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Open-/Closed-loop Active Learning for Data-driven Predictive Control

Shilun Feng, Dawei Shi, Yang Shi, Kaikai Zheng

TL;DR

Open-/Closed-loop Active Learning for Data-driven Predictive Control presents a data-efficient framework for learning unknown LTI systems with bounded disturbances by shrinking the set of admissible models through matrix-ellipsoid ASEs. It introduces a two-stage open-loop data acquisition (formation and contraction) and a closed-loop learning criterion integrated with an adaptive tube-based predictive controller, proving ASE contraction, recursive feasibility, and stability. Key contributions include a contraction condition for ASE volume, an input-design strategy to minimize volume in the open-loop phase, and a data-selection rule in the closed-loop phase that preserves contraction and stability guarantees. Numerical experiments across scalar, higher-dimensional, and aerospace-like systems demonstrate reduced conservatism, improved tracking, and robustness to noise.

Abstract

An important question in data-driven control is how to obtain an informative dataset. In this work, we consider the problem of effective data acquisition of an unknown linear system with bounded disturbance for both open-loop and closed-loop stages. The learning objective is to minimize the volume of the set of admissible systems. First, a performance measure based on historical data and the input sequence is introduced to characterize the upper bound of the volume of the set of admissible systems. On the basis of this performance measure, an open-loop active learning strategy is proposed to minimize the volume by actively designing inputs during the open-loop stage. For the closed-loop stage, a closed-loop active learning strategy is designed to select and learn from informative closed-loop data. The efficiency of the proposed closed-loop active learning strategy is proved by showing that the unselected data cannot benefit the learning performance. Furthermore, an adaptive predictive controller is designed in accordance with the proposed data acquisition approach. The recursive feasibility and the stability of the controller are proved by analyzing the effect of the closed-loop active learning strategy. Finally, numerical examples and comparisons illustrate the effectiveness of the proposed data acquisition strategy.

Open-/Closed-loop Active Learning for Data-driven Predictive Control

TL;DR

Open-/Closed-loop Active Learning for Data-driven Predictive Control presents a data-efficient framework for learning unknown LTI systems with bounded disturbances by shrinking the set of admissible models through matrix-ellipsoid ASEs. It introduces a two-stage open-loop data acquisition (formation and contraction) and a closed-loop learning criterion integrated with an adaptive tube-based predictive controller, proving ASE contraction, recursive feasibility, and stability. Key contributions include a contraction condition for ASE volume, an input-design strategy to minimize volume in the open-loop phase, and a data-selection rule in the closed-loop phase that preserves contraction and stability guarantees. Numerical experiments across scalar, higher-dimensional, and aerospace-like systems demonstrate reduced conservatism, improved tracking, and robustness to noise.

Abstract

An important question in data-driven control is how to obtain an informative dataset. In this work, we consider the problem of effective data acquisition of an unknown linear system with bounded disturbance for both open-loop and closed-loop stages. The learning objective is to minimize the volume of the set of admissible systems. First, a performance measure based on historical data and the input sequence is introduced to characterize the upper bound of the volume of the set of admissible systems. On the basis of this performance measure, an open-loop active learning strategy is proposed to minimize the volume by actively designing inputs during the open-loop stage. For the closed-loop stage, a closed-loop active learning strategy is designed to select and learn from informative closed-loop data. The efficiency of the proposed closed-loop active learning strategy is proved by showing that the unselected data cannot benefit the learning performance. Furthermore, an adaptive predictive controller is designed in accordance with the proposed data acquisition approach. The recursive feasibility and the stability of the controller are proved by analyzing the effect of the closed-loop active learning strategy. Finally, numerical examples and comparisons illustrate the effectiveness of the proposed data acquisition strategy.
Paper Structure (23 sections, 14 theorems, 110 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 23 sections, 14 theorems, 110 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix Ellipsoids and the Matrix S-Lemma
  4. Admissible System Ellipsoids
  5. Adaptive Tube-based Data-driven Predictive Control
  6. Problem Formulation
  7. Measure and Overapproximation of ASEs
  8. Measure of ASEs with Square Data Matrix
  9. Overapproximation of General ASEs
  10. Open-loop Active Learning for Data-driven Modeling
  11. Learning for An ASE with Small Volume
  12. Learning for ASEs with Decreasing Bound of Volume
  13. Adaptive Data-driven Predictive Control
  14. Design of Error Set and Terminal Constraint Set
  15. Closed-Loop Active Learning for Contracting ASEs
  16. ...and 8 more sections

Key Result

Lemma 1

A matrix ellipsoid is measurable. For $\mathcal{E}_m$ defined in eq:matrix_ellipsoid_1, a measure can be defined as $\mu(\mathcal{E}_{m})\!:=\!m(\mathcal{V}(\mathcal{E}_{m}))$, where $\!\mathcal{V}(\mathcal{E}_{m})\!:=\!\{\text{vec}(s) \mid s\!\in\!\mathcal{E}_{m}\}\!$ and $m(\mathcal{V}(\mathcal{E}

Figures (6)

  • Figure 1: Schematic diagram of the proposed ATDPC schemes.
  • Figure 2: Volume of the ASE obtained by different approaches.
  • Figure 4: Ratio of feasible problems using different approaches.
  • Figure 5: Closed-loop state trajectories of different controllers and the tracking reference.
  • Figure 7: Ratio of feasible problems under different noise levels.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1: 2021_Trade-off
  • Lemma 1: see 2021_Trade-off
  • Lemma 2: 2022_SLemma
  • Remark 1
  • Lemma 3
  • Remark 2
  • Lemma 4
  • proof
  • Remark 3
  • Lemma 5
  • ...and 24 more