Stabilization of Linear Systems Over Gaussian Networks

TL;DR

This paper investigates the conditions for stabilizing linear systems over Gaussian relay networks, deriving both necessary and sufficient criteria, and identifying optimal policies under various network configurations.

Contribution

It provides new necessary and sufficient conditions for stabilization over Gaussian networks, and characterizes optimal policies for different system and channel setups.

Findings

Linear policies are optimal or asymptotically optimal in certain scenarios.

Minimum capacity requirements are achieved by linear time-varying policies.

Non-linear policies can meet the fundamental capacity lower bound in noiseless source and parallel channel settings.

Abstract

The problem of remotely stabilizing a noisy linear time invariant plant over a Gaussian relay network is addressed. The network is comprised of a sensor node, a group of relay nodes and a remote controller. The sensor and the relay nodes operate subject to an average transmit power constraint and they can cooperate to communicate the observations of the plant's state to the remote controller. The communication links between all nodes are modeled as Gaussian channels. Necessary as well as sufficient conditions for mean-square stabilization over various network topologies are derived. The sufficient conditions are in general obtained using delay-free linear policies and the necessary conditions are obtained using information theoretic tools. Different settings where linear policies are optimal, asymptotically optimal (in certain parameters of the system) and suboptimal have been identified. For the case with noisy multi-dimensional sources controlled over scalar channels, it is shown that linear time varying policies lead to minimum capacity requirements, meeting the fundamental lower bound. For the case with noiseless sources and parallel channels, non-linear policies which meet the lower bound have been identified.