• Media type: E-Book; Electronic Thesis; Doctoral Thesis
  • Title: Monotone dynamical systems, graphs, and stability of large-scale interconnected systems ; Monotone dynamische Systeme, Graphen und Stabilität großer gekoppelter Systeme
  • Contributor: Rüffer, Björn S. [Author]
  • Published: Universität Bremen; Fachbereich 03: Mathematik/Informatik (FB 03), 2007-09-19
  • Language: English
  • Keywords: decay sets ; large-scale systems ; graphs ; dynamical systems ; small-gain theorem ; monotone systems ; input-to-state stability
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  • Description: For a class of monotone operators T on the positive orthant of n-dimensional Euclidean space we introduce the concept of decay sets. These consist of points x satisfying T(x)<x. Considering the induced dynamical system x(k 1)=T(x(k)), we establish results relating stability properties of the origin, order conditions on T, and topological properties of decay sets. In particular, we construct paths in the decay sets and derive a quasi-invertibility result of the operator (Id-T).These results are applied to derive generalized small-gain type conditions for the input-to-state stability(ISS) of large-scale interconnections of (individually input-to-state stable) control systems: The interconnection topology together with the ISS gains yields a monotone operator with an inherent graph structure. We provide trajectory estimate based small-gain theorems and also construct ISS Lyapunov functions for the composite system. It is also shown how an algorithm due to Eaves can be used to numerically verify the small-gain condition.
  • Access State: Open Access