A Passivity Analysis Tool for Linear Clustered Multi-Agent Systems

Passivity of a large-scale interconnected system is often broken down to the passivity of the individual subsystems that compose it. Nevertheless, there are cases in which the individual elements are not all passive, yet the overall large-scale system is. In such scenarios, we need to directly solve very large problems to conclude on the passivity. This letter proposes a methodology to analyze passivity based on the topology of the multi-agent system. In many cases, large multi-agent systems are formed by interconnected clusters, which are groups of agents densely interconnected. The clusters are sparsely interconnected with each other and this leads to a time-scale separation with a fast dynamics inside the clusters and a slow one between them. The purpose of this letter is twofold. First, we exploit the time-scale separation property inherent to such a system to provide a computationally efficient alternative to analyze its passivity. Second, we provide insight into how robust its passivity is with respect to the inter- and intra-cluster agent interactions. To achieve this, we consider the singular perturbation framework with respect to the ratio of the strength of the controls between and within the clusters, and rely on the connection between positive realness, passivity, and multi-input multi-output system phase. We consider agents with identical linear time-invariant dynamics. The method is illustrated on a numerical example.