Differential Privacy, Entropy and Security in Distributed Control of Cyber Physical Systems

The concept of differential privacy stems from the study of private query of datasets. In this work, we apply this concept to discrete-time, linear distributed control systems in which agents need to maintain privacy of certain preferences, while sharing information for better system-level performance. The system has N agents operating in a shared environment that couples their dynamics. We show that for stable systems the performance grows as O(T3/Ne2), where T is the time horizon and e is the differential privacy parameter. Next, we study lower-bounds in terms of the Shannon entropy of the minimal mean square estimate of the system's private initial state from noisy communications between an agent and the server. We show that for any of noise-adding differentially private mechanism, then the Shannon entropy is at least nN(1ln(e/2)), where n is the dimension of the system, and t he lower bound is achieved by a Laplace-noise-adding mechanism. Finally, we study the problem of keeping the objective functions of individual agents differentially private in the context of cloud-based distributed optimization. The result shows a trade-off between the privacy of objective functions and the performance of the distributed optimization algorithm with noise.