Stable and robust neural network controllers

Abstract

Neural networks are expressive function approimators that can be employed for state estimation in control problems. However, control systems with machine learning in the loop often lack stability proofs and performance guarantees, which are crucial for safety-critical applications. In this work, a feedback controller using a feedforward neural network of arbitrary size to estimate unknown dynamics is suggested. The controller is designed for solving a general trajectory tracking problem for a broad class of two-dimensional nonlinear systems. The controller is proven to stabilize the closed-loop system, such that it is input-to-state and finite-gain ${{\mathcal{L}}_p}$-stable from the neural network estimation error to the tracking error. Furthermore, the controller is proven to make the tracking error globally and exponentially converge to a ball centered at the origin. When the neural network estimate is updated discretely, or the state measurements are affected by bounded noise, the convergence bound is shown to be dependent on the Lipschitz constant of the neural network estimator. In light of this, we demonstrate how regularization techniques can be beneficial when utilizing deep learning in control. Experiments on simulated data confirm the theoretical results.