Contraction Theory for Dynamical Systems on Hilbert Spaces

Cisneros-Velarde, Pedro; Jafarpour, Saber; Bullo, Francesco

doi:10.48550/arxiv.2010.01219

Cited by 1 publication

(1 citation statement)

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“…Due to all these useful properties, extensions of contraction theory have been considered in many different settings. These include, but are not limited to, stochastic contraction (Gaussian white noise [13,16,17,34], Poisson shot noise and Lévy noise [35]), contraction for discrete and hybrid nonlinear systems [7,8,13,17,37,42], partial contraction [11], transverse contraction [43], incremental stability analysis of nonlinear estimation (the Extended Kalman Filter (EKF) [44], nonlinear observers [16,45], Simultaneous Localization And Mapping (SLAM) [46]), generalized gradient descent based on geodesical convexity [47], contraction on Finsler and Riemannian manifolds [48][49][50], contraction on Banach and Hilbert spaces for PDEs [51][52][53], non-Euclidean contraction [54], contracting learning with piecewise-linear basis functions [55], incremental quadratic stability analysis [56], contraction after small transients [57], immersion and invariance stabilizing controller design [58,59], and Lipschitz-bounded neural networks for robustness and stability guarantees [60][61][62].…”

Section: Contraction Theory (Sec 2)mentioning

confidence: 99%

Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

Tsukamoto,

Chung,

Slotine

2021

Preprint

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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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