Diversity and Interaction Quality of a Heterogeneous Multi-Agent System Applied to a Synchronization Problem

In this paper, scalable controller design to achieve output synchronization for a heterogeneous discrete-time nonlinear multi-agent system is considered. The agents are assumed to exhibit potentially nonlinear dynamics but share linear common oscillatory modes. In a distributed control architecture, scalability is ensured by designing a small number of distinguished controllers, significantly fewer than the number of agents, even when agent diversity is high. Our findings indicate that the number of controllers required can be effectively determined by the number of strongly connected components of the underlying graph. The study in this paper builds on the recently developed phase theory of matrices and systems. First, we employ the concept of matrix phase, specifically the phase alignability of a collection of matrices, to quantify agent diversity. Next, we use matrix phase, particularly the essential phase of the graph Laplacian, to evaluate the interaction quality among the agents. Based on these insights, we derive a sufficient condition for the solvability of the synchronization problem, framed as a trade-off between the agent diversity and the interaction quality. In the process, a controller design procedure based on Lyapunov analysis is provided, which produces low gain, component-wise synchronizing controllers when the solvability condition is satisfied. Numerical examples are given to illustrate the effectiveness of the proposed design procedure. Furthermore, we consider cases where the component-wise controller design problem is unsolvable. We propose alternative strategies involving the design of a small inventory of controllers, which can still achieve synchronization effectively by employing certain clustering methods to manage heterogeneity.