  We study the linear quadratic Gaussian (LQG) control problem, in which the
controller's observation of the system state is such that a desired cost is
unattainable. To achieve the desired LQG cost, we introduce a communication
link from the observer (encoder) to the controller. We investigate the optimal
trade-off between the improved LQG cost and the consumed communication
(information) resources, measured with the conditional directed information,
across all encoding-decoding policies. The main result is a semidefinite
programming formulation for that optimization problem in the finite-horizon
scenario, which applies to time-varying linear dynamical systems. This result
extends a seminal work by Tanaka et al., where the only information the
controller knows about the system state arrives via a communication channel, to
the scenario where the controller has also access to a noisy observation of the
system state. As part of our derivation to show the optimiality of an encoder
that transmits a memoryless Gaussian measurement of the state, we show that the
presence of the controller's observations at the encoder can not reduce the
minimal directed information. For time-invariant systems, where the optimal
policy may be time-varying, we show in the infinite-horizon scenario that the
optimal policy is time-invariant and can be computed explicitly from a solution
of a finite-dimensional semidefinite programming. The results are demonstrated
via examples that show that even low-quality measurements can have a
significant impact on the required communication resources.
