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The Relationship Between Augmented Lyapunov-Krasovskii Functionals and Estimated Inequalities

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Abstract

Different estimated inequalities and augmented Lyapunov-Krasovskii functionals (LKFs) play important role in assessing the stability of time-delay systems. In this technical note, three categories of estimated inequalities are introduced, in which either all matrices, some matrices or no matrices are free. Then, the internal relationship among the three categories of estimated inequalities is fully revealed. Next, an optimal method is provided for selecting the estimated inequalities and constructing the Lyapunov-Krasovskii functionals. That is, the inequalities and the functionals should tailor for each other (see Table 2), which is proved theoretically. Finally, a numerical example is presented to verify the results.

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Authors and Affiliations

  1. School of Science, Hunan Institute of Technology, Hengyang, 421002, China

    Daixi Liao

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Shouming Zhong

  3. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541006, China

    Jun Cheng

  4. School of Information Sciences and Engineering, Chengdu University, Chengdu, 610106, China

    Kaibo Shi & Can Zhao

  5. School of Science, Chongqing University of Technology, Chongqing, 400054, China

    Shaohua Long

Authors
  1. Daixi Liao
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  2. Shouming Zhong
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  3. Jun Cheng
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  4. Kaibo Shi
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  5. Shaohua Long
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  6. Can Zhao
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Correspondence to Daixi Liao.

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The authors declare that there is no competing financial interest or personal relationship that could have appeared to influence the work reported in this paper.

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Daixi Liao received his B.S. and M.S. degrees from the School of Mathematics and Computational Science from Xiangtan University, in 2003 and 2009, respectively. He received a Ph.D. degree from the University of Electronic Science and Technology of China, Chengdu, China, in 2019. His current research interests include time-delay systems, stability theorem, and networked control systems.

Shouming Zhong was born on November 5, 1955. He graduated from University of Electronic Science and Technology of China, majoring applied mathematics on differential equation. He is a professor of the School of Mathematical Sciences, University of Electronic Science and Technology of China, on June 1997. He is Director of Chinese Mathematical Biology Society, the chair of Biomathematics in Sichuan, Editors of Journal of Biomathematics. He has reviewed for many Journals, such as Journal of Theory and Application on Control, Journal of Automation, Journal of Electronics, and Journal of Electronics Science. His research interests include stability theorem and its application research of the differential system, the robustness control, neural network and biomathematics.

Jun Cheng received his B.S. degree from the Hubei University for Nationalities, Hubei, China, and a Ph.D. degree in instrumentation science and technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2015. From 2015 to 2019, he was a staff with the Hubei Minzu University. He was a Visiting Scholar with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, from 2013 to 2014, and the Department of Electrical Engineering, Yeungnam University, Gyeongsan, Korea, in 2016 and 2018. Since 2019, he has been with the Guangxi Normal University, Guilin, China, where he is currently a Professor with the College of Mathematics and Statistics. His current research interests include analysis and synthesis for stochastic hybrid systems, networked control systems, robust control, and nonlinear systems. Prof. Cheng has been a recipient of the Highly Cited Researcher Award listed by Clarivate Analytics in 2019 and 2020. He is an Associate Editor of the International Journal of Control, Automation, and Systems.

Kaibo Shi was born in Anhui, China. He received his Ph.D. degree from the School of Automation Engineering at the University of Electronic Science and Technology of China. He is a professor of the School of Information Sciences and Engineering, Chengdu University. His current research interests include stability theorem, robustness stability, robust control, sampled-data control, synchronization, Lurie chaotic system, stochastic systems, and neural networks. He is a very active reviewer for many international journals.

Shaohua Long was born in Hunan, China. He received his B.S. and M.S. degrees from the School of Mathematics and Computational Science from Xiangtan University, in 2003 and 2007, respectively. He received a Ph.D. degree in applied mathematics from University of Electronic Science and Technology of China, Sichuan, China, in 2013. He now is working at Chonqing University of Technology, Chongqing, China. His current research interests include robust control, singular systems, filtering, Markovian jump systems, and time-delay systems.

Can Zhao was born in Fuyang, China. He received his B.S. degree from Fuyang Normal University, Fuyang, China, in 2015. He received a Ph.D. degree in applied mathematics from University of Electronic Science and Technology of China, Sichuan, China, in 2021. His current research interests include stability theorem, neural networks, time-delay system, and synchronization.

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Liao, D., Zhong, S., Cheng, J. et al. The Relationship Between Augmented Lyapunov-Krasovskii Functionals and Estimated Inequalities. Int. J. Control Autom. Syst. (2024). https://doi.org/10.1007/s12555-022-1233-y

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  • DOI: https://doi.org/10.1007/s12555-022-1233-y

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