  We study transmission power budget minimization of battery-powered nodes in a
remote state estimation problem over multi-hop wireless networks. Communication
links between nodes are subject to fading, thereby generating random dropouts.
Relay nodes help to transmit measurements from distributed sensors to an
estimator node. Hopping through each relay node introduces a unit delay.
Motivated by the need for estimators with low computational and implementation
cost, we propose a jump estimator whose modes depend on a Markovian parameter
that describes measurement transmission outcomes over a finite interval. It is
well known that transmission power helps to increase the reliability of
measurement transmissions, at the expense of reducing the life-time of the
nodes' battery. Motivated by this, we derive a tractable iterative procedure,
based on semi-definite programming, to design a finite set of filter gains, and
associated power control laws to minimize the energy budget while guaranteeing
an estimation performance level. This procedure allows us to tradeoff the
complexity of the filter implementation with performance and energy use.
