Discrete-time Optimal Covariance Steering via Semidefinite Programming

TL;DR

This paper introduces an efficient semidefinite programming approach for solving the optimal covariance steering problem in stochastic discrete-time linear systems, ensuring exact solutions with reduced computational effort.

Contribution

It presents a lossless convex relaxation method for covariance steering, applicable to various constraints and boundary conditions, improving computational efficiency over existing methods.

Findings

The relaxation is proven to be lossless in all cases.

The method achieves exact solutions efficiently.

Comparative studies show reduced computational resources.

Abstract

This paper addresses the optimal covariance steering problem for stochastic discrete-time linear systems subject to probabilistic state and control constraints. A method is presented for efficiently attaining the exact solution of the problem based on a lossless convex relaxation of the original non-linear program using semidefinite programming. Both the constrained and the unconstrained versions of the problem with either equality or inequality terminal covariance boundary conditions are addressed. We first prove that the proposed relaxation is lossless for all of the above cases. A numerical example is then provided to illustrate the method. Finally, a comparative study is performed in systems of various sizes and steering horizons to illustrate the advantages of the proposed method in terms of computational resources compared to the state of the art.