Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems

TL;DR

The paper addresses stabilization of nonlinear, nonholonomic systems by recasting finite-horizon control as density trajectory matching using DDPMs, while ensuring the reverse diffusion is deterministic. It develops two algorithms (one with a drift-based forward process and another via nonholonomic score matching) and provides rigorous results for driftless and LTI settings, including invertibility of key operators and exponential convergence in driftless cases. Numerical experiments on a 5D driftless system, a unicycle model (with and without obstacles), and a 4D linear system validate performance, obstacle-aware planning, and the feasibility of diffusion-inspired control in both nonlinear and linear settings. The work offers a scalable, dynamics-aware alternative to classical nonlinear control methods by integrating system dynamics into the diffusion framework to synthesize stabilizing trajectories under state-space constraints.

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.

Figures (5)

• Figure 1: (a) DDPMs in two dimensions, transforming the data distribution $p_{\text{target}}$ to $p_n$ and learning to reverse the process. (b) Reformulation from the classical control problem to the density control problem.
• Figure 2: Experiments with a five-dimensional nonlinear system \ref{['eqn:fived-sys']}. (a) KL divergence between forward and reverse processes for algorithms 1 and 2. (b) Position of samples at final time step for algorithms 1 and 2.
• Figure 3: Experiments with unicycle dynamics \ref{['eqn:unicycle']} with $p_\text{target} = \mathcal{N}(4,0.2I)$. (a) Final estimated KL divergence for different numbers of measurement instances $N$ vs. training iterations: shows that more measurement instances are required to achieve better feedback control when going from one Gaussian distribution to another, (b) Final KL divergence vs training iterations for different number of training samples: shows that the neural network can identify the controller with a sufficiently large number of training samples and state measuring instances.
• Figure 4: Experiments with unicycle dynamics \ref{['eqn:unicycle']} with obstacles in the environment with $p_\text{target} = \mathcal{N}(4,0.2I)$. The evolution of particles at different time instances in the control horizon of 10 seconds is depicted in the figure. It is evident that particles make use of the space between the obstacles to stabilize $p_\text{target}$.
• Figure 5: Experiments with four dimensional linear system \ref{['eqn:DI']}

Theorems & Definitions (30)

• Definition 4.2
• Proposition 4.5: Properties of the Nonholonomic Fokker-Planck equations
• Proposition 4.6
• proof
• Lemma 4.7: Exact tracking of positive densities
• Lemma 4.8: Tracking the Holonomic Fokker-Planck Equation
• proof
• Theorem 4.9
• Theorem 4.10
• proof