑dimensional system, demonstrates that the reduced‑order observer achieves comparable convergence speed and privacy protection with substantially fewer transmitted variables.

In summary, the paper makes several notable contributions: (1) it establishes mean‑square and almost‑sure consensus conditions for observer‑based linear MASs using backstepping and supermartingale theory; (2) it proves that the separation principle remains valid under decaying Laplace noise; (3) it provides a joint design methodology for observer gain, control gain, and noise to guarantee ε‑DP at every step; (4) it relaxes the exponential‑decay requirement on noise, allowing slower decay rates; (5) it derives sufficient conditions for simultaneous consensus and differential privacy for both full‑order and reduced‑order observers; and (6) it supplies explicit design formulas for achieving a desired privacy budget (\varepsilon^\star). The results broaden the applicability of differential privacy to dynamic, output‑feedback control of linear MASs and open new avenues for privacy‑preserving distributed control in practical networked systems.