An approximate solution to the decentralized two-controller infinite-horizon scalar LQG problem: Part I- fast dynamics

TL;DR

This paper addresses the decentralized scalar LQG problem with fast dynamics, proposing nonlinear controllers that achieve near-optimal performance by leveraging insights from wireless communication theory.

Contribution

It introduces a set of finite-dimensional nonlinear controllers that are approximately optimal for the fast dynamics case in decentralized LQG problems, supported by new mathematical tools.

Findings

Nonlinear controllers outperform linear ones in fast dynamics scenarios.

Proposed controllers achieve within a constant ratio of the optimal cost.

New mathematical tools like geometric slicing and approximate-comb-lattice model are developed.

Abstract

We consider scalar decentralized average-cost infinite-horizon LQG problems with two controllers, focusing on the fast dynamics case when the (scalar) eigenvalue of the system is large. It is shown that the best linear controllers' performance can be an arbitrary factor worse than the optimal performance. We propose a set of finite-dimensional nonlinear controllers, and prove that the proposed set contains an easy-to-find approximately optimal solution that achieves within a constant ratio of the optimal quadratic cost. The insight for nonlinear strategies comes from revealing the relationship between information flow in control and wireless information flow. More precisely, we discuss a close relationship between the high-SNR limit in wireless communication and fast-dynamics case in decentralized control, and justify how the proposed nonlinear control strategy can be understood as exploiting the generalized degree-of-freedom gain in wireless communication theory. For a rigorous justification of this argument, we develop new mathematical tools and ideas. To reveal the relationship between infinite-horizon problems and generalized MIMO Witsenhausen's counterexamples, we introduce the idea of geometric slicing. To analyze the nonlinear strategy performance, we introduce an approximate-comb-lattice model for the relevant random variables.