  Event-triggered control is often argued to lower the average triggering rate
compared to time-triggered control while still achieving a desired control
goal, e.g., the same performance level. However, this property, often called
consistency, cannot be taken for granted and can be hard to analyze in many
settings. In particular, although numerous decentralized event-triggered
control schemes have been proposed in the past years, their performance
properties with respect to time-triggered control remain mostly unexplored. In
this paper, we therefore examine the performance properties of event-triggered
control (relative to time-triggered control) for a single-integrator consensus
problem with a level-triggering rule. We consider the long-term average
quadratic deviation from consensus as a performance measure. For this setting,
we show that enriching the information the local controllers use improves the
performance of the consensus algorithm but renders a previously consistent
event-triggered control scheme inconsistent. In addition, we do so while
deploying optimal control inputs which we derive for both information cases and
all triggering schemes. With this insight, we can furthermore explain the
relationship between two contrasting consistency results from the literature on
decentralized event-triggered control. We support our theoretical findings with
simulation results.
