Fractional controller based on a robust PI^α observer for uncertain fractional systems

ABSTRACT

In this paper, we present a fractional sliding mode controller based on a robust $P{I}^{\alpha }$ observer for fractional systems with bounded disturbances. This controller is introduced to achieve closed-loop Mittag–Leffler stability for uncertain fractional systems; meanwhile, the robust $P{I}^{\alpha }$ observer is designed such that a ${\mathcal{L}}_2$ stability is obtained via quadratic Lyapunov functions and the tuning is achieved with linear matrix inequality approach. The performance and stability of the proposed controller are analysed and demonstrated through a generalised $n\alpha$-differentiator. Finally, to illustrate the effectiveness of the proposed approach a comparative analysis against the fractional Super-Twisting is carried out.

KEYWORDS:

• Fractional sliding mode controller
• robust $P{I}^{\alpha }$ observer
• fractional systems
• stability analysis
• chattering reduction
• $n\alpha$-differentiator

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Sergio M. Delfín-Prieto http://orcid.org/0000-0001-5241-4668

Notes

1. To present a less cumbersome notation, all fractional integral $\left({}_0{\mathcal{I}}_t^{\alpha }\right)$ and derivatives $\left({}_0^C{\mathcal{D}}_t^{\alpha }\right)$ are defined with lower limit equal to zero.

Additional information

Notes on contributors

Iván Trejo-Zúñiga

Iván Trejo-Zúñiga was born in Querétaro city, México, in 1988. He earned his Ph.D. from the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV-IPN) in 2018. Currently, he is a full time professor at the Mechatronics Department of the Technological University of San Juan del Río. His research interests include nonlinear observers design and fault diagnosis on integer and fractional order systems.

Sergio M. Delfín-Prieto

Sergio M. Delfín-Prieto was born in Nanchital, Veracruz, Mexico, in 1986. He received his B.Sc degree in Automation Engineering and the M.Sc degree in Instrumentation and Control from Autonomous University of Queretaro (UAQ), Mexico, in 2010 and 2014, respectively. He earns his Ph.D. in Automatic Control from the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV-IPN), Mexico City, in 2018. His research interests are observers design, control of fractional order systems, secure communications, synchronisation, and chaos.

Rafael Martínez-Guerra

Rafael Martínez-Guerra was born in Mexico City in 1959, he earns his Ph.D. from Universidad Autonoma Metropolitana in 1996. Currently, he is a researcher at the Automatic Control Department of the CINVESTAV-IPN (Center of Research and Advanced Studies of IPN) and member of the National System Researchers since 1992 (level II, currently). He is the author and co-author of more 71 papers in International Journals and more than 125 contributions in International Conference in Proceedings. He is the co-author of 7 books (5 Springer). His main research interests are in the areas of Nonlinear Systems, Differential Geometric and Differential Algebraic Methods, Fractional Nonlinear Observers, Fractional Fault Detection and Diagnosis Problem, Secure Communications, Fractional Synchronisation and Chaos.