Atangana, A., & Baleanu, D. (2016). New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model. Thermal Science, 20(2), 763-769. doi:10.48550/arxiv.1602.03408
Azarmi, R., Tavakoli-Kakhki, M., Sedigh, A. K., & Fatehi, A. (2016). Robust Fractional Order PI Controller Tuning Based on Bode’s Ideal Transfer Function. IFAC-PapersOnLine, 49(9), 158-163. doi:10.1016/j.ifacol.2016.07.519
Baranowski, J., Bauer, W., Zagórowska, M., Dziwiński, T., & Piątek, P. (2015, August 24-27). Time-domain Oustaloup approximation. In: Proceedings of the 2015 20th international Conference on Methods and Models in Automation and Robotics (MMAR) (pp. 116-120). doi:10.1109/MMAR.2015.7283857
Boubaker, O., & Jafary, S. (Eds.) (2018) (n.d.). Recent Advances in Chaotic Systems and Synchronization From Theory to Real World Applications. Academic Press.
Caponetto, R., Dongola, G., Fortuna, L., & Petráš, I. (2010). Fractional Order Systems: Modeling and Control Applications. World Scientific. doi:10.1142/7709
Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent—II. Geophysical Journal International, 13(5), 529-539. doi:10.1111/J.1365-246X.1967.TB02303.X
Chen, Y. Q., & Moore, K. L. (2002). Discretization schemes for fractional-order differentiators and integrators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(3), 363-367. doi:10.1109/81.989172
Chen, Y., Petras, I., & Xue, D. (2009, June 10-12). Fractional order control - A tutorial. In: Proceedings of the 2009 American Control Conference (pp. 1397-1411). doi:10.1109/ACC.2009.5160719
Cortés-Romero, J., Luviano-Juárez, A., & Sira-Ramírez, H. (2013). A Delta Operator Approach for the Discrete-Time Active Disturbance Rejection Control on Induction Motors. Mathematical Problems in Engineering, 2013, 572026. doi:10.1155/2013/572026
Dokuyucu, M. A. (2020). Caputo and Atangana-Baleanu-Caputo Fractional Derivative Applied to Garden Equation. Turkish Journal of Science, 5(1), 1-7.
Ganguli, S., Kaur, G., & Sarkar, P. (2021). Global heuristic methods for reduced-order modelling of fractional-order systems in the delta domain: a unified approach. Ricerche di Matematica. doi:10.1007/s11587-021-00644-7
Gao, J., Chai, S., Shuai, M., Zhang, B., & Cui, L. (2018, July 25-27). Detecting False Data Injection Attack on Cyber-Physical System Based on Delta Operator. In: Proceedings of the 2018 37th Chinese Control Conference (CCC) (pp. 5961-5966). doi:10.23919/ChiCC.2018.8483314
Khanra, M., Pal, J., & Biswas, K. (2010, November 29-December 01). Rational approximation of fractional operator — A comparative study. In: Proceedings of the 2010 International Conference on Power, Control and Embedded Systems (pp. 1-5). doi:10.1109/ICPCES.2010.5698677
Khattri, S. K. (2009). New close form approximations of ln(1 + x). The Teaching of Mathematics, 12(1), 7-14.
