Semi-global stabilisation of fractional-order linear systems with actuator saturation by output feedback

Abstract

This paper considers the problem of the semi-global asymptotic stabilisation of fractional-order (FO) linear systems subject to actuator saturation by output feedback. To solve the problem, a family of observer-based linear output feedback laws, parameterised in a positive scalar, is proposed by means of the low gain feedback design technique. The design applies to FO linear systems that are stabilisable and detectable, but not exponentially unstable. For such an FO system under the proposed observer-based linear low gain output feedback, the peak value of the control input for a given initial condition can be made arbitrarily small to avoid actuator saturation by decreasing the value of the parameter towards zero and thus semi-global asymptotic stabilisation is achieved. To obtain these results, we establish the properties of low gain feedback, derive asymptotic expansions and the bounds of high-order derivatives of the Mittag-Leffler (ML) functions to estimate the state responses of FO linear systems, and explicitly construct a Hermitian matrix to satisfy a linear matrix inequality (LMI) stability condition for FO linear systems. The results in this paper extend the corresponding results for integer-order (IO) linear systems.

Keywords:

• Actuator saturation
• Caputo derivative
• fractional-order linear systems
• low gain feedback
• output feedback
• semi-global stabilisation

Disclosure statement

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

Data availability statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Additional information

Notes on contributors

Jie Xu

Jie Xu received her Master of Science degree in control theory and control engineering in 2009 from Shanghai Jiao Tao University, Shanghai, China, where she is currently pursuing her Ph.D. degree in control science and control engineering with the Department of Automation. Her current research interests include fractional-order systems, robust control, and control systems with actuator saturation.

Zongli Lin

Zongli Lin is the Ferman W. Perry Professor in the School of Engineering and Applied Science and a Professor of Electrical and Computer Engineering at University of Virginia. He received his B.S. degree in mathematics and computer science from Xiamen University, Xiamen, China, in 1983, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in 1989, and his Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, Washington, in 1994. His current research interests include nonlinear control, robust control, and control applications. He was an Associate Editor of IEEE Transactions on Automatic Control (2001-2003), IEEE/ASME Transactions on Mechatronics (2006-2009) and IEEE Control Systems Magazine (2005-2012). He was elected a member of the Board of Governors of the IEEE Control Systems Society (2008-2010, 2019-2021) and chaired the IEEE Control Systems Society Technical Committee on Nonlinear Systems and Control (2013-2015). He has served on the operating committees of several conferences. He was the program chair of the 2018 American Control Conference, a general chair of the 16th IEEE International Conference on Control and Automation, held in 2020, and a general chair of the 13th and 16th International Symposium on Magnetic Bearings, held in 2012 and 2018, respectively. He currently serves on the editorial boards of several journals and book series, including Automatica, Systems & Control Letters, and Springer/Birkhauser book series Control Engineering. He is a Fellow of IEEE, IFAC, and AAAS (American Association for the Advancement of Science).