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Steady State Covariance Steering via Sparse Intervention

Yosuke Inoue, Masaki Inoue

TL;DR

Numerical simulations demonstrate that the proximal gradient-based algorithm effectively identifies sparse, structurally-constrained interventions to achieve precise covariance steering.

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.

Steady State Covariance Steering via Sparse Intervention

TL;DR

Numerical simulations demonstrate that the proximal gradient-based algorithm effectively identifies sparse, structurally-constrained interventions to achieve precise covariance steering.

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.
Paper Structure (7 sections, 2 theorems, 19 equations, 4 figures, 1 algorithm)

This paper contains 7 sections, 2 theorems, 19 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Proximal Gradient-based Algorithm
  4. Gradient Characterization
  5. Optimization Algorithm
  6. Numerical Simulation
  7. Conclusion

Key Result

Proposition 1

For an $n$-dimensional random variable, the KL divergence between the two zero-mean Gaussian distributions $\mathcal{N}(0_n, \Sigma_1)$ and $\mathcal{N}(0_n, \Sigma_2)$ is expressed as

Figures (4)

  • Figure 1: Covariance ellipsoid of the reference covariance matrix $\Sigma_{\mathrm{ref}}$ and scatter plot of $x$ projected onto the principal component space (Without control).
  • Figure 2: Covariance ellipsoid of the reference covariance matrix $\Sigma_{\mathrm{ref}}$ and scatter plot of $x$ projected onto the principal component space (With control).
  • Figure 3: Evolution of $J$ against the number of iterations.
  • Figure 4: Relationship between the regularization parameter $\lambda$, the number of non-zero elements in the intervention matrix $U$ (blue plot), and the final objective value $J$ (orange plot).

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • Theorem 1
  • Proof
  • Definition 2