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Safe and Stable Control via Lyapunov-Guided Diffusion Models
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Proposes S²Diff, a model-based diffusion planner guided by learned control Lyapunov barrier functions (CLBFs) that jointly learns certificates from diffusion-sampled trajectories, yielding safe and stable control for nonlinear systems without QP or control-affine assumptions.
Problem
Ensuring both safety and stability in control of general nonlinear dynamical systems, where QP-based certificate methods require control-affine dynamics, slack variables, and suffer from coupled training instabilities.
Approach
Treat the safe-stable-cost trajectory distribution as a Gibbs measure and sample it with model-based diffusion guided by a learned neural CLBF; soft Lie-derivative penalties replace slack variables, and diffusion-sampled trajectories are reused to refine the CLBF, with an Almost Lyapunov theorem giving almost-sure exponential decay up to a buffer term.
Key results
- S²Diff achieves 98.75% average safety rate vs 78.75% for rCLBF-QP, 66.25% for MPC, and 73.75% for MBD across the five control-affine tasks.
- On the non-control-affine F-16, S²Diff reduces terminal distance to 47.61 vs 68.34 for MBD, with inference 257.2 ms vs 611.3 ms.
- Lie derivative violation region volume is below 2.4% of the state space across benchmark tasks (e.g., 1.6% for 2D Quad).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use S^2Diff instead of rCLBF-QP | control policy · Average Performance above | Safety rate: 78.75% -> 98.75% | Helped |
| use S^2Diff instead of rCLBF-QP | control policy · Average Performance above | ||x - x_star||: 0.384 -> 0.226 | Helped |
| use S^2Diff instead of rCLBF-QP | control policy · Average Performance above | Eval. time (ms): 10.06 -> 45.64 | Hurt |
| use S^2Diff instead of MBD | control policy · F-16 (non-control-affine) | ||x - x_star||: 68.34 \pm 32.77 -> 47.61 \pm 12.45 | Helped |
| use S^2Diff instead of MBD | control policy · F-16 (non-control-affine) | Eval. time (ms): 611.3\pm13.7 -> 257.2\pm5.5 | Helped |
| set stability temperature gamma_2 to 0.1 rather than 0.5 (ablation) | S^2Diff · Neural Lander | Safety rate: 35% -> 100% | Helped |
| set stability temperature gamma_2 to 0.1 rather than 0.5 (ablation) | S^2Diff · Neural Lander | ||x - x_star||: 0.18 \pm 0.08 -> 0.06 \pm 0.02 | Helped |
| set stability temperature gamma_2 to 0.01 rather than 0.1 (ablation) | S^2Diff · Neural Lander | ||x - x_star||: 0.06 \pm 0.02 -> 0.12 \pm 0.02 | Hurt |
| set stability temperature gamma_2 to 0.5 rather than 0.1 (ablation) | S^2Diff · Neural Lander | Safety rate: 100% -> 35% | Hurt |
| use balanced automatic and discretized Lie derivative penalties rather than only the discretized penalty (ablation) | S^2Diff CLBF loss · Neural Lander | ||x - x_star||: 0.21 \pm 0.09 -> 0.06 \pm 0.02 | Helped |
| use only the discretized Lie derivative penalty rather than balanced automatic and discretized penalties (ablation) | S^2Diff CLBF loss · Neural Lander | Safety rate: 100% -> 85% | Hurt |
| use both automatic and discretized Lie derivative penalties rather than neither penalty (ablation) | S^2Diff CLBF loss · Neural Lander | Safety rate: 10% -> 100% | Helped |
Surprising findings
- On the inverted pendulum and car kinematic tasks, rCLBF-QP is faster (about 4-10 ms) than S²Diff (about 23-38 ms), and on Car (Slip) MPC achieves lower terminal error than S²Diff.
Limitations
- Inference is slower than QP-based controllers; the authors suggest policy distillation as a fix.
- Theoretical guarantees rely on a small violation volume and regularity assumptions (bounded gradients, Lipschitz policy), which are not verified in practice beyond empirical violation estimates.
- Some results are mixed: MPC and rCLBF-QP outperform S²Diff on certain individual metrics (e.g., Car Slip, Segway speed).
Experimental setup
Datasets: trajectories sampled from simulated dynamical systems
Benchmarks: inverted pendulum car kinematic tracking car slip tracking Segway neural lander 2D quadrotor 3D quadrotor F-16
Metrics: safety rate terminal distance to equilibrium evaluation time (ms) violation rate
Baselines: rCLBF-QP MPC MBD
Builds on: Model-Based Diffusion (MBD) Diffuser control Lyapunov barrier function (CLBF) rCLBF-QP Almost Lyapunov theory
Closest papers by meaning
- DiRecT: Safe Diffusion-Based Planning via Receding-Horizon Denoising (NeurIPS 2026 · similarity 0.88)
- SafeDiffuser: Safe Planning with Diffusion Probabilistic Models (ICLR 2025 · similarity 0.88)
- From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control (ICML 2025 · similarity 0.87)
- Safely Learning Controlled Stochastic Dynamics (NeurIPS 2025 · similarity 0.87)
- Analytical Lyapunov Function Discovery: An RL-based Generative Approach (ICML 2025 · similarity 0.86)
- Diffusion Policy Policy Optimization (ICLR 2025 · similarity 0.86)
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