The construction of augmented Lyapunov-Krasovskii functionals and the estimation of their derivatives in stability analysis of time-delay systems: a survey

Abstract

The Bessel-Legendre inequality plays an important role in stability analysis of linear systems with time-varying delays. However, various integral vectors inherited from the Bessel-Legendre inequality bring some challenging issues, such as the construction of suitable augmented Lyapunov-Krasovskii functionals (LKFs); high-degree polynomial estimation on the derivative of the chosen LKF. This paper offers an overview of recent developments on these issues. First, an instructive review on the construction of suitable augmented LKFs catering for the use of the Bessel-Legendre inequality is given in detail. Second, an in-depth analysis and insightful understanding of high-degree polynomial inequalities on closed intervals are made. Third, recent results on reciprocally convex combination inequalities are briefly discussed. Finally, several challenging problems are presented for future research.

Keywords:

• Time-delay systems
• stability
• Lyapunov-Krasovskii functional
• reciprocally convex combination
• high-degree polynomial inequality

Data availability statement

The authors confirm that data and materials that support the results or analyses presented in this paper are freely available upon request.

Disclosure statement

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

Additional information

Notes on contributors

Xian-Ming Zhang

Xian-Ming Zhang received the M.Sc. degree in Applied Mathematics and the Ph.D. degree in Control Theory and Control Engineering from Central South University, Changsha, China, in 1992 and 2006, respectively.In 1992, he joined Central South University, where he was an Associate Professor with the School of Mathematics and Statistics. From 2007 to 2014, he was a Post-Doctoral Research Fellow and a Lecturer with the School of Engineering and Technology, Central Queensland University, Rockhampton, QLD, Australia. From 2014 to 2016, he was a Lecturer with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. In 2016, he joined the Swinburne University of Technology, Melbourne, VIC, Australia, where he is currently an Associate Professor with the School of Science, Computing and Engineering Technologies. His current research interests include H-infinity filtering, event-triggered control systems, networked control systems, neural networks, multi-agent systems, offshore platforms and time-delay systems.Associate Professor Zhang was a recipient of the second National Natural Science Award in China in 2013, jointly with Prof. M. Wu and Prof. Y. He. He was also the recipient of The 2020 IEEE Transactions on Industrial Informatics Outstanding Paper Award, The 2019 IEEE Systems, Man, and Cybernetics Society Andrew P. Sage Best Transactions Paper Award, and The 2016, 2020 IET Control Theory and Applications Premium Award.He is an Associate Editor of the IEEE TRANSACTIONS ON CYBERNETICS, Journal of the Franklin Institute, International Journal of Control, Automation, and Systems, Neurocomputing, and Neural Processing Letters.

Qing-Long Han

Qing-Long Han received the B.Sc. degree in Mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Sc. and Ph.D. degrees in Control Engineering from East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively.Professor Han is Pro Vice-Chancellor (Research Quality) and a Distinguished Professor at Swinburne University of Technology, Melbourne, Australia. He held various academic and management positions at Griffith University and Central Queensland University, Australia. His research interests include networked control systems, multi-agent systems, time-delay systems, smart grids, unmanned surface vehicles and neural networks.Professor Han is a Highly Cited Researcher in both Engineering and Computer Science (Clarivate Analytics). He was one of Australia's Top 5 Lifetime Achievers (Research Superstars) in Engineering and Computer Science (The Australian's 2020 Research Magazine). He was the recipient of The 2021 Norbert Wiener Award (the Highest Award in systems science and engineering, and cybernetics) and The 2021 M. A. Sargent Medal (the Highest Award of the Electrical College Board of Engineers Australia). He was the recipient of The 2020 IEEE Systems, Man, and Cybernetics (SMC) Society Andrew P. Sage Best Transactions Paper Award, The 2020 IEEE Transactions on Industrial Informatics Outstanding Paper Award and The 2019 IEEE SMC Society Andrew P. Sage Best Transactions Paper Award.Professor Han is a Member of the Academia Europaea (The Academy of Europe). He is a Fellow of The Institute of Electrical and Electronics Engineers (FIEEE) and a Fellow of The Institution of Engineers Australia (FIEAust). He has served as an AdCom Member of IEEE Industrial Electronics Society (IES), a Member of IEEE IES Fellow Committee and Chair of IEEE IES Technical Committee on Networked Control Systems. He is Co-Editor-in-Chief of IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, Co-Editor of Australian Journal of Electrical and Electronic Engineering, an Associate Editor for 12 international journals, including the IEEE TRANSACTIONS ON CYBERNETICS, the IEEE INDUSTRIAL ELECTRONICS MAGAZINE, the IEEE/CAA JOURNAL OF AUTOMATICA SINICA, Control Engineering Practice, and Information Sciences, and a Guest Editor for 13 Special Issues.

Xiaohua Ge

Xiaohua Ge received the B.Eng. degree in electronics and information engineering from Nanchang Hangkong University, Nanchang, China, in 2008, the M.Eng. degree in control theory and control engineering from Hangzhou Dianzi University, Hangzhou, China, in 2011, and the Ph.D. degree in computer engineering from Central Queensland University, Rockhampton, QLD, Australia, in 2014.From 2011 to 2013, he was a Research Assistant with the Centre for Intelligent and Networked Systems, Central Queensland University, where he was a Research Fellow in 2014. From 2015 to 2017, he was a Research Fellow with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. He is currently a Senior Lecturer with the School of Science, Computing and Engineering Technologies, Swinburne University of Technology, Melbourne, VIC, Australia. His research interests include networked, event-triggered, secure, and intelligent control and estimation, and their applications in autonomous vehicles and connected vehicles.Dr. Ge received The 2019 IEEE Systems, Man, and Cybernetics Society Andrew P. Sage Best Transactions Paper Award, The 2020 IEEE/CAA Journal of Automatica Sinica Outstanding Reviewer Award, and The 2017 IEEE Transactions on Cybernetics Outstanding Reviewer Award. He is an Associate Editor of the IEEE TRANSACTION ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS and a Guest Editor for several Special Issues.