  Real-time remote estimation is critical for mission-critical applications
including industrial automation, smart grid and tactile Internet. In this
paper, we propose a hybrid automatic repeat request (HARQ)-based real-time
remote estimation framework for linear time-invariant (LTI) dynamic systems.
Considering the estimation quality of such a system, there is a fundamental
tradeoff between the reliability and freshness of the sensor's measurement
transmission. We formulate a new problem to optimize the sensor's online
transmission control policy for static and Markov fading channels, which
depends on both the current estimation quality of the remote estimator and the
current number of retransmissions of the sensor, so as to minimize the
long-term remote estimation mean squared error (MSE). This problem is
non-trivial. In particular, it is challenging to derive the condition in terms
of the communication channel quality and the LTI system parameters, to ensure a
bounded long-term estimation MSE. We derive an elegant sufficient condition of
the existence of a stationary and deterministic optimal policy that stabilizes
the remote estimation system and minimizes the MSE. Also, we prove that the
optimal policy has a switching structure, and accordingly derive a
low-complexity suboptimal policy. Numerical results show that the proposed
optimal policy significantly improves the performance of the remote estimation
system compared to the conventional non-HARQ policy.
