Robust stability checking of a class of linear interval fractional order system using lyapunov inequality

IFAC Proceedings Volumes (IFAC-PapersOnline)

Volumn 2, Issue PART 1, 2006, Pages 89-94

  Ahn, Hyosung ^a   Chen, YangQuan ^a   Podlubny, Igor ^b  

^a Utah State University   (United States)

^b TECHNICAL UNIVERSITY OF KO ICE   (Slovakia)

Author keywords

Fractional order linear dynamic systems; Interval linear time invariant systems; Interval matrix; Robust stability

Indexed keywords

APPLICATION PROGRAMS; EIGENVALUES AND EIGENFUNCTIONS; INVARIANCE; LINEAR CONTROL SYSTEMS; LINEAR SYSTEMS; LYAPUNOV METHODS; NUMERICAL METHODS; PERTURBATION TECHNIQUES; STABILITY; TIME VARYING CONTROL SYSTEMS;

FRACTIONAL ORDER LINEAR DYNAMIC SYSTEMS; FRACTIONAL-ORDER SYSTEMS; INTERVAL LINEAR TIME INVARIANT SYSTEMS; INTERVAL MATRIX; LINEAR TIME INVARIANT; LYAPUNOV INEQUALITY; MATRIX PERTURBATION THEORY; ROBUST STABILITY;

MATRIX ALGEBRA;

EID: 79961123541     PISSN: 14746670     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.3182/20060719-3-pt-4902.00015     Document Type: Conference Paper

References (35)

