TY - GEN
T1 - Nonlinear small-gain condition covering iISS systems
T2 - 45th IEEE Conference on Decision and Control 2006, CDC
AU - Ito, Hiroshi
AU - Jiang, Zhong Ping
PY - 2006
Y1 - 2006
N2 - This paper presents conditions for stability of interconnected nonlinear systems consisting of integral input-to-state stable subsystems. Both the necessity and the sufficiency are investigated using a Lyapunov formulation. The ultimate result is a necessary and sufficient condition for integral input-to-state stability (iISS) of the interconnection, which is believed to be the first of its kind. This paper derives two sufficient conditions. One is the iISS-ISS small-gain condition which has been proposed earlier by one of the authors. Its sufficiency is strengthened to cover situations more general than the earlier result. The other condition is a universal representation of the nonlinear small-gain condition which has been a popular tool for dealing with input-to-state stable (ISS) subsystems. This paper proves that, for establishing iISS property of the interconnection, the universal representation can deal with iISS subsystems as much as the iISS-ISS small-gain condition if supply rate functions are smooth everywhere and analytic at zero. Furthermore, two necessary conditions this paper derives demonstrate that the nonlinear small-gain condition in the universal form is also necessary for the iISS property.
AB - This paper presents conditions for stability of interconnected nonlinear systems consisting of integral input-to-state stable subsystems. Both the necessity and the sufficiency are investigated using a Lyapunov formulation. The ultimate result is a necessary and sufficient condition for integral input-to-state stability (iISS) of the interconnection, which is believed to be the first of its kind. This paper derives two sufficient conditions. One is the iISS-ISS small-gain condition which has been proposed earlier by one of the authors. Its sufficiency is strengthened to cover situations more general than the earlier result. The other condition is a universal representation of the nonlinear small-gain condition which has been a popular tool for dealing with input-to-state stable (ISS) subsystems. This paper proves that, for establishing iISS property of the interconnection, the universal representation can deal with iISS subsystems as much as the iISS-ISS small-gain condition if supply rate functions are smooth everywhere and analytic at zero. Furthermore, two necessary conditions this paper derives demonstrate that the nonlinear small-gain condition in the universal form is also necessary for the iISS property.
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U2 - 10.1109/cdc.2006.377605
DO - 10.1109/cdc.2006.377605
M3 - Conference contribution
AN - SCOPUS:39649097615
SN - 1424401712
SN - 9781424401710
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 355
EP - 360
BT - Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC
PB - Institute of Electrical and Electronics Engineers Inc.
Y2 - 13 December 2006 through 15 December 2006
ER -