Feedback Control Over Lossy SNR-Limited Channels: Linear Encoder-Decoder-Controller Design

In this paper, we consider the problem of encoding and decoding codesign for linear feedback control of a scalar, possibly unstable, stochastic linear system when the sensed signal is to be transmitted over a finite capacity communication channel. In particular, we consider a limited capacity channel which transmits quantized data and is subject to packet losses. We first characterize the optimal strategy when perfect channel feedback is available, i.e., the transmitter has perfect knowledge of the packet loss history. This optimal scheme, innovation forwarding hereafter, is reminiscent of differential pulse-code modulation schemes adapted to deal with state space models, and is strictly better than a scheme which simply transmits the measured data, called measurement forwarding (MF) hereafter. Comparison in terms of control cost as well as of critical regimes, i.e., regimes where the cost is not finite, are provided. We also consider and compare two popular suboptimal schemes from the existing literature, based on 1) state estimate forwarding and 2) measurement forwarding, which ignore quantization effects in the associated estimator and controller design. In particular, it is shown that surprisingly the suboptimal MF strategy is always better then the suboptimal state forwarding strategy for small signal-to-quantization-noise-ratios.