### Abstract

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.

Original language | English (US) |
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Title of host publication | Proceedings of the IEEE Conference on Decision and Control |

Pages | 4158-4164 |

Number of pages | 7 |

DOIs | |

State | Published - 2012 |

Event | 51st IEEE Conference on Decision and Control, CDC 2012 - Maui, HI, United States Duration: Dec 10 2012 → Dec 13 2012 |

### Other

Other | 51st IEEE Conference on Decision and Control, CDC 2012 |
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Country | United States |

City | Maui, HI |

Period | 12/10/12 → 12/13/12 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*(pp. 4158-4164). [6426994] https://doi.org/10.1109/CDC.2012.6426994

**A cyclic small-gain condition and an equivalent matrix-like criterion for iISS networks.** / Ito, Hiroshi; Jiang, Zhong-Ping; Dashkovskiy, Sergey N.; Ruffer, Bjorn S.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the IEEE Conference on Decision and Control.*, 6426994, pp. 4158-4164, 51st IEEE Conference on Decision and Control, CDC 2012, Maui, HI, United States, 12/10/12. https://doi.org/10.1109/CDC.2012.6426994

}

TY - GEN

T1 - A cyclic small-gain condition and an equivalent matrix-like criterion for iISS networks

AU - Ito, Hiroshi

AU - Jiang, Zhong-Ping

AU - Dashkovskiy, Sergey N.

AU - Ruffer, Bjorn S.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874280579&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874280579&partnerID=8YFLogxK

U2 - 10.1109/CDC.2012.6426994

DO - 10.1109/CDC.2012.6426994

M3 - Conference contribution

AN - SCOPUS:84874280579

SP - 4158

EP - 4164

BT - Proceedings of the IEEE Conference on Decision and Control

ER -