A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stability

TL;DR

This paper provides a theoretical overview of Neural Contraction Metrics (NCM), demonstrating their ability to guarantee stability and robustness in learning-based control of nonlinear systems through convex optimization.

Contribution

It introduces a formal framework for NCM, showing how they can be used to ensure stability and robustness in control systems with learning components.

Findings

NCM can bound the distance between target and perturbed trajectories.

NCM control guarantees exponential decrease of trajectory deviations.

The approach enables real-time, robust learning-based control for nonlinear systems.

Abstract

This paper presents a theoretical overview of a Neural Contraction Metric (NCM): a neural network model of an optimal contraction metric and corresponding differential Lyapunov function, the existence of which is a necessary and sufficient condition for incremental exponential stability of non-autonomous nonlinear system trajectories. Its innovation lies in providing formal robustness guarantees for learning-based control frameworks, utilizing contraction theory as an analytical tool to study the nonlinear stability of learned systems via convex optimization. In particular, we rigorously show in this paper that, by regarding modeling errors of the learning schemes as external disturbances, the NCM control is capable of obtaining an explicit bound on the distance between a time-varying target trajectory and perturbed solution trajectories, which exponentially decreases with time even under the presence of deterministic and stochastic perturbation. These useful features permit simultaneous synthesis of a contraction metric and associated control law by a neural network, thereby enabling real-time computable and probably robust learning-based control for general control-affine nonlinear systems.