Kothari, K., Mehta, U., & Prasad, R. (2019). Fractional-Order System Modeling and its Applications. Journal of Engineering Science and Technology Review, 12, 1-10. doi:10.25103/jestr.126.01
Krishna, B. T. (2011). Studies on fractional order differentiators and integrators: A survey. Signal Processing, 91(3), 386-426. doi:10.1016/j.sigpro.2010.06.022
Krishna, B. T. (2015, December 18-20). Design of Fractional order differintegrators using reduced order s to z transforms. In: Proceedings of the 2015 IEEE 10th International Conference on Industrial and Information Systems (ICIIS) (pp. 469-473). doi:10.1109/ICIINFS.2015.7399057
Keyser, R. D., & Muresan, C. I. (2016, October 09-12). Analysis of a new continuous-to-discrete-time operator for the approximation of fractional order systems. In: Proceedings of the 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC) (pp. 3211-3216). doi:10.1109/SMC.2016.7844728
Lamrabet, O., Tissir, E. H., & Haoussi, F. E. (2020, June 09-11). Controller design for delta operator time-delay systems subject to actuator saturation. In: Proceedings of the 2020 International Conference on Intelligent Systems and Computer Vision (ISCV). doi:10.1109/ISCV49265.2020.9204303
Maione, G. (2011). High-Speed Digital Realizations of Fractional Operators in the Delta Domain. IEEE Transactions on Automatic Control, 56(3), 697-702. doi:10.1109/TAC.2010.2101134
Middleton, R., & Goodwin, G. (1986). Improved finite word length characteristics in digital control using delta operators. IEEE Transactions on Automatic Control, 31(11), 1015-1021. doi:10.1109/TAC.1986.1104162
Middleton, R. H. & Goodwin, G. C. (1990a). Digital control and estimation: a unified approach. Prentice Hall.
Middleton, R. H., & Goodwin, G. C. (1990b). Digital Control and Estimation: A Unified Approach (Prentice Hall Information and System Sciences Series). Prentice Hall.
Miller, K. S., & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Willey Blackwell.
Nakagawa, M., & Sorimachi, K. (1992). Basic Characteristics of a Fractance Device. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 75, 1814-1819.
Oldham, K. B., & Spanier, J. (Eds.) (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier.
Oustaloup, A. (1995). La dérivation non entière: Théorie, synthèse et applications. Hermes.
Pan, I., & Das, S. (2013). Intelligent Fractional Order Systems and Control. Studies in Computational Intelligence (SCI, volume 438), Springer. doi:10.1007/978-3-642-31549-7
Podlubny, I. (1999). Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers. IEEE Transactions on Automatic Control, 44(1), 208-214. doi:10.1109/9.739144
Quezada-Téllez, L. A., Franco-Pérez, L., & Fernandez-Anaya, G. (2020). Controlling Chaos for a Fractional-Order Discrete System. IEEE Open Journal of Circuits and Systems, 1, 263-269. doi:10.1109/OJCAS.2020.3033154
Sarkar, P., Shekh, R. R., & Iqbal, A. (2016, November 09-11). A unified approach for reduced order modeling of fractional order system in delta domain. In: Proceedings of the 2016 International Automatic Control Conference (CACS) (pp. 257-262). doi:10.1109/CACS.2016.7973920
Skaar, S. B., Michel, A. N., & Miller, R. K. (1988). Stability of viscoelastic control systems. IEEE Transactions on Automatic Control, 33(4), 348-357. doi:10.1109/9.192189
Sun, H., Abdelwahab, A., & Onaral, B. (1984a). Linear approximation of transfer function with a pole of fractional power. IEEE Transactions on Automatic Control, 29(5), 441-444. doi:10.1109/TAC.1984.1103551
Sun, H. H., Onaral, B., & Tso, Y.-Y. (1984b). Application of the Positive Reality Principle to Metal Electrode Linear Polarization Phenomena. IEEE Transactions on Biomedical Engineering, BME-31(10), 664-674. doi:10.1109/TBME.1984.325317
Swarnakar, J., Sarkar, P., Dey, M., & Singh, L. J. (2017, December 21-23). Rational approximation of fractional order system in delta domain — A unified approach. In: Proceedings of the 2017 IEEE Region 10 Humanitarian Technology Conference (R10-HTC) (pp. 144-150). doi:10.1109/R10-HTC.2017.8288926
Tabatadze, V., Karaçuha, K., & Veliyev, E. I. (2020). The solution of the plane wave diffraction problem by two strips with different fractional boundary conditions. Journal of Electromagnetic Waves and Applications, 34(7), 881-893. doi:10.1080/09205071.2020.1759461
Vinagre, B. M., Podlubny, I., Hernández, A., & Feliu, V. (2000) Some approximations of fractional order operators used in control theory and applications. J. Fractional Calculus Appl. Anal., 4, 47-66.
