An ISS small gain theorem for general networks

  • Sergey Dashkovskiy
  • Björn S. Rüffer
  • Fabian R. Wirth
Original Article

DOI: 10.1007/s00498-007-0014-8

Cite this article as:
Dashkovskiy, S., Rüffer, B.S. & Wirth, F.R. Math. Control Signals Syst. (2007) 19: 93. doi:10.1007/s00498-007-0014-8

Abstract

We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than 1. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated lower-dimensional discrete time dynamical system.

Keywords

Interconnected systems Input-to-state stability Small gain theorem Large-scale systems Monotone maps 

Copyright information

© Springer-Verlag London Limited 2007

Authors and Affiliations

  • Sergey Dashkovskiy
    • 1
  • Björn S. Rüffer
    • 1
  • Fabian R. Wirth
    • 2
  1. 1.Universität Bremen, Zentrum für TechnomathematikBremenGermany
  2. 2.The Hamilton InstituteNUI MaynoothMaynoothIreland

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