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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Two Cases of Digraph Structures Corresponding to Minimal Positive Realisation of Fractional Continuous-Time Linear Systems of Commensurate Order

Journal of Applied Nonlinear Dynamics 6(2) (2017) 265--282 | DOI:10.5890/JAND.2017.06.011

Konrad Andrzej Markowski

Faculty of Electrical Engineering, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland

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Abstract

The positive and minimal realisation problem for fractional continuous-time linear single-input and single-output (SISO) systems is formulated. Method based on the one-dimensional digraph for finding a positive and minimal realisation of a given proper transfer function is proposed. Two special cases of the digraph structure are given. Sufficient conditions for the existence of a positive minimal realisation of a given proper transfer function systems are established. The algorithm for computation of a positive minimal realisation is proposed and illustrated with a numerical example. The algorithm is based on a parallel computing method to gain needed speed and computational power for such a solution.

References

  1. [1]  Luenberger, D.G. (1979), Introduction to Dynamic Systems: Theory, Models, and Applications, New York: Wiley, ch. Positive linear systems.
  2. [2]  Sontag, E.D. (2005), Molecular systems biology and control, European Journal of Control, 11(4-5), 396-435.
  3. [3]  Haddad, W.M. and Chellaboina, V.S. (2005), Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems, Nonlinear Analysis Real World Applications, 6(1), 35-65.
  4. [4]  Hofbauer, J. and Sigmund, K.(1998), Evolutionary Games and Population Dynamics, Cambridge University Press, cambridge Books Online. [Online]. Available: http://dx.doi.org/10.1017/CBO9781139173179
  5. [5]  Shorten, R., Wirth, F., and Leith, D. (2006), A positive systems model of tcp-like congestion control: asymptotic results, IEEE/ACM Transactions on Networking, 14(3), 616-629.
  6. [6]  Shorten, R., King, C., Wirth, F., and Leith, D. (2007), Modelling tcp congestion control dynamics in drop-tail environments, Automatica, 43(3), 441-449.
  7. [7]  Jong, H.D. (2000), Modeling and Simulation of Genetic Regulatory Systems: A Literature Review, INRIA, Tech. Rep. RR-4032, [Online]. Available: https://hal.inria.fr/inria-00072606
  8. [8]  John, C.P.S. and Jacquez, A. (1993), Qualitative theory of compartmental systems, SIAM Review, 35(1), 43-79. [Online]. Available: http://www.jstor.org/stable/2132525
  9. [9]  Benvenuti, L. and Farina, L. (2002), Positive and compartmental systems, IEEE Transactions on Automatic Control, (47), 370-373.
  10. [10]  Berman, A.,Neumann, M., and Stern, R.J. Nonnegative Matrices in Dynamic Systems, (1989), New York: Wiley.
  11. [11]  Seneta, E. (1981), Non-negative Matrices and Markov Chains, New York, NY: Springer New York, ch. Inhomogeneous Products of Non-negative Matrices, 80-111. [Online]. Available: http://dx.doi.org/10.1007/ 0-387-32792-4_3
  12. [12]  Kaczorek, T. (1985), Two-dimensional Linear Systems, London: Springer Verlag.
  13. [13]  Farina, L. and Rinaldi, S. (2000), Positive linear systems: theory and applications, New York: Wiley- Interscience, Series on Pure and Applied Mathematics.
  14. [14]  Kaczorek, T. (2001), Positive 1D and 2D systems, London: Springer Verlag.
  15. [15]  Benvenuti, L., DeSantis, A., and Farina, L. (2003), Positive Systems, ser. Lecture Notes on Control and Information Sciences 294, Berlin: Springer-Verlag, 2003.
  16. [16]  Farina, L. (2002), Positive systems in the state space approach: main issues and recent results, Citeseer.
  17. [17]  Benvenuti, L. and Farina, L. (2004), A tutorial on the positive realization problem, IEEE Transactions on Automatic Control, 49(5), 651-664, 2004.
  18. [18]  Kaczorek, T. and Sajewski, L. (2014), The Realization Problem for Positive and Fractional Systems, Berlin: Springer International Publishing. [Online]. Available: http://dx.doi.org/10.1007/978-3-319-04834-5
  19. [19]  Hryniów, K. and Markowski, K.A. (2014), Parallel digraphs-building algorithm for polynomial realisations, in Proceedings of 2014 15th International Carpathian Control Conference (ICCC), 174-179. [Online]. Available: http://dx.doi.org/10.1109/CarpathianCC.2014.6843592
  20. [20]  Markowski, K.A. and Hryniów, K. (2015), Solving minimal realisation problem of positive two-dimensional hybrid systems in the form of second fornasini-marchesini model, in Proceedings of 2nd IEEE International Conference on Cybernetics, CD-ROM, Gdynia, Poland, June 24-26, 2015, 172-177. [Online]. Available: http: //dx.doi.org/10.1109/CYBConf.2015.7175927
  21. [21]  Kaczorek, T. (2011), Selected Problems of Fractional Systems Theory, Berlin: Springer Verlag.
  22. [22]  Sajewski, L. (2012), Positive realization of fractional continuous-time linear systems with delays, Pomiary Automatyka Robotyka, (5), 413-417.
