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Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems

Ruikun Zhou, Yiming Meng, Zhexuan Zeng, Jun Liu

TL;DR

An algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency, potentially as low as 10Hz or less is proposed.

Abstract

Koopman operator theory has gained significant attention in recent years for identifying discrete-time nonlinear systems by embedding them into an infinite-dimensional linear vector space. However, providing stability guarantees while learning the continuous-time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman-based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high-accuracy Koopman generator learning for capturing transient system transitions and physics-informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.

Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems

TL;DR

An algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency, potentially as low as 10Hz or less is proposed.

Abstract

Koopman operator theory has gained significant attention in recent years for identifying discrete-time nonlinear systems by embedding them into an infinite-dimensional linear vector space. However, providing stability guarantees while learning the continuous-time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman-based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high-accuracy Koopman generator learning for capturing transient system transitions and physics-informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.

Paper Structure

This paper contains 15 sections, 15 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman Operators and the Infinitesimal Generator
  4. Concept of Stability
  5. Problem Formulation
  6. Learning the Generator and Identifying the Vector Field via Koopman Resolvent
  7. Stability Certificates for the Unknown Systems
  8. Data-Driven Algorithm for Deriving Stability Certificates
  9. Learning the Koopman Generator
  10. Learning the Lyapunov Function
  11. The Choice of the Dictionary
  12. Numerical Experiments
  13. Reversed Van der Pol Oscillator
  14. Two-machine Power System
  15. CONCLUSION

Figures (2)

  • Figure 1: The learned Lyapunov function and corresponding certified ROA estimates, where the blue curve is the one using the proposed method. The magenta dashed one is the verified largest ROA estimate with $V_P(x)$, while the green dot-dashed line is with the neural network-based approach proposed in zhou2022neural. The red dot-dashed circle denotes $\mathcal{X}$ on which we collect the data for computing the Koopman generator.
  • Figure 2: The learned Lyapunov function and corresponding certified ROA estimates, where the red dot-dashed circle denotes $\mathcal{X}$ on which we collect the data for computing the Koopman generator. The blue curve is the certified ROA estimate using the proposed method, while the magenta dashed line is the one with $V_P(x)$.

Theorems & Definitions (1)

  • proof