  We propose and demonstrate a nonlinear control method that can be applied to
unknown, complex systems where the controller is based on a type of artificial
neural network known as a reservoir computer. In contrast to many modern
neural-network-based control techniques, which are robust to system
uncertainties but require a model nonetheless, our technique requires no prior
knowledge of the system and is thus model-free. Further, our approach does not
require an initial system identification step, resulting in a relatively simple
and efficient learning process. Reservoir computers are well-suited to the
control problem because they require small training data sets and remarkably
low training times. By iteratively training and adding layers of reservoir
computers to the controller, a precise and efficient control law is identified
quickly. With examples on both numerical and high-speed experimental systems,
we demonstrate that our approach is capable of controlling highly complex
dynamical systems that display deterministic chaos to nontrivial target
trajectories.
