A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stability
This work addresses stability guarantees for learning-based control in nonlinear systems, offering a theoretical framework with potential applications in robotics and autonomous systems, though it is incremental in building on contraction theory.
The paper tackles the problem of ensuring stability in learning-based control for nonlinear systems by introducing a Neural Contraction Metric (NCM), which provides formal robustness guarantees and explicit bounds on trajectory errors that exponentially decrease over time under perturbations.
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.