Connection between cooperative positive systems and integral input-to-state stability of large-scale systems

Authors:
Björn S. Rüffer;Christopher M. Kellett;Steven R. Weller
Affiliations:
Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, VIC 3010, Australia;School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, NSW 2308, Australia;School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, NSW 2308, Australia
Venue:
Automatica (Journal of IFAC)
Year:
2010

Citing 4
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Abstract

We consider a class of continuous-time cooperative systems evolving on the positive orthant R"+^n. We show that if the origin is globally attractive, then it is also globally stable and, furthermore, there exists an unbounded invariant manifold where trajectories strictly decay. This leads to a small-gain-type condition which is sufficient for global asymptotic stability (GAS) of the origin. We establish the following connection to large-scale interconnections of (integral) input-to-state stable (ISS) subsystems: If the cooperative system is (integral) ISS, and arises as a comparison system associated with a large-scale interconnection of (i)ISS systems, then the composite nominal system is also (i)ISS. We provide a criterion in terms of a Lyapunov function for (integral) input-to-state stability of the comparison system. Furthermore, we show that if a small-gain condition holds then the classes of systems participating in the large-scale interconnection are restricted in the sense that certain iISS systems cannot occur. Moreover, this small-gain condition is essentially the same as the one obtained previously by Dashkovskiy, Ruffer, and Wirth (2007, in press) for large-scale interconnections of ISS systems.