Skip to main content
SpringerLink
Log in

An Accuracy Estimation for a Non Integer Order, Discrete, State Space Model of Heat Transfer Process

  • Conference paper
  • First Online:
Book cover Automation 2017 (ICA 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 550))

Included in the following conference series:

Abstract

In the paper an accuracy analysis for non integer order, discrete, state space model of heat transfer process in one dimensional plant is presented. The proposed model is a discrete version of time - continuous, non integer order, state space model proposed previously by Authors. The discretization of integro/differential operator was done with the use of backward difference method. The accuracy and convergence of the discussed model was considered as a function of model order and memory length necessary to proper estimation of non integer order operator. Tests were done with the use of PLC and SCADA based experimental system. Results of experiments show that the proposed, discrete model assures the good performance in the sense of MSE cost function, but its size is relatively high.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Al Aloui, M.A.: Dicretization methods of fractional parallel PID controllers. In: Proceedings of 6th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2009, pp. 327–330 (2009)

    Google Scholar 

  2. Berger, H.: Automating with STEP 7 in STL and SCL: SIMATIC S7–300/400 Programmable Controllers. SIEMENS (2012)

    Google Scholar 

  3. Berger, H.: (2013) Automating with SIMATIC: Controllers. Software, Programming, Data. SIEMENS (2013)

    Google Scholar 

  4. Caponetto R., Dongola G., Fortuna l., Petras I.: Fractional Order Systems. In: Modeling and Control Applications, World Scientific Series on Nonlinear Science, Series A, vol. 72. World Scientific Publishing (2010)

    Google Scholar 

  5. Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circ. Syst.- I: Fundam. Theo. Appl. 49(3), 363–367 (2002)

    Google Scholar 

  6. Das, S.: Functional fractional calculus for system identification and controls (2008)

    Google Scholar 

  7. Das, S., Pan, I.: Intelligent Fractional Order Systems and Control. An Introduction. SCI, vol. 438. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  8. Douambi, A., Charef, A., Besancon, A.: Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function. Int. J. Appl. Math. Comp. Sci. 17(4), 455–462 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Dzieliński, A., Sierociuk, D., Sarwas, G.: Some applications of fractional order calculus. Bull. Pol. Acad. Sci. Tech. Sci. 58(4), 583–592 (2010)

    MATH  Google Scholar 

  10. Frey, G., Litz, L.: Formal methods in PLC programming. In: Proceedings of the IEEE Conference on Systems Man and Cybrenetics, SMC 2000, Nashville, 8–11 October 2000, pp. 2431–2436 (2000)

    Google Scholar 

  11. Isermann, R., Muenchhof, M.: Identification of Dynamic Systems. An Introduction with Applications. Springer, Heidelberg (2011)

    Book  Google Scholar 

  12. John, K.H., Tiegelkamp, M.: IEC 61131–3: Programming Industrial Automation Systems. Concepts and Programming Languages, Requirements for Programming Systems, Decision Making Aids. Springer, Heidelberg (2010)

    Google Scholar 

  13. Kaczorek, T.: Selected Problems in Fractional Systems Theory. LNCIS, vol. 411. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  14. Kaczorek, T., Rogowski K.: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok (2014)

    Google Scholar 

  15. Lewis, R.W.: Programming industrial control systems using IEC 1131–3 IEE 1998 (1998)

    Google Scholar 

  16. Luo, Y., Chen, Y.Q., Wang, C.Y., Pi, Y.G.: Tuning fractional order proportional integral controllers for fractional order systems. J. Process Control 20(2010), 823–831 (2010)

    Article  Google Scholar 

  17. Mitkowski W., Oprzedkiewicz K.: Optimal sample time estimation for the finite-dimensional discrete dynamic compensator implemented at the “soft PLC” platform. In: 23rd IFIP TC 7 Conference on System Modelling and Optimization, Cracow, Poland, July 23–27 (2007). Book of abstracts Korytowski, A., Mitkowski, W., Szymkat, M. (eds.): AGH University of Science and Technology. Faculty of Electrical Engineering, Automatics, Computer Science and Electronics, Krakow, pp. 77–78 (2007). ISBN 978-83-88309-0

    Google Scholar 

  18. Mitkowski, W., Oprzedkiewicz, K.: Fractional-order \(P2D^\beta \) controller for uncertain parameter DC motor. In: Mitkowski, W., Kacprzyk, J., Baranowski, J.(eds.) Advances in the Theory and Applications of Non-integer Order Systems. LNEE, vol. 257, pp. 249–259. Springer, Heidelberg (2013). 10.1007/978-3-319-00933-9_23

  19. Mozyrska, D., Pawluszewicz, E.: Fractional discrete-time linear control systems with initialisation. Int. J. Control 2011, 1–7 (2011)

    MATH  Google Scholar 

  20. Oprzedkiewicz, K., Chochol, M., Bauer, W., Meresinski, T.: Modeling of elementary fractional order plants at PLC SIEMENS platform. In: Latawiec, K.J., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in Modelling and Control of Non-integer-Order Systems. LNEE, vol. 320, pp. 265–273. Springer, Cham (2015). doi:10.1007/978-3-319-09900-2_25

