  The distributed observer design problem holds significant importance in cases
in which the output information of a system is decentralized across different
subsystems. Each subsystem has a local observer and access to one part of the
measurement outputs and information exchanged through communication networks.
This paper focuses on the design of distributed observer with jointly connected
directed switching networks. The problem presents challenges due to passive
switching modes and the open-loop unboundedness that results from local
observability. To overcome these challenges, we develop a network
transformation mapping method whereby each local observer can classify itself
into an independent subgraph based on independent judgment. Next, an observable
decomposition and reorganization method is developed for the digraph case to
ensure that each subgraph possesses independent dynamic properties. Asymptotic
omniscience is then proven using a developed recursive proof method. This paper
includes many previous results as special cases, because most are only suitable
for undirected switching topologies or fast-switching cases. An adaptive
coupling gain design is proposed to simplify the calculation and verification
of conditions that guarantee asymptotic omniscience. Finally, simulation
results with the power system show the validity of the developed theory.
