Even for known nonlinear dynamical systems, feedback controller synthesis is a difficult problem that often requires leveraging the particular structure of the dynamics to induce a stable closed-loop system. For general nonlinear models, including those fit to data, there may not be enough known structure to reliably synthesize a stabilizing feedback controller. In this paper, we discuss a state-dependent nonlinear tracking controller formulation based on a state-dependent Riccati equation for general nonlinear control-affine systems. This formulation depends on a nonlinear factorization of the system of vector fields defining the control-affine dynamics, which always exists under mild smoothness assumptions. We propose a method for learning this factorization from a finite set of data. On a variety of simulated nonlinear dynamical systems, we empirically demonstrate the efficacy of learned versions of this controller in stable trajectory tracking. Alongside our learning method, we evaluate recent ideas in jointly learning a controller and stabilizability certificate for known dynamical systems; we show experimentally that such methods can be frail in comparison.

翻译：即使是已知的非线性动力系统，反馈控制器合成也是一个难题，通常需要利用动力学的特定结构来诱导稳定的闭环系统。对于一般的非线性模型（包括那些从数据中拟合的模型），可能没有足够已知的结构来可靠地合成一个稳定化反馈控制器。本文针对一般非线性控制仿射系统，讨论了一种基于状态相关Riccati方程的状态相关非线性跟踪控制器公式。该公式依赖于定义控制仿射动力学的向量场系统的非线性分解，在温和的平滑性假设下，这种分解总是存在的。我们提出了一种从有限数据集中学习这种分解的方法。在各种模拟的非线性动力系统上，我们通过实验证明了该控制器学习版本在稳定轨迹跟踪中的有效性。除了我们的学习方法外，我们还评估了最近关于为已知动力系统联合学习控制器和稳定化证书的思路；实验表明，与此相比，此类方法可能较为脆弱。