Denoising Diffusion-Based Control of Nonlinear Systems
This work addresses control problems for nonlinear systems, offering a novel application of diffusion models, but it appears incremental as it adapts existing generative methods to a specific domain.
The paper tackles controlling nonlinear dynamical systems by framing it as a generative task using Denoising Diffusion Probabilistic Models (DDPMs), achieving exact tracking under controllability conditions and validating results through numerical experiments on various systems.
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.