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.

MSC codes

  1. 93A15
  2. 34D20
  3. 47H07

Keywords

  1. nonlinear systems
  2. input-to-state stability
  3. interconnected systems
  4. large scale systems
  5. Lipschitz ISS Lyapunov function
  6. small gain condition

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Information & Authors

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Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 4089 - 4118
ISSN (online): 1095-7138

History

Submitted: 14 January 2009
Accepted: 24 March 2010
Published online: 12 May 2010

MSC codes

  1. 93A15
  2. 34D20
  3. 47H07

Keywords

  1. nonlinear systems
  2. input-to-state stability
  3. interconnected systems
  4. large scale systems
  5. Lipschitz ISS Lyapunov function
  6. small gain condition

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