This paper presents a novel approach to generating stabilizing controllers
for a large class of dynamical systems using diffusion models. The core
objective is to develop stabilizing control functions by identifying the
closest asymptotically stable vector field relative to a predetermined manifold
and adjusting the control function based on this finding. To achieve this, we
employ a diffusion model trained on pairs consisting of asymptotically stable
vector fields and their corresponding Lyapunov functions. Our numerical results
demonstrate that this pre-trained model can achieve stabilization over
previously unseen systems efficiently and rapidly, showcasing the potential of
our approach in fast zero-shot control and generalizability.

使用扩散模型为广泛的动力系统生成稳定控制器，并基于预先训练的模型在未见系统上实现快速的零样本控制和泛化能力。

具有扩散模型的流形引导的李亚普诺夫控制

Manifold-Guided Lyapunov Control with Diffusion Models

This paper revisits a classical challenge in the design of stabilizing
controllers for nonlinear systems with a norm-bounded input constraint. By
extending Lin-Sontag's universal formula and introducing a generic
(state-dependent) scaling term, a unifying controller design method is
proposed. The incorporation of this generic scaling term gives a unified
controller and enables the derivation of alternative universal formulas with
various favorable properties, which makes it suitable for tailored control
designs to meet specific requirements and provides versatility across different
control scenarios. Additionally, we present a constructive approach to
determine the optimal scaling term, leading to an explicit solution to an
optimization problem, named optimization-based universal formula. The resulting
controller ensures asymptotic stability, satisfies a norm-bounded input
constraint, and optimizes a predefined cost function. Finally, the essential
properties of the unified controllers are analyzed, including smoothness,
continuity at the origin, stability margin, and inverse optimality. Simulations
validate the approach, showcasing its effectiveness in addressing a challenging
stabilizing control problem of a nonlinear system.

通过扩展 Lin-Sontag 的通用公式，并引入一种通用的（状态相关的）缩放项，本文提出了一种统一的控制器设计方法，该方法具有多种有利特性的替代通用公式的推导，适用于满足特定要求的定制控制设计，并可在不同控制场景中具备灵活性。此外，我们还提出了一种构造性方法来确定最优的缩放项，从而得到了一个显式解决方案的优化问题，命名为基于优化的通用公式。最终得到的控制器确保了渐近稳定性，满足了范数有界输入限制，并优化了预定义的成本函数。最后，分析了统一控制器的基本特性，包括光滑性、原点处的连续性、稳定裕度和反向最优性。仿真验证了该方法，在解决非线性系统的挑战性稳定控制问题方面的有效性。

使用有界控制输入统一控制器设计以稳定非线性系统

Unifying Controller Design for Stabilizing Nonlinear Systems with  Norm-Bounded Control Inputs

Quadrotor stabilizing controllers often require careful, model-specific
tuning for safe operation. We use reinforcement learning to train policies in
simulation that transfer remarkably well to multiple different physical
quadrotors. Our policies are low-level, i.e., we map the rotorcrafts' state
directly to the motor outputs. The trained control policies are very robust to
external disturbances and can withstand harsh initial conditions such as
throws. We show how different training methodologies (change of the cost
function, modeling of noise, use of domain randomization) might affect flight
performance. To the best of our knowledge, this is the first work that
demonstrates that a simple neural network can learn a robust stabilizing
low-level quadrotor controller (without the use of a stabilizing PD controller)
that is shown to generalize to multiple quadrotors.

使用强化学习训练神经网络，学习低级别的四轴飞行器控制策略，无需使用稳定的 PD 控制器，可以推广到多种四轴飞行器。