Robust ISS of uncertain discrete-time singularly perturbed systems with disturbances

ABSTRACT

This paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stability and stabilisation of uncertain discrete-time SPSs are also obtained as the particular cases of robust ISS and robust ISS by feedback. We first find a sufficient condition by using the fixed-point principle in terms of linear matrix inequalities (LMIs) to guarantee that the considered system is always standard discrete-time SPSs subject to uncertainty and disturbances. Then, the full systems could decompose into the continuous-time uncertain slow subsystem with disturbance and discrete-time uncertain fast subsystems with disturbance, respectively. Based on the two-time-scale decomposition technique, sufficient condition in terms of LMIs is given such that the full systems are uniformly standard and robust ISS simultaneously. In addition, a state feedback controller is constructed by using the LMI approach such that the resulting closed-loop systems are robust ISS. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.

KEYWORDS:

• Discrete-time singularly perturbed systems
• two-time-scale
• input-to-state stability (ISS)
• linear matrix inequalities (LMIs)
• robust stability

Acknowledgements

The authors would like to thank the Associate Editor and anonymous reviewers for their valuable comments and suggestions that certainly improved the quality of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by the National Natural Science Foundation of China [61703447, 11471118], the Research Foundation of the Henan Higher Education Institutions of China [18A110039], the Science and Technology Commission of Shanghai Municipality (STCSM) [13dz2260400, 18dz2271000], Shanghai Key Laboratory of PMMP and also supported by the Science Foundation of Zhoukou Normal University [ZKNUC2017020, ZKNUB2201806].

Notes on contributors

Mehvish Naz

Mehvish Naz received the degree of Master of Science in Mathematics and degree of Master of Philosophy in Applied Mathematics from Quaid-e-Azam University, Islamabad, Pakistan in 2006 and 2008, respectively. She is currently a Ph.D. student at East China Normal University, Shanghai, China. She has been working as a Lecturer at COMSATS University, Islamabad, Pakistan. Her research interests are singular perturbation method and robust control systems.

Wei Liu

Wei Liu received his Ph.D. in Mathematics from East Normal University, China, 2016 and M.S. degree from the same department of the East Normal University, China, in 2010. Since 2010, he has been with Zhoukou Normal University, Zhoukou, China. His research interests include singularly perturbed systems, robust control, networked control systems, and nonlinear systems.

Zhiming Wang

Zhiming Wang received both his Master at East China Normal University (ECNU) in Applied Mathematics and PhD. in systems synthesis and control. Now he is full professor of the School of Mathematical Sciences, ECNU, Shanghai, China. He served as director of Shanghai Nonlinear Analysis Association (SNAA), Shanghai SIAM and China Singular Perturbation Association respectively. He worked at Evelyn Hone College, Lusaka, Zambia, the Department of Electronic and Electric Engineering of Melbourne University in Australia and The Mathematics Department of St. Olaf College in United States. His resent research interests include control theory, singular perturbation, geometric method, dynamics systems.