  This work studies the performance of an event-based control approach, namely
level-triggered sampling, in a standard multidimensional single-integrator
setup. We falsify a conjecture from the literature that the deployed $p$-norm
in the triggering condition supposedly has no impact on the performance of the
sampling scheme in that setting. In particular, we show for the considered
setup that the usage of the maximum norm instead of the Euclidean norm induces
a performance deterioration of level-triggered sampling for sufficiently large
system dimensions, when compared to periodic control at the same sampling rate.
Moreover, we investigate the performance for other $p$-norms in simulation and
observe that it degrades with increasing $p$. In addition, our findings reveal
the previously unknown role of the triggering rule in the cause of a recently
discovered phenomenon: Previous work has shown for a single-integrator
consensus setup that the commonly observed performance advantage of event-based
control over periodic control can be lost in distributed settings with a
cooperative control goal. In our work, we obtain similar results for a
non-cooperative setting only by adjusting the norm in the level-triggered
sampling scheme. We therefore demonstrate that the performance degradation
found in the distributed setting originates from the triggering rule and not
from the considered cooperative control goal.
