On the Delayed Scaled Consensus Problems

In this note, we study the scaled consensus (tracking) problems, wherein all agents reach agreement, but with different assigned ratios in the asymptote. Based on the nearest neighbor-interaction rules, the scaled consensus processes are characterized with and without time delay. We consider both the signal transmission and signal processing delays and calculate the final scaled consensus values. When the underlying communication network contains a spanning tree, it is found that the scaled consensus can be achieved independent of the transmission delays while the specified consensus values in the asymptote depend on the initial history of the agents over a period of time. This phenomenon is in sharp contrast to the case of processing delays, where large delays are likely to jeopardize the consensus behavior, but the scaled consensus values once achieved are the same as the undelayed case.