Reducing the LQG Cost with Minimal Communication

TL;DR

This paper explores how to minimize communication in LQG control by optimally balancing control cost and information flow, providing a semidefinite programming approach for finite-horizon systems and explicit solutions for infinite-horizon cases.

Contribution

It extends existing work by formulating a semidefinite programming approach to optimize the trade-off between control performance and communication resources in LQG systems.

Findings

Optimal policies are time-invariant in infinite-horizon scenarios.

Memoryless Gaussian measurements are optimal for encoding.

Low-quality measurements significantly reduce communication needs.

Abstract

We study the linear quadratic Gaussian (LQG) control problem, in which the controller's observation of the system state is such that a desired cost is unattainable. To achieve the desired LQG cost, we introduce a communication link from the observer (encoder) to the controller. We investigate the optimal trade-off between the improved LQG cost and the consumed communication (information) resources, measured with the conditional directed information, across all encoding-decoding policies. The main result is a semidefinite programming formulation for that optimization problem in the finite-horizon scenario, which applies to time-varying linear dynamical systems. This result extends a seminal work by Tanaka et al., where the only information the controller knows about the system state arrives via a communication channel, to the scenario where the controller has also access to a noisy observation of the system state. As part of our derivation to show the optimiality of an encoder that transmits a memoryless Gaussian measurement of the state, we show that the presence of the controller's observations at the encoder can not reduce the minimal directed information. For time-invariant systems, where the optimal policy may be time-varying, we show in the infinite-horizon scenario that the optimal policy is time-invariant and can be computed explicitly from a solution of a finite-dimensional semidefinite programming. The results are demonstrated via examples that show that even low-quality measurements can have a significant impact on the required communication resources.