Steering state statistics with output feedback

TL;DR

This paper explores how to control the distribution of a linear stochastic system's state using output feedback, establishing conditions under which the state covariance can be steered to a desired distribution.

Contribution

It extends previous work by providing necessary and sufficient conditions for steering the state covariance with output feedback, involving the Kalman filter error covariance.

Findings

Steering the state covariance is possible if the terminal covariance exceeds the Kalman filter error covariance.

The work characterizes the conditions for covariance steering with partial observations.

Provides a theoretical framework linking control, estimation, and distribution steering.

Abstract

Consider a linear stochastic system whose initial state is a random vector with a specified Gaussian distribution. Such a distribution may represent a collection of particles abiding by the specified system dynamics. In recent publications, we have shown that, provided the system is controllable, it is always possible to steer the state covariance to any specified terminal Gaussian distribution using state feedback. The purpose of the present work is to show that, in the case where only partial state observation is available, a necessary and sufficient condition for being able to steer the system to a specified terminal Gaussian distribution for the state vector is that the terminal state covariance be greater (in the positive-definite sense) than the error covariance of a corresponding Kalman filter.