  This paper presents a new deep learning-based framework for robust nonlinear
estimation and control using the concept of a Neural Contraction Metric (NCM).
The NCM uses a deep long short-term memory recurrent neural network for a
global approximation of an optimal contraction metric, the existence of which
is a necessary and sufficient condition for exponential stability of nonlinear
systems. The optimality stems from the fact that the contraction metrics
sampled offline are the solutions of a convex optimization problem to minimize
an upper bound of the steady-state Euclidean distance between perturbed and
unperturbed system trajectories. We demonstrate how to exploit NCMs to design
an online optimal estimator and controller for nonlinear systems with bounded
disturbances utilizing their duality. The performance of our framework is
illustrated through Lorenz oscillator state estimation and spacecraft optimal
motion planning problems.
