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.

本文研究了初始值的差分隐私与具有高斯过程和传感器噪声的一般线性动力系统的可观测性之间的关系，其中某些初始值是隐私敏感的，其余的被假定为公开的。首先，分别建立了保持全局敏感初值的差分隐私性和不可观测性的充分必要条件，以显示它们的独立性质。具体来说，我们表明，通过一组敏感初始状态减少的可观察性矩阵不仅表征了噪声的结构特性以实现差分隐私，而且还影响可实现的隐私级别，而不可观察性依赖于这种减少的可观察性矩阵的等级. 下一个，探索了所考虑的线性系统的固有网络性质，其中每个单独的状态对应于一个节点，状态和输出矩阵引发交互和感知图，从而形成网络系统。在这种网络视角下，将先前建立的结果扩展到本地节点的初始值，以研究它们的差异隐私和与其可观察性的联系。此外，还表明差分初始值隐私的定性性质是 将先前建立的结果扩展到本地节点的初始值，以研究它们的差异隐私和与其可观察性的联系。此外，还表明差分初始值隐私的定性性质是 将先前建立的结果扩展到本地节点的初始值，以研究它们的差异隐私和与其可观察性的联系。此外，还表明差分初始值隐私的定性性质是generically preserved or generically lost，这与局部意义上的不可观察性相同，而与全局意义上的不可观察性有细微的区别，即完全保留或一般丢失。最后，重新审视了一种隐私保护共识算法，以说明所建立结果的有效性。