# American Institute of Mathematical Sciences

April  2021, 8(2): 153-163. doi: 10.3934/jcd.2021007

## Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm

 Faculty of Electrical Engineering, Bialystok University of Technology, Wiejska 45D, 15-351 Biaƚystok, Poland

* Corresponding author: Andrzej Ruszewski

Received  October 2020 Revised  December 2020 Published  April 2021 Early access  March 2021

Fund Project: This work was supported by National Science Centre in Poland under Grant No. 2017/27/B/ST7/02443

The shuffle algorithm is applied to analysis of the fractional descriptor discrete-time linear systems. Using the shuffle algorithm the singularity of the fractional descriptor linear system is eliminated and the system is decomposed into dynamic and static parts. Procedures for computation of the solution and dynamic and static parts of the system are proposed. Sufficient conditions for the positivity of the fractional descriptor discrete-time linear systems are established.

Citation: Tadeusz Kaczorek, Andrzej Ruszewski. Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm. Journal of Computational Dynamics, 2021, 8 (2) : 153-163. doi: 10.3934/jcd.2021007
##### References:
 [1] R. Bru, C. Coll and E. Sanchez, Structural properties of positive linear time-invariant difference-algebraic equations, Linear Algebra Appl., 349 (2002), 1-10.  doi: 10.1016/S0024-3795(02)00277-X. [2] R. Bru, C. Coll, S. Romero-Vivo and E. Sáanchez, Some problems about structural properties of positive descriptor systems, Positive Systems. Lecture Notes in Control and Information Science, Springer, Berlin, 294 (2003), 233–240. doi: 10.1007/978-3-540-44928-7_32. [3] M. Busƚowicz and T. Kaczorek, Simple conditions for practical stability of positive fractional discrete-time linear systems, Int. J. Appl. Math. Comput. Sci., 19 (2009), 263-269.  doi: 10.2478/v10006-009-0022-6. [4] L. Dai, Singular Control Systems, Lectures Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1989. doi: 10.1007/BFb0002475. [5] J. D. Dixon, J. C. Poland, I. S. Pressman and L. Ribes, The shuffle algorithm and Jordan blocks, Linear Algebra Appl., 142 (1990), 159-165.  doi: 10.1016/0024-3795(90)90264-D. [6] M. Dodig and M. Stošić, Singular systems, state feedbacks problems, Linear Algebra Appl., 431 (2009), 1267-1292.  doi: 10.1016/j.laa.2009.04.024. [7] G.-R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0. [8] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000. doi: 10.1002/9781118033029. [9] R. K. H. Galvão, K. H. Kienitz and S. Hadjiloucas, Conversion of descriptor representations to state-space form: An extension of the shuffle algorithm, Internat. J. Control, 91 (2018), 2199-2213.  doi: 10.1080/00207179.2017.1336671. [10] R. Herrmann, Fractional Calculus; An Introduction for Physicists, World Scientific Publishing, Singapore, 2018. doi: 10.1142/11107. [11] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, London, 2002. doi: 10.1007/978-1-4471-0221-2. [12] T. Kaczorek, Selected Problems of Fractional System Theory, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6. [13] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Springer, New York, 2015. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [15] V. Kučera and P. Zagalak, Fundamental theorem of state feedback for singular systems, Automatica J. IFAC, 24 (1988), 653-658.  doi: 10.1016/0005-1098(88)90112-4. [16] F. L. Lewis, A survey of linear singular systems, Circuits Systems Signal Process., 5 (1986), 3-36.  doi: 10.1007/BF01600184. [17] D. G. Luenberger, Time-invariant descriptor systems, Automatica, 14 (1978), 473-480.  doi: 10.1016/0005-1098(78)90006-7. [18] D. G. Luenberger, Dynamic equations in descriptor form, IEEE Trans. Autom. Contr., 22 (1977), 312-321.  doi: 10.1109/tac.1977.1101502. [19] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, J. Wiley, New York, 1993. [20] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [21] P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing; Series in Computer Vision, World Scientific Publishing, Hackensack, New York, 2016. doi: 10.1142/9833. [22] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. [23] J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, London, 2007. doi: 10.1007/978-1-4020-6042-7. [24] Ƚ. Sajewski, Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller, Bull. Pol. Acad. Sci. Tech., 65 (2017), 709-714. [25] HG. Sun, Y. Zhang, D. Baleanu, W. Chen and YQ. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul, 64 (2018), 213-231. [26] E. Virnik, Stability analysis of positive descriptor systems, Linear Algebra Appl., 429 (2008), 2640-2659.  doi: 10.1016/j.laa.2008.03.002. [27] B. J. West, Nature's Patterns and the Fractional Calculus; Fractional Calculus in Applied Sciences and Engineering, De Gruyter, Berlin, 2017. doi: 10.1515/9783110535136.

