  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.
