On interconnected systems, passivity and some generalisations

Abstract

A sufficient condition for the stability of large-scale interconnections of N linear time-variant systems is presented. Such a condition represents important extensions to passivity criteria and ensures stability by means of the existence of a positive definite (full-block) matrix P which is a common solution to Lyapunov equations involving a diagonal stacking of the N systems and the interconnection structure matrix. An experimental methodology for the verification of the sufficient condition also is proposed, based on evolutionary computation techniques. Applications of the new stability results are provided through illustrative examples, which are developed using particle swarm optimisation and genetic algorithms.

Keywords:

• passivity
• large-scale interconnections
• system stability
• evolutionary computation based techniques

Acknowledgements

This work was in part supported by Science Foundation Ireland grant 11/PI/1177 and ‘Departamento de Postgrado y Postítulo de la Vicerrectoría de Asuntos Académicos’, University of Chile. Parts of this work have been submitted to the 52nd IEEE Conference on Decision and Control (Ordóñez-Hurtado, Griggs and Shorten, 2013).

Notes

Uniform asymptotic stability is defined in Sastry (1999, Chapter 5) and Sun and Ge (2005, Chapter 2.7).

denotes maximum singular value.