Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Data-Driven Stochastic Output-Feedback Predictive Control

Guanru Pan, Ruchuan Ou, Timm Faulwasser

TL;DR

This article provides sufficient conditions for the recursive feasibility of the proposed output-feedback scheme based on a data-driven design of the terminal ingredients of the OCP, and provides a robustness analysis of the closed-loop performance.

Abstract

The fundamental lemma by Jan C. Willems and co-authors enables the representation of all input-output trajectories of a linear time-invariant system by measured input-output data. This result has proven to be pivotal for data-driven control. Building on a stochastic variant of the fundamental lemma, this paper presents a data-driven output-feedback predictive control scheme for stochastic Linear Time-Invariant (LTI) systems. The considered LTI systems are subject to non-Gaussian disturbances about which only information about their first two moments is known. Leveraging polynomial chaos expansions, the proposed scheme is centered around a data-driven stochastic Optimal Control Problem (OCP). Through tailored online design of initial conditions, we provide sufficient conditions for the recursive feasibility of the proposed output-feedback scheme based on a data-driven design of the terminal ingredients of the OCP. Furthermore, we provide a robustness analysis of the closed-loop performance. A numerical example illustrates the efficacy of the proposed scheme.

On Data-Driven Stochastic Output-Feedback Predictive Control

TL;DR

This article provides sufficient conditions for the recursive feasibility of the proposed output-feedback scheme based on a data-driven design of the terminal ingredients of the OCP, and provides a robustness analysis of the closed-loop performance.

Abstract

The fundamental lemma by Jan C. Willems and co-authors enables the representation of all input-output trajectories of a linear time-invariant system by measured input-output data. This result has proven to be pivotal for data-driven control. Building on a stochastic variant of the fundamental lemma, this paper presents a data-driven output-feedback predictive control scheme for stochastic Linear Time-Invariant (LTI) systems. The considered LTI systems are subject to non-Gaussian disturbances about which only information about their first two moments is known. Leveraging polynomial chaos expansions, the proposed scheme is centered around a data-driven stochastic Optimal Control Problem (OCP). Through tailored online design of initial conditions, we provide sufficient conditions for the recursive feasibility of the proposed output-feedback scheme based on a data-driven design of the terminal ingredients of the OCP. Furthermore, we provide a robustness analysis of the closed-loop performance. A numerical example illustrates the efficacy of the proposed scheme.
Paper Structure (22 sections, 10 theorems, 92 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 10 theorems, 92 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic Output Feedback Predictive Control
  4. Data-Driven Uncertainty Propagation
  5. Basics of Polynomial Chaos Expansions
  6. Data-Driven Uncertainty Propagation via PCE
  7. Data-Driven Stochastic Optimal Control
  8. Data-Driven Stochastic Output Feedback
  9. Recursive Feasibility with Backup Initial Condition
  10. Online Selection of Initial Conditions
  11. Design of the Projection $T_k$
  12. Proposed Predictive Control Algorithm
  13. Analysis of Closed-Loop Properties
  14. Data-Driven Design of Terminal Ingredients
  15. Robustness Analysis with Estimated Disturbances
  16. ...and 7 more sections

Key Result

Lemma 1

The pair $(\widetilde{A},\left[\widetilde{B},\widetilde{E}\right])$ from eq:extended_dyna is controllable regardless of the ARX system matrices $\Phi$ and $D$. $\square$

Figures (2)

  • Figure 1: Histograms of the output $Y^2$ from 1000 closed-loop realization trajectories of Scheme II).
  • Figure 2: 20 different closed-loop realization trajectories of Algorithm \ref{['alg:datadrivenSMPC']}. Blue-dashed line: $y^1=-1$; red-dashed line: $y^1=1$. Left: Scheme I) with measured disturbances; right: Scheme II) with estimated disturbances.

Theorems & Definitions (24)

  • Definition 1
  • Lemma 1
  • Remark 1: Disturbances and measurement noise
  • Remark 2: Witsenhausen's point of view
  • Definition 2: Exact PCE representation muehlpfordt18comments
  • Lemma 2: Exact PCEs by first two moments
  • Lemma 3: PCE for chance constraints Pan2023
  • Definition 3: Persistency of excitation Willems2005
  • Lemma 4: Stochastic fundamental lemma Pan21sFaulwasser2022
  • Proposition 1: Recursive feasibility with updated basis
  • ...and 14 more