  Control design for general nonlinear robotic systems with guaranteed
stability and/or safety in the presence of model uncertainties is a challenging
problem. Recent efforts attempt to learn a controller and a certificate (e.g.,
a Lyapunov function or a contraction metric) jointly using neural networks
(NNs), in which model uncertainties are generally ignored during the learning
process. In this paper, for nonlinear systems subject to bounded disturbances,
we present a framework for jointly learning a robust nonlinear controller and a
contraction metric using a novel disturbance rejection objective that certifies
a universal $\mathcal L_\infty$ gain bound using NNs for user-specified
variables. The learned controller aims to minimize the effect of disturbances
on the actual trajectories of state and/or input variables from their nominal
counterparts while providing certificate tubes around nominal trajectories that
are guaranteed to contain actual trajectories in the presence of disturbances.
Experimental results demonstrate that our framework can generate tighter tubes
and a controller that is computationally efficient to implement.
