Attentively efficient controllers for event-triggered feedback systems

State dependent event-triggered systems sample the system state when the difference between the current state and the last sampled state exceeds a state-dependent threshold. These systems exhibit the efficient attentiveness property when the the length of the inter-sampling interval increases monotonically as the sampled state approaches the equilibrium. The efficient attentiveness property may partly explain why event-triggered systems sometimes exhibit inter-sampling intervals that are much longer than those found in comparably performing periodically sampled control systems. This paper establishes sufficient conditions under which an event-triggered system is attentively efficient. These conditions depend on the relative rates of growth in the class K functions used in dissipative characterizations of the input-to-state stability (ISS) property. Since these functions determine the type of controller used by the system, these results suggest that a suitable choice of controller has a greatly increase the inter-sampling intervals seen in event-triggered control systems. In other words, the design of attentively efficient event-triggers with sufficiently long sampling intervals may really be an issue of nonlinear controller design.