A Differential Lyapunov Framework for Contraction Analysis

Forni, Fulvio; Sepulchre, Rodolphe

doi:10.1109/tac.2013.2285771

Cited by 302 publications

(452 citation statements)

References 52 publications

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“…The application of contraction analysis to the coupled oscillator model (1) yields the incremental exponential stability (24) in 2 -type metrics (Chung and Slotine, 2010, Theorem 7) or in ∞ -type metrics (Forni and Sepulchre, 2014, Example 6). Choi et al (2011a, Theorem 4.1) report the incremental stability (24) in an 1 -metric.…”

Section: Jacobian Analysismentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

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Section: Jacobian Analysismentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…These metrics are non-quadratic and could be considered through the differential framework recently developed in [2].…”

Section: ) Lyapunov Functions and Contracting Metricsmentioning

confidence: 99%

A spectral operator-theoretic framework for global stability

Mauroy

Mezić

2013

52nd IEEE Conference on Decision and Control

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Abstract-The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

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“…Loosely speaking, this means that: (i) the states of any two solutions, whose initial conditions are 'close' to each other, remain 'close' to each other for all positive time; (ii) that the states of any two solutions converge towards each other as time proceeds. Related stability notions are those of convergent systems (e.g., [4], [15]) and contraction (e.g., [5], [12]). In the current paper, the focus is on novel definitions of incremental asymptotic stability for hybrid systems in the formalism of [8], although we project that this work will also support further developments on establishing sufficient conditions for contraction and novel definitions of convergence for hybrid systems.…”

Section: Introductionmentioning

confidence: 99%

Definitions of incremental stability for hybrid systems

Biemond

Heemels

Wouw

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-The analysis of incremental stability properties typically involves measuring the distance between any pair of solutions of a given dynamical system, corresponding to different initial conditions, at the same time instant. This approach is not directly applicable for hybrid systems in general. Indeed, hybrid systems generate solutions that are defined with respect to hybrid times, which consist of both the continuous time elapsed and the discrete time, that is the number of jumps the solution has experienced. Two solutions of a hybrid system do not a priori have the same time domain, and we may therefore not be able to compare them at the same hybrid time instant. To overcome this issue, we invoke graphical closeness concepts. We present definitions for incremental stability depending on whether incremental asymptotic stability properties hold with respect to the hybrid time, the continuous time, or the discrete time, respectively. Examples are provided throughout the paper to illustrate these definitions, and the relations between these three incremental stability notions are investigated. The definitions are shown to be consistent with those available in the literature for continuous-time and discrete-time systems.

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A Differential Lyapunov Framework for Contraction Analysis

Cited by 302 publications

References 52 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

A spectral operator-theoretic framework for global stability

Definitions of incremental stability for hybrid systems

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