Small Gain Theorems for Large Scale Systems and Construction of ISS Lyapunov Functions

Authors:
Sergey N. Dashkovskiy;Björn S. Rüffer;Fabian R. Wirth
Affiliations:
dsn@math.uni-bremen.de;brueffer@unimelb.edu.au;wirth@mathematik.uni-wuerzburg.de
Venue:
SIAM Journal on Control and Optimization
Year:
2010

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Abstract

We consider interconnections of $n$ nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS; the cases of summation, maximization, and separation with respect to external gains are obtained as corollaries.