Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for the asymptotic stability

• Autorzy: Stanisławski R. , Latawiec K. J.
• Języki publikacji: EN

Abstrakty

EN

This paper presents a series of new results on the asymptotic stability of discrete-time fractional difference (FD) state space systems and their finite-memory approximations called finite FD (FFD) and normalized FFD (NFFD) systems. In Part I, new, general, necessary and sufficient stability conditions are introduced in a unified form for FD/FFD/NFFD-based systems. In Part II, an original, simple, analytical stability criterion is offered for FD-based systems, and the result is used to develop simple, efficient, numerical procedures for testing the asymptotic stability for FFD-based and, in particular, NFFD-based systems. Consequently, the so-called f-poles and f-zeros are introduced for FD-based system and their closed-loop stability implications are discussed.

Słowa kluczowe

EN

Grünwald-Letnikov fractional difference; discrete-time fractional order system; stability conditions; Cauchy’s argument principle

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2013
• Tom: Vol. 61, nr 2
• Strony: 353--361
• Opis fizyczny: Bibliogr. 39 poz., rys., wykr.

Twórcy

autor

Stanisławski R.

• Institute of Control and Computer Engineering, Opole University of Technology, 31 Sosnkowskiego St., 45-272 Opole, Poland

autor

Latawiec K. J.

• Institute of Control and Computer Engineering, Opole University of Technology, 31 Sosnkowskiego St., 45-272 Opole, Poland

