A RegionDependent Gain Condition for Asymptotic Stability
Abstract
A sufficient condition for the stability of a system resulting from the interconnection of dynamical systems is given by the small gain theorem. Roughly speaking, to apply this theorem, it is required that the gains composition is continuous, increasing and upper bounded by the identity function. In this work, an alternative sufficient condition is presented for the case in which this criterion fails due to either lack of continuity or the bound of the composed gain is larger than the identity function. More precisely, the local (resp. nonlocal) asymptotic stability of the origin (resp. global attractivity of a compact set) is ensured by a regiondependent small gain condition. Under an additional condition that implies convergence of solutions for almost all initial conditions in a suitable domain, the almost global asymptotic stability of the origin is ensured. Two examples illustrate and motivate this approach.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.02851
 Bibcode:
 2015arXiv150202851S
 Keywords:

 Mathematics  Dynamical Systems;
 Computer Science  Systems and Control;
 Mathematics  Optimization and Control
 EPrint:
 Automatica, vol. 52, pp. 309316, 2015