Abstract
In the paper the stability problems of fractional discrete-time linear scalar systems with pure delay are considered. Using the classical D-partition method, the necessary and sufficient condition for practical and asymptotic stability are given. The considerations are illustrated by numerical examples.
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Busłowicz, M.: Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix. Bull. Pol. Acad. Sci. Tech. Sci. 60, 809–814 (2012)
Busłowicz, M.: Stability conditions for linear continuous-time fractional-order state-delayed systems. Bull. Pol. Acad. Sci. Tech. Sci. 64, 3–7 (2016)
Busłowicz, M., Kaczorek, T.: Simple conditions for practical stability of linear positive fractional discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 19, 263–269 (2009)
Busłowicz, M., Ruszewski, A.: Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems. Bull. Pol. Acad. Sci. Tech. Sci. 61, 779–786 (2013)
Dzieliński, A., Sierociuk, D.: Stability of discrete fractional state-space systems. J. Vibr. Control 14, 1543–1556 (2008)
Gryazina, E.N., Polyak, B.T., Tremba, A.A.: D-decomposition technique state-of-the-art. Autom. Remote Control 69(12), 1991–2026 (2008)
Kaczorek, T.: Practical stability of positive fractional discrete-time systems. Bull. Pol. Acad. Sci. Tech. Sci. 56, 313–317 (2008)
Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)
Kaczorek, T.: A new approach to the realization problem for fractional discrete-time linear systems. Bull. Pol. Acad. Sci. 64, 9–14 (2016)
Kaczorek, T., Ostalczyk, P.: Responses comparison of the two discrete-time linear fractional state-space models. Fractional Calc. Appl. Anal. 19, 789–805 (2016)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-Order Systems and Controls. Springer, London (2010)
Ostalczyk, P.: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. Int. J. Appl. Math. Comput. Sci. 22, 533–538 (2012)
Ostalczyk, P.: Discrete Fractional Calculus: Applications in Control and Image Processing. Series in Computer Vision. World Scientific Publishing, Singapore (2016)
Ruszewski, R.: Stability conditions of fractional discrete-time scalar systems with pure delay. Pomiary Automatyka Robotyka 17, 340–344 (2013)
Ruszewski, R.: Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model. Arch. Control Sci. 17, 340–344 (2016)
Ruszewski, R.: Stability analysis for the new model of fractional discrete-time linear state-space systems. In: Babiarz, A. et al. (eds.) Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 407, pp. 381–389. Springer, Heidelberg (2017)
Ruszewski, R., Busłowicz, M.: Practical and asymptotic stability of fractional discrete-time scalar systems with multiple delays. In: Malinowski, K., et al. (eds.) Recent Advances in Control and Automation, pp. 183–192. Academic Publishing House Exit, Warsaw (2014)
Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds.): Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, London (2007)
Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems, Part I: New necessary and sufficient conditions for asymptotic stability. Bull. Pol. Acad. Sci. 61, 353–361 (2013)
Stanisławski, R.: New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions. Fractional Calc. Appl. Anal. 20, 243–259 (2017)
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This work was supported by the National Science Centre in Poland under the work No. 2014/13/B/ST7/03467.
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Ruszewski, A. (2019). Stability Analysis of Fractional Discrete-Time Linear Scalar Systems with Pure Delay. In: Ostalczyk, P., Sankowski, D., Nowakowski, J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham. https://doi.org/10.1007/978-3-319-78458-8_8
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DOI: https://doi.org/10.1007/978-3-319-78458-8_8
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