Low gain magic: Control of linear systems with distributed infinite input delays

Significance 

Time delays are intrinsic features for various practical systems in engineering, biology, chemistry, economics, mechanics, physics, physiology, population dynamics, and so on. Unfortunately, systems with time delays are difficult to deal with because they belong to the infinite dimensional differential functional equations. Most of existing works on time-delayed systems focus on bounded time-delays, and various approaches to stability analysis and controller synthesis have been developed. Recently, there are increasing interests in systems with infinite delays as infinite delays indeed exist in some real-world systems, such as the HIV spread model, the oscillator model, the traffic flow model, the neural network model, and so on. Infinite delays, also known as unbounded delays, are more general but also more difficult to deal with. Infinite delays can be classified as either time-varying infinite delays or distributed infinite delays. The major challenges in dealing with infinite delays, especially in control of systems with infinite delays, include limitations of analysis and synthesis tools, the sensitivity of solutions to initial conditions, and mathematical complexity.

A group of researchers at the City University of Hong Kong, Department of Biomedical Engineering: Dr. Xiang Xu, Dr. Lu Liu and Professor Gang Feng investigated the stabilization problem of systems with distributed infinite delays at the input. They developed two low gain feedback controllers, for two different classes of linear systems with distributed infinite input delays. They considered two cases where the first case comprised of unstable eigenvalues at the imaginary axis while the second case consisted of unstable eigenvalues at the origin. Furthermore, they showed that the proposed low gain controllers could be used to stabilize those linear systems with a distributed infinite input delay. Their research work is published in the research journal, Automatica.

The authors observed that the linear system could be globally stabilized asymptotically by using the low gain controllers. Consequently, the use of low gain feedback in solving stabilization problems of linear systems with distributed infinite delays at the input was efficiently confirmed. This was attributed to the fact that the results were based on the stability results of the systems with infinite delays. Furthermore, unlike in the previous studies, the authors noted that the existing results in the literature concerning the bounded distributed input delays as well as constant input delays were both considered special with regard to the new results obtained.

The study by City University of Hong Kong scientists is the first to successfully solve stabilization problems of linear systems with distributed infinite input delays based on low gain feedback controllers. The effectiveness and efficiency of the proposed low gain controllers are validated through numerical simulations in two engineering systems. Their new method provides a powerful and useful basis in further study of systems with infinite delays, and also provides a great potential for their applications in real-world systems.

P.S The authors acknowledge the support by the research Grants Council of Hong Kong under grant CityU-11206817.

About the author

Xiang Xu received the Bachelor of Engineering degree in Automation from Nanjing University of Science and Technology, China in 2014 and the Ph.D. degree in City University of Hong Kong, Hong Kong in 2018. He is currently a Postdoctoral Fellow in City University of Hong Kong. His research interests include multi-agent systems and time-delay systems.

About the author

Lu Liu received her Ph.D. degree in 2008 in the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong SAR. From 2009 to 2012, she was an Assistant Professor in The University of Tokyo, Japan, and then a Lecturer in The University of Nottingham, United Kingdom. After that, she joined City University of Hong Kong, Hong Kong SAR, where she is currently an Associate Professor at Department of Biomedical Engineering. Her research interests are primarily in multi-agent systems, control theory and applications and biomedical devices.

About the author

Gang Feng received the B.Eng and M.Eng. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992.

He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992-1999. He was awarded an Alexander von Humboldt Fellowship in 1997, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007, and Changjiang chair professorship from Education Ministry of China in 2009.

His current research interests include multi-agent systems and control, intelligent systems and control, and networked systems and control. Prof. Feng is an IEEE Fellow, has been an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Fuzzy Systems, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, Journal of Systems Science and Complexity, and Journal of Control Theory and Applications.

Reference

Xu, X., Liu, L., & Feng, G. (2018). Stabilization of linear systems with distributed infinite input delays: A low gain approach. Automatica, 94, 396-408.

Go To Automatica

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