  We present a new deep learning-based adaptive control framework for nonlinear
systems with multiplicatively-separable parametric uncertainty, called an
adaptive Neural Contraction Metric (aNCM). The aNCM uses a neural network model
of an optimal adaptive contraction metric, the existence of which guarantees
asymptotic stability and exponential boundedness of system trajectories under
the parametric uncertainty. In particular, we exploit the concept of a Neural
Contraction Metric (NCM) to obtain a nominal provably stable robust control
policy for nonlinear systems with bounded disturbances, and combine this policy
with a novel adaptation law to achieve stability guarantees. We also show that
the framework is applicable to adaptive control of dynamical systems modeled
via basis function approximation. Furthermore, the use of neural networks in
the aNCM permits its real-time implementation, resulting in broad applicability
to a variety of systems. Its superiority to the state-of-the-art is illustrated
with a simple cart-pole balancing task.
