# Limits on the Network Sensitivity Function for Multi-Agent Systems on a Graph

### Stefania Tonetti, Richard M Murray

CDS Technical Report 2009.001

This report explores the tradeoffs and limits of performance in feedback control of interconnected multi-agent systems, focused on the network sensitivity functions. We consider the interaction topology described by a directed graph and we prove that the sensitivity transfer functions between every pair of agents, arbitrarily connected, can be derived using a version of the Mason's Direct Rule. Explicit forms for special types of graphs are presented. An analysis of the role of cycles points out that these structures influence and limit considerably the performance of the system. The more the cycles are equally distributed among the formation, the better performance the system can achieve, but they are always worse than the single agent case. We also prove the networked version of Bode's integral formula, showing that it still holds for multi-agent systems.

- Technical Report: http://resolver.caltech.edu/CaltechCDSTR:2009.001
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