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Data-Driven Stabilization of Continuous-Time LTI Systems from Noisy Input-Output Data

Alessandro Bosso, Marco Borghesi, Andrea Iannelli, Bowen Yi, Giuseppe Notarstefano

Abstract

We present an approach to compute stabilizing controllers for continuous-time linear time-invariant systems directly from an input-output trajectory affected by process and measurement noise. The proposed output-feedback design combines (i) an observer of a non-minimal realization of the plant and (ii) a feedback law obtained from a linear matrix inequality (LMI) that depends solely on the available data. Under a suitable interval excitation condition and knowledge of a noise energy bound, the feasibility of the LMI is shown to be necessary and sufficient for stabilizing all non-minimal realizations consistent with the data. We further provide a condition for the feasibility of the LMI related to the signal-to-noise ratio, guidelines to compute the noise energy bound, and numerical simulations that illustrate the effectiveness of the approach.

Data-Driven Stabilization of Continuous-Time LTI Systems from Noisy Input-Output Data

Abstract

We present an approach to compute stabilizing controllers for continuous-time linear time-invariant systems directly from an input-output trajectory affected by process and measurement noise. The proposed output-feedback design combines (i) an observer of a non-minimal realization of the plant and (ii) a feedback law obtained from a linear matrix inequality (LMI) that depends solely on the available data. Under a suitable interval excitation condition and knowledge of a noise energy bound, the feasibility of the LMI is shown to be necessary and sufficient for stabilizing all non-minimal realizations consistent with the data. We further provide a condition for the feasibility of the LMI related to the signal-to-noise ratio, guidelines to compute the noise energy bound, and numerical simulations that illustrate the effectiveness of the approach.

Paper Structure

This paper contains 23 sections, 5 theorems, 74 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Data-Driven Stabilization from Noisy Input--Output Data
  4. Non-Minimal Realization for Data-Driven Control
  5. Observer-Based Control Design
  6. Non-Minimal Realizations Consistent with the Dataset
  7. Data-Driven Stabilization via the Matrix S-Lemma
  8. Feasibility of the LMI
  9. Remarks on the Noise Bound Computation
  10. Bound on the Filtered Process Noise
  11. Bound on the Filtered Measurement Noise
  12. Numerical Results
  13. Scalar System with Process and Measurement Noise
  14. Unstable Batch Reactor with Process Noise
  15. Conclusion
  16. ...and 8 more sections

Key Result

Lemma 1

Under Assumption hyp:ctrb and given $F$, $G$, and $L$ as in eq:FGL, there exist full-rank matrices $\Pi \in {\mathbb{R}}^{np \times \mu}$, $H \in {\mathbb{R}}^{p \times \mu}$ that satisfy and, in addition, ensure the following properties:

Figures (2)

  • Figure 1: Scalar example. Top: measured output $y(\cdot)$ and noise signals $w(\cdot)$ and $v(\cdot)$. Bottom: representation of $\boldsymbol{\mathcal{E}}$ (blue), ${\mathbb{R}}^{3}\backslash\boldsymbol{\mathcal{C}}(P, K)$ (orange), $\Theta^\star = [0\; 1.5\; 0.5]$ (red dot), and $\hat{\Theta} = [-0.0054\; 1.6106\; 0.5395]$ (blue dot).
  • Figure 2: Batch reactor example. Each value of $\delta_w$ corresponds to $200$ simulation runs. Left axis: box plot of $\rho$ as defined in \ref{['eq:rho']}. Right axis: percentage of times the LMI \ref{['eq:LMI']} is feasible.

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Corollary 1
  • Proposition 1