A relaxed criterion for contraction theory: application to an underwater vehicle observer

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On the contrary to Lyapunov theory, contraction theory studies system behavior independently from a specific attractor, thus leading to simpler computations when verifying exponential convergence of nonlinear systems. To check the contraction property, a condition of negativity on the Jacobian of the system has to be fulfilled. In this paper, attention is paid to results for which the negativity condition can be relaxed, i.e. the maximum eigenvalue of the Jacobian may take zero or positive values. In this issue, we present a theorem and a corollary which sufficient conditions enable to conclude when the Jacobian is not uniformly negative definite but fulfils some weaker conditions. Intended as an illustrative example, a nonlinear underwater vehicle observer, which Jacobian is not uniformly negative definite, is presented and proven to be exponentially convergent using the new criterion.

Original languageEnglish
Publication date2003
Publication statusPublished - 2003
EventEuropean Control Conference (ECC'03) - Cambridge, United Kingdom
Duration: 24. Aug 2010 → …


ConferenceEuropean Control Conference (ECC'03)
Country/TerritoryUnited Kingdom
Period24/08/2010 → …


  • contraction theory
  • Exponential convergence
  • nonlinear observers
  • autonomous underwater vehicles


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