Continuous-Time Channel Gain Control for Minimum-Information Kalman-Bucy Filtering

TL;DR

This paper develops a framework for controlling the channel gain in continuous-time Kalman-Bucy filtering to optimize estimation accuracy and information transfer, introducing novel control strategies and optimization techniques.

Contribution

It formulates a new optimal control problem for channel gain in continuous-time filtering, deriving necessary conditions and proposing an SDP heuristic for solution.

Findings

Optimal scalar gain is piece-wise constant with at most two switches.

Derived formulas for estimation error and mutual information based on channel gain.

Proposed SDP heuristic effectively approximates the optimal gain.

Abstract

We consider the problem of estimating a continuous-time Gauss-Markov source process observed through a vector Gaussian channel with an adjustable channel gain matrix. For a given (generally time-varying) channel gain matrix, we provide formulas to compute (i) the mean-square estimation error attainable by the classical Kalman-Bucy filter, and (ii) the mutual information between the source process and its Kalman-Bucy estimate. We then formulate a novel "optimal channel gain control problem" where the objective is to control the channel gain matrix strategically to minimize the weighted sum of these two performance metrics. To develop insights into the optimal solution, we first consider the problem of controlling a time-varying channel gain over a finite time interval. A necessary optimality condition is derived based on Pontryagin's minimum principle. For a scalar system, we show that the optimal channel gain is a piece-wise constant signal with at most two switches. We also consider the problem of designing the optimal time-invariant gain to minimize the average cost over an infinite time horizon. A novel semidefinite programming (SDP) heuristic is proposed and the exactness of the solution is discussed.