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Data-Driven Covariance Steering with Output Feedback

Dimitrios Moustroufis, Panagiotis Tsiotras

Abstract

This paper addresses the problem of output-feedback covariance steering for stochastic, discrete-time, linear, time-invariant systems without knowledge of the system model. We employ a controllable, non-minimal state representation constructed from past inputs and outputs and convert the problem to one in state-feedback form. In this representation, the induced disturbance becomes temporally correlated, which requires explicit propagation of the cross-covariance between the state and disturbance processes. To handle the lack of a system model, we leverage persistently exciting data collected offline and formulate the mean and covariance steering problems using an indirect and a direct approach, respectively. The indirect formulation requires an estimate of the mean dynamics model, while the direct formulation relies on an estimate of the noise realization in the collected data. To this end, we present an estimation method suitable to handle temporally correlated noise, enabling consistent identification of both components. Using a convex relaxation, we convert the covariance steering problem to a semidefinite program that can be solved efficiently. We conduct numerical simulations to evaluate the performance of the developed framework.

Data-Driven Covariance Steering with Output Feedback

Abstract

This paper addresses the problem of output-feedback covariance steering for stochastic, discrete-time, linear, time-invariant systems without knowledge of the system model. We employ a controllable, non-minimal state representation constructed from past inputs and outputs and convert the problem to one in state-feedback form. In this representation, the induced disturbance becomes temporally correlated, which requires explicit propagation of the cross-covariance between the state and disturbance processes. To handle the lack of a system model, we leverage persistently exciting data collected offline and formulate the mean and covariance steering problems using an indirect and a direct approach, respectively. The indirect formulation requires an estimate of the mean dynamics model, while the direct formulation relies on an estimate of the noise realization in the collected data. To this end, we present an estimation method suitable to handle temporally correlated noise, enabling consistent identification of both components. Using a convex relaxation, we convert the covariance steering problem to a semidefinite program that can be solved efficiently. We conduct numerical simulations to evaluate the performance of the developed framework.

Paper Structure

This paper contains 19 sections, 3 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Problem Statement
  5. Non-minimal State Space Representation
  6. Non-minimal State Distribution Steering Problem
  7. Data-Driven System Representation
  8. Persistency of Excitation and Multi-Output Systems
  9. Total Least Squares Method
  10. Instrumental Variables Method
  11. Data-Driven Distribution Steering
  12. Data-driven Mean Steering
  13. Data-driven Covariance Steering
  14. Numerical Example
  15. Model Estimation
  16. ...and 4 more sections

Key Result

Lemma 1

The samples $\tilde{\zeta}_k$ are zero-mean, Gaussian, identically distributed, but not independent. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure E1: Model estimation error.
  • Figure E2: Terminal mean error.
  • Figure E3: Terminal covariance ellipses.
  • Figure E4: Output trajectories in phase plane.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2