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Learning-based Adaptive Control via Contraction Theory.

04 Mar 2021-arXiv: Learning-
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.

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Topics: Adaptive control (64%), Artificial neural network (56%), Contraction (operator theory) (54%) ... show more
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8 results found


Open accessPosted Content
31 Dec 2020-
Abstract: High-performance feedback control requires an accurate model of the underlying dynamical system which is often difficult, expensive, or time-consuming to obtain. Online model learning is an attractive approach that can handle model variations while achieving the desired level of performance. However, most model learning methods developed within adaptive nonlinear control are limited to certain types of uncertainties, called matched uncertainties, because the certainty equivalency principle can be employed in the design phase. This work develops a universal adaptive control framework that extends the certainty equivalence principle to nonlinear systems with unmatched uncertainties through two key innovations. The first is introducing parameter-dependent storage functions that guarantee closed-loop tracking of a desired trajectory generated by an adapting reference model. The second is modulating the learning rate so the closed-loop system remains stable during the learning transients. The analysis is first presented under the lens of contraction theory, and then expanded to general Lyapunov functions which can be synthesized via feedback linearization, backstepping, or optimization-based techniques. The proposed approach is more general than existing methods as the uncertainties can be unmatched and the system only needs to be stabilizable. The developed algorithm can be combined with learned feedback policies, facilitating transfer learning and bridging the sim-to-real gap. Simulation results showcase the method

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Topics: Nonlinear control (61%), Feedback linearization (59%), Adaptive control (59%) ... show more

6 Citations


Open accessJournal ArticleDOI: 10.1109/LRA.2021.3091019
Hiroyasu Tsukamoto1, Soon-Jo Chung1Institutions (1)
25 Feb 2021-
Abstract: This letter presents Learning-based Autonomous Guidance with RObustness and Stability guarantees (LAG-ROS), which provides machine learning-based nonlinear motion planners with formal robustness and stability guarantees, by designing a differential Lyapunov function using contraction theory. LAG-ROS utilizes a neural network to model a robust tracking controller independently of a target trajectory, for which we show that the Euclidean distance between the target and controlled trajectories is exponentially bounded linearly in the learning error, even under the existence of bounded external disturbances. We also present a convex optimization approach that minimizes the steady-state bound of the tracking error to construct the robust control law for neural network training. In numerical simulations, it is demonstrated that the proposed method indeed possesses superior properties of robustness and nonlinear stability resulting from contraction theory, whilst retaining the computational efficiency of existing learning-based motion planners.

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Topics: Robustness (computer science) (62%), Stability (learning theory) (61%), Lyapunov function (57%) ... show more

5 Citations


Open accessPosted Content
02 Oct 2021-arXiv: Learning
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.

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Topics: Stability (learning theory) (56%), Lyapunov function (56%), Artificial neural network (55%) ... show more

3 Citations


Open accessJournal ArticleDOI: 10.1016/J.ARCONTROL.2021.10.001
Abstract: 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. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is therefore to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

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Topics: Exponential stability (60%), Contraction (operator theory) (57%), Linear system (56%) ... show more

2 Citations


Journal ArticleDOI: 10.1109/LCSYS.2021.3095427
01 Jan 2022-
Abstract: This letter develops an adaptive actuator failure compensation method for nonlinear systems with unmatched parametric uncertainty based on contraction metrics. The proposed method, which is constructed by benefiting from the recent achievements on contraction metrics based adaptive control techniques, ensures the closed-loop stability and asymptotic tracking of the desired trajectory in the presence of actuator failures. In particular, a sufficient convex condition is derived for constructing a valid metric, by which a quadratic program-based controller is obtained to determine the inputs of the actuators. The introduced method is more general than the common adaptive actuator failure compensation methods, as it does not require the system to have an identical relative degree for all inputs and be transformable into the parametric strick-feedback or feedback linearization form. Besides, it can be enriched with learning-based algorithms and common robust modifications. Simulation results are presented to verify the effectiveness of the proposed controller.

