Feedback Coding Schemes for Control Over Gaussian Networks

The presence of inexpensive and powerful sensing and communication devices has made it possible to deploy large scale distributed systems for a variety of applications. Interactions among different components of such a system include communication of information and controlling dynamical processes, among others. Thus, it is important to blend ideas from information theory and control theory to address problems at the core of such distributed systems.

This dissertation looks at communication scenarios, in which there are feedback channels available for enhancing communication and control performance over noisy forward links. Such feedback can considerably increase the reliability or reduce the complexity of coding schemes that approach capacity. Moreover, transmission schemes with feedback can be used to stabilize unstable plants over communication channels, where a sensor transmits the plant state information to a remotely placed controller. We develop transmission schemes for a certain class of Gaussian networks with feedback and apply these schemes to obtain sufficient and, in some cases, necessary conditions for stabilizing a plant via a remotely placed controller.

Specifically, we develop coding schemes for a Gaussian relay channel and a Gaussian product channel with feedback. The coding scheme for the Gaussian relay channel with feedback is based on distributed stochastic approximation algorithms. We consider two topologies for the relay channel: (i) a cascade of two Gaussian point to point channels, and (ii) a Gaussian relay channel. Further, we use the proposed coding scheme to mean square stabilize an unstable discrete-time linear time invariant (LTI) system over a Gaussian relay channel and derive sufficient conditions for stabilizability. For the Gaussian product channel, we concentrate on the feedback stabilization problem. It is known that linear coding schemes may lead to overly restrictive stabilizability conditions in such scenarios. We develop a non-linear quantization-based coding scheme and present the resulting stabilizability conditions. When these conditions are satisfied with equality, the proposed coding scheme transmits data across the product channel at a rate equal to the capacity of the channel; thus, the conditions are necessary as well. We combine coding schemes for the relay channel and the product channel to consider more general network examples, whose achievability and stability results can be obtained using the simpler relay and product channel results derived earlier.