We propose a novel approach based on Denoising Diffusion Probabilistic Models (DDPMs) to control nonlinear dynamical systems. DDPMs are the state-of-art of generative models that have achieved success in a wide variety of sampling tasks. In our framework, we pose the feedback control problem as a generative task of drawing samples from a target set under control system constraints. The forward process of DDPMs constructs trajectories originating from a target set by adding noise. We learn to control a dynamical system in reverse such that the terminal state belongs to the target set. For control-affine systems without drift, we prove that the control system can exactly track the trajectory of the forward process in reverse, whenever the the Lie bracket based condition for controllability holds. We numerically study our approach on various nonlinear systems and verify our theoretical results. We also conduct numerical experiments for cases beyond our theoretical results on a physics-engine.

翻译：我们提出了一种基于去噪扩散概率模型（DDPMs）的非线性动力系统控制新方法。DDPMs作为生成模型的前沿技术，已在各类采样任务中取得显著成功。在我们的框架中，将反馈控制问题建模为在控制系统约束下从目标集合中抽取样本的生成任务。DDPMs的前向过程通过添加噪声构建出始于目标集合的轨迹。我们学习逆向控制动力系统，使得终端状态最终落入目标集合。对于无漂移的控制仿射系统，我们证明了当基于李括号的可控性条件成立时，控制系统能够精确追踪逆向过程中的轨迹。我们通过多种非线性系统进行了数值研究，验证了理论结果。此外，我们还在物理引擎上进行了超出理论范围的数值实验。