Tracking and Control of Gauss–Markov Processes over Packet-Drop Channels with Acknowledgments

Abstract

We consider the problem of tracking the state of Gauss-Markov processes over rate-limited erasureprone links. We concentrate first on the scenario in which several independent processes are seen by a single observer. The observer maps the processes into finite-rate packets that are sent over the erasureprone links to a state estimator, and are acknowledged upon packet arrivals. The aim of the state estimator is to track the processes with zero delay and with minimum mean square error (MMSE). We show that, in the limit of many processes, greedy quantization with respect to the squared error distortion is optimal.That is, there is no tension between optimizing the MMSE of the process in the current time instant and that of future times. For the case of packet erasures with delayed acknowledgments, we connect the problem to that of compression with side information that is known at the observer and may be known at the state estimator -where the most recent packets serve as side information that may have been erased, and demonstrate that the loss due to a delay by one time unit is rather small. For the scenario where only one process is tracked by the observer-state estimator system, we further show that variable-length coding This work was done, in part, while A. Khina and V. Kostina were visiting the Simons Institute for the Theory of Computing.techniques are within a small gap of the many-process outer bound. We demonstrate the usefulness of the proposed approach for the simple setting of discrete-time scalar linear quadratic Gaussian control with a limited data-rate feedback that is susceptible to packet erasures. Index TermsState tracking, state estimation, networked control, packet erasures, source coding with side information, sequential coding of correlated sources, successive refinement. I. INTRODUCTIONTracking the state of a system from noisy and possibly partially observable measurements is of prime importance in many estimation scenarios, and serves as an important building block in many control setups.The recent rapid growth in wireless connectivity and its ad hoc distributed nature, while offering a plethora of new and exciting possibilities, introduces new design challenges for control over such media.These challenges include, among others, the need to track processes with minimal error over digital links of limited data rate which could be prone to (packet) erasures, and joint processing and reconstruction of distributed processes.An important scenario, often encountered in practice, depicted in Fig. 1, is that of a multi-track system that tracks several processes over a single shared communication link. In this scenario, at each time instant, several processes are observed by a single observer. The observer, in turn, collects the measured states of these processes into a single vector state or frame, and maps them into finite-rate packets, which are sent to the state-estimator over a channel which is prone to packet erasures. The state estimator tracks the latest states of the different processes, by...

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“…[ 39 ] provides an asymptotically optimal scheme that roughly uses Gaussian codebooks to quantize the innovation errors between the encoder’s observation and the decoder’s estimation. Our work is closest to [ 39 ] and our encoding scheme shares similarities with the predictive Differential Pulse Code Modulation (DPCM) based scheme employed in [ 39 ]. However, our work differs in an important aspect from all the works mentioned in this research thread: we take transmission delays into account.…”

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