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Data-driven input-to-state stabilization

Hailong Chen, Andrea Bisoffi, Claudio De Persis

TL;DR

The paper tackles data-driven input-to-state stabilization for nonlinear input-affine systems with polynomial dynamics by leveraging noisy data to certify ISS for all models consistent with the data. It develops two complementary SOS-based design pipelines: a nonconvex, bilinear (biconvex) approach and a convex SOS formulation that yields ISS controllers and ISS-Lyapunov functions by exploiting a structured parametrization. An ellipsoidal overapproximation of the data-consistent model set enables tractable guarantees for actuator and process disturbances, and model-based specialization demonstrates the framework’s versatility. A numerical example confirms feasibility and illustrates the tradeoffs between the two design paths, including computation times and conservatism. The work advances data-driven control by providing constructive, verifiable ISS certificates in settings with uncertain dynamics derived from data.

Abstract

For the class of nonlinear input-affine systems with polynomial dynamics, we consider the problem of designing an input-to-state stabilizing controller with respect to typical exogenous signals in a feedback control system, such as actuator and process disturbances. We address this problem in a data-based setting when we cannot avail ourselves of the dynamics of the actual system, but only of data generated by it under unknown bounded noise. For all dynamics consistent with data, we derive sum-of-squares programs to design an input-to-state stabilizing controller, an input-to-state Lyapunov function and the corresponding comparison functions. This numerical design for input-to-state stabilization seems to be relevant not only in the considered data-based setting, but also in a model-based setting. Illustration of feasibility of the provided sum-of-squares programs is provided on a numerical example.

Data-driven input-to-state stabilization

TL;DR

The paper tackles data-driven input-to-state stabilization for nonlinear input-affine systems with polynomial dynamics by leveraging noisy data to certify ISS for all models consistent with the data. It develops two complementary SOS-based design pipelines: a nonconvex, bilinear (biconvex) approach and a convex SOS formulation that yields ISS controllers and ISS-Lyapunov functions by exploiting a structured parametrization. An ellipsoidal overapproximation of the data-consistent model set enables tractable guarantees for actuator and process disturbances, and model-based specialization demonstrates the framework’s versatility. A numerical example confirms feasibility and illustrates the tradeoffs between the two design paths, including computation times and conservatism. The work advances data-driven control by providing constructive, verifiable ISS certificates in settings with uncertain dynamics derived from data.

Abstract

For the class of nonlinear input-affine systems with polynomial dynamics, we consider the problem of designing an input-to-state stabilizing controller with respect to typical exogenous signals in a feedback control system, such as actuator and process disturbances. We address this problem in a data-based setting when we cannot avail ourselves of the dynamics of the actual system, but only of data generated by it under unknown bounded noise. For all dynamics consistent with data, we derive sum-of-squares programs to design an input-to-state stabilizing controller, an input-to-state Lyapunov function and the corresponding comparison functions. This numerical design for input-to-state stabilization seems to be relevant not only in the considered data-based setting, but also in a model-based setting. Illustration of feasibility of the provided sum-of-squares programs is provided on a numerical example.
Paper Structure (20 sections, 8 theorems, 82 equations, 3 figures, 3 tables)

This paper contains 20 sections, 8 theorems, 82 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Auxiliary results
  5. Problem formulation and motivation
  6. Data-driven representation with noisy data
  7. Overapproximation of the set $\mathcal{I}$ of consistent matrices
  8. Motivation
  9. Main results
  10. ISS with respect to actuator disturbances
  11. ISS with respect to process disturbances
  12. Construction of convex SOS programs for ISS
  13. Model-based specialization
  14. Numerical illustration
  15. Establishing ISS with biconvex programs
  16. ...and 5 more sections

Key Result

Lemma 2.6

\newlabellem:classk0 Consider $\alpha \colon \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ defined as for some integer $N \ge 1$. If the scalars $c_1, c_2, \dots, c_N$ satisfy $c_1 \geq 0$, $c_2 \geq 0$, …, $c_N \geq 0$ and $c_1+c_2+\dots+c_N>0$, then $\alpha$ belongs to class $\mathcal{K}_{\infty}$.

Figures (3)

  • Figure 1: Data-collection experiment for the considered polynomial system.
  • Figure 2: Left: evolution of $\dot{V}^{\textup{b,a}}(\cdot)$ (solid) and $-\alpha_3^{\textup{b,a}}(|x(\cdot)|)+\alpha_4^{\textup{b,a}}(|w(\cdot)|)$ (dotted). Right: evolution of $\dot{V}^{\textup{b,p}}(\cdot)$ (solid) and $-\alpha_3^{\textup{b,p}}(|x(\cdot)|)+\alpha_4^{\textup{b,p}}(|d(\cdot)|)$ (dotted).
  • Figure 3: Left: evolution of $\dot{V}^{\textup{c,a}}(\cdot)$ (solid) and $-\alpha_3^{\textup{c,a}}(|x(\cdot)|)+\alpha_4^{\textup{c,a}}(|w(\cdot)|)$ (dotted). Right: evolution of $\dot{V}^{\textup{c,p}}(\cdot)$ (solid) and $-\alpha_3^{\textup{c,p}}(|x(\cdot)|)+\alpha_4^{\textup{c,p}}(|d(\cdot)|)$ (dotted).

Theorems & Definitions (21)

  • Definition 2.1: isidori1999nonlinear
  • Definition 2.2: isidori1999nonlinear
  • Definition 2.3: isidori1999nonlinear
  • Lemma 2.6: chen2023data
  • Proof 1
  • Remark 3.1
  • Theorem 3.4
  • Proof 2
  • Lemma 3.5
  • Proof 3
  • ...and 11 more