In this article, we consider composite networks formed from the Kronecker product of smaller networks. We find the observability and controllability properties of the product network from those of its constituent smaller networks. The overall network is modeled as a Linear-Structure-Invariant (LSI) dynamical system where the underlying matrices have a fixed zero/non-zero structure but the non-zero elements are potentially time-varying. This approach allows to model the system parameters as free variables whose values may only be known within a certain tolerance. We particularly look for minimal sufficient conditions ¹¹ We emphasize that a minimal sufficient condition is not necessarily a necessary and sufficient condition. In fact, it implies that among all sufficient conditions that may result in an event, this condition is the least conservative but usually is not necessary; see [1] for details. on the observability and controllability of the composite network, which have a direct application in distributed estimation and in the design of networked control systems. The methodology in this article is based on the structured systems analysis and graph theory, and therefore, the results are generic, i.e., they apply to almost all non-zero choices of free parameters. We show the controllability/observability results for composite product networks resulting from full structural-rank systems and self-damped networks. We provide an illustrative example of estimation based on Kalman filtering over a composite network to verify our results.

中文翻译：

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通过Kronecker产品的复合网络的结构可观性/可控性的最小充分条件

在本文中，我们考虑由较小网络的Kronecker产品形成的复合网络。我们从组成较小的网络中发现产品网络的可观察性和可控性。整个网络被建模为线性结构不变（LSI）动力学系统，其中基础矩阵具有固定的零/非零结构，但非零元素可能随时间变化。这种方法允许将系统参数建模为自由变量，其自由值只能在一定公差范围内知道。我们特别寻求最低限度的充分条件 ¹¹ 我们强调，最低限度的充分条件不一定是必要条件和充分条件。实际上，这意味着在可能导致事件发生的所有充分条件中，该条件是最不保守的，但通常不是必需的。有关详细信息，请参见[1]。复合网络的可观察性和可控制性，在分布式估计和网络控制系统的设计中具有直接的应用。本文中的方法基于结构化系统分析和图论，因此结果是通用的，即，它们适用于几乎所有非零的自由参数选择。我们显示了由完整的结构等级系统和自阻尼网络导致的复合产品网络的可控性/可观察性结果。