On integral input-to-state stability for a feedback interconnection of parameterised discrete-time systems

Abstract

This paper addresses integral input-to-state stability (iISS) for a feedback interconnection of parameterised discrete-time systems involving two subsystems. Particularly, we give a construction for a smooth iISS Lyapunov function for the whole system from the sum of nonlinearly weighted Lyapunov functions of individual subsystems. Motivations for such a construction are given. We consider two main cases. The first one investigates iISS for the whole system when both subsystems are iISS. The second one gives iISS for the interconnected system when one of subsystems is allowed to be input-to-state stable. The approach is also valid for both discrete-time cascades and a feedback interconnection of iISS and static systems. Examples are given to illustrate the effectiveness of the results.

Keywords:

• input-to-state stability
• integral input-to-state stability
• 0-global asymptotic stability
• parameterised discrete-time systems
• small-gain conditions

Acknowledgements

The first author is grateful to Sergey Dashkovskiy and Mojtaba Mahzoon for helpful discussions.

Notes

1. By swapping λ₁(.) with λ₂(.) in the above arguments, it is clear that the same arguments hold for the case of k ∈ (0, 1).

Additional information

Notes on contributors

N. Noroozi

Navid Noroozi was born in Isfahan, Iran, in 1983. He received his BSc and MSc degrees in 2005 and 2009, respectively. He is currently a Ph.D. student in Shiraz University, Shiraz, Iran. His research interests include nonlinear control theory, hybrid dynamical systems and computational methods.

A. Khayatian

Alireza Khayatian received his BSc degree in electrical engineering from Isfahan Technical University, Iran, in 1985, and MSc degree also in electrical engineering from Tarbiat Modarres University, Iran, in 1988. He worked as a research assistant in the Electromechanical Systems Laboratory at the Georgia Institute of Technology, where he received his Ph.D. degree in control engineering in 1993. Since 1993, he has been a faculty member with the Department of Electrical Engineering, Shiraz University, Shiraz, Iran, and his research interests include nonlinear control, estimation and robotics.

S. Ahmadizadeh

Saeed Ahmadizadeh received his B.S. and M.S. degree from Shiraz University and Iran University of Science and Technology (IUST), respectively. He is currently pursuing a Ph.D. degree at the Department of Electrical and Electronics Engineering at the University of Melbourne, Australia. His research interests are in the areas of nonlinear and hybrid systems, robust control/filter design, active noise and vibration control.

H.R. Karimi

Hamid Reza Karimi is a professor in control systems at the Faculty of Engineering and Science of the University of Agder in Norway. His research interests are in the areas of nonlinear systems, networked control systems, robust control/filter design, time-delay systems, wavelets and vibration control of flexible structures with an emphasis on applications in engineering. Dr Karimi is a senior member of IEEE and is serving as the chairman of the IEEE chapter on control systems at the IEEE Norway section. He is also serving as an editorial board member for some international journals, such as Mechatronics, Neurocomputing, Information Sciences, Asian Journal of Control, Journal of Franklin Institute, Journal of Systems and Control Engineering and International Journal of Control, Automation and Systems, among others. He is a member of the IEEE Technical Committee on Systems with Uncertainty, IFAC Technical Committee on Robust Control and IFAC Technical Committee on Automotive Control.