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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

Method for Finding a set of (A,B,C,D) Realizations for Single-Input Multiple-Output / Multiple-Input Single-Output One-dimensional Continuous-time Fractional Systems

Journal of Environmental Accounting and Management 8(1) (2019) 97--108 | DOI:10.5890/JAND.2019.03.008

Konrad Andrzej Markowski, Krzysztof Hryniów

Warsaw University of Technology, Faculty of Electrical Engineering Institute of Control and Industrial Electronics, Koszykowa Street No 75, 00-662 Warsaw, Poland

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Abstract

In the paper presented is a method allowing for determination of a set of (A,B,C,D) realizations for fractional-order dynamic systems. Proposed method is an extension of previously proposed algorithm that was used to determine realizations of fractional-order 1D single-input single-output (SISO) dynamic systems for both single-input multipleoutput (SIMO) and multiple-input single-output (MISO) fractional dynamic systems. The main advantage of the method over canonical forms is that the algorithm finds a set of realizations, not just a single realization. Also, the solutions found tend to be minimal in terms of size of state matrix A. Additionally, the method allows for the possibility of obtaining a set of state matrices directly from digraph form of the system and can be efficiently used as GPGPU computer algorithm. Proposed method is presented in pseudo-code and illustrated with example.

References

  1. [1]  Luenberger, D.G. (1979), Introduction to Dynamic Systems: Theory, Models, and Applications, chap. Positive linear systems, Wiley.
  2. [2]  Benvenuti, L. and Farina, L. (2004), A tutorial on the positive realization problem, IEEE Transactions on Automatic Control , 49, 651-664.
  3. [3]  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.
  4. [4]  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 , pp. 576-585.
  5. [5]  Nishimoto, K. (1984), Fractional Calculus, Decartess Press.
  6. [6]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press.
  7. [7]  Das, S. (2011), Functional Fractional Calculus, Springer Berlin Heidelberg.
  8. [8]  Ortigueira, M.D. (2011), Fractional Calculus for Scientists and Engineers, Academic Press.
  9. [9]  Petras, I., Sierociuk, D., and Podlubny, I. (2012), Identification of parameters of a half-order system, Signal Processing, IEEE Transactions on, 60, 5561-5566.
  10. [10]  Podlubny, I., Skovranek, T., and Datsko, B. (2014), Recent advances in numerical methods for partial fractional differential equations. Control Conference (ICCC), 2014 15th International Carpathian, pp. 454-457, IEEE.
  11. [11]  Martynyuk, V. and Ortigueira,M. (2015), Fractionalmodel of an electrochemical capacitor. Signal Processing, 107, 355-360.
  12. [12]  Machado, J. and Lopes, A.M. (2015), Fractional state space analysis of temperature time series. Fractional Calculus and Applied Analysis, 18, 1518-1536.
  13. [13]  Machado, J., Mata, M.E., and Lopes, A.M. (2015), Fractional state space analysis of economic systems. Entropy, 17, 5402-5421.
  14. [14]  Muresan, C.I., Dulf, E.H., and Prodan, O. (2016), A fractional order controller for seismic mitigation of structures equipped with viscoelastic mass dampers. Journal of Vibration and Control , 22, 1980-1992, DOI: 10.1177/1077546314557553.
  15. [15]  Muresan, C.I., Dutta, A., Dulf, E.H., Pinar, Z., Maxim, A., and Ionescu, C.M. (2016), Tuning algorithms for fractional order internal model controllers for time delay processes. International Journal of Control , 89, 579-593, DOI: 10.1080/00207179.2015.1086027.
  16. [16]  Markowski, K.A. (2017), Fractional kinetics of compartmental systems. First approach with use digraph-based method, Proc. SPIE, 10445, (in press).
  17. [17]  Kaczorek, T. (1987), Realization problem for general model of two-dimensional linear systems, Bulletin of the Polish Academy of Sciences, Technical Sciences, 35, 633-637.
  18. [18]  Bisiacco, M., Fornasini, E., and Marchesini, G. (1989), Dynamic regulation of 2D systems: A state-space approach, Linear Algebra and Its Applications, 122-124, 195-218.
  19. [19]  Xu, L., Wu, Q., Lin, Z., Xiao, Y., and Anazawa, Y. (2004), Futher results on realisation of 2D filters by Fornasini-Marchesini model, 8th International Conference on Control, Automation, Robotics and Vision, Kunming, China, 6-9th December, pp. 1460-1464.
  20. [20]  Xu, L., Wu, L., Wu, Q., Lin, Z., and Xiao, Y. (2005), On realization of 2D discrete systems by Fornasini Marchesini model, International Journal of Control, Automation, and Systems, 4, 631-639.
  21. [21]  Kaczorek, T. (2007), Positive realization of 2D general model, Logistyka, nr 3, 1-13.
  22. [22]  Hryniów, K. and Markowski, K.A. (2014), Parallel digraphs-building algorithm for polynomial realisations. Proceedings of 2014 15th International Carpathian Control Conference (ICCC), pp. 174-179.
  23. [23]  Hryniów, K. and Markowski, K.A. (2015), Optimisation of digraphs-based realisations for polynomials of one and two variables. Szewczyk, R., Zieliński, C., and Kaliczyńska, M. (eds.), Progress in Automation, Robotics and Measuring Techniques, vol. 350 of Advances in Intelligent Systems and Computing, pp. 73-83, Springer International Publishing.
  24. [24]  Markowski, K.A. and Hryniów, K. (2017), Finding a Set of (A, B, C, D) Realisations for Fractional One- Dimensional Systems with Digraph-Based Algorithm, vol. 407, pp. 357-368, Springer International Publishing.
  25. [25]  Bang-Jensen, J. and Gutin, G. (2009), Digraphs: Theory, Algorithms and Applications (2nd Edition), Springer-Verlag.
  26. [26]  Hryniów, K. and Markowski, K.A. (2015), Optimisation of digraphs creation for parallel algorithm for finding a complete set of solutions of characteristic polynomial. 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 2015 , pp. 1139-1144.
  27. [27]  Hryniów, K. and Markowski, K.A. (2016), Parallel digraphs-building computer algorithm for finding a set of characteristic polynomial realisations of dynamic system, Journal of Automation, Mobile Robotics & Intelligent Systems (JAMRIS), 10, 38-51.
  28. [28]  Hryniów, K. and Markowski, K.A. (2015), Digraphs minimal positive stable realisations for fractional onedimensional systems, Domek, S. and Dworak, P. (eds.), Theoretical Developments and Applications of Non- Integer Order Systems, vol. 357 of Lecture Notes in Electrical Engineering, pp. 105-118, Springer International Publishing.
  29. [29]  Markowski, K.A. (2016), Digraphs structures corresponding to minimal realisation of fractional continuous-time linear systems with all-pole and all-zero transfer function, 2016 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR).
  30. [30]  Markowski, K.A. (2017), 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, 265-282, DOI: 10.5890/JAND.2017.06.011.
  31. [31]  Hryniów, K. and Markowski, K.A. (2015), Classes of digraph structures corresponding to characteristic polynomials. Szewczyk, R., Zieliński, C., and Kaliczyńska, M. (eds.), Challenges in Automation, Robotics and Measurement Techniques, vol. 440 of Advances in Intelligent Systems and Computing, pp. 329-339, Springer International Publishing.

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