Time scale state feedback h-stabilisation of linear systems under Lipschitz-type disturbances

Abstract

This paper studies the h-stabilisation problem of certain classes of perturbed systems on time scales. Sufficient conditions for the control law design are proposed to ensure the h-stability of the closed-loop dynamical system under Lipschitz-type disturbances. Using the Gronwall inequality approach with time scale theory, the h-stability of the closed-loop system is investigated in non-uniform time domains with bounded graininess. Some numerical examples are provided to show the feasibility of the obtained results using the proposed approach for systems evolving on some arbitrary time scales.

Keywords:

• Time scale theory
• $\mathit{\text{h}}$-stability
• $\mathit{\text{h}}$-stabilisation
• nonlinear dynamic systems
• Gronwall inequalities

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST) through the base research fund under [grant number BAS/1/1627-01-01].

Notes on contributors

Bacem Ben Nasser

Bacem Ben Nasser received his Master and Ph.D. degrees in Mathematics from Faculty of Sciences of Sfax, University of Sfax, Tunisia, in 2012 and 2016 respectively. In 2017, he joined the Higher Institute of Applied Sciences and Technology of Kairouan, University of Kairouan, Tunisia as assistant professor of Applied Mathematics. He is currently a postdoctoral researcher at the King Abdullah University of Science and Technology (KAUST), Kingdom of Saudi Arabia. His current research interests concerns the analysis and control design for nonlinear and hybrid dynamical systems including time scales theory and modeling on non-uniform time domains.

Mohamed Djemai

Mohamed Djemai, IEEE Senior member, is currently full professor at University Polytechnic Hauts-de France, Valenciennes, France since 2008. He received his B.Sc. in electrical engineering from ENP-Alger, Algeria in 1991, his M.Sc. (DEA) and PhD in Control and Signal System from University of Paris-Sud, France in 1992, and January 1996 respectively. He joined ENSEA in September 2000 as associate professor and he was deputy director of ECSLaboratory. He is member of 2 IFAC TC-2.1 control system, and TC-1.3 on Discrete Event and Hybrid Systems, and 2 IEEE TC on hybrid systems and TC on VSS & SMC. Prof. Djemaï was visiting professor at Norhumbria University (2010–2013). His research interests are mainly related to nonlinear control systems, observation, and fault detection theory on hybrid system including variable structure systems and time scale systems, with applications to power systems, robot and vehicles. He published more than 80 journals and 135 Conf. papers in his area of research.

Michael Defoort

Michael Defoort is currently an associate professor at University Polytechnic Hauts-de France, Valenciennes, France since 2009. He received his Ph. D. degree in Automatic Control from Ecole Centrale de Lille, France, in 2007. From 2007 to 2008, he was a Research Fellow with the Department of System Design Engineering, Keio University, Japan. From 2008 to 2009, he was a Research Fellow with Ecole des Mines de Douai, France. He is a member of IFAC TC 1.5, Networked Systems, IFAC TC 2.5. Discrete Event and Hybrid Systems and IFAC TC 1.3. Robust Control. He serves as a Subject Editor of the Journal of the Franklin Institute, as a Subject Editor of Nonlinear Dynamics and as an Associate Editor of IEEE/ASME Transactions on Mechatronics. His research interests include nonlinear control, observation and fault detection with applications to power systems, multi-agent systems and vehicles.

Taous-Meriem Laleg-Kirati

Taous-Meriem Laleg-Kirati is an associate professor in the division of Computer, Electrical and Mathematical Sciences and Engineering at KAUST and member of the Computational Bioscience Research Center (CBRC). She joined KAUST in 2011 after being a postdoctoral fellow and a researcher at the French Institute for research in Computer Sciences and Control Systems (INRIA) in Bordeaux. She received her Ph.D. in Applied Mathematics in 2008 from INRIA and Versailles University. Professor Laleg's work is in the general area of mathematical control theory, systems modeling, signal processing and their applications. Her primary research goals are directed towards developing effective estimation methods and algorithms to understand complex systems, extract hidden information and design control and monitoring strategies. Her research projects are motivated by real world problems in engineering and biomedical fields. She is an IEEE senior member, a member of the IEEE Control Conference Editorial Board, an associate editor of the IEEE access Journal and several control conferences including the European Control Conference. She is also member of the international federation of automatic control (IFAC) technical committee on Biological and Medical systems (TC8.2) and Modeling and Control of Environmental Systems (TC8.3).