  This work attempts to discuss the observability of linear time-invariant
systems with event-triggered measurements. A new notion of observability,
namely, $\epsilon$-observability is defined with parameter $\epsilon$, which
relates to the worst-case performance of inferring the initial state based on
not only the received measurement but also the implicit information in the
event-triggering conditions at no-event instants. A criterion is developed to
test the proposed $\epsilon$-observability of discrete-time linear systems,
based on which an iterative event-triggered set-membership observer is designed
to evaluate a set containing all possible values of the state. The proposed
set-membership observer is designed as the outer approximation of the
ellipsoids predicted based on previous state estimates and the ellipsoids
inferred by fusing the received measurement and communication conditions, which
is optimal in the sense of trace at each step and is proved to be
asymptotically bounded. The efficiency of the proposed event-triggered
set-membership state observer is verified by numerical experiments.
