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A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stability.
TLDR
The Neural Contraction Metric (NCM) as mentioned in this paper is a neural network model of an optimal contraction metric and corresponding differential Lyapunov function, which is a necessary and sufficient condition for incremental exponential stability of non-autonomous nonlinear system trajectories.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.read more
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Journal ArticleDOI
Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview
TL;DR: Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other as mentioned in this paper.
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Learning-based Adaptive Control via Contraction Theory.
TL;DR: In this article, a deep learning-based adaptive control framework for nonlinear systems with multiplicatively separable parametrization, called aNCM -for adaptive Neural Contraction Metric is presented.
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Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview.
TL;DR: Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other.
Proceedings ArticleDOI
Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems
TL;DR: Physics-informed machine learning (PIML) as mentioned in this paper is a set of methods and tools that systematically integrate machine learning algorithms with physical constraints and abstract mathematical models developed in scientific and engineering domains.
References
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Understanding deep learning requires rethinking generalization.
TL;DR: This article showed that deep neural networks can fit a random labeling of the training data, and that this phenomenon is qualitatively unaffected by explicit regularization, and occurs even if the true images are replaced by completely unstructured random noise.