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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.
About: This article is published in Annual Reviews in Control.The article was published on 2021-11-05 and is currently open access. It has received 19 citations till now. The article focuses on the topics: Exponential stability & Contraction (operator theory).
Citations
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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.
Abstract: We present a deep learning-based adaptive control framework for nonlinear systems with multiplicatively separable parametrization, called aNCM - for adaptive Neural Contraction Metric. The framework utilizes a deep neural network to approximate a stabilizing adaptive control law parameterized by an optimal contraction metric. The use of deep networks permits real-time implementation of the control law and broad applicability to a variety of systems, including systems modeled with basis function approximation methods. We show using contraction theory that aNCM ensures exponential boundedness of the distance between the target and controlled trajectories even under the presence of the parametric uncertainty, robustly to the learning errors caused by aNCM approximation as well as external additive disturbances. Its superiority to the existing robust and adaptive control methods is demonstrated in a simple cart-pole balancing task.

9 citations

Journal ArticleDOI
TL;DR: In this article , the problem of constant output regulation for a class of input-affine multi-input multi-output nonlinear systems was studied, and sufficient conditions for the design of a state-feedback control law able to make the resulting closed-loop system incrementally stable, uniformly with respect to the references and the disturbances were provided.
Abstract: In this article, we study the problem of constant output regulation for a class of input-affine multi-input multi-output nonlinear systems, which do not necessarily admit a normal form. We allow the references and the disturbances to be arbitrarily large and the initial conditions of the system to range in the full-state space. We cast the problem in the contraction framework, and we rely on the common approach of extending the system with an integral action processing the regulation error. We then present sufficient conditions for the design of a state-feedback control law able to make the resulting closed-loop system incrementally stable, uniformly with respect to the references and the disturbances. Such a property ensures uniqueness and attractiveness of an equilibrium on which output regulation is obtained. To this end, we develop an incremental version of the forwarding (mod $\lbrace L_gV\rbrace$ ) approach. Finally, we provide a set of sufficient conditions for the design of a pure (small-gain) integral-feedback control. The proposed approach is also specialized for two classes of systems that are linear systems having a Lipschitz nonlinearity and a class of minimum-phase systems whose zero dynamics are incrementally stable.

6 citations

Posted Content
TL;DR: 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.

4 citations

Journal ArticleDOI
TL;DR: A comprehensive survey of certificate learning for control can be found in this article , where the authors provide an accessible introduction to the theory and practice of certificate-learning for control, both for those who wish to apply these tools to practical robotics problems and to those who want to dive more deeply into the theory of learning for controlling.
Abstract: Learning-enabled control systems have demonstrated impressive empirical performance on challenging control problems in robotics, but this performance comes at the cost of reduced transparency and lack of guarantees on the safety or stability of the learned controllers. In recent years, new techniques have emerged to provide these guarantees by learning certificates alongside control policies—these certificates provide concise data-driven proofs that guarantee the safety and stability of the learned control system. These methods not only allow the user to verify the safety of a learned controller but also provide supervision during training, allowing safety and stability requirements to influence the training process itself. In this article, we provide a comprehensive survey of this rapidly developing field of certificate learning. We hope that this article will serve as an accessible introduction to the theory and practice of certificate learning, both to those who wish to apply these tools to practical robotics problems and to those who wish to dive more deeply into the theory of learning for control.

3 citations

Proceedings ArticleDOI
08 Jun 2022
TL;DR: In this paper , a comprehensive contraction analysis for recurrent neural networks is presented, which establishes quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes.
Abstract: Critical questions in dynamical neuroscience and machine learning are related to the study of recurrent neural networks and their stability, robustness, and computational efficiency. These properties can be simultaneously established via a contraction analysis.This paper develops a comprehensive contraction theory for recurrent neural networks. First, for non-Euclidean ℓ 1 /ℓ logarithmic norms, we establish quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes. Second, for locally Lipschitz maps (e.g., arising as activation functions), we show that their one-sided Lipschitz constant equals the essential supremum of the logarithmic norm of their Jacobian. Third and final, we apply these general results to classes of recurrent neural circuits, including Hopfield, firing rate, Persidskii, Lur’e and other models. For each model, we compute the optimal contraction rate and corresponding weighted non-Euclidean norm via a linear program or, in some special cases, via a Hurwitz condition on the Metzler majorant of the synaptic matrix. Our non-Euclidean analysis establishes also absolute, connective, and total contraction properties.

2 citations

References
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Proceedings ArticleDOI
27 Jun 2016
TL;DR: In this article, the authors proposed a residual learning framework to ease the training of networks that are substantially deeper than those used previously, which won the 1st place on the ILSVRC 2015 classification task.
Abstract: Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers—8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.

123,388 citations

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TL;DR: This book provides a clear and simple account of the key ideas and algorithms of reinforcement learning, which ranges from the history of the field's intellectual foundations to the most recent developments and applications.
Abstract: Reinforcement learning, one of the most active research areas in artificial intelligence, is a computational approach to learning whereby an agent tries to maximize the total amount of reward it receives when interacting with a complex, uncertain environment. In Reinforcement Learning, Richard Sutton and Andrew Barto provide a clear and simple account of the key ideas and algorithms of reinforcement learning. Their discussion ranges from the history of the field's intellectual foundations to the most recent developments and applications. The only necessary mathematical background is familiarity with elementary concepts of probability. The book is divided into three parts. Part I defines the reinforcement learning problem in terms of Markov decision processes. Part II provides basic solution methods: dynamic programming, Monte Carlo methods, and temporal-difference learning. Part III presents a unified view of the solution methods and incorporates artificial neural networks, eligibility traces, and planning; the two final chapters present case studies and consider the future of reinforcement learning.

37,989 citations

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TL;DR: In this article, the focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them, and a comprehensive introduction to the subject is given. But the focus of this book is not on the optimization problem itself, but on the problem of finding the appropriate technique to solve it.
Abstract: Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

33,341 citations

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01 Jan 1985
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

23,986 citations

Journal ArticleDOI
TL;DR: It is rigorously established that standard multilayer feedforward networks with as few as one hidden layer using arbitrary squashing functions are capable of approximating any Borel measurable function from one finite dimensional space to another to any desired degree of accuracy, provided sufficiently many hidden units are available.

18,794 citations

Trending Questions (1)
What is contraction theory?

Contraction theory is an analytical tool used to study the stability of nonlinear systems by defining a contraction metric that characterizes the exponential stability of solution trajectories with respect to each other.