Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems

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.