Modern data-driven control applications call for flexible nonlinear models
that are amenable to principled controller synthesis and realtime feedback.
Many nonlinear dynamical systems of interest are control affine. We propose two
novel classes of nonlinear feature representations which capture control affine
structure while allowing for arbitrary complexity in the state dependence. Our
methods make use of random features (RF) approximations, inheriting the
expressiveness of kernel methods at a lower computational cost. We formalize
the representational capabilities of our methods by showing their relationship
to the Affine Dot Product (ADP) kernel proposed by Casta\~neda et al. (2021)
and a novel Affine Dense (AD) kernel that we introduce. We further illustrate
the utility by presenting a case study of data-driven optimization-based
control using control certificate functions (CCF). Simulation experiments on a
double pendulum empirically demonstrate the advantages of our methods.

现代数据驱动控制应用需要灵活的非线性模型，以便于合理的控制器合成和实时反馈。我们提出了两个新颖的非线性特征表示类，可以捕捉控制仿射结构，同时允许在状态依赖上具有任意复杂性。我们的方法利用了随机特征（RF）逼近，以较低的计算成本继承了核方法的表达能力。通过展示它们与 Castaneda 等人提出的仿射点积（ADP）核和我们引入的新颖仿射稠密（AD）核之间的关系，我们对我们方法的表现能力进行了正式化。通过使用控制证书函数（CCF）的数据驱动基于优化的控制的案例研究，进一步说明了实用性。在双摆实验中进行的模拟实验实证了我们方法的优势。