Vinagre, B. M., Chen, Y. Q., & Petráš, I. (2003). Two direct Tustin discretization methods for fractional-order differentiator/integrator. Journal of the Franklin Institute, 340(5), 349-362. doi:10.1016/j.jfranklin.2003.08.001
Xue, D., Zhao, C., & Chen, Y. (2006, June 25-28). A Modified Approximation Method of Fractional Order System. In: Proceedings of the 2006 International Conference on Mechatronics and Automation (pp. 1043-1048). doi:10.1109/ICMA.2006.257769
Yumuk, E., Güzelkaya, M., & Eksin, İ. (2019). Analytical fractional PID controller design based on Bode’s ideal transfer function plus time delay. ISA Transactions, 91, 196-206. doi:10.1016/J.ISATRA.2019.01.034
Yumuk, E., Güzelkaya, M., & Eksin, İ. (2022). A robust fractional-order controller design with gain and phase margin specifications based on delayed Bode’s ideal transfer function. Journal of the Franklin Institute, 359(11), 5341-5353. doi:10.1016/J.JFRANKLIN.2022.05.033
Zhao, Y., & Zhang, D. (2017, May 24-26). H∞ fault detection for uncertain delta operator systems with packet dropout and limited communication. In: Proceedings of the 2017 American Control Conference (ACC) (4772-4777). doi:10.23919/ACC.2017.7963693
Discretization of Fractional Order Operator in Delta Domain
The fractional order operator is the backbone of the fractional order system (FOS). The fractional order operator (FOO) is generally represented as s^(±μ) (0<μ<1). Discrete time FOS can be obtained through the discretization of the fractional order operator. The FOO is the general form of either fractional order differentiator (FOD) or integrator (FOI) depending upon the values of μ. Out of the two discretization methods, direct discretization outperforms the method of indirect discretization. The mapping between the continuous time and discrete time domain is done with the development of generating function. Continuous fraction expansion (CFE) is used expand the generating function for the rational approximation of the FOO. There is an inherent problem associated with the discretization of FOO in discrete z-domain particularly at very fast sampling rate. In the other hand, discretization using delta operator parameterization provides the continuous time and discrete time results in hand to hand, when the continuous time systems are sampled at very fast sampling rate and circumventing the problem with shift operator parameterization at fast sampling rate. In this work, a new generating function is proposed to discretize the FOO using the Gauss-Legendre 3-point quadrature rule and generating function is expanded using the CFE to form rational approximation of the FOO in delta domain. The benchmark fractional order systems are considered in this work for the simulation purpose and comparison of results are made to prove the efficacy of the proposed method using MATLAB.
Atangana, A., & Baleanu, D. (2016). New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model. Thermal Science, 20(2), 763-769. doi:10.48550/arxiv.1602.03408
Azarmi, R., Tavakoli-Kakhki, M., Sedigh, A. K., & Fatehi, A. (2016). Robust Fractional Order PI Controller Tuning Based on Bode’s Ideal Transfer Function. IFAC-PapersOnLine, 49(9), 158-163. doi:10.1016/j.ifacol.2016.07.519
Baranowski, J., Bauer, W., Zagórowska, M., Dziwiński, T., & Piątek, P. (2015, August 24-27). Time-domain Oustaloup approximation. In: Proceedings of the 2015 20th international Conference on Methods and Models in Automation and Robotics (MMAR) (pp. 116-120). doi:10.1109/MMAR.2015.7283857
Boubaker, O., & Jafary, S. (Eds.) (2018) (n.d.). Recent Advances in Chaotic Systems and Synchronization From Theory to Real World Applications. Academic Press.
