Simple Conditions for Practical Stability of Positive Fractional Discrete-Time Linear Systems

Busłowicz, Mikołaj; Kaczorek, Tadeusz

doi:10.2478/v10006-009-0022-6

Abstract

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.

References

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.
Search in Google Scholar
Busłowicz, M. (2008b). Frequency domain method for stability 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.
Search in Google Scholar
Busłowicz, M. (2008c). Robust stability of convex combination of two fractional degree characteristic polynomials, Acta Mechanica et Automatica 2(2): 5-10.
Search in Google Scholar
Busłowicz, M. (2008d). Practical robust stability of positive fractional scalar discrete-time systems, Zeszyty Naukowe Politechniki Saląskiej: Automatyka 151: 25-30, (in Polish).
Search in Google Scholar
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): 2611-2618.10.1016/j.sigpro.2006.02.011
Search in Google Scholar
Das, S. (2008). Functional Fractional Calculus for System Identification and Controls, Springer, Berlin.
Search in Google Scholar
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.
Search in Google Scholar
Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, New York, NY.10.1002/9781118033029
Search in Google Scholar
Gałkowsk, i 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).
Search in Google Scholar
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.
Search in Google Scholar
Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer-Verlag, London.10.1007/978-1-4471-0221-2
Search in Google Scholar
Kaczorek, T. (2007a). Reachability and controllability to zero of positive fractional discrete-time systems, Machine Intelligence and Robotic Control 6(4): 139-143.10.23919/ECC.2007.7068247
Search in Google Scholar
Kaczorek, T. (2007b). Reachability and controllability to zero of cone fractional linear systems, Archives of Control Sciences 17(3): 357-367.10.23919/ECC.2007.7068247
Search in Google Scholar
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): 471-475.10.2478/v10006-007-0039-7
Search in Google Scholar
Kaczorek, T. (2008a). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science 18(2): 223-228.10.2478/v10006-008-0020-0
Search in Google Scholar
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): 769-787.10.3166/jesa.42.769-787
Search in Google Scholar
Kaczorek, T. (2008c). Fractional 2D linear systems, Journal of Automation, Mobile Robotics and Intelligent Systems 2(2): 5-9.
Search in Google Scholar
Kaczorek, T. (2008d). Positive different orders fractional 2D linear systems, Acta Mechanica et Automatica 2(2): 51-58.
Search in Google Scholar
Kaczorek, T. (2008e). Practical stability of positive fractional discrete-time systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 313-317.
Search in Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Search in Google Scholar
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.
Search in Google Scholar
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.10.1007/978-1-4020-6042-7
Search in Google Scholar
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).
Search in Google Scholar
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.
Search in Google Scholar

Simple Conditions for Practical Stability of Positive Fractional Discrete-Time Linear Systems

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