1. M. Axtell and E. M. Bise. Fractional calculus applications in control systems. In Proceedings of the IEEE 1990 Nat. Aerospace and Electronics Conf., pages 563-566, New York, USA, 1990.
2. C. Bonnet and J. R. Partington. Coprime factorizations and stability of fractional differential systems. Systems & Control Letters, 41, 2000.
3. M. Caputo. Linear models of dissipation whose q is almost frequency independent-II. Geophys. J. R. Astronom. Soc., 13:529-539, 1967.
4. YangQuan Chen, Dingyü Xue, and Huifang Dou. Fractional calculus and biomimetic control. In Proc. of the First IEEE Int. Conf. on Robotics and Biomimetics (RoBio04), pages robio2004-347, Shengyang, China, August 2004. IEEE.
5. YangQuan Chen, Hyo-Sung Ahn, and Igor Pudlubny. Robust stability check of fractional order linear time invariant systems with interval uncertainties. In Proc. of the IEEE Conference on Mechatronics and Automation, pages 210-215, Niagara Falls, Canada, July 2005a.
6. YangQuan Chen, Hyosung Ahn, and Dingyü Xue. Robust controllability of interval fractional order linear time invariant systems. In Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, pages 1-9, Long Beach, CA., Sep. 24-28 2005b.
7. Lokenath Debnath. A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technol., 35(4):487-501, 2004.
8. Sgaul Gutman and Eliahu I. Jury. A general theory for matrix root-clustering in subregions of the complex plane. IEEE Trans. Automatic Control, 26(4):853-863, 1981.
9. G. I. Hargreaves. Interval analysis in matlab. Numerical Analysis Report No. 416, The University of Manchester, December 2002.
10. David Hertz. The eigenvalues and stability of Hermitian interval matrices. IEEE Trans. on Circuit and System-I: Fundamental Theory and Applications, 39(6), 1992.
11. Boris J. Lurie. Three-parameter tunable tilt-integral-derivative (TID) controller. US Patent US5371670, 1994.
12. J. A. Tenreiro Machado. Special Issue on Fractional Calculus and Applications. Nonlinear Dynamics (Guest Editor), 29:1-385, March 2002.
13. S. Manabe. The non-integer integral and its application to control systems. JIEE (Japanese Institute of Electrical Engineers) Journal, 80 (860):589-597, 1960.
14. S. Manabe. The non-integer integral and its application to control systems. ETJ of Japan, 6(3-4):83-87, 1961.
15. D. Matignon. Stability result on fractional differential equations with applications to control processing. In IMACS-SMC Proceedings, pages 963-968, Lille, France, July 1996.
16. D. Matignon. Stability properties for generalized fractional differential systems. In ESAIM: Proceedings, vol. 5, pages 145-158. URL: http://www.emath.fr/Maths/Proc/Vol.5/, Dec. 1998a.
17. D. Matignon. Generalized fractional differential and difference equations: stability properties and modelling issues. In Proc. of the Mathematical Theory of Networks and Systems symposium (MTNS'98), pages 503-506, Padova, Italy, July 1998b.
18. D. Matignon. Stability properties for generalized fractional differential systems. In D. Matignon and G. Montseny, editors, Proceedings of the Colloquium FDS'98: Fractional Differential Systems: Models, Methods and Applications, number 5, pages 145-158, Paris, December 1998c.
19. D. Matignon and B. d'Andréa Novel. Some results on controllability and observability of finite-dimensional fractional differential systems. In Proc. of the Computational Engineering in Systems and Application Multiconference, volume 2, pages 952-956, Lille, France, July 1996. IMACS, IEEE-SMC.
20. B. P. Molinari. The stabilizing solution of the discrete algebraic Riccati equation. IEEE Trans. Automatic Control, 20(3):393-399, 1975.
21. Mathieu Moze and Jocelyn Sabatier. LMI tools for stability analysis of fractional systems. In Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, pages 1-9, Long Beach, CA., Sep. 24-28 2005.
22. Manuel Duarte Ortigueira and J. A. Tenreiro Machado. Special Issue on Fractional Signal Processing and Applications. Signal Processing (Guest Editors), 83(11):2285-2480, Nov. 2003.
23. A. Oustaloup. Fractional order sinusoidal oscilators: Optimization and their use in highly linear FM modulators. IEEE Transactions on Circuits and Systems, 28(10):1007-1009, 1981. (Pubitemid 12441063)
24. A. Oustaloup, B. Mathieu, and P. Lanusse. The CRONE control of resonant plants: application to a flexible transmission. European Journal of Control, 1(2), 1995.
25. A. Oustaloup, X. Moreau, and M. Nouillant. The CRONE suspension. Control Engineering Practice, 4(8):1101-1108, 1996. (Pubitemid 126378651)
26. Ivo Petráš, YangQuan Chen, and Blas M. Vinagre. Robust stability test for interval fractional order linear systems, volume 208-210, chapter 6.5: Vincent D. Blondel and Alexander Megretski (Eds.), Unsolved problems in the mathematics of systems and control. Princeton University Press, July 2004.
27. Ivo Petráš, YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. Stability of linear time invariant systems with interval fractional orders and interval coefficients. In Proc. of the International Conference on Computation Cybernetics (ICCC04), pages 1-4, Viena Technical University, Viena, Austria., 8/30-9/1 2005.
28. I. Podlubny. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, 1999a.
29. μ-controllers. IEEE Trans. Automatic Control, 44(1):208-214, 1999b.
30. H.F. Raynaud and A. Zergaïnoh. State-space representation for fractional order controllers. Automatica, 36:1017-1021, 2000.
31. W. J. Rugh. Linear systems theory. Prentice-Hall, Englewood Cliffs, NJ., 1993.
32. S. B. Skaar, A. N. Michel, and R. K. Miller. Stability of viscoelastic control systems. IEEE Transactions on Automatic Control, 33 (4):348-357, April 1988.
33. Mathukumalli Vidyasagar. A characterization of eAt and a constructive proof of the controllability criterion. IEEE Trans. Automatic Control, 60(2):370-371, 1971.
34. Blas M. Vinagre and YangQuan Chen. Lecture notes on fractional calculus applications in automatic control and robotics. In Blas M. Vinagre and YangQuan Chen, editors, The 41st IEEE CDC2002 Tutorial Workshop # 2, pages 1-310. [Online] http://mechatronics.ece.usu.edu/foc/cdc02 tw2 ln.pdf, Las Vegas, Nevada, USA, 2002.
35. Dingyü Xue and YangQuan Chen. A comparative introduction of four fractional order controllers. In Proc. of The 4th IEEE World Congress on Intelligent Control and Automation (WCICA02), pages 3228-3235, Shanghai, China, June 2002. IEEE.