Bibliografia

• [1] H. Delavari, A. Ranjbar, R. Ghaderi, and S. Momani, “Fractional order control of a coupled tank”, Nonlinear Dynamics 61 (3), 383-397 (2010).
• [2] I. Petr´aˇs and B. Vinagre, “Practical application of digital fractional-order controller to temperature control”, Acta MontanisticaSlovaca 7 (2), 131-137 (2002).
• [3] D. Riu, N. Reti´ere, and M. Ivanes, “Turbine generator modeling by non-integer order systems”, IEEE Int. Conf. on ElectricMachines and Drives, Cambridge 1, 185-187 (2001).
• [4] V. Zaborowsky and R. Meylaov, “Informational network traffic model based on fractional calculus”, Proc. Int. Conf. Info-techand Info-net (ICII’2001) 1, 58-63 (2001).
• [5] C.H. Lubich, “Discretized fractional calculus”, SIAM J. onMathematical Analysis 17 (3), 704-719 (1986).
• [6] M.D. Ortigueira, “Introduction to fractional linear systems ii: Discrete-time case”, IEE Proc. on Vision, Image and SignalProcessing 147 (1), 71-78 (2000).
• [7] P. Ostalczyk, “The non-integer difference of the discrete-time function and its application to the control system synthesis”, Int. J. Systems Science 31 (12), 1551-1561 (2000).
• [8] 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).
• [9] I. Petr´aˇs, L. Dorˇc´ak, and I. Koˇstial, “The modelling and analysis of fractional-order control systems in discrete domain”, Proc. Int. Carpatian Control Conf. 1, 257-260 (2000).
• [10] A. Dzieliński and D. Sierociuk, “Stability of discrete fractional order state-space systems”, J. Vibration and Control 14 (9-10), 1543-1556 (2008).
• [11] T. Kaczorek, “Practical stability of positive fractional discretetime linear systems”, Bull. Pol. Ac.: Tech. 56 (4), 313-317 (2008).
• [12] D. Sierociuk and A. Dzieliński, “Fractional Kalman filter algorithm for states, parameters and order of fractional system estimation”, Int. J. Applied Mathematics and Computer Science 16 (1), 101-112 (2006).
• [13] R. Stanisławski and K.J. Latawiec, “Normalized finite fractional differences: the computational and accuracy breakthroughs”, Int. J. Applied Mathematics and Computer Science 22 (4), 907-919 (2012).
• [14] M. Busłowicz and T. Kaczorek, “Simple conditions for practical stability of positive fractional discrete-time linear systems”, Int. J. Applied Mathematics and Computer Science 19 (2), 263-269 (2009).
• [15] C. Monje, Y. Chen, B. Vinagre, D. Xue, and V. Feliu, Fractional-order Systems and Controls, Springer-Verlag, London, 2010.
• [16] I. Podlubny, Fractional Differential Equations, Academic Press, Orlando, 1999.
• [17] R. Stanisławski, “Identification of open-loop stable linear systems using fractional orthonormal basis functions”, Proc. 14thInt. Conf. on Methods and Models in Automation and Robotics 1, 935-985 (2009).
• [18] R. Stanisławski, W. Hunek, and K.J. Latawiec, “Finite approximations of a discrete-time fractional derivative”, 16th Int. Conf.on Methods and Models in Automation and Robotics 1, 142-145 (2011).
• [19] R. Stanisławski and K.J. Latawiec, “Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions”, Proc. 15th Int. Conf. on Methods and Models in Automation and Robotics 1, 411-414 (2010).
• [20] I. Kheirzad, M.S. Tavazoei, and A.A. Jalali, “Stability criteria for a class of fractional order systems”, Nonlinear Dynamics 61 (1-2), 153-161 (2010).
• [21] C.-p. Li and Z.-g. Zhao, “Asymptotical stability analysis of linear fractional differential systems”, J. Shanghai University(English Edition) 13 (3), 197-206 (2009).
• [22] D. Matignon, “Stability results for fractional differential equations with applications to control processing”, ComputationalEngineering in Systems Applications 2, 963-968 (1996).
• [23] D. Matignon, “Stability properties for generalized fractional differential systems”, ESAIM: Proc. 5, 145-158 (1998).
• [24] I. Petrá˘s, “Stability of fractional-order systems with rational orders: A survey”, Fractional Calculus and Applied Analysis 12 (3), 269-298 (2009).
• [25] J. Sabatier, M. Moze, and C. Farges, “On stability of fractional order systems”, Third IFAC Workshop on Fractional Differentiationand Its Applications FDA’08 1, CD-ROM (2008).
• [26] S. Guermah, S. Djennoune, and M. Bettayeb, “A new approach for stability analysis of linear discrete-time fractional-order systems”, New Trends in Nanotechnology and Fractional CalculusApplications 1, 151-162 (2010).
• [27] S.B. Stojanovic and D.L. Debeljkovic, “Simple stability conditions of linear discrete time systems with multiple delay”, Serbian J. Electrical Engineering 7 (1), 69-79 (2010).
• [28] M. Busłowicz, “Robust stability of positive discrete-time linear systems of fractional order”, Bull. Pol. Ac.: Tech. 58 (4), 567–572 (2010).
• [29] T. Kaczorek, “Practical stability and asymptotic stability of positive fractional 2d linear systems”, Asian J. Control 12 (2), 200–207 (2010).
• [30] T. Kaczorek, “New stability tests of positive standard and fractional linear systems”, Circuits and Systems 2 (4), 261–268 (2011).
• [31] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 2011.
• [32] K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, New York, 1993.
• [33] K. Oldham and J. Spanier, The Fractional Calculus, Academic Press, Orlando, 1974.
• [34] K.J. Latawiec, R. Stanisławski, W.P. Hunek, and M. Łukaniszyn, “Adaptive finite fractional difference with a timevarying forgetting factor”, Proc. 17th Int. Conf. on Methods and Models in Automation and Robotics 1, 64–69 (2012).
• [35] R. Stanisławski, Advances in Modeling of Fractional Difference Systems – New Accuracy, Stability and Computational Results, Opole University of Technology Press, Opole, 2013, (to be published).
• [36] R. Stanisławski, “New Laguerre filter approximators to the Gr¨unwald-Letnikov fractional difference”, Mathematical Problems in Engineering 2012, 732917 (2012).
• [37] G. Shilov, Linear Algebra, ser. Dover Books on Advanced Mathematics, Dover Publications, Mineola, 1977.
• [38] B.L. Willis, “Eigenvalues by row operations”, University of Nebraska at Kearney, Karney, [Online]. Available: http://www.unk.edu/uploadedFiles/facstaff/profiles/willisb/eigens-by-row(1).pdf (2005).
• [39] R. Lopez, Advanced Engineering Mathematics, Addison Wesley Publishing Company, Boston, 2001.