Caponetto, R., Dongola, G., Fortuna, L., & Petráš, I. (2010). Fractional Order Systems: Modeling and Control Applications. World Scientific. doi:10.1142/7709
Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent—II. Geophysical Journal International, 13(5), 529-539. doi:10.1111/J.1365-246X.1967.TB02303.X
Chen, Y. Q., & Moore, K. L. (2002). Discretization schemes for fractional-order differentiators and integrators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(3), 363-367. doi:10.1109/81.989172
Chen, Y., Petras, I., & Xue, D. (2009, June 10-12). Fractional order control - A tutorial. In: Proceedings of the 2009 American Control Conference (pp. 1397-1411). doi:10.1109/ACC.2009.5160719
Cortés-Romero, J., Luviano-Juárez, A., & Sira-Ramírez, H. (2013). A Delta Operator Approach for the Discrete-Time Active Disturbance Rejection Control on Induction Motors. Mathematical Problems in Engineering, 2013, 572026. doi:10.1155/2013/572026
Dokuyucu, M. A. (2020). Caputo and Atangana-Baleanu-Caputo Fractional Derivative Applied to Garden Equation. Turkish Journal of Science, 5(1), 1-7.
Ganguli, S., Kaur, G., & Sarkar, P. (2021). Global heuristic methods for reduced-order modelling of fractional-order systems in the delta domain: a unified approach. Ricerche di Matematica. doi:10.1007/s11587-021-00644-7
Gao, J., Chai, S., Shuai, M., Zhang, B., & Cui, L. (2018, July 25-27). Detecting False Data Injection Attack on Cyber-Physical System Based on Delta Operator. In: Proceedings of the 2018 37th Chinese Control Conference (CCC) (pp. 5961-5966). doi:10.23919/ChiCC.2018.8483314
Khanra, M., Pal, J., & Biswas, K. (2010, November 29-December 01). Rational approximation of fractional operator — A comparative study. In: Proceedings of the 2010 International Conference on Power, Control and Embedded Systems (pp. 1-5). doi:10.1109/ICPCES.2010.5698677
Khattri, S. K. (2009). New close form approximations of ln(1 + x). The Teaching of Mathematics, 12(1), 7-14.
Kothari, K., Mehta, U., & Prasad, R. (2019). Fractional-Order System Modeling and its Applications. Journal of Engineering Science and Technology Review, 12, 1-10. doi:10.25103/jestr.126.01
Krishna, B. T. (2011). Studies on fractional order differentiators and integrators: A survey. Signal Processing, 91(3), 386-426. doi:10.1016/j.sigpro.2010.06.022
Krishna, B. T. (2015, December 18-20). Design of Fractional order differintegrators using reduced order s to z transforms. In: Proceedings of the 2015 IEEE 10th International Conference on Industrial and Information Systems (ICIIS) (pp. 469-473). doi:10.1109/ICIINFS.2015.7399057
Keyser, R. D., & Muresan, C. I. (2016, October 09-12). Analysis of a new continuous-to-discrete-time operator for the approximation of fractional order systems. In: Proceedings of the 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC) (pp. 3211-3216). doi:10.1109/SMC.2016.7844728
Lamrabet, O., Tissir, E. H., & Haoussi, F. E. (2020, June 09-11). Controller design for delta operator time-delay systems subject to actuator saturation. In: Proceedings of the 2020 International Conference on Intelligent Systems and Computer Vision (ISCV). doi:10.1109/ISCV49265.2020.9204303
Maione, G. (2011). High-Speed Digital Realizations of Fractional Operators in the Delta Domain. IEEE Transactions on Automatic Control, 56(3), 697-702. doi:10.1109/TAC.2010.2101134
Middleton, R., & Goodwin, G. (1986). Improved finite word length characteristics in digital control using delta operators. IEEE Transactions on Automatic Control, 31(11), 1015-1021. doi:10.1109/TAC.1986.1104162
Middleton, R. H. & Goodwin, G. C. (1990a). Digital control and estimation: a unified approach. Prentice Hall.
Middleton, R. H., & Goodwin, G. C. (1990b). Digital Control and Estimation: A Unified Approach (Prentice Hall Information and System Sciences Series). Prentice Hall.
Miller, K. S., & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Willey Blackwell.
Nakagawa, M., & Sorimachi, K. (1992). Basic Characteristics of a Fractance Device. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 75, 1814-1819.
Oldham, K. B., & Spanier, J. (Eds.) (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier.
Oustaloup, A. (1995). La dérivation non entière: Théorie, synthèse et applications. Hermes.
