A novel methodology to solve the CQLF problem for a finite set of stable second-order systems

Abstract

A new methodology to provide conclusive information about the existence/non-existence of a common quadratic Lyapunov function (CQLF) for a finite set of stable second-order systems is presented. Despite the high complexity of the CQLF problem, even in the case of N second-order systems, the results presented in this paper have a very simple and intuitive theoretical support, including topics such as classical intersection of convex sets and properties of convex linear combinations. Illustrative examples to show the performance of the proposed methodology are provided.

Keywords:

• common quadratic Lyapunov functions
• system stability
• intersection of convex hulls
• evolutionary computation
• particle swarm optimisation

Acknowledgements

The author wishes to thank to Robert Shorten and Wynita Griggs for their useful discussions about the content of the paper, and to the anonymous reviewers for their assertive comments.

Notes

1. As an abuse of notation, we are going to call a CQLF both the function (2(2) ), and the associated matrix P.

2. To compute such a CQLF, computing methods as LMI Toolbox (Gahinet et al., 1994), Gradient (Liberzon & Tempo, 2004) and PSO (Ordóñez-Hurtado & Duarte-Mermoud, 2012) are suggested.

3. To compute such a CQLF, computing methods as LMI Toolbox (Gahinet et al., 1994), Gradient (Liberzon & Tempo, 2004) and PSO (Ordóñez-Hurtado & Duarte-Mermoud, 2012) are suggested.

Additional information

Funding

This work was supported by Science Foundation Ireland [grant 11/PI/1177].