Steady State Covariance Steering via Sparse Intervention

TL;DR

This paper introduces a method for steering the steady state covariance of linear systems through sparse structural interventions, utilizing a proximal gradient algorithm that minimizes KL divergence with analytical gradient expressions.

Contribution

It presents a novel analytical expression for the KL divergence gradient and a proximal gradient algorithm for sparse covariance steering via structural intervention.

Findings

Effective identification of sparse interventions

Accurate covariance steering demonstrated in simulations

Analytical gradient expression improves optimization efficiency

Abstract

This paper addresses the steady state covariance steering for linear dynamical systems via structural intervention on the system matrix. We formulate the covariance steering problem as the minimization of the Kullback-Leibler (KL) divergence between the steady state and target Gaussian distributions. To solve the problem, we develop a solution method, hereafter referred to as the proximal gradient-based algorithm, of promoting sparsity in the structural intervention by integrating the objective into a proximal gradient framework with L1 regularization. The main contribution of this paper lies in the analytical expression of the KL divergence gradient with respect to the intervention matrix: the gradient is characterized by the solutions to two Lyapunov equations related to the state covariance equation and its adjoint. Numerical simulations demonstrate that the proximal gradient-based algorithm effectively identifies sparse, structurally-constrained interventions to achieve precise covariance steering.