Stability conditions for linear continuous-time fractional-order state-delayed systems

• Autorzy: Busłowicz M.

Identyfikatory

DOI

10.1515/bpasts-2016-0001

• Języki publikacji: EN

Abstrakty

EN

The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.

Słowa kluczowe

EN

linear system; fractional; continuous-time; state delay; asymptotic stability

PL

systemy liniowe; stabilność asymptotyczna; opóźnienie czasowe

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2016
• Tom: Vol. 64, nr 1
• Strony: 3--7
• Opis fizyczny: Bibliogr. 24 poz., rys., wykr.

Twórcy

autor

Busłowicz M.

• Faculty of Electrical Engineering, Bialystok University of Technology, 45D Wiejska St., 15-351 Bialystok, Poland

Bibliografia

• [1] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2008.
• [2] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin, 2011.
• [3] Y. Luo and Y-Q. Chen, Fractional Order Motion Controls, John Wiley & Sons Ltd, Chichester, 2013.
• [4] 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.
• [5] D. Matignon, “Stability result on fractional differential equations with applications to control processing”, IMACS-SMC Proc. 1, 963-968 (1996).
• [6] M.S. Tavazoei and M. Haeri, “Note on the stability of fractional order systems”, Mathematics and Computers in Simulation 79, 1566-1576 (2009).
• [7] M. Busłowicz, “Stability of state-space models of linear continuous-time fractional order systems”, Acta Mechanica et Automatica 5, 15-22 (2011).
• [8] 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).
• [9] R. Stanisławski and K.J. Latawiec, “Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems”, Bull. Pol. Ac.: Tech. 61 (2), 363-370 (2013).
• [10] M. Busłowicz and A. Ruszewski, “Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems”, Bull. Pol. Ac.: Tech. 61 (4), 353-361 (2013).
• [11] T. Kaczorek, “Practical stability of positive fractional discretetime systems”, Bull. Pol. Ac.: Tech. 56 (4), 313-317 (2008).
• [12] T. Kaczorek, “New stability tests of positive standard and fractional linear systems”, Circuits and Systems 2, 261-268 (2011).
• [13] M. Busłowicz, “Stability of linear continuous-time fractional order systems with delays of the retarded type”, Bull. Pol. Ac.: Tech. 56, 319-324 (2008).
• [14] M. Shi and Z. Wang, “An effective analytical criterion for stability testing of fractional-delay systems”, Automatica 47, 2001-2005 (2011).
• [15] A.R. Fioravanti, C. Bonnet, H. Özbay, and S. Niculescu, “A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems”, Automatica 48, 2824-2830 (2012).
• [16] A. Mesbahi and M. Haeri, “Stability of linear time invariant fractional delay systems of retarded type in the space of delay parameters”, Automatica 49, 1287-1294 (2013).
• [17] E. Kaslik and S. Sivasundaram, “Analytical and numerical methods for the stability analysis of linear fractional delay differential equations”, J. Computational and Applied Mathematics 236, 4027-4041 (2012).
• [18] H. Li, S.M. Zhong, and H.B. Li, “Stability analysis of fractional order systems with time delay”, Int. J. Mathematical, Computational Science and Engineering 8 (4), 400-403 (2014).
• [19] H. Zhang, D. Wu, J. Cao, and H. Zhang, “Stability analysis for fractional-order linear singular delay differential systems”, Hindawi Publishing Corporation, Discrete Dynamics in Nature and Society ID 850279, http://dx.doi.org/10.1155/2014/850279 (2014).
• [20] M. Busłowicz, “Simple stability criterion for a class of delay differential system”, Int. J. Systems Science 18 (5), 993-995 (1987).
• [21] M. Barszcz and A.W. Olbrot, “Stability criterion for a linear differential-difference system”, IEEE Trans. Autom. Control AC-24 (2), 368-369 (1979).
• [22] T. Mori and E. Noldus, “Stability criteria for linear differential difference systems”, Int. J. Systems Science 15 (1), 87-94 (1984).
• [23] 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).
• [24] T. Kaczorek, Theory of Control, PWN, Warsaw, 1977 (in Polish).

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.