  Real-time remote estimation is critical for mission-critical applications
including industrial automation, smart grid, and the 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. When a failed transmission occurs, the sensor can either
retransmit the previous old measurement such that the receiver can obtain a
more reliable old measurement, or transmit a new but less reliable measurement.
To design the optimal decision, we formulate a new problem to optimize the
sensor's online decision policy, i.e., to retransmit or not, depending 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 not clear what the condition is in terms of the communication
channel quality and the LTI system parameters, to ensure that the long-term
estimation MSE can be bounded. We give a 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 derive a low-complexity suboptimal policy. Our
numerical results show that the proposed optimal policy notably improves the
performance of the remote estimation system compared to the conventional
non-HARQ policy.
