Differential initial-value privacy and observability of linear dynamical systems

This paper studies the relationship between the differential privacy of initial values and the observability for general linear dynamical systems with Gaussian process and sensor noises, where certain initial values are privacy-sensitive and the rest is assumed to be public. First of all, necessary and sufficient conditions are established for preserving the differential privacy and unobservability of the global sensitive initial values, respectively to show their independent properties. Specifically, we show that the observability matrix reduced by the set of sensitive initial states not only characterizes the structural property of noises for achieving the differential privacy, but also affects the achievable privacy levels, while the unobservability relies on the rank of such reduced observability matrix. Next, the inherent network nature of the considered linear system is explored, where each individual state corresponds to a node and the state and output matrices induce interaction and sensing graphs, leading to a network system. Under this network perspective, the previously established results are extended for initial values of local nodes to study their differential privacy and connections with their observability. Moreover, it is shown that the qualitative property of the differential initial-value privacy is either preserved generically or lost generically, which is the same as the unobservability in the local sense, while subject to a subtle difference from the unobservability in the global sense that is either preserved fully or lost generically. Finally, a privacy-preserving consensus algorithm is revisited to illustrate the effectiveness of the established results.