Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems

Bulletin of the Polish Academy of Sciences: Technical Sciences

Volumn 61, Issue 4, 2013, Pages 779-786

  Buslowicz M ^a   Ruszewski, A ^a  

^a BIALYSTOK UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

Asymptotic stability; Discrete time; Fractional; Linear system; Practical stability

Indexed keywords

ASYMPTOTIC STABILITY; DIGITAL CONTROL SYSTEMS; DISCRETE TIME CONTROL SYSTEMS; LINEAR SYSTEMS; MATHEMATICAL TECHNIQUES; MATRIX ALGEBRA;

CONTINUOUS-TIME LINEAR SYSTEMS; DISCRETE TIME LINEAR SYSTEMS; DISCRETE TIME SYSTEM; DISCRETE-TIME; FRACTIONAL; PARAMETRIC DESCRIPTION; PRACTICAL STABILITY; STATE SPACE SYSTEMS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84896320941     PISSN: 02397528     EISSN: None     Source Type: Journal    
DOI: 10.2478/bpasts-2013-0084     Document Type: Article

References (32)

1. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
2. C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, and V. Feliu-Batlle, Fractional-order Systems and Controls Fundamentals and Applications, Springer, London, 2010.
3. M.M. Meerschaert and C. Tadjeran, "Finite difference approximations for fractional advection-dispersion flow equations", J. Computational and Applied Mathematics 172 (1), 65-77 (2004).
4. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
5. L. Debnath, "Recent applications of fractional calculus to science and engineering", Int. J. Mathematics and Mathematical Sciences 54, 3413-3442 (2003).
6. A. Dzieliński, D. Sierociuk, and G. Sarwas, "Some applications of fractional order calculus", Bull. Pol. Ac.: Tech. 58 (4), 583-592 (2010).
7. P. Lino and G. Maione, "Loop-shaping and easy tuning of fractional-order proportional integral controllers for position servo systems", Asian J. Control 15 (3), 796-805 (2012).
8. Y. Luo and Y-Q. Chen, Fractional Order Motion Controls, John Wiley & Sons Ltd, Chichester, 2013.
9. Y-H. Chang, C-I. Wu, H-W. Lin, H-C. Chen, and C-W. Chang, "Fractional order integral sliding-mode flux observer for direct field-oriented induction machines", Int. J. Innovative Computing, Information and Control 8 (7A), 4851-4198, (2012).
10. R. Melicio, V.M.F. Mendes, and J.P.S. Catalao, "Power converter topologies and fractional-order controllers: Wind energy applications", Proc. Int. Conf. on Power Electronics, Electrical Drives and Motion 1, 1334-1338 (2010).
11. D. Matignon, "Stability result on fractional differential equations with applications to control processing", Proc. IMACSSMC 2, 963-968 (1996).
12. I. Petras, "Stability of fractional-order systems with rational orders: a survey", Fractional Calculus & Applied Analysis. Int. J. Theory and Applications 12 (3), 269-298 (2009).
13. M.S. Tavazoei and M. Haeri, "Note on the stability of fractional order systems", Mathematics and Computers in Simulation 79 (5), 1566-1576 (2009).
14. M. Busłowicz, "Stability of linear continuous-time fractional order systems with delays of the retarded type", Bull. Pol. Ac.: Tech. 56 (4), 319-324 (2008).
15. M. Busłowicz, "Stability analysis of linear continuous-time fractional systems of commensurate order", J. Automation, Mobile Robotics and Intelligent Systems 3 (1), 16-21 (2009).
16. M. Busłowicz, "Stability of state-space models of linear continuous-time fractional order systems", Acta Mechanica et Automatica 5 (2), 15-22 (2011).
17. M. Busłowicz, "Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders", Bull. Pol. Ac.: Tech. 60 (2), 279-284 (2012).
18. J. Sabatier, M. Moze, and C. Farges, "LMI stability conditions for fractional order systems", Computers and Mathematics with Applications 59 (5), 1594-1609 (2010).
19. A. Dzieliński and D. Sierociuk, "Stability of discrete fractional order state-space systems", J. Vibration and Control 14, 1543-1556 (2008).
20. S. Guermah, S. Djennoune, and M. Bettayeb, "A new approach for stability analysis of linear discrete-time fractional-order systems", in New Trends in Nanotechnology and Fractional Calculus Applications, ed. D Baleanu, pp. 151-162, Springer, Berlin, 2010.
21. R. Stanisławski, W.P. Hunek, and K.J. Latawiec, "Finite approximations of a discrete-time fractional derivative", Proc. 16th Int. Conf. Methods and Models in Automation and Robotics 1, 142-145 (2011).
22. M. Busłowicz, "Practical stability of scalar discrete-time linear systems of fractional order", in Automation of Discrete Processes: Theory and Applications, eds. A. Świerniak and J. Krystek, vol. 1, pp. 31-40, Printing House of Silesian University of Technology, Gliwice, 2012, (in Polish).
23. M. Busłowicz, "Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix", Bull. Pol. Ac.: Tech. 60 (4), 809-814 (2012).
24. P. Ostalczyk, "Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains", Int. J. Applied Mathematics and Computer Science 22 (3), 533-538 (2012).
25. M. Busłowicz and T. Kaczorek, "Simple conditions for practical stability of linear positive fractional discrete-time linear systems", Int. J. Applied Mathematics and Computer Science 19 (2), 263-269 (2009).
26. T. Kaczorek, "Practical stability of positive fractional discretetime systems", Bull. Pol. Ac.: Tech. 56 (4), 313-317 (2008).
27. T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin, 2011.
28. T. Kaczorek, "Practical stability and asymptotic stability of positive fractional 2D linear systems", Asian J. Control 12 (2), 200-207 (2010).
29. E.N. Gryazina, B.T. Polyak, and A.A. Tremba, "Ddecomposition technique state-of-The-art", Automation and Remote Control 69 (12), 1991-2026 (2008).
30. J. Ackermann, Sampled-data Control Systems, Springer, Berlin, 1985.
31. R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, Singapore 2000.
32. R. Stanisławski and K. J. Latawiec, "Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for asymptotic stability", Bull. Pol. Ac.: Tech., 61 (2), 353-361 (2013).