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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

Tadeusz Kaczorek and  Andrzej Ruszewski ,

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.

Keywords: Fractional, descriptor, discrete-time, system, solution, decomposition, dynamic, static part.
Mathematics Subject Classification: Primary: 93C05; Secondary: 93C55, 26A33.
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.  Google Scholar

[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.  Google Scholar

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L. Dai, Singular Control Systems, Lectures Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1989. doi: 10.1007/BFb0002475.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G.-R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0.  Google Scholar

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L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000. doi: 10.1002/9781118033029.  Google Scholar

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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.  Google Scholar

[10]

R. Herrmann, Fractional Calculus; An Introduction for Physicists, World Scientific Publishing, Singapore, 2018. doi: 10.1142/11107.  Google Scholar

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T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, London, 2002. doi: 10.1007/978-1-4471-0221-2.  Google Scholar

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T. Kaczorek, Selected Problems of Fractional System Theory, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6.  Google Scholar

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T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Springer, New York, 2015.  Google Scholar

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A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[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.  Google Scholar

[16]

F. L. Lewis, A survey of linear singular systems, Circuits Systems Signal Process., 5 (1986), 3-36.  doi: 10.1007/BF01600184.  Google Scholar

[17]

D. G. Luenberger, Time-invariant descriptor systems, Automatica, 14 (1978), 473-480.  doi: 10.1016/0005-1098(78)90006-7.  Google Scholar

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D. G. Luenberger, Dynamic equations in descriptor form, IEEE Trans. Autom. Contr., 22 (1977), 312-321.  doi: 10.1109/tac.1977.1101502.  Google Scholar

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K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, J. Wiley, New York, 1993. Google Scholar

[20] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.   Google Scholar
[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.  Google Scholar

[22] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[26]

E. Virnik, Stability analysis of positive descriptor systems, Linear Algebra Appl., 429 (2008), 2640-2659.  doi: 10.1016/j.laa.2008.03.002.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Dai, Singular Control Systems, Lectures Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1989. doi: 10.1007/BFb0002475.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G.-R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0.  Google Scholar

[8]

L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000. doi: 10.1002/9781118033029.  Google Scholar

[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.  Google Scholar

[10]

R. Herrmann, Fractional Calculus; An Introduction for Physicists, World Scientific Publishing, Singapore, 2018. doi: 10.1142/11107.  Google Scholar

[11]

T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, London, 2002. doi: 10.1007/978-1-4471-0221-2.  Google Scholar

[12]

T. Kaczorek, Selected Problems of Fractional System Theory, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6.  Google Scholar

[13]

T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Springer, New York, 2015.  Google Scholar

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[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.  Google Scholar

[16]

F. L. Lewis, A survey of linear singular systems, Circuits Systems Signal Process., 5 (1986), 3-36.  doi: 10.1007/BF01600184.  Google Scholar

[17]

D. G. Luenberger, Time-invariant descriptor systems, Automatica, 14 (1978), 473-480.  doi: 10.1016/0005-1098(78)90006-7.  Google Scholar

[18]

D. G. Luenberger, Dynamic equations in descriptor form, IEEE Trans. Autom. Contr., 22 (1977), 312-321.  doi: 10.1109/tac.1977.1101502.  Google Scholar

[19]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, J. Wiley, New York, 1993. Google Scholar

[20] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.   Google Scholar
[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.  Google Scholar

[22] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[26]

E. Virnik, Stability analysis of positive descriptor systems, Linear Algebra Appl., 429 (2008), 2640-2659.  doi: 10.1016/j.laa.2008.03.002.  Google Scholar

[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.  Google Scholar

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