#### Chaos and order in event-triggered control

##### Gabriel de Albuquerque Gleizer, Manuel Mazo

A major recent development in control theory, event-triggered control (ETC) is claimed to provide enormous reductions in sampling frequency when compared to periodic sampling. However, little is formally known about its generated traffic. This work shows that ETC can exhibit very complex traffic, sometimes even chaotic, especially when the triggering condition is aggressive in reducing communications. First, we characterize limit traffic patterns by observing invariant lines and planes through the origin, as well as their attractivity. Then, we present methods to compute limit metrics, such as limit average and limit inferior inter-sample time (IST), with considerations to the robustness of such metrics. In particular, for periodic ETC (PETC) the methods are based on symbolic abstractions and can obtain rigorous bounds for these metrics, sometimes the exact values, as well as metrics that measure chaos in the traffic patterns. The developed analyses and techniques allow us to show that, if PETC stabilizes a periodic sampling pattern, than this pattern globally stabilizes the system; in addition, we show a case of PETC with chaotic sampling whose limit average IST is higher than what can be attained with periodic sampling, proving for the first time ETC's superiority over periodic sampling.

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