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##### References:
 [1] R. Bru, C. Coll and E. Sanchez, Structural properties of positive linear time-invariant difference-algebraic equations, Linear Algebra Appl., 349 (2002), 1-10.  doi: 10.1016/S0024-3795(02)00277-X. [2] R. Bru, C. Coll, S. Romero-Vivo and E. Sáanchez, Some problems about structural properties of positive descriptor systems, Positive Systems. Lecture Notes in Control and Information Science, Springer, Berlin, 294 (2003), 233–240. doi: 10.1007/978-3-540-44928-7_32. [3] M. Busƚowicz and T. Kaczorek, Simple conditions for practical stability of positive fractional discrete-time linear systems, Int. J. Appl. Math. Comput. Sci., 19 (2009), 263-269.  doi: 10.2478/v10006-009-0022-6. [4] L. Dai, Singular Control Systems, Lectures Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1989. doi: 10.1007/BFb0002475. [5] J. D. Dixon, J. C. Poland, I. S. Pressman and L. Ribes, The shuffle algorithm and Jordan blocks, Linear Algebra Appl., 142 (1990), 159-165.  doi: 10.1016/0024-3795(90)90264-D. [6] M. Dodig and M. Stošić, Singular systems, state feedbacks problems, Linear Algebra Appl., 431 (2009), 1267-1292.  doi: 10.1016/j.laa.2009.04.024. [7] G.-R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0. [8] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000. doi: 10.1002/9781118033029. [9] R. K. H. Galvão, K. H. Kienitz and S. Hadjiloucas, Conversion of descriptor representations to state-space form: An extension of the shuffle algorithm, Internat. J. Control, 91 (2018), 2199-2213.  doi: 10.1080/00207179.2017.1336671. [10] R. Herrmann, Fractional Calculus; An Introduction for Physicists, World Scientific Publishing, Singapore, 2018. doi: 10.1142/11107. [11] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, London, 2002. doi: 10.1007/978-1-4471-0221-2. [12] T. Kaczorek, Selected Problems of Fractional System Theory, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6. [13] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Springer, New York, 2015. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [15] V. Kučera and P. Zagalak, Fundamental theorem of state feedback for singular systems, Automatica J. IFAC, 24 (1988), 653-658.  doi: 10.1016/0005-1098(88)90112-4. [16] F. L. Lewis, A survey of linear singular systems, Circuits Systems Signal Process., 5 (1986), 3-36.  doi: 10.1007/BF01600184. [17] D. G. Luenberger, Time-invariant descriptor systems, Automatica, 14 (1978), 473-480.  doi: 10.1016/0005-1098(78)90006-7. [18] D. G. Luenberger, Dynamic equations in descriptor form, IEEE Trans. Autom. Contr., 22 (1977), 312-321.  doi: 10.1109/tac.1977.1101502. [19] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, J. Wiley, New York, 1993. [20] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [21] P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing; Series in Computer Vision, World Scientific Publishing, Hackensack, New York, 2016. doi: 10.1142/9833. [22] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. [23] J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, London, 2007. doi: 10.1007/978-1-4020-6042-7. [24] Ƚ. Sajewski, Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller, Bull. Pol. Acad. Sci. Tech., 65 (2017), 709-714. [25] HG. Sun, Y. Zhang, D. Baleanu, W. Chen and YQ. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul, 64 (2018), 213-231. [26] E. Virnik, Stability analysis of positive descriptor systems, Linear Algebra Appl., 429 (2008), 2640-2659.  doi: 10.1016/j.laa.2008.03.002. [27] B. J. West, Nature's Patterns and the Fractional Calculus; Fractional Calculus in Applied Sciences and Engineering, De Gruyter, Berlin, 2017. doi: 10.1515/9783110535136.
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