On the estimation of the domain of attraction for discrete-time switched and hybrid nonlinear systems

Abstract

This paper addresses the estimation of the domain of attraction for discrete-time nonlinear systems where the vector field is subject to changes. First, the paper considers the case of switched systems, where the vector field is allowed to arbitrarily switch among the elements of a finite family. Second, the paper considers the case of hybrid systems, where the state space is partitioned into several regions described by polynomial inequalities, and the vector field is defined on each region independently from the other ones. In both cases, the problem consists of computing the largest sublevel set of a Lyapunov function included in the domain of attraction. An approach is proposed for solving this problem based on convex programming, which provides a guaranteed inner estimate of the sought sublevel set. The conservatism of the provided estimate can be decreased by increasing the size of the optimisation problem. Some numerical examples illustrate the proposed approach.

Keywords:

• domain of attraction
• nonlinear system
• switched system
• hybrid system

Acknowledgements

The authors would like to thank the editors and the reviewers for their useful comments.

Additional information

Funding

This work is supported in part by the Research Grants Council of Hong Kong [grant number HKU711213E].

Notes on contributors

Chuen Kit Luk

Chuen Kit Luk received the Beng degree in computer engineering from the University of Hong Kong. He is currently a PhD student in the Department of Electrical and Electronic Engineering, the University of Hong Kong. His research interests include hybrid systems, switched systems, domain of attraction, LMI (linear matrix inequality), and SOS (sum of squares).

Graziano Chesi

Graziano Chesi received the Laurea in information engineering from the University of Florence in 1997 and the PhD in systems engineering from the University of Bologna in 2001. He was with the Department of Information Engineering of the University of Siena during 2000–2006 and then he joined the Department of Electrical and Electronic Engineering of the University of Hong Kong. He was a visiting scientist at the Department of Engineering of the University of Cambridge during 1999–2000 and at the Department of Information Physics and Computing of the University of Tokyo during 2001–2004. Dr Chesi has served as associate editor for Automatica, BMC Research Notes, the European Journal of Control, the IEEE Transactions on Automatic Control, and Systems & Control Letters. Also, he has served as guest editor of the special issues on positive polynomials in control, systems biology, and visual servoing for various journals. He is the founder and chair of the Technical Committee on Systems with Uncertainty of the IEEE Control Systems Society. He is author of the book ‘Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems’ (Springer, 2009) and editor of the book ‘Visual Servoing via Advanced Numerical Methods’ (Springer, 2010). He is first author in more than 100 technical publications.