  Distributed state estimation is examined for a sensor network tasked with
reconstructing a system's state through the use of a distributed and
event-triggered observer. Each agent in the sensor network employs a deep
neural network (DNN) to approximate the uncertain nonlinear dynamics of the
system, which is trained using a multiple timescale approach. Specifically, the
outer weights of each DNN are updated online using a Lyapunov-based gradient
descent update law, while the inner weights and biases are trained offline
using a supervised learning method and collected input-output data. The
observer utilizes event-triggered communication to promote the efficient use of
network resources. A nonsmooth Lyapunov analysis shows the distributed
event-triggered observer has a uniformly ultimately bounded state
reconstruction error. A simulation study is provided to validate the result and
demonstrate the performance improvements afforded by the DNNs.
