On Grünwlad-Letinkov Fractional Operator with Measurable Order on Continuous-Discrete Time Scale

Pawłuszewicz, Ewa; Koszewnik, Andrzej; Burzyński, Piotr

doi:10.2478/ama-2020-0023

Abstract

Considering experimental implementation control laws on digital tools that measurement cards are discharged every time unit one can see that time of simulations is partially continuous and partially discrete. This observation provides the motivation for defining the Grünvald-Letnikov fractional operator with measurable order defined on continuous-discrete time scale. Some properties of this operator are discussed. The simulation analysis of the proposed approach to the Grünwald-Letnikov operator with the measurement functional order is presented.

References

1. Alagoz B.B., Tepljakov A., Ates A. (2019) Time-domain identification of one noninteger order plus time delay models from step response measurements, International Journal of Modeling, Simulation and Scientific Computing, Vol. 10, No. 1, 1941011-1–1941011-22.
Search in Google Scholar Back to article
2. Alagz B.B., Alisoy H. (2018) Estimation of reduced order equivalent circuit model parametres of batteries from noisy current and voltage measurements, Balkan Journal of Electrical & Computer Engineering, Vol. 6, No. 4, 224–231.
Search in Google Scholar Back to article
3. Balaska H., Ladaci S., Djouambi A., Schulte H., Bourouba B. (2020) Fractional order tube model reference adaptive control for a class of fractional order linear systems International Journal of Applied Mathematics and Computer Science,, Vol. 30, No. 3, 501–515
Search in Google Scholar Back to article
4. Bohner M., Petrson A. (2002) Dynamic Equations on Time Scales: A survey, Journal of Computational and Applied Mathematics, Vol. 141, No. 1–2, 1–26.
Search in Google Scholar Back to article
5. Buslowicz M. Nartowicz T. (2009) Design of fractional order controller for a class of plants with delay, Measurement Automation and Robotics, Vol. 2, 398–405.
Search in Google Scholar Back to article
6. Coimbra C.,(2003), Mechanics with variable-order differential operators, Annual Physics, Vol. 12, 692-703.
Search in Google Scholar Back to article
7. Djennoune S., Bettayeb M., Al-Saggaf U.M. (2019) Synchronization of fractional order discrete-time chaotic systems by exact state reconstructor: application to secure communication, International Journal of Applied Mathematics and Computer Science, Vol. 29, No. 1, 179–194.
Search in Google Scholar Back to article
8. Janczak J., Kondratiuk M., Pawluszewicz E. (2016) Testing of adaptive non-uniform sampling switch algorithm with real-time simulation-in-the-loop, Control and Cybernetics, Vol. 45, No. 3, 317–328.
Search in Google Scholar Back to article
9. Kavuran G., Yeroğlu C., Ates A. Alagoz B.B. (2017) Effects of fractional order integration on ASDM signals, Int. J. Dynam. Control Vol. 5, 10–17
Search in Google Scholar Back to article
10. Kondratiuk M., Ambroziak L., Pawluszewicz E, Janczak J. (2018) Discrete PID algorithm with non-uniform sampling Practical implementation in control system, AIP Conference Proceedings 2029, 020029, doi: 10.1063/1.5066491.10.1063/1.5066491
Search in Google Scholar Back to article
11. Koszewnik A., Ostaszewski M., Pawłuszewicz E., Radgowski P. (2018) Performance Assessment of the Tilt Fractional Order Integral Derivative Regulator for Control Flow Rate in Festo MPSR©PA Compact Workstation, Proceedings of 23rd International Conference on Methods and Models in Automation and Robotics, Poland.10.1109/MMAR.2018.8486080
Search in Google Scholar Back to article
12. Koszewnik A., Pawluszewicz E., Nartowicz T., (2016), Fractional order controller to control pomp in Festo MPS® PA Compact Workstation, Proceedings of the 17th International Carpathian Control Conference (ICCC 2016), 364–367.10.1109/CarpathianCC.2016.7501124
Search in Google Scholar Back to article
13. Lorenzo C.F., Hartley T.T. (2002) Variable order and distributed order fractional operators, Nonlinear Dynamics, Vol. 29, 57–98.
Search in Google Scholar Back to article
14. Ortigueira M., Torres D.M.F., Trujillo J. (2016) Exponents and Laplace transforms on non-uniform time scale, Communications in Nonlinear Science and Numerical Simulations, Vol. 39, 252–270.
