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

• Published in: SIAM journal on control and optimization Vol. 48; no. 6; pp. 4089 - 4118
• Main Authors: Dashkovskiy, Sergey N., Rüffer, Björn S., Wirth, Fabian R.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Applied sciences; Computer science; control theory; systems; Construction; Control system analysis; Control theory. Systems; Exact sciences and technology; Gain; International Space Station; Lyapunov functions; Mathematical functions; Matrix; Maximization; Networks; Nonlinear systems; Operators; Optimization; Studies; System theory; Systems stability; Maximization; 34D20; Interconnected power system; Non linear control; input-to-state stability; Non linear system; 93A15; Scaling function; nonlinear systems; Lipschitz ISS Lyapunov function; small gain condition; Input to state stability; Small gain theorem; Lipschitz function; large scale systems; 47H07; interconnected systems; Lyapunov function; Large scale system
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/090746483

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. [PUBLICATION ABSTRACT]