Neural Networks in the Loop: Learning with Stability and Robustness Guarantees

The increasing use of neural networks (NNs) in control systems for robots, autonomous systems, and other safety-critical applications will demand confidence in their safety, stability, and robustness. The aim of this article is to review recently developed NN model structures (both static and dynamic) with built-in certification of robustness in various forms, including bounded sensitivity to adversarial disturbances (Lipschitzness), dynamical stability, and robust invertibility. In doing so, we aim to help strengthen connections between control theory and machine learning. We present a unifying overview of tools to guarantee such robustness properties while retaining model expressivity and computational efficiency of training and inference. Expressivity is maintained via sophisticated certification methods building on the theory of integral quadratic constraints in robust control, while computational efficiency is maintained via direct parameterizations, which enable learning of robust models via standard unconstrained optimization methods such as gradient descent, without any auxiliary constraints or projections. We then show how such robust NN models can be used in building blocks within architectures for learning control system components such as physics-informed nonlinear observers (state estimators) with guaranteed convergence, control policy parameterizations with guaranteed stability and robustness, and Lyapunov functions and their variants, such as storage and value functions.