Remote Estimation With Noisy Measurements Subject to Packet Loss and Quantization Noise

Abstract

In this paper, we consider the problem of designing coding and decoding schemes to estimate the state of a scalar stable stochastic linear system subject to noisy measurements and in the presence of a wireless communication channel between the sensor and the estimator. In particular, we consider a communication channel which is prone to packet loss and includes quantization noise due to its limited capacity. We study two scenarios: the first with channel feedback and the second with no channel feedback. More specifically, in the first scenario the transmitter is aware of the quantization noise and the packet loss history of the channel, while in the second scenario the transmitter is aware of the quantization noise only. We show that in the first scenario, the optimal strategy among all possible linear encoders corresponds to the transmission of the Kalman filter innovation, similar to the differential pulse-code modulation (DPCM) technique used in digital communications. In the second scenario, we show that there is a critical packet loss probability above which it is better to transmit the state rather than the innovation. We also propose a heuristic strategy based on the transmission of a convex combination of the state and the Kalman filter innovation which is shown to provide a performance close to the one obtained with channel feedback.

Linear encoder-decoder-controller design over channels with packet loss and quantization noise

“…In this work we extend the results of [19] and [20] by considering the possibility to pre-process the raw measurement at the transmitter. We show that the optimal strategy when full channel feedback is available at the transmitter is to send the difference from the estimated state at the transmitter and the predicted state at the receiver as in [19] and to build a Kalman filter and a state feedback with constant gain at the receiver as in [20].…”

“…We show that the optimal strategy when full channel feedback is available at the transmitter is to send the difference from the estimated state at the transmitter and the predicted state at the receiver as in [19] and to build a Kalman filter and a state feedback with constant gain at the receiver as in [20]. However, although the performance is improved as compared to strategy proposed in [20], the stability region is the same.…”

“…Obviously, this strategy is rather different from the one that transmits the complete state estimate. Recently, in [19] we have shown that a strategy inspired by DPCM, is optimal in the context of remote estimation when considering both packet loss and quantization limitations for a scalar stable system when full channel feedback is present. However, the optimal strategy in the presence of imperfect channel feedback remains elusive and only sensible heuristic have been proposed in [19].…”

“…Recently, in [19] we have shown that a strategy inspired by DPCM, is optimal in the context of remote estimation when considering both packet loss and quantization limitations for a scalar stable system when full channel feedback is present. However, the optimal strategy in the presence of imperfect channel feedback remains elusive and only sensible heuristic have been proposed in [19]. In the context of closed loop unstable control system, simultaneous analysis of packet loss and quantization has been studied in [20] assuming that the transmitter simply forwards a quantized version of the raw measurement.…”

“…However, although the performance is improved as compared to strategy proposed in [20], the stability region is the same. In the imperfect channel scenario, we propose a number of heuristics similarly to those proposed in [19]. However, differently from remote estimation, in closed loop systems the effect of packet loss between the transmitter and the controller appears implicitly in the measurements observed at the transmitter even in the absence of any channel feedback.…”