Abstract
In this paper, we study the asymptotic stability of three fractional systems by the method of universal adaptive stabilization. Moreover, when α ∈ (2, 3] and λ > 0, Mittag–Leffler function E α( − λk α) is shown to be Nussbaum function. Finally, two simulation results are provided to illustrate the concepts.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ilchmann A (1993) Non-identifier-based high-gain adaptive control. Springer-Verlag, New York
Shankar S, Bodson M (1989) Adapative control-stability convergence and robustness. Prentice Hall, Englewood Cliffs, New Jersey
Axtell M, Bise ME (1990) Fractional calculus applications in control systems. In: Proceeding of the IEEE 1990 national aerospace electronics conference, New York, pp 563–566
Podlubny I (1994) Fractional-order systems and fractional-order controllers. In Proceedings of the Conference Internationale Francophone d’Automatique. UEF-03-94, Inst Exp Phys, Slovak Acad Sci Kosice
Oustaloup A (1988) From fractality to non integer derivation through recursivity, a property common to these two concepts: a fundamental idea for a new process control strategy. In: Proceedings of the 12th IMACS world conference, Paris, France 3:203–208, July 1988
Yan Li, YangQuan Chen, Yongcan Cao (2008) Fractional order universal adaptive stabilization. In: Proceedings of the 3rd IFAC workshop on fractional differentiation and its applications, Ankara, Turkey, November 2008
Igor Podlubny (1999) Fractional differential equations. Academic Press, San Diego
Bagley RL, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30:133–155, February 1986
Gorenflo R, Mainardi R (1996) Fractional oscillations and Mittag-Leffler functions. [Online Available:] http://citeseer.ist.psu.edu/gorenflo96fractional.html
Wagner DA, Morman Jr KN (1994) A viscoelastic analogy for solving 2-D electromagnetic problems. Finite Elem Anal Des 15:Februrary 1994
Townley S (1992) Generic properties of universal adaptive stabilization schemes. In: Proceedings of the 31st IEEE conference on decision and control 4:3630–3631, September 1992
Jianlong Zhang, Petros A Ioannou (2006) Non-identifier based adaptive control scheme with guaranteed stability. In: Proceedings of the 2006 American control conference, pp 5456–5461, June 2006
Schuster H, Westermaier C, Schröder D (2006) Non-identifier-based adaptive tracking control for a two-mass system. In: Proceedings of the 2006 American control conference, pp 190–195
Hans Schuster, Christian Westermaier, Dierk Schröder (2006) Non-identifier-based adaptive control for a mechatronic system achieving stability and steady state accuracy. In: Proceedings of the 2006 IEEE, pp 1819–1824, October 2006
Hao Lei, Wei Lin, Bo Yang (2007) Adaptive robust stabilzation of a family of uncertain nonlinear systems by output feedback: the non-polynomial case. In: Proceedings of the 2007 American control conference, July 2007
Hilfer R and Seybold HJ (2006) Computation of the generalized Mittag-Leffler fucntion and its inverse in the complex plane. Integr Transf Spec Funct 17(9):637–652, September 2006
Kenneth S Miller, Stefan G Samko (2001) Completely monotonic functions. Integr Transf Spec Funct 12(4):389–402
Shayok Mukhopadhyay, Yan Li, YangQuan Chen (2008) Experimental studies of a fractional order universal adaptive stabilizer. In: Proceedings of the 2008 IEEE/ASME international conference on mechatronic and embedded systems and applications, Beijing, China, October 2008
Acknowledgment
Yan Li would like to thank Y. Cao for polishing an early version of [6].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Li, Y., Chen, Y. (2010). Stability Analysis of Fractional Order Universal Adaptive Stabilization. In: Baleanu, D., Guvenc, Z., Machado, J. (eds) New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3293-5_31
Download citation
DOI: https://doi.org/10.1007/978-90-481-3293-5_31
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3292-8
Online ISBN: 978-90-481-3293-5
eBook Packages: EngineeringEngineering (R0)