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Topics: Adaptive control (59%), Feedback linearization (56%), Control theory (55%) ... show more

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43 results found


Journal ArticleDOI: 10.1162/NECO.1997.9.8.1735
Sepp Hochreiter1, Jürgen Schmidhuber2Institutions (2)
01 Nov 1997-Neural Computation
Abstract: Learning to store information over extended time intervals by recurrent backpropagation takes a very long time, mostly because of insufficient, decaying error backflow. We briefly review Hochreiter's (1991) analysis of this problem, then address it by introducing a novel, efficient, gradient based method called long short-term memory (LSTM). Truncating the gradient where this does not do harm, LSTM can learn to bridge minimal time lags in excess of 1000 discrete-time steps by enforcing constant error flow through constant error carousels within special units. Multiplicative gate units learn to open and close access to the constant error flow. LSTM is local in space and time; its computational complexity per time step and weight is O. 1. Our experiments with artificial data involve local, distributed, real-valued, and noisy pattern representations. In comparisons with real-time recurrent learning, back propagation through time, recurrent cascade correlation, Elman nets, and neural sequence chunking, LSTM leads to many more successful runs, and learns much faster. LSTM also solves complex, artificial long-time-lag tasks that have never been solved by previous recurrent network algorithms.

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49,735 Citations


Journal ArticleDOI: 10.1109/TSMC.1983.6313077
01 Sep 1983-
Abstract: It is shown how a system consisting of two neuronlike adaptive elements can solve a difficult learning control problem. The task is to balance a pole that is hinged to a movable cart by applying forces to the cart's base. It is argued that the learning problems faced by adaptive elements that are components of adaptive networks are at least as difficult as this version of the pole-balancing problem. The learning system consists of a single associative search element (ASE) and a single adaptive critic element (ACE). In the course of learning to balance the pole, the ASE constructs associations between input and output by searching under the influence of reinforcement feedback, and the ACE constructs a more informative evaluation function than reinforcement feedback alone can provide. The differences between this approach and other attempts to solve problems using neurolike elements are discussed, as is the relation of this work to classical and instrumental conditioning in animal learning studies and its possible implications for research in the neurosciences.

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Topics: Robot learning (59%), Unsupervised learning (58%), Adaptive control (55%) ... show more

3,112 Citations


Open accessProceedings Article
15 Feb 2018-
Abstract: One of the challenges in the study of generative adversarial networks is the instability of its training. In this paper, we propose a novel weight normalization technique called spectral normalization to stabilize the training of the discriminator. Our new normalization technique is computationally light and easy to incorporate into existing implementations. We tested the efficacy of spectral normalization on CIFAR10, STL-10, and ILSVRC2012 dataset, and we experimentally confirmed that spectrally normalized GANs (SN-GANs) is capable of generating images of better or equal quality relative to the previous training stabilization techniques.

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2,626 Citations


Journal ArticleDOI: 10.1109/72.165588
Abstract: A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture uses a network of Gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibit by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example systems. >

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Topics: Adaptive control (58%), Lyapunov function (57%), Gaussian (53%) ... show more

2,137 Citations


Open accessBook
01 Jan 1987-
Abstract: Part 1 Hypergeometric Functions and Elliptic Integrals: Some Basic Topics In Analytical Dynamics The Problem of Two Bodies Two-Body Orbits and the Initial-Value Problem Solving Kepler's Equation Two-Body Orbital Boundary Value Problem Solving Lambert's Problem Appendices Part 2 Non-Keplerian Motion: Patched-Conic Orbits and Perturbation Methods Variation of Parameters Two-Body Orbital Transfer Numerical Integration of Differential Equations The Celestial Position Fix Space Navigation Appendices

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Topics: Lambert's problem (60%), Elliptic boundary value problem (58%), Free boundary problem (57%) ... show more

1,997 Citations


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