Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915222534488064 |
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| author | Zhou, Ruikun Meng, Yiming Zeng, Zhexuan Liu, Jun |
| author_facet | Zhou, Ruikun Meng, Yiming Zeng, Zhexuan Liu, Jun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02807 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems Zhou, Ruikun Meng, Yiming Zeng, Zhexuan Liu, Jun Systems and Control Machine Learning Dynamical Systems 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. |
| title | Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems |
| topic | Systems and Control Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2412.02807 |