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Convex computation of regions of attraction from data using Sums-of-Squares programming

Oumayma Khattabi, Matteo Tacchi, Sorin Olaru

TL;DR

This work tackles the challenge of computing the finite-horizon Region of Attraction (RoA) for unknown autonomous systems using only data. It introduces a data-driven, moment-SOS framework that encodes uncertainty from a Lipschitz bound into a differential inclusion and solves a convex program to obtain certified outer FH-RoA estimates, along with a method for worst-case inner approximations. The approach relaxes the need for a polynomial model of the dynamics and demonstrates convergence of data-driven certificates to the true RoA as data quality and quantity improve, with compelling 1D and 2D numerical examples. The results highlight how data placement and the chosen Lipschitz bound influence conservatism and accuracy, and they outline scalability improvements via set partitioning and alternative polynomial bases for SOS computations.

Abstract

This paper focuses on the analysis of the Region of Attraction (RoA) for unknown autonomous dynamical systems. A data-driven approach based on the moment-Sum-of-Squares (SoS) hierarchy is proposed, enabling novel RoA outer approximations despite the reduced information on the dynamics. The main contribution consists of bypassing the system model and, hence, the recurring constraint on its polynomial structure. Numerical experiments showcase the influence of data on learned approximating sets, highlighting the potential of this method.

Convex computation of regions of attraction from data using Sums-of-Squares programming

TL;DR

This work tackles the challenge of computing the finite-horizon Region of Attraction (RoA) for unknown autonomous systems using only data. It introduces a data-driven, moment-SOS framework that encodes uncertainty from a Lipschitz bound into a differential inclusion and solves a convex program to obtain certified outer FH-RoA estimates, along with a method for worst-case inner approximations. The approach relaxes the need for a polynomial model of the dynamics and demonstrates convergence of data-driven certificates to the true RoA as data quality and quantity improve, with compelling 1D and 2D numerical examples. The results highlight how data placement and the chosen Lipschitz bound influence conservatism and accuracy, and they outline scalability improvements via set partitioning and alternative polynomial bases for SOS computations.

Abstract

This paper focuses on the analysis of the Region of Attraction (RoA) for unknown autonomous dynamical systems. A data-driven approach based on the moment-Sum-of-Squares (SoS) hierarchy is proposed, enabling novel RoA outer approximations despite the reduced information on the dynamics. The main contribution consists of bypassing the system model and, hence, the recurring constraint on its polynomial structure. Numerical experiments showcase the influence of data on learned approximating sets, highlighting the potential of this method.

Paper Structure

This paper contains 16 sections, 3 theorems, 47 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Data-driven region of attraction
  3. Problem statement
  4. Finite time region of attraction
  5. Illustration on a toy example
  6. The optimisation framework
  7. Linear program on functions
  8. Toy example revisited
  9. Worst-case inner approximations
  10. Numerical results
  11. Analysis of data position's influence
  12. Numerical example in $\mathbb{R}^2$
  13. Computational challenges and proposed resolution
  14. Conclusion
  15. The sums-of-squares hierarchy
  16. ...and 1 more sections

Key Result

Lemma 1

Let $(v,w)$ be feasible for problem eq:LP and define Recalling the definition eq:dataROA of $X_0$, it holds $X_0 \subset \hat{X}_0(w)$. Moreover, considering a minimising sequence $(v_\epsilon, w_\epsilon)$ s.t. where $w^\star$ is the optimal value of problem eq:LP, it holds

Figures (9)

  • Figure 1: Toy example with 3 data points
  • Figure 2: Identification of the FH-RoA with 3 data points
  • Figure 3: Identification of the FH-RoA with 5 data points
  • Figure 4: LP based FH-RoA approximation with 3 data points.
  • Figure 5: LP based FH-RoA approximation with 5 data points
  • ...and 4 more figures

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 6 more