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Data-Driven Robust Covariance Control for Uncertain Linear Systems

Joshua Pilipovsky, Panagiotis Tsiotras

TL;DR

This work develops a data-driven covariance steering framework for unknown discrete-time linear systems with Gaussian disturbances. By leveraging Willems' Fundamental Lemma and persistence of excitation, it expresses control policies directly from input–state data, and employs maximum-likelihood methods to bound noise realizations, forming a robust convex optimization (RC) problem to ensure terminal covariance constraints. The approach is validated on a double integrator, showing competitive performance against model-based controllers and robustness to model perturbations. The results enable end-to-end data-driven CS with provable robustness, offering practical relevance for safe, distribution-focused control in uncertain systems.

Abstract

The theory of covariance control and covariance steering (CS) deals with controlling the dispersion of trajectories of a dynamical system, under the implicit assumption that accurate prior knowledge of the system being controlled is available. In this work, we consider the problem of steering the distribution of a discrete-time, linear system subject to exogenous disturbances under an unknown dynamics model. Leveraging concepts from behavioral systems theory, the trajectories of this unknown, noisy system may be (approximately) represented using system data collected through experimentation. Using this fact, we formulate a direct data-driven covariance control problem using input-state data. We then propose a maximum likelihood uncertainty quantification method to estimate and bound the noise realizations in the data collection process. Lastly, we utilize robust convex optimization techniques to solve the resulting norm-bounded uncertain convex program. We illustrate the proposed end-to-end data-driven CS algorithm on a double integrator example and showcase the efficacy and accuracy of the proposed method compared to that of model-based methods

Data-Driven Robust Covariance Control for Uncertain Linear Systems

TL;DR

This work develops a data-driven covariance steering framework for unknown discrete-time linear systems with Gaussian disturbances. By leveraging Willems' Fundamental Lemma and persistence of excitation, it expresses control policies directly from input–state data, and employs maximum-likelihood methods to bound noise realizations, forming a robust convex optimization (RC) problem to ensure terminal covariance constraints. The approach is validated on a double integrator, showing competitive performance against model-based controllers and robustness to model perturbations. The results enable end-to-end data-driven CS with provable robustness, offering practical relevance for safe, distribution-focused control in uncertain systems.

Abstract

The theory of covariance control and covariance steering (CS) deals with controlling the dispersion of trajectories of a dynamical system, under the implicit assumption that accurate prior knowledge of the system being controlled is available. In this work, we consider the problem of steering the distribution of a discrete-time, linear system subject to exogenous disturbances under an unknown dynamics model. Leveraging concepts from behavioral systems theory, the trajectories of this unknown, noisy system may be (approximately) represented using system data collected through experimentation. Using this fact, we formulate a direct data-driven covariance control problem using input-state data. We then propose a maximum likelihood uncertainty quantification method to estimate and bound the noise realizations in the data collection process. Lastly, we utilize robust convex optimization techniques to solve the resulting norm-bounded uncertain convex program. We illustrate the proposed end-to-end data-driven CS algorithm on a double integrator example and showcase the efficacy and accuracy of the proposed method compared to that of model-based methods
Paper Structure (16 sections, 8 theorems, 53 equations, 2 figures)

This paper contains 16 sections, 8 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Notation
  4. Data-Driven Covariance Steering
  5. Problem Reformulation
  6. Data-Driven Parameterization
  7. Direct Data-Driven Covariance Steering
  8. Indirect Data-Driven Mean Steering
  9. Uncertainty Quantification
  10. Maximum Likelihood Estimation of Noise Realization
  11. Uncertainty Error Bounds
  12. Robust DD-CS
  13. Problem Formulation and Robust LMI Counterpart
  14. Numerical Example
  15. Conclusion
  16. ...and 1 more sections

Key Result

theorem 1

Using the data-driven parameterization of the feedback gains eq:WFL_gains, Problem problem:3 may be relaxed as the convex program where $\Xi_{0,T} := [\boldsymbol{\xi}_0, \ldots, \boldsymbol{\xi}_{T-1}]\in\mathbb{R}^{n\times T}$ is the Hankel matrix of the (unknown) disturbances, and $\boldsymbol{\xi}_k \sim \mathcal{N}(0,\Sigma_{\boldsymbol{\xi}})$, where $\Sigma_{\boldsymbol{\xi}} := DD^\interc

Figures (2)

  • Figure 1: Comparison of model-based and data-driven covariance steering controllers for (a) perfect system knowledge, and (b) imperfect system knowledge. The robust data-driven solutions are not only adaptable to any linear dynamics due to sampling from the true system but also achieve a desirable terminal covariance.
  • Figure 2: Model-based (a) and robust data-driven (b) covariance control designs for perturbed disturbance model $D + \Delta D$.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Remark 1
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • Corollary 1
  • Lemma 1
  • ...and 9 more