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Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview

Tsukamoto, Hiroyasu and Chung, Soon-Jo and Slotine, Jean-Jaques E. (2021) Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview. Annual Reviews in Control, 52 . pp. 135-169. ISSN 1367-5788. doi:10.1016/j.arcontrol.2021.10.001.

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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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Tsukamoto, Hiroyasu0000-0002-6337-2667
Chung, Soon-Jo0000-0002-6657-3907
Slotine, Jean-Jaques E.0000-0002-7161-7812
Additional Information:© 2021 Elsevier. Received 18 June 2021, Revised 30 September 2021, Accepted 1 October 2021, Available online 5 November 2021, Version of Record 7 December 2021. This work was in part funded by the Jet Propulsion Laboratory, California Institute of Technology , and benefited from discussions with Nick Boffi, Winfried Lohmiller, Brett Lopez, Ian Manchester, Quang Cuong Pham, Sumeet Singh, Stephen Tu, and Patrick Wensing. We also thank Chuchu Fan and Guanya Shi. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Subject Keywords:Contraction theory; Learning-based control; Data-driven control; Nonlinear stability; Robust control and estimation; Optimal control and estimation; Adaptive control
Record Number:CaltechAUTHORS:20220308-454020000
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Official Citation:Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jaques E. Slotine, Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview, Annual Reviews in Control, Volume 52, 2021, Pages 135-169, ISSN 1367-5788,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113797
Deposited By: George Porter
Deposited On:09 Mar 2022 16:04
Last Modified:09 Mar 2022 16:04

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