An ISS small gain theorem for general networks

By Sergey N. Dashkovskiy, Björn S. Rüffer, Fabian R. Wirth. Math. Control. Signals Syst., 2005

418 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/c92b0c02560fc0731749a8f82032f87d42ac873e

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In Control and Systems Engineering

Citations in the corpus

D-Stability and Delay-Independent Stability of Homogeneous Cooperative Systems (2010, background): “Another result of this type for discrete time systems appeared in [12].”
Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods (2010, methodology): “The first results about the ISS property for the delay-free case were given for two coupled continuous systems in [15] and for an arbitrarily large number (n ∈ N) of coupled continuous systems in [16], using a small-gain argument which is a condition on the interconnection structure of the system.”
Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space (2010, neutral): “To this end note that the implication”
Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions (2014, methodology): “Input-to-state stability (ISS) [37] unified these two concepts and provided powerful tools for stability analysis of ordinary differential equations (ODEs) such as ISS Lyapunov functions [38], methods for design of nonlinear control systems [25, 10], and the ISS small-gain theorem [20, 19] which has”