A cyclic small-gain condition and an equivalent matrix-like criterion for iISS networks

• Published in: 2012 IEEE 51st IEEE Conference on Decision and Control (CDC) pp. 4158 - 4164
• Main Authors: Ito, H., Zhong-Ping Jiang, Dashkovskiy, S. N., Ruffer, B. S.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01-12-2012
• Subjects: Educational institutions; Indium tin oxide; Lyapunov methods; Nickel; Stability criteria; USA Councils; Vectors
• ISBN: 9781467320658; 146732065X
• ISSN: 0191-2216
• DOI: 10.1109/CDC.2012.6426994

This paper considers nonlinear dynamical networks consisting of individually iISS (integral input-to-state stable) subsystems which are not necessarily ISS (input-to-state stable). Stability criteria for internal and external stability of the networks are developed in view of both necessity and sufficiency. For the sufficiency, we show how we can construct a Lyapunov function of the network explicitly under the assumption that a cyclic small-gain condition is satisfied. The cyclic small-gain condition is shown to be equivalent to a matrix-like condition. The two conditions and their equivalence precisely generalize some central ISS results in the literature. Moreover, the necessity of the matrix-like condition is established. The allowable number of non-ISS subsystems for stability of the network is discussed through several necessity conditions.