Abstract
Many systems appearing in practice exhibit a hybrid nature, that is, a coupling between continuous dynamics and discrete events. Special cases of such systems are those in which the linear subsystems are switched according to time or according to state and/or input. In the paper a method for approximation of some kinds of switched fractional linear systems is proposed and their stability is analyzed.
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References
1. Busłowicz M.: Robust stability of positive discrete-time linear systems of fractional order. Bulletin of Polish Academy of Sciences, Technical Sciences, Vol. 58, No. 4, pp. 567–572, 2010.
2. Busłowicz M.: Stability of state-space models of linear continuous-time fractional order systems. Acta Mechanica et Automatica, Vol. 5, No. 2, pp. 15–22, 2011.
3. Busłowicz M., Ruszewski A.: Robust stability check of fractional discrete-time linear systems with interval uncertainties. In: Latawiec K., Łukaniszyn M., Stanisławski R. (eds.): Advances in Modelling and Control of Non-integer Order Systems. LN in Electrical Engineering 320, Springer, pp. 199–208, 2014.
4. Chen Y. Q., Ahnand H.S., Podlubny I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Processing, Vol. 86, No. 1, pp. 2611–2618, 2006.
5. Domek S.: Piecewise Affine Representation of Discrete in Time, Non-integer Order Systems. In: Mitkowski W., Kacprzyk J., Baranowski J. (eds.): Advances in the Theory and Applications of Non-integer Order Systems. LN in Electrical Engineering 257, Springer, pp. 149–160, 2013.
6. Domek S.: Fractional-order differential calculus in model predictive control. West Pomeranian University of Technology Academic Press, Szczecin, 2013.
7. Domek S.: Switched state model predictive control of fractional-order nonlinear discrete-time systems. In: Pisano A., Caponetto R. (eds.): Advances in Fractional Order Control and Estimation, Asian J. Control, Special Issue, Vol. 15, No. 3, pp. 658–668, 2013.
8. Dzieliński A., Sierociuk D.: Stability of discrete fractional order state-space systems. Journal of Vibration and Control, Vol. 14, No. 9–10, pp. 1543–1556, 2008.
9. Fang L., Lin H., Antsaklis P. J.: Stabilization and performance analysis for a class of switched systems. In: Proc. 43rd IEEE Conf. Decision Control, Atlantis, pp. 1179–1180, 2004.
10. Geyer T., Torrisi F., Morari M.: Optimal complexity reduction of polyhedral piecewise affine systems. Automatica, Vol. 44, No. 7, pp. 1728–1740, 2008.
11. Kaczorek T.: New stability tests of positive standard and fractional linear systems. Circuits and Systems, Vol. 2, No. 4, pp. 261–268, 2011.
12. Kaczorek T.: Selected Problems of Fractional Systems Theory. Springer, Berlin, 2011.
13. Lin H., Antsaklis P. J.: Switching Stabilization and L2 Gain Performance Controller Synthesis for Discrete-Time Switched Linear Systems. Proc. 45th IEEE Conference on Decision & Control, San Diego, CA, USA, pp. 2673–2678, 2006.
14. Lin H., Antsaklis P. J.: Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Trans. on Automatic Control, Vol. 54, No. 2, pp. 308–322, 2009.
15. Macias M., Sierociuk D.: An alternative recursive fractional variable-order derivative definition and its analog validation., Proc. Inter. Conf. on Fractional Differentiation and its Applications (ICFDA), Catania, pp. 1–6, 2014.
16. Mäkilä P. M., Partington J. R.: On linear models for nonlinear systems. Automatica, Vol. 39, pp. 1–13, 2003.
17. Monje C. A., Chen Y. Q., Vinagre B. M., Xue D., Feliu V.: Fractional order systems and controls. Springer-Verlag, London, 2010.
18. Moze M., Sabatier J., and Oustaloup A.: LMI characterization of fractional systems stability. In: Sabatier J., Agrawal O. P., Tenreiro Machado J. A. (eds.): Advances in Fractional Calculus: Theoretical developments and applications in physics and engineering, Springer, pp. 419–434, 2007.
19. Ostalczyk P.: The non-integer difference of the discrete-time function and its application to the control system synthesis. Int. J. Syst. Sci., vol. 31, no. 12, pp. 1551–1561, 2000.
20. Petrás I.: Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab. In: Assi A. (ed.): Engineering Education and Research Using MATLAB, InTech, Rijeka, Shanghai 2011.
21. Podlubny I.: Fractional Differential Equations, San Diego, Academic Press, 1999.
22. Shevitz D., Paden B.: Lapunov stability theory of nonsmooth systems. IEEE Trans. on Automatic Control, Vol. 39, No. 9, pp. 1910–1914, 1994.
23. Stanisławski R.: Advances in modeling of fractional difference systems − new accuracy, stability and computational results. Oficyna Wydawnicza Politechniki Opolskiej, Opole, 2013.
24. Stanisławski R., Latawiec K. L., Łukaniszyn M., Gałek M.: Time-domain approximations to the Grünwald-Letnikov difference with application to modeling of fractional-order state space systems. Proc. 20th Int. Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 579–584, 2015.
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Domek, S. (2017). Approximation and stability analysis of some kinds of switched fractional linear systems. In: Mitkowski, W., Kacprzyk, J., Oprzędkiewicz, K., Skruch, P. (eds) Trends in Advanced Intelligent Control, Optimization and Automation. KKA 2017. Advances in Intelligent Systems and Computing, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-60699-6_43
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DOI: https://doi.org/10.1007/978-3-319-60699-6_43
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