On the Robustness of Nominally Well-Posed Event-Triggered Controllers

In this letter, we show that: 1) if the Krasovskii regularization of a hybrid system <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> has complete and discrete solutions, then <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> has solutions with arbitrarily small separation between jumps under the influence of admissible state perturbations; 2) if <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> is nominally well-posed and does not have complete discrete solutions, then it does not have solutions with vanishing time between jumps (such as Zeno solutions); 3) if, in addition, there exists a compact set <inline-formula> <tex-math notation="LaTeX">$\mathcal {A}$ </tex-math></inline-formula> such that all maximal solutions to <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">$\mathcal {A}$ </tex-math></inline-formula> are complete, discrete and remain in <inline-formula> <tex-math notation="LaTeX">$\mathcal {A}$ </tex-math></inline-formula>, then all solutions converging to <inline-formula> <tex-math notation="LaTeX">$\mathcal {A}$ </tex-math></inline-formula> have vanishing time between jumps. The results in this letter demonstrate that a good practice to avoid solutions with arbitrarily fast sampling in Event-Triggered Control (ETC) is to ensure that the closed-loop system is nominally well-posed and that it does not have complete discrete solutions.