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Manifold-Guided Lyapunov Control with Diffusion Models

Amartya Mukherjee, Thanin Quartz, Jun Liu

TL;DR

This paper presents a novel approach to generating stabilizing controllers for a large class of dynamical systems using diffusion models trained on pairs consisting of asymptotically stable vector fields and their corresponding Lyapunov functions.

Abstract

This paper presents a novel approach to generating stabilizing controllers for a large class of dynamical systems using diffusion models. The core objective is to develop stabilizing control functions by identifying the closest asymptotically stable vector field relative to a predetermined manifold and adjusting the control function based on this finding. To achieve this, we employ a diffusion model trained on pairs consisting of asymptotically stable vector fields and their corresponding Lyapunov functions. Our numerical results demonstrate that this pre-trained model can achieve stabilization over previously unseen systems efficiently and rapidly, showcasing the potential of our approach in fast zero-shot control and generalizability.

Manifold-Guided Lyapunov Control with Diffusion Models

TL;DR

This paper presents a novel approach to generating stabilizing controllers for a large class of dynamical systems using diffusion models trained on pairs consisting of asymptotically stable vector fields and their corresponding Lyapunov functions.

Abstract

This paper presents a novel approach to generating stabilizing controllers for a large class of dynamical systems using diffusion models. The core objective is to develop stabilizing control functions by identifying the closest asymptotically stable vector field relative to a predetermined manifold and adjusting the control function based on this finding. To achieve this, we employ a diffusion model trained on pairs consisting of asymptotically stable vector fields and their corresponding Lyapunov functions. Our numerical results demonstrate that this pre-trained model can achieve stabilization over previously unseen systems efficiently and rapidly, showcasing the potential of our approach in fast zero-shot control and generalizability.
Paper Structure (23 sections, 4 theorems, 34 equations, 7 figures, 1 algorithm)

This paper contains 23 sections, 4 theorems, 34 equations, 7 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM FORMULATION
  3. Lyapunov Stability
  4. Diffusion Models
  5. Diffusion on Manifolds
  6. Problem Formulation
  7. MANIFOLD-GUIDED LYAPUNOV CONTROL (MGLC)
  8. Proposed Method
  9. Convergence Proofs
  10. Algorithm
  11. IMPLEMENTATION
  12. Dataset Generation
  13. Diffusion model
  14. Controller
  15. NUMERICAL RESULTS
  16. ...and 8 more sections

Key Result

Theorem 1

If there exists a continuously differentiable scalar function $V(x)$ satisfying $V(0)=0$, $V(x)>0$ if $x\neq 0$, and $\nabla V(x)\cdot f(x,u(x))<0$, for a given feedback controller $u$, then the system is asymptotically stable (or Lyapunov-stable) at the origin, and $V$ is called a Lyapunov function

Figures (7)

  • Figure 1: A schematic view of MGLC. Our proposed method introduces a new manifold $\mathcal{W}$ corresponding to the set of vector fields that can be returned by modifying our control input function.
  • Figure 2: An overview of our model in the 2D control setting. We train the generator to output images with three channels. The first two channels are the scalar elements of an asymptotically stable vector field, $f_1(x,y)$ and $f_2(x,y)$. The third channel is a Lyapunov function $V(x,y)$ that verifies the stability of the vector field.
  • Figure 3: Visualization of MGLC for the inverted pendulum environment
  • Figure 4: Trajectories of the controlled inverted pendulum system with 100 different randomly sampled initial conditions
  • Figure 5: Trajectories of the controlled damped Duffing oscillator system with 100 different randomly sampled initial conditions
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2: Lie Derivatives
  • Theorem 1
  • Definition 3: Tweedie's estimate efron2011tweediechung2022improving
  • Proposition 1: Probabilistic Concentration chung2022improvinghe2023manifold
  • Proposition 2
  • proof
  • Remark 1
  • Theorem 2
  • proof