Covariance Control of Discrete-Time Gaussian Linear Systems Using Affine Disturbance Feedback Control Policies

TL;DR

This paper introduces a new affine disturbance feedback control policy for finite-horizon covariance steering in discrete-time Gaussian linear systems, simplifying the problem into tractable optimization forms.

Contribution

It proposes a novel parametrization that reduces covariance steering to SDP and DCP problems, improving computational efficiency over existing methods.

Findings

Reduces hard-constrained covariance steering to a semi-definite program.

Transforms soft-constrained covariance steering into a difference of convex functions program.

Demonstrates computational advantages through theoretical analysis and simulations.

Abstract

In this paper, we present a new control policy parametrization for the finite-horizon covariance steering problem for discrete-time Gaussian linear systems (DTGLS) which can reduce the latter stochastic optimal control problem to a tractable optimization problem. The covariance steering problem seeks for a feedback control policy that will steer the state covariance of a DTGLS to a desired positive definite matrix in finite time. We consider two different formulations of the covariance steering problem, one with hard terminal LMI constraints (hard-constrained covariance steering) and another one with soft terminal constraints in the form of a terminal cost which corresponds to the squared Wasserstein distance between the actual terminal state (Gaussian) distribution and the desired one (soft-constrained covariance steering). We propose a solution approach that relies on the affine disturbance feedback parametrization for both problem formulations. We show that this particular parametrization allows us to reduce the hard-constrained covariance steering problem into a semi-definite program (SDP) and the soft-constrained covariance steering problem into a difference of convex functions program(DCP). Finally, we show the advantages of our approach over other covariance steering algorithms in terms of computational complexity and computation time by means of theoretical analysis and numerical simulations.