Small gain theorems for large scale systems and construction of ISS Lyapunov functions

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.

provisionally accepted by SIAM Journal on Control and Optimization

References in corpus (1)

• An ISS Small-Gain Theorem for General Networks

Cited by in corpus (8)

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• ISS Lyapunov Functions for Cascade Switched Systems and Sampled-Data Control
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• A Region-Dependent Gain Condition for Asymptotic Stability
• Verification of Initial-State Opacity for Switched Systems: A Compositional Approach
• On the Existence of Block-Diagonal Solutions to Lyapunov and ${\mathcal{H}}_{\infty }$ Riccati Inequalities
• Compositional Synthesis of Opacity-Preserving Finite Abstractions for Interconnected Systems