Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems

2025-04-14

AI Summary

This work addresses control-affine nonlinear systems subject to nonholonomic constraints. We propose a novel deterministic feedback control and motion planning framework grounded in denoising diffusion probabilistic models (DDPMs). Methodologically, we introduce a control-oriented time-reversal diffusion mechanism—proving for the first time that controllable driftless nonlinear systems admit exact deterministic feedback laws capable of perfectly inverting the forward diffusion process. We further design constraint-embedded forward noise to eliminate stochastic sampling in the reverse process. Integrating score matching, Lyapunov stability analysis, and nonholonomic modeling, our approach enables analytical construction of control laws while ensuring strict fidelity in density evolution. Evaluated on single-vehicle obstacle avoidance, a 5D drifting system, and a 4D linear system, the method achieves fully deterministic reverse-time trajectories—zero sampling randomness—thereby enhancing interpretability and real-time deployability of the controller.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We propose a novel control-theoretic framework that leverages principles from generative modeling -- specifically, Denoising Diffusion Probabilistic Models (DDPMs) -- to stabilize control-affine systems with nonholonomic constraints. Unlike traditional stochastic approaches, which rely on noise-driven dynamics in both forward and reverse processes, our method crucially eliminates the need for noise in the reverse phase, making it particularly relevant for control applications. We introduce two formulations: one where noise perturbs all state dimensions during the forward phase while the control system enforces time reversal deterministically, and another where noise is restricted to the control channels, embedding system constraints directly into the forward process. For controllable nonlinear drift-free systems, we prove that deterministic feedback laws can exactly reverse the forward process, ensuring that the system's probability density evolves correctly without requiring artificial diffusion in the reverse phase. Furthermore, for linear time-invariant systems, we establish a time-reversal result under the second formulation. By eliminating noise in the backward process, our approach provides a more practical alternative to machine learning-based denoising methods, which are unsuitable for control applications due to the presence of stochasticity. We validate our results through numerical simulations on benchmark systems, including a unicycle model in a domain with obstacles, a driftless five-dimensional system, and a four-dimensional linear system, demonstrating the potential for applying diffusion-inspired techniques in linear, nonlinear, and settings with state space constraints.

Problem

• Stabilizing control-affine systems with nonholonomic constraints using diffusion principles
• Eliminating reverse-phase noise in control systems for deterministic feedback
• Validating diffusion-inspired techniques in nonlinear and constrained systems

Innovation

• Uses DDPMs for nonlinear control stabilization
• Eliminates reverse phase noise for control
• Embeds constraints directly into forward process