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 propose a novel nonlinear tracking controller formulation based on a state-dependent Riccati equation for general nonlinear control-affine systems. Our formulation depends on a nonlinear factorization of the system of vector fields defining the control-affine dynamics, which we show always exists under mild smoothness assumptions. We discuss how this factorization can be learned from a finite set of data. On a variety of simulated nonlinear dynamical systems, we demonstrate the efficacy of learned versions of our controller in stable trajectory tracking. Alongside our method, we evaluate recent ideas in jointly learning a controller and stabilizability certificate for known dynamical systems; we show empirically that such methods can be data-inefficient in comparison.

翻译：即使是已知的非线性动力系统，反馈控制器综合仍然是一个难题，通常需要利用动力学特有的结构来构建稳定的闭环系统。对于一般的非线性模型（包括基于数据拟合的模型），可能缺乏足够已知的结构来可靠地综合镇定反馈控制器。本文针对一般非线性控制仿射系统，提出一种基于状态相关Riccati方程的新型非线性跟踪控制器构造方法。我们的方法依赖于定义控制仿射动力学的向量场系统的非线性分解，并证明在温和的光滑性假设下该分解始终存在。我们进一步讨论了如何从有限数据集中学习这种分解。通过在多种仿真非线性动力系统上的实验，验证了所提出控制器学习版本在稳定轨迹跟踪中的有效性。与我们的方法并行，我们评估了近期针对已知动力系统联合学习控制器与可镇定性证明的思路；实验表明，此类方法在数据效率上相对较低。