Manifold-Guided Lyapunov Control with Diffusion Models

Khái niệm cốt lõi

Using diffusion models to generate stabilizing controllers efficiently and rapidly for nonlinear systems.

Tóm tắt

• Introduction
  • Challenges in constructing Lyapunov functions.
  • Diffusion models for generative purposes.
• Manifold-Guided Lyapunov Control
  • Training a diffusion model for stabilizing controllers.
  • Introducing Manifold Guided Lyapunov Control (MGLC).
• Preliminaries and Problem Formulation
  • Definitions of asymptotic stability and Lie derivatives.
  • Introduction to diffusion models and Tweedie's estimate.
• Diffusion on Manifolds
  • Guided diffusion and conditional sampling.
  • Utilizing manifolds for improved estimates.
• Problem Formulation
  • Formulation of nonlinear control systems.
• Algorithm
  • Step-by-step process of MGLC.
• Implementation
  • Dataset generation and training of diffusion model.
  • Controller design and numerical results.
• Numerical Results
  • Application of MGLC to various systems.
• Conclusions and Future Works
  • Summary of the approach and potential future extensions.

Thống kê

Our proposed solution significantly reduces the computational cost of finding a Lyapunov function.
The diffusion model was trained on a dataset of 2,000 pairs of (f, V) in 2D space.
The training time for the diffusion model was approximately 8 hours.

Trích dẫn

"Generative models, such as diffusion models, map inputs from a noisy space to the data space."
"Manifold Guided Lyapunov Control (MGLC) is a novel approach to feedback control problems based on diffusion models."

Thông tin chi tiết chính được chắt lọc từ

Manifold-Guided Lyapunov Control with Diffusion Models

by Amartya Mukh... lúc arxiv.org 03-27-2024

https://arxiv.org/pdf/2403.17692.pdf

Yêu cầu sâu hơn

How can the MGLC approach be extended to high-dimensional systems

To extend the Manifold-Guided Lyapunov Control (MGLC) approach to high-dimensional systems, we can leverage models like latent diffusion. Latent diffusion models have shown promise in handling high-dimensional data by learning a latent space representation that captures the underlying structure of the data. By incorporating latent variables into the diffusion process, we can effectively model complex systems with high-dimensional state spaces. This extension would involve training a diffusion model on high-dimensional data and using the latent space representation to guide the control design process. Additionally, perfect encoders can be employed to improve convergence guarantees in high-dimensional systems by ensuring that the latent space captures the essential features of the data distribution accurately.

What are the implications of using diffusion models for transfer learning in control problems

The implications of using diffusion models for transfer learning in control problems are significant. By leveraging pre-trained generative diffusion models, we can efficiently generate Lyapunov functions and stabilizing controllers for nonlinear systems. This approach simplifies the control problem by identifying stable vector fields relative to a predetermined manifold and adjusting the control function based on this information. The use of diffusion models enables rapid stabilization of unseen systems, showcasing the potential for fast zero-shot control and generalizability. Furthermore, the ability to transfer knowledge from the pre-trained diffusion model to new control problems reduces the computational effort required to identify stabilizing controllers, making it a valuable tool for a wide range of control applications.

How can perfect encoders improve convergence guarantees in high-dimensional systems

Perfect encoders can significantly improve convergence guarantees in high-dimensional systems by ensuring that the latent space representation captures the essential features of the data distribution accurately. In the context of diffusion models, perfect encoders can help in mapping high-dimensional data to a lower-dimensional latent space in a way that preserves the underlying structure of the data. By using perfect encoders, we can reduce information loss during the encoding process and enhance the quality of the latent representations. This, in turn, leads to more accurate and stable control designs, especially in complex high-dimensional systems where capturing the right features is crucial for effective control. Perfect encoders play a vital role in improving the performance and robustness of diffusion models in control applications.