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Safe and Stable Control via Lyapunov-Guided Diffusion Models

Xiaoyuan Cheng, Xiaohang Tang, Yiming Yang

TL;DR

This work tackles the challenge of achieving safety and stability in diffusion-based control policies. It introduces Safe and Stable Diffusion (S^2Diff), a model-based diffusion planning framework that learns certificate functions inspired by Almost Lyapunov theory and uses diffusion sampling to generate trajectory-level policies without gradient-based QP constraints or control-affine assumptions. A probabilistic CLBF guided diffusion process ensures safety and stability by shaping a target distribution over trajectories and updating the CLBF from sampled data, with theoretical guarantees of almost-sure exponential convergence outside small dissipation-violating regions. Empirically, S^2Diff outperforms gradient-based certificate methods and model-based diffusion baselines across a range of nonlinear dynamical systems, demonstrating higher safety rates, better stability, and favorable evaluation times. The approach provides a flexible path to robust, safe learning-based control with potential for extension to richer neural certificate representations and faster inference through distillation.

Abstract

Diffusion models have made significant strides in recent years, exhibiting strong generalization capabilities in planning and control tasks. However, most diffusion-based policies remain focused on reward maximization or cost minimization, often overlooking critical aspects of safety and stability. In this work, we propose Safe and Stable Diffusion ($S^2$Diff), a model-based diffusion framework that explores how diffusion models can ensure safety and stability from a Lyapunov perspective. We demonstrate that $S^2$Diff eliminates the reliance on both complex gradient-based solvers (e.g., quadratic programming, non-convex solvers) and control-affine structures, leading to globally valid control policies driven by the learned certificate functions. Additionally, we uncover intrinsic connections between diffusion sampling and Almost Lyapunov theory, enabling the use of trajectory-level control policies to learn better certificate functions for safety and stability guarantees. To validate our approach, we conduct experiments on a wide variety of dynamical control systems, where $S^2$Diff consistently outperforms both certificate-based controllers and model-based diffusion baselines in terms of safety, stability, and overall control performance.

Safe and Stable Control via Lyapunov-Guided Diffusion Models

TL;DR

This work tackles the challenge of achieving safety and stability in diffusion-based control policies. It introduces Safe and Stable Diffusion (S^2Diff), a model-based diffusion planning framework that learns certificate functions inspired by Almost Lyapunov theory and uses diffusion sampling to generate trajectory-level policies without gradient-based QP constraints or control-affine assumptions. A probabilistic CLBF guided diffusion process ensures safety and stability by shaping a target distribution over trajectories and updating the CLBF from sampled data, with theoretical guarantees of almost-sure exponential convergence outside small dissipation-violating regions. Empirically, S^2Diff outperforms gradient-based certificate methods and model-based diffusion baselines across a range of nonlinear dynamical systems, demonstrating higher safety rates, better stability, and favorable evaluation times. The approach provides a flexible path to robust, safe learning-based control with potential for extension to richer neural certificate representations and faster inference through distillation.

Abstract

Diffusion models have made significant strides in recent years, exhibiting strong generalization capabilities in planning and control tasks. However, most diffusion-based policies remain focused on reward maximization or cost minimization, often overlooking critical aspects of safety and stability. In this work, we propose Safe and Stable Diffusion (Diff), a model-based diffusion framework that explores how diffusion models can ensure safety and stability from a Lyapunov perspective. We demonstrate that Diff eliminates the reliance on both complex gradient-based solvers (e.g., quadratic programming, non-convex solvers) and control-affine structures, leading to globally valid control policies driven by the learned certificate functions. Additionally, we uncover intrinsic connections between diffusion sampling and Almost Lyapunov theory, enabling the use of trajectory-level control policies to learn better certificate functions for safety and stability guarantees. To validate our approach, we conduct experiments on a wide variety of dynamical control systems, where Diff consistently outperforms both certificate-based controllers and model-based diffusion baselines in terms of safety, stability, and overall control performance.

Paper Structure

This paper contains 53 sections, 8 theorems, 75 equations, 12 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Certificate-Based Method.
  3. Related Work.
  4. Preliminary
  5. Problem Definition.
  6. Method
  7. Probabilistic Formulation and Diffusion Sampling
  8. CLBF Update via Sampled Trajectories
  9. Theoretical Results
  10. Experiment
  11. Control Performance Comparison
  12. Tasks.
  13. Baseline Algorithms.
  14. Evaluation Metric.
  15. Result Analysis.
  16. ...and 38 more sections

Key Result

Theorem 3.1

Let $\mathcal{X}$ be a compact state space and consider the continuously differentiable dynamical system $f$ in Equation eq:dynamics. Let $V:\mathcal{X}\to\mathbb{R}^+$ be a smooth positive definite function. Assume that there exist constants $\lambda>0$ and $\epsilon>0$, and a connected, non-self-o Then, there exist positive constants $\lambda_1$ and $M$, with $0 < \lambda_1 < \lambda$, such that

Figures (12)

  • Figure 1: Overview of $S^2$Diff. Bottom to top: as guidance function improves, vector fields align better with the goal, Lyapunov landscapes get smooth, and trajectories converge more reliably. Right to left: the generated policy is guided by the Lyapunov function, and diffusion sampled trajectories are used to update the Lyapunov function.
  • Figure 2: Benchmark control tasks for safety and stability.
  • Figure 3: Left: CLBFs learned by Gradient-based method (left-1) vs. Diffusion Sampling (left-2) for inverted pendulum . Right: Contour maps along different axes of the CLBF learned by $S^2$Diff for the high-dimensional, non-control-affine F-16 with non-convex constraints. The smooth level sets across 2D projections highlight the CLBF’s expressiveness and its ability to capture complex, constrained dynamics.
  • Figure 4: Control trajectories of a 2D quadrotor with four methods including ours ($S^2$Diff). The $\circ$ and $\times$ mark the start and end points, respectively. Green lines denote safe states; red lines indicate constraint violations. $S^2$Diff achieves higher safety and stability, effectively handling non-convex constraints where baselines struggle.
  • Figure 5: Comparison of control performance and Lyapunov behavior on the F-16 system. (a) $S^2$Diff achieves improved tracking compared to MBD. (b) The scalar value of the CLBF under $S^2$Diff shows a nearly monotonic decrease, demonstrating Almost Lyapunov stability.
  • ...and 7 more figures

Theorems & Definitions (17)

  • Definition 2.1: CLBF romdlony2016stabilization
  • Remark 2.2
  • Theorem 3.1: Safety and Stability with Almost Sure Guarantees
  • Definition B.1: Safe and Stable Control Policy
  • Remark B.2
  • Definition B.3: Lie Derivative
  • Proposition B.4
  • Lemma B.5: Monte-Carlo Estimation of Score
  • proof
  • Lemma C.1
  • ...and 7 more