### Resumé

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.

Originalsprog | Engelsk |
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Publikationsdato | 2003 |

Status | Udgivet - 2003 |

Begivenhed | European Control Conference (ECC'03) - Cambridge, StorbritannienVarighed: 24. aug. 2010 → … |

### Konference

Konference | European Control Conference (ECC'03) |
---|---|

Land | Storbritannien |

By | Cambridge |

Periode | 24/08/2010 → … |

### Fingeraftryk

### Citer dette

*A relaxed criterion for contraction theory: application to an underwater vehicle observer*. Afhandling præsenteret på

*European Control Conference (ECC'03)*, Cambridge, Storbritannien.

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*European Control Conference (ECC'03)*, Cambridge, Storbritannien, 24/08/2010, .

**A relaxed criterion for contraction theory: application to an underwater vehicle observer.** / Jouffroy, Jerome.

*European Control Conference (ECC'03)*, Cambridge, Storbritannien.

Publikation: Konferencebidrag uden forlag/tidsskrift › Paper › Forskning › peer review

TY - CONF

T1 - A relaxed criterion for contraction theory: application to an underwater vehicle observer

AU - Jouffroy, Jerome

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - contraction theory

KW - Exponential convergence

KW - nonlinear observers

KW - autonomous underwater vehicles

M3 - Paper

ER -

*European Control Conference (ECC'03)*, Cambridge, Storbritannien.