Pan, I., & Das, S. (2013). Intelligent Fractional Order Systems and Control. Studies in Computational Intelligence (SCI, volume 438), Springer. doi:10.1007/978-3-642-31549-7
Podlubny, I. (1999). Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers. IEEE Transactions on Automatic Control, 44(1), 208-214. doi:10.1109/9.739144
Quezada-Téllez, L. A., Franco-Pérez, L., & Fernandez-Anaya, G. (2020). Controlling Chaos for a Fractional-Order Discrete System. IEEE Open Journal of Circuits and Systems, 1, 263-269. doi:10.1109/OJCAS.2020.3033154
Sarkar, P., Shekh, R. R., & Iqbal, A. (2016, November 09-11). A unified approach for reduced order modeling of fractional order system in delta domain. In: Proceedings of the 2016 International Automatic Control Conference (CACS) (pp. 257-262). doi:10.1109/CACS.2016.7973920
Skaar, S. B., Michel, A. N., & Miller, R. K. (1988). Stability of viscoelastic control systems. IEEE Transactions on Automatic Control, 33(4), 348-357. doi:10.1109/9.192189
Sun, H., Abdelwahab, A., & Onaral, B. (1984a). Linear approximation of transfer function with a pole of fractional power. IEEE Transactions on Automatic Control, 29(5), 441-444. doi:10.1109/TAC.1984.1103551
Sun, H. H., Onaral, B., & Tso, Y.-Y. (1984b). Application of the Positive Reality Principle to Metal Electrode Linear Polarization Phenomena. IEEE Transactions on Biomedical Engineering, BME-31(10), 664-674. doi:10.1109/TBME.1984.325317
Swarnakar, J., Sarkar, P., Dey, M., & Singh, L. J. (2017, December 21-23). Rational approximation of fractional order system in delta domain — A unified approach. In: Proceedings of the 2017 IEEE Region 10 Humanitarian Technology Conference (R10-HTC) (pp. 144-150). doi:10.1109/R10-HTC.2017.8288926
Tabatadze, V., Karaçuha, K., & Veliyev, E. I. (2020). The solution of the plane wave diffraction problem by two strips with different fractional boundary conditions. Journal of Electromagnetic Waves and Applications, 34(7), 881-893. doi:10.1080/09205071.2020.1759461
Vinagre, B. M., Podlubny, I., Hernández, A., & Feliu, V. (2000) Some approximations of fractional order operators used in control theory and applications. J. Fractional Calculus Appl. Anal., 4, 47-66.
Vinagre, B. M., Chen, Y. Q., & Petráš, I. (2003). Two direct Tustin discretization methods for fractional-order differentiator/integrator. Journal of the Franklin Institute, 340(5), 349-362. doi:10.1016/j.jfranklin.2003.08.001
Xue, D., Zhao, C., & Chen, Y. (2006, June 25-28). A Modified Approximation Method of Fractional Order System. In: Proceedings of the 2006 International Conference on Mechatronics and Automation (pp. 1043-1048). doi:10.1109/ICMA.2006.257769
Yumuk, E., Güzelkaya, M., & Eksin, İ. (2019). Analytical fractional PID controller design based on Bode’s ideal transfer function plus time delay. ISA Transactions, 91, 196-206. doi:10.1016/J.ISATRA.2019.01.034
Yumuk, E., Güzelkaya, M., & Eksin, İ. (2022). A robust fractional-order controller design with gain and phase margin specifications based on delayed Bode’s ideal transfer function. Journal of the Franklin Institute, 359(11), 5341-5353. doi:10.1016/J.JFRANKLIN.2022.05.033
Zhao, Y., & Zhang, D. (2017, May 24-26). H∞ fault detection for uncertain delta operator systems with packet dropout and limited communication. In: Proceedings of the 2017 American Control Conference (ACC) (4772-4777). doi:10.23919/ACC.2017.7963693
Dolai, S. K., Mondal, A., & Sarkar, P. (2022). Discretization of Fractional Order Operator in Delta Domain. Gazi University Journal of Science Part A: Engineering and Innovation, 9(4), 401-420. https://doi.org/10.54287/gujsa.1167156