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Data-Driven Synthesis of Robust Positively Invariant Sets from Noisy Data

Chi Wang, David Angeli

Abstract

This paper develops a method to construct robust positively invariant (RPI) tube sets from finite noisy input-state data of an unknown linear time-invariant (LTI) system, yielding tubes that can be directly embedded in tube-based robust data-driven predictive control. Data-consistency uncertainty sets are constructed under process/measurement noise with polytopic/ellipsoidal bounds. In the measurement-noise case, we provide a deterministic and data-consistent procedure to certify the induced residual bound from data. Based on these sets, a robustly stabilizing state-feedback gain is certified via a common quadratic contraction, which in turn enables constructive polyhedral/ellipsoidal RPI tube computation. Numerical examples quantify the conservatism induced by noisy data and the employed certification step.

Data-Driven Synthesis of Robust Positively Invariant Sets from Noisy Data

Abstract

This paper develops a method to construct robust positively invariant (RPI) tube sets from finite noisy input-state data of an unknown linear time-invariant (LTI) system, yielding tubes that can be directly embedded in tube-based robust data-driven predictive control. Data-consistency uncertainty sets are constructed under process/measurement noise with polytopic/ellipsoidal bounds. In the measurement-noise case, we provide a deterministic and data-consistent procedure to certify the induced residual bound from data. Based on these sets, a robustly stabilizing state-feedback gain is certified via a common quadratic contraction, which in turn enables constructive polyhedral/ellipsoidal RPI tube computation. Numerical examples quantify the conservatism induced by noisy data and the employed certification step.
Paper Structure (15 sections, 7 theorems, 70 equations, 2 figures)

This paper contains 15 sections, 7 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem setup
  5. Process noise
  6. Measurement noise
  7. Data-consistency sets and robust stabilizing controller
  8. Data-consistent set
  9. Robust stabilizing controller
  10. Robust positively invariant sets
  11. Example
  12. Conclusion
  13. Data-consistent induced gain bounds
  14. Data-driven certification of $\gamma_{\mathrm P}^\ast$
  15. Data-driven certification of $\gamma_{\mathrm E}^\ast$

Key Result

Lemma 1

For any $v_k,v_{k+1}\in\mathcal{V}$ and any matrix $A$, the residual $\eta=v_{k+1}-Av_k$ satisfies Consequently, for any $\gamma\ge \|A\|_{\mathcal{V}}$,

Figures (2)

  • Figure 1: Relative tube-size gap (percentage) w.r.t. the model-based tube computed using $(A^\ast,B^\ast)$, over a grid of $(T,\bar{v})$.
  • Figure 2: lossless vertex LMIs vs. lossy S-procedure under the same data.

Theorems & Definitions (17)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • Proposition 2
  • proof
  • Remark 1
  • Lemma 3
  • ...and 7 more