  23. [23]  Markowski, K.A. and Hryniów, K. (2015), Digraphs minimal positive stable realisations for fractional onedimensional systems, in Proceedings of 7th Conference on Non-integer Order Calculus and its Applications7th Conference on Non-integer Order Calculus and its Applications, 105-118. [Online]. Available: http://dx. doi.org/10.1007/978-3-319-23039-9_9
  24. [24]  Markowski, K.A. (2015), Deretmination of minimal positive realisation of one-dimensional continuous-time fractional linear system, in Procedings of the International Symposium on Fractional Signal and Systems, R.Both-Rusu, Ed. U.T. Press, 7-12.
  25. [25]  Markowski, K.A. (2016), Determination of minimal realisation of one-dimensional continuous-time fractional linear system, International Journal of Dynamics and Control, 5, 40-50.
  26. [26]  Markowski, K.A. (2017), Challenges in Automation, Robotics and Measurement Techniques: Proceedings of AUTOMATION-2016, March 2-4, 2016, Warsaw, Poland. Cham: Springer International Publishing, 2016, ch. Realisation of Positive Continuous-Time Linear Systems Consisting of n Subsystems with Different Fractional Order, 363-373.
  27. [27]  Nishimoto, K. (1984), Fractional Calculus, Koriama: Decartess Press.
  28. [28]  Das, S.V (2011), Functional Fractional Calculus, New York: Springer Berlin Heidelberg. [Online]. Available: http://dx.doi.org/10.1007/978-3-642-20545-3
  29. [29]  Miller, K. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differenctial Equations, New York: Willeys.
  30. [30]  Ortigueira, M.D. (2011), Fractional Calculus for Scientists and Engineers, Springer Netherlands: Academic Press. [Online]. Available: 10.1007/978-94-007-0747-4
  31. [31]  Podlubny, I. (1999), Fractional Differential Equations, San Diego: Academic Press.
  32. [32]  Ortigueira, M.D., Rivero, M., and Trujillo, J.J. (2015), Steady-state response of constant coefficient discretetime differential systems, Journal of King Saud University-Science.
  33. [33]  Martynyuk, V. and Ortigueira,M. (2015), Fractional model of an electrochemical capacitor, Signal Processing, 107, 355-360.
  34. [34]  Lopes, A.M. and Machado, J.A.T. (2015), Visualizing control systems performance: A fractional perspective, Advances in Mechanical Engineering, 7(12), 1687814015619831.
  35. [35]  Machado, J. and Lopes, A.M. (2015), Fractional state space analysis of temperature time series, Fractional Calculus and Applied Analysis, 18(6), 1518-1536.
  36. [36]  Machado, J., Mata, M.E., and Lopes, A.M. (2015), Fractional state space analysis of economic systems, Entropy, 17(8), 5402-5421.
  37. [37]  Podlubny, I., Skovranek, T., and Datsko, B. (2014), Recent advances in numerical methods for partial fractional differential equations, in Control Conference (ICCC), 2014 15th International Carpathian, IEEE, 454- 457.
  38. [38]  Petras, I., Sierociuk, D., and Podlubny, I. (2012), Identification of parameters of a half-order system, Signal Processing, IEEE Transactions on, 60(10), 5561-5566.
  39. [39]  Ortigueira, M.D., Simos, T.E., Psihoyios, G., Tsitouras, C., and Anastassi, Z. (2012), Two-sided discrete fractional derivatives and systems, AIP Conference Proceedings-American Institute of Physics, 1479(1), 1416.
  40. [40]  Magin, R., Ortigueira, M.D., Podlubny, I., and Trujillo, J. (2011), On the fractional signals and systems, Signal Processing, 91(3), 350-371.
  41. [41]  Caputo, M. (1967), Linear models of dissipation whose q is almost frequency independent-ii, Geographical Journal International, 13(5), 529-539.
  42. [42]  Bang-Jensen, J. and Gutin, G. (2009), Digraphs: Theory, Algorithms and Applications, London: Springer- Verlag.
  43. [43]  Wallis, W.D. (2007), A Beginner’s Guide to Graph Theory, Biiokh¨auser.
  44. [44]  Fornasini, E. and Valcher, M.E. (1997), Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs, Linear Algebra and Its Applications, 263, 275-310.
  45. [45]  Fornasini, E. and Valcher, M.E. (2005), Controllability and reachability of 2D positive systems: a graph theoretic approach, IEEE Transaction on Circuits and Systems I, (52), 576-585.
  46. [46]  Hryniów, K. and Markowski, K.A. (2016), Challenges in Automation, Robotics and Measurement Techniques: Proceedings of AUTOMATION-2016, Warsaw, Poland, Cham: Springer International Publishing, 2016, ch. Classes of Digraph Structures Corresponding to Characteristic Polynomials, 329-339. [Online]. Available: http://dx.doi.org/10.1007/978-3-319-29357-8_30
  47. [47]  Hryniów, K. and Markowski, K.A. (2015), Digraphs minimal realisations of state matrices for fractional positive systems, in Progress in Automation, Robotics and Measuring Techniques, ser. Advances in Intelligent Systems and Computing, R.Szewczyk, C.Zielinski, and M.Kaliczynska, Eds, Springer International Publishing, 350, 63-72. [Online]. Available: http://dx.doi.org/10.1007/978-3-319-15796-2\textunderscore7
  48. [48]  Hryniów, K. and Markowski, K.A.(2016), Parallel Digraphs-Building Computer Algorithm For Finding a Set of Characteristic Polynomial Realisations of Dynamic System, in Journal of Automation, Mobile Robotics & Intelligent Systems (JAMRIS), Journal of Automation, Mobile Robotics & Intelligent Systems (JAMRIS), 10(3), 38-51.

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