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:即使对于已知的非线性动态系统,反馈控制器合成也是一个困难的问题,往往需要利用动态的特定结构来诱发稳定的闭环系统。对于一般的非线性模型,包括适合数据的模型,可能没有足够的已知结构来可靠地合成稳定的反馈控制器。在本文中,我们提议以国家依赖的Riccati方程式为基础,为一般的非线性控制系统提出一个新的非线性跟踪控制器配方。我们的配方取决于矢量字段系统的非线性因子化来定义控制-情感动态,我们总是在温和的假设下显示这种动态存在。我们讨论如何从有限的数据集中学习这种因子化。在各种模拟的非线性动态系统中,我们展示了我们掌握的控制器在稳定的轨迹跟踪中所学过版本的功效。我们用我们的方法评价了在共同学习已知动态系统的控制器和可稳定性证书方面的最新想法;我们从经验上表明,这种方法在比较时可以达到数据效率。