Performance implications of different $p$-norms in level-triggered sampling

TL;DR

This paper investigates how different $p$-norms used in level-triggered sampling affect control performance, revealing that the choice of norm can cause performance deterioration, especially with larger system dimensions and higher $p$-values.

Contribution

It demonstrates that the $p$-norm in the triggering condition significantly impacts performance, challenging previous assumptions of norm invariance, and clarifies the triggering rule's role in performance degradation.

Findings

Maximum norm causes performance deterioration at large dimensions.

Performance degrades with increasing $p$-norms in simulations.

Triggering rule, not the control goal, causes performance issues.

Abstract

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.