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Digraphs Minimal Realisations of State Matrices for Fractional Positive Systems

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Part of the Advances in Intelligent Systems and Computing book series (AISC,volume 350)

Abstract

This paper presents a method of the determination of characteristic polynomial realisations of the fractional positive system. The algorithm finds a complete set of all possible realisations instead of only a few realisations. In addition, all realisations in the set are minimal. The proposed method uses a parallel computing algorithm based on a digraphs theory which is used to gain much needed speed and computational power for a numeric solution. The presented procedure has been illustrated with a numerical example.

Keywords

  • fractional systems
  • positive
  • digraphs
  • algorithm

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  • DOI: 10.1007/978-3-319-15796-2_7
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References

  1. Benvenuti, L., Farina, L.: Positive and compartmental systems. IEEE Transactions on Automatic Control (47), 370–373 (2002)

    Google Scholar 

  2. Benvenuti, L., Farina, L.: A tutorial on the positive realization problem. IEEE Transactions on Automatic Control 49(5), 651–664 (2004)

    CrossRef  MathSciNet  Google Scholar 

  3. Berman, A., Neumann, M., Stern, R.J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York (1989)

    MATH  Google Scholar 

  4. Farina, L., Rinaldi, S.: Positive linear systems: theory and applications. Wiley-Interscience, Series on Pure and Applied Mathematics, New York (2000)

    CrossRef  Google Scholar 

  5. Fornasini, E., Valcher, M.E.: Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs. Linear Algebra and Its Applications 263, 275–310 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Fornasini, E., Valcher, M.E.: On the positive reachability of 2D positive systems. In: Benvenuti, L., De Santis, A., Farina, L. (eds.) Positive Systems. LNCIS, vol. 294, pp. 297–304. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

  7. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge Univ. Press (1991)

    Google Scholar 

  8. Hryniów, K., Markowski, K.A.: Parallel digraphs-building algorithm for polynomial realisations. In: Proceedings of 2014 15th International Carpathian Control Conference (ICCC), pp. 174–179 (2014), http://dx.doi.org/10.1109/CarpathianCC.2014.6843592

  9. Hryniów, K., Markowski, K.A.: Reachability index calculation by parallel digraphs-building. In: 19th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, September 2-5, pp. 808–813 (2014), http://dx.doi.org/10.1109/MMAR.2014.6957460

  10. Hryniów, K., Markowski, K.A.: Digraphs-building of complete set of minimal characteristic polynomial realisations as means for solving minimal realisation problem of nD systems. International Journal of Control (Submitted to)

    Google Scholar 

  11. Kaczorek, T.: Two-dimensional Linear Systems. Springer, London (1985)

    MATH  Google Scholar 

  12. Kaczorek, T.: Positive 1D and 2D systems. Springer, London (2001)

    Google Scholar 

  13. Kaczorek, T.: Positive realization for 2D systems with delays. In: Proceedings of 2007 International Workshop on Multidimensional (nD) Systems, pp. 137–141. IEEE (2007)

    Google Scholar 

  14. Kaczorek, T.: Realization problem for positive 2D hybrid systems. COMPEL 27(3), 613–623 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)

    CrossRef  MATH  Google Scholar 

  16. Luenberger, D.G.: Positive linear systems. In: Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley, New York (1979)

    Google Scholar 

  17. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differenctial Equations. Willeys, New York (1993)

    Google Scholar 

  18. Nishimoto, K.: Fractional Calculus. Decartess Press, Koriama (1984)

    MATH  Google Scholar 

  19. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

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Correspondence to Krzysztof Hryniów .

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Hryniów, K., Markowski, K.A. (2015). Digraphs Minimal Realisations of State Matrices for Fractional Positive Systems. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Progress in Automation, Robotics and Measuring Techniques. ICA 2015. Advances in Intelligent Systems and Computing, vol 350. Springer, Cham. https://doi.org/10.1007/978-3-319-15796-2_7

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  • DOI: https://doi.org/10.1007/978-3-319-15796-2_7

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15795-5

  • Online ISBN: 978-3-319-15796-2

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