### Resumé

Contraction theory is a recent tool enabling to study the stability of nonlinear systems trajectories with respect to one another, and therefore belongs to the class of incremental stability methods. In this paper, we extend the original definition of contraction theory to incorporate in an explicit manner the control input of the considered system. Such an extension, called *universal contraction*, is quite analogous in spirit to the well-known Input-to-State Stability (ISS). It serves as a simple formulation of incremental ISS, external stability, and detectability in a *differential *setting. The hierarchical combination result of contraction theory is restated in this framework, and a differential small-gain theorem is derived from results already available in Lyapunov theory.

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) |
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Land | Storbritannien |

By | Cambridge |

Periode | 24/08/2010 → … |

### Fingeraftryk

### Citer dette

*A simple extension of contraction theory to study incremental stability properties*. 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 simple extension of contraction theory to study incremental stability properties.** / Jouffroy, Jerome.

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

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

TY - CONF

T1 - A simple extension of contraction theory to study incremental stability properties

AU - Jouffroy, Jerome

PY - 2003

Y1 - 2003

N2 - Contraction theory is a recent tool enabling to study the stability of nonlinear systems trajectories with respect to one another, and therefore belongs to the class of incremental stability methods. In this paper, we extend the original definition of contraction theory to incorporate in an explicit manner the control input of the considered system. Such an extension, called universal contraction, is quite analogous in spirit to the well-known Input-to-State Stability (ISS). It serves as a simple formulation of incremental ISS, external stability, and detectability in a differential setting. The hierarchical combination result of contraction theory is restated in this framework, and a differential small-gain theorem is derived from results already available in Lyapunov theory.

AB - Contraction theory is a recent tool enabling to study the stability of nonlinear systems trajectories with respect to one another, and therefore belongs to the class of incremental stability methods. In this paper, we extend the original definition of contraction theory to incorporate in an explicit manner the control input of the considered system. Such an extension, called universal contraction, is quite analogous in spirit to the well-known Input-to-State Stability (ISS). It serves as a simple formulation of incremental ISS, external stability, and detectability in a differential setting. The hierarchical combination result of contraction theory is restated in this framework, and a differential small-gain theorem is derived from results already available in Lyapunov theory.

M3 - Paper

ER -

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