用扩散模型思想控制非线性系统的状态分布，实现精准路径规划。
Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems
- 通过前向扩散探索可达状态空间，再设计确定性反向反馈控制
- 在无漂移和LTI系统上理论保证密度演化可逆，实现稳定收敛
- 适合复杂障碍环境下的机器人路径规划与密度控制任务
本文提出一种确定性扩散框架，用于控制非线性控制仿射系统的概率密度，并在无漂移和线性时不变（LTI）动力学下提供理论保证。核心思路是先以白噪声激励系统，使前向扩散过程探索状态空间的可达区域；随后设计一个确定性反馈律，作为去噪机制将系统引导回目标集上的期望目标分布。该去噪阶段生成反馈控制器，驱动系统到达目标集。在此框架中，控制综合转化为构建一个确定性的逆过程，以重现状态密度的期望演化。我们推导了可控无漂移和LTI系统存在此类确定性逆实现的条件，并证明所得反馈律为非线性控制提供了可计算的替代方案，即将密度控制视为对目标集控制的松弛。在带障碍物的单轮车模型、五维无漂移系统及四维LTI系统上的数值实验验证了该方法在扩散启发下的可靠密度控制能力。
原文摘要 · Abstract (English)
In this paper, we propose a deterministic diffusion-based framework for controlling the probability density of nonlinear control-affine systems, with theoretical guarantees for drift-free and linear time-invariant (LTI) dynamics. The central idea is to first excite the system with white noise so that a forward diffusion process explores the reachable regions of state space, and then to design a deterministic feedback law that acts as a denoising mechanism driving the system back toward a desired target distribution supported on the target set. This denoising phase provides a feedback controller that steers the control system to the target set. In this framework, control synthesis reduces to constructing a deterministic reverse process that reproduces the desired evolution of state densities. We derive existence conditions ensuring such deterministic realizations of time-reversals for controllable drift-free and LTI systems, and show that the resulting feedback laws provide a tractable alternative to nonlinear control by viewing density control as a relaxation of controlling a system to target sets. Numerical studies on a unicycle model with obstacles, a five-dimensional driftless system, and a four-dimensional LTI system demonstrate reliable diffusion-inspired density control.
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