Search in Google Scholar Back to article
15. Ortigueira M.D. (1997) Fractional discrete-time linear systems, Proceedings of the EEICASSP, Munich, Germany, IEEE New York, Vol. 3, 2241–2244.
Search in Google Scholar Back to article
16. Ostalczyk P. (2012) Variable-fractional-order discrete PID controller, IEEE Proceedings of the 17th International Conference on Methods and Models in Automation and Robotics, MMAR 2012, Miedzyzdroje, Poland, 534–539.10.1109/MMAR.2012.6347829
Search in Google Scholar Back to article
17. Ostalczyk P., Duch P. Brzezinski D.W., Sankowski D. (2015) Order functions selection in the variable-, fractional-order PID controller in: Advances in modelling and control of non-integer-order systems, Eds. Latawiec K.J., Lukaniszyn M., Stanislawski R., 159–170.
Search in Google Scholar Back to article
18. Patniak S., Hollkamp J.P., Semperlotti A. (2002) Applications of variable-order fractional operators: a review, Proceedings of the Royal Society A, 476, 1–32.
Search in Google Scholar Back to article
19. Pawluszewicz E., Koszewnik A., (2019), Markov parameters of the input-output map for discrete-time order systems with Grünwlad-Letnikov h-difference operator, Proceedings of the 24th International Conference on Methods and Models in Automation and Robotics (MMAR), 456–459.10.1109/MMAR.2019.8864668
Search in Google Scholar Back to article
20. Podlubny, I., Dorcak, L., Misanek, J. (1995) Application of fractional order derivatives to calculation of heat load intensity change in blast furnace walls, Transactions of Technical University of Kosice, Vol. 5, 137–144.
Search in Google Scholar Back to article
21. Samko S.G., Ross B. (1993) Integration and differentiation to a variable fractional order, Journal Integral Transforms and Special Functions, Vol. 1, No. 4, 277–300.
Search in Google Scholar Back to article
22. Sierociuk D., Macias M. (2013) Comparison of variable fractional order PID controller for different types of variable order derivatives, Proceedings of the 14th International Carpathian Control Conference ICCC 2013, Rytro, Poland, 334–339.10.1109/CarpathianCC.2013.6560565
Search in Google Scholar Back to article
23. Sierociuk D., Malesza W., Macias M. (2013) On a new definition of fractional variable-order derivative, Proceedings of the 14th International Carpathian Control Conference ICCC 2013, Rytro, Poland, 339–345.10.1109/CarpathianCC.2013.6560566
Search in Google Scholar Back to article
24. Sierociuk D., Malesza W., Macias M. (2015) Deviation, interpolation and analog modelling of fractional variable order derivative definitions, Applied Mathematics and Modelling, Vol. 39, 3876–3888.
Search in Google Scholar Back to article
25. Stanislawski R., Latawiec K. (2012) Normalized finite fractional differences: computational and accuracy breakthrough, International Journal of Applied Mathematics and Computer Science, Vol. 22, No. 4, 907–919.
Search in Google Scholar Back to article
26. Tepljakov A. (2017) Fractional-order modeling and control of dynamic systems, Springer-Verlag.10.1007/978-3-319-52950-9
Search in Google Scholar Back to article
27. Tepljakov A., Alagoz B.B. et al. (2018) FOPID controllers and their industrial applications: a survey of recent results, IFAC Papers On Line 51-4, 25–30.10.1016/j.ifacol.2018.06.014
Search in Google Scholar Back to article
28. Tepljakov A., Petlekov E., Belikov J. (2012) A flexible Matlab tool for optimal fractional-order PID controller design subject to specifications, Proceedings of the 31st Chinese Control Conference, 4698–4703.
Search in Google Scholar Back to article
29. Valerio D., Sa da Costa J. (2001) Variable-order fractional derivatives and their numerical approximations, Signal Processing, Vol. 91, 470–483.
Search in Google Scholar Back to article
30. Wang J.C. (1987) Realizations of generalized Warburg impedance with RC ladder networks and transmission lines, Journal of Electro-chemical Society, Vol. 134, No. 8, 1915–1920.
Search in Google Scholar Back to article

On Grünwlad-Letinkov Fractional Operator with Measurable Order on Continuous-Discrete Time Scale

Abstract

Paradigm

My account