    Google Scholar 

  21. Oprzedkiewicz, K., Mitkowski, W., Gawin, E.: Application of fractional order transfer functions to modeling of high - order systems. In: MMAR 2015: 20th International Conference on Methods and Models in Automation and Robotics, 24–27, Mędzyzdroje, Poland: Program, Abstracts, Proceedings (CD), Szczecin: ZAPOL Sobczyk Sp. j. (2015)+ CD. 978–1, ISBN: 978-1-4799-8701-6-4799-8700-9, ISBN: 978-83-7518-756-4, S. 127. Full text CD, pp. 1169-1174, August 2015

    Google Scholar 

  22. Oprzędkiewicz, K., Mitkowski, W., Gawin, E.: An estimation of accuracy of oustaloup approximation. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds.) Challenges in Automation, Robotics and Measurement Techniques. AISC, vol. 440, pp. 299–307. Springer, Cham (2016). doi:10.1007/978-3-319-29357-8_27

    Chapter  Google Scholar 

  23. Oprzedkiewicz, K., Mitkowski, W., Gawin, E.: Parameter identification for non integer order, state space models of heat plant. In: MMAR 2016: 21th International Conference on Methods and Models in Automation and Robotics, 29 August, 2016, Międzyzdroje, Poland, pp. 184–188, September 2016, ISBN: 978-1-5090-1866-6, ISBN: 978-83-7518-791-5

    Google Scholar 

  24. Oprzedkiewicz K., Gawin E.: Non integer order, state space model for one dimensional heat transfer process. Arch. Control Sci., 26(2), 261–275 (2016). ISSN 1230–2384, https://www-1degruyter-1com-1atoz.wbg2.bg.agh.edu.pl/downloadpdf/j/acsc.2016.26.issue-2/acsc-2016-0015/acsc-2016-0015.xml

  25. Ostalczyk, P.: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. Int. J. Appl. Math. Comput. Sci. 22(3), 533–538 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ostalczyk, P.: Discrete Fractional Calculus. Applications in control and image processing. Series in Computer Vision, vol. 4. World Scientific Publishing (2016)

    Google Scholar 

  27. Padula, F., Visioli, A.: Tuning rules for optimal PID and fractional-order PID controllers. J. Process Control 21, 69–81 (2011)

    Article  MATH  Google Scholar 

  28. Petras, I.: Fractional order feedback control of a DC motor. J. Electr. Eng. 60(3), 117–128 (2009)

    Google Scholar 

  29. Petras, I.: Realization of fractional order controller based on PLC and its utilization to temperature control, Transfer inovácií 14/2009 (2009)

    Google Scholar 

  30. Petras, I.: Tuning and implementation methods for fractional-order controllers. Frac. Calc. Appl. Anal. 15(2), 2012 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Petras, I.: http://people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod1.m

  32. Petras, I.: http://people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod2.m

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

    MATH  Google Scholar 

  34. Sierociuk, D., Macias, M.: New recursive approximation of fractional order derivative and its application to control. In: Proceedings of 17th International Carpathian Control Conference (ICCC), pp. 673–678 (2016)

    Google Scholar 

  35. Stanislawski, R., Latawiec, K.J., Lukaniszyn, M.: A comparative analysis of laguerre-based approximators to the grünwald-Letnikov fractional-order difference. Mathematical Problems in Engineering, vol. 2015, Article ID 512104, 10 p. (2015). http://dx.doi.org/10.1155/2015/512104

  36. Valerio, D., da Costa, J.S.: Tuning of fractional PID controllers with Ziegler-Nichols-type rules. Sig. Process. 86(2006), 2771–2784 (2006)

    Article  MATH  Google Scholar 

  37. Vinagre, B.M., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fractional Calc. Appl. Anal. 3(3), 231–248 (2000)

    MathSciNet  MATH  Google Scholar 

  38. Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340, 349–362 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The paper was sponsored by AGH University grant no 11.11.120.817.

Author information

Authors and Affiliations

  1. AGH University, A. Mickiewicza 30, 30-59, Krakow, Poland

    Krzysztof Oprzedkiewicz & Wojciech Mitkowski

  2. State Higher Vocational School in Tarnow, A. Mickiewicza 8, 33-100, Tarnow, Poland

    Edyta Gawin

Authors
  1. Krzysztof Oprzedkiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wojciech Mitkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edyta Gawin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krzysztof Oprzedkiewicz .

Editor information

Editors and Affiliations

  1. Industrial Research Institute for Automation and Measurements PIAP, Warsaw, Poland

    Roman Szewczyk

  2. Industrial Research Institute for Automation and Measurements PIAP, Warsaw, Poland

    Cezary Zieliński

  3. Industrial Research Institute for Automation and Measurements PIAP, Warsaw, Poland

    Małgorzata Kaliczyńska

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Oprzedkiewicz, K., Mitkowski, W., Gawin, E. (2017). An Accuracy Estimation for a Non Integer Order, Discrete, State Space Model of Heat Transfer Process. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Automation 2017. ICA 2017. Advances in Intelligent Systems and Computing, vol 550. Springer, Cham. https://doi.org/10.1007/978-3-319-54042-9_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-54042-9_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-54041-2

  • Online ISBN: 978-3-319-54042-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation