Simple conditions for practical stability of positive fractional discrete-time linear systems

Busłowicz, M. ; Kaczorek, T.

International Journal of Applied Mathematics and Computer Science, Tome 19 (2009) no. 2, pp. 263-269.

Résumé

In the paper the problem of practical stability of linear positive discrete-time systems of fractional order is addressed. New simple necessary and sufficient conditions for practical stability and for practical stability independent of the length of practical implementation are established. It is shown that practical stability of the system is equivalent to asymptotic stability of the corresponding standard positive discrete-time systems of the same order. The discussion is illustrated with numerical examples.

Détail

Keywords: linear system, positive, discrete-time, fractional, stability, practical stability
Mots-clés : system liniowy, system dyskretno-czasowy, system ułamkowy, stateczność, stateczność praktyczna

Bibliographie

[1] Busłowicz, M. (2008a). Stability of linear continuous-time fractional systems of commensurate order, Pomiary, Automatyka, Robotyka: 475-484, (on CD-ROM, in Polish); Journal of Automation, Mobile Robotics and Intelligent Systems 3(1): 16--21.

[2] Busłowicz, M. (2008b). Frequency domain method for stabilitym analysis of linear continuous-time fractional systems, in K. Malinowski and L. Rutkowski (Eds.), Recent Advances in Control and Automation, Academic Publishing House EXIT, Warsaw, pp. 83-92.

[3] Busłowicz, M. (2008c). Robust stability of convex combination of two fractional degree characteristic polynomials, Acta Mechanica et Automatica 2(2): 5-10.

[4] Busłowicz, M. (2008d). Practical robust stability of positive fractional scalar discrete-time systems, Zeszyty Naukowe Politechniki Śląskiej: Automatyka 151: 25-30, (in Polish).

[5] Chen, Y.-Q., Ahn, H.-S. and Podlubny, I. (2006). Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal Processing 86(10): 2611-2618.

[6] Das, S. (2008). Functional Fractional Calculus for System Identification and Controls, Springer, Berlin.

[7] Dzieliński, A. and Sierociuk, D. (2006). Stability of discrete fractional state-space systems, Proceedings of the 2-nd IFAC Workshop on Fractional Differentiation and Its Applications, IFAC FDA'06, Porto, Portugal, pp. 518-523.

[8] Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, New York, NY.

[9] Gałkowski K. and Kummert, A. (2005). Fractional polynomials and nD systems, Proceedings of the IEEE International Symposium on Circuits and Systems, ISCAS'2005, Kobe, Japan, (on CD-ROM).

[10] Gałkowski, K., Bachelier, O. and Kummert, A. (2006). Fractional polynomial and nD systems-A continuous case, Proceedings ot the IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 2913-2917.

[11] Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer-Verlag, London.

[12] Kaczorek, T. (2007a). Reachability and controllability to zero of positive fractional discrete-time systems, Machine Intelligence and Robotic Control 6(4): 139-143.

[13] Kaczorek, T. (2007b). Reachability and controllability to zero of cone fractional linear systems, Archives of Control Sciences 17(3): 357-367.

[14] Kaczorek, T. (2007c). Choice of the forms of Lyapunov functions for positive 2D Roesser model, International Journal of Applied Mathematics and Computer Science 17(4): 471-475.

[15] Kaczorek, T. (2008a). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science 18(2): 223-228.

[16] Kaczorek, T. (2008b). Reachability and controllability to zero tests for standard and positive fractional discrete-time systems, Journal of Automation and System Engineering 42(6-7-8): 769-787.

[17] Kaczorek, T. (2008c). Fractional 2D linear systems, Journal of Automation, Mobile Robotics and Intelligent Systems 2(2): 5-9.

[18] Kaczorek, T. (2008d). Positive different orders fractional 2D linear systems, Acta Mechanica et Automatica 2(2): 51-58.

[19] Kaczorek, T. (2008e). Practical stability of positive fractional discrete-time systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 313-317.

[20] Kilbas, A. A., Srivastava, H.M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

[21] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.

[22] Sabatier, J., Agrawal, O. P. and Machado, J. A. T. (Eds.) (2007). Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, London.

[23] Sierociuk, D. (2007). Estimation and Control of Discrete Dynamical Systems of Fractional Order in State Space, Ph.D. thesis, Faculty of Electrical Engineering, Warsaw University of Technology, Warsaw, (in Polish).

[24] Vinagre, B. M., Monje, C. A. and Calderon, A. J. (2002). Fractional order systems and fractional order control actions, Proceedings of the IEEE CDC Conference Tutorial Workshop: Fractional Calculus Applications in Automatic Control and Robotics, Las Vegas, NY, pp. 15-38.