Arindam Mondal, Sujay Kumar Dolai, Prasanta Sarkar

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The fractional order operator ( , ) plays the pivotal role for the realization of fractional orders systems (FOS). For the realization of the FOS, fractional order operator (FOO) needs to be realized either in discrete or continuous time domain. Discrete time rational approximation of FOO in the -domain fails to provide meaningful information at fast sampling interval. Moreover, domain rational transfer function becomes highly sensitive with respect to its coefficients variation resulting to the poor finite word length effects for digital realization. In the other hand delta operator parameterized system allows to develop unification of continuous and discrete time formulations leading to the development of a unified framework for digital realization at fast sampling interval. The discrete time approximation of the FOO in delta domain is found to be robust to its coefficient variation in comparison to the shift operator based discretization of FOO. In this paper, discrete -operator parameterization is proposed for the digital realization using direct discretization of FOO. As a result, superior finite word length effect is observed for the realization of the FOO in discrete delta domain. Fractional order operator with different orders ( ) are considered for the realization purpose using the proposed method and the results obtained using MATLAB are presented for validation.


Delta domain, delta operator parameterization, finite word length effects, fractional order operator (FOO), fractional order system (FOS)

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A. Oustaloup, La Commande CRONE. Commande Robuste d’Ordre non Entièr. Paris, France: Editions Hermès, 1991.

I. Podlubny, Fractional Differential Equations. San Diego, CA: Academic Press, 1999.

Y. Q. Chen, I. Petrá and D. Xue, “Fractional order control - A tutorial,” In Proceedings of the American Control Conference, 2009, pp. 1397–1411.

J. A. T. Machado, "Analysis and design of fractional-order digital control systems", Syst. Anal. Model. Simul., vol. 27, pp. 107-122, 1997.

N. Engheta, "Fractional calculus and fractional paradigm in electromagnetic theory", In Proceedings of the International Conference on Mathematical Methods in Electromagnetic Theory (MMET 98), 1998, pp. 43-49.

H. H. Sun, A. A. Abdelwahab and B. Onaral, "Linear approximation of transfer function with a pole of fractional power", IEEE Trans. Autom. Control, vol. 29, pp. 441–444, 1984.

A. Oustaloup, F. Levron, B. Mathieu and F. M. Nanot, "Frequency band complex noninteger differentiator: characterization and synthesis", IEEE Trans. Circuits Sysemt I: Fundam. Theory Appl., vol. 47, no. 1, pp. 25-39, Jan. 2000.

B. M. Vinagre, I. Podlubny, A. Hernandez and V. Feliu, "Some approximations of fractional-order operators used in control theory and applications", J. Frac. Calc. Appl. Anal., vol. 3, no. 3, pp. 231-248, 2000.

Y. Q. Chen, B. M. Vinagre and I. Podlubny, "Continued fraction expansion approaches to discretizing fractional-order derivatives. An expository review", Nonlin. Dynam., Spec. Issue Frac. Derivatives Appl., vol. 38, no. 1-2, pp. 155-170, Dec. 2004.

Y. Q. Chen and K. L. Moore, "Discretization schemes for fractional order differentiators and integrators", IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., vol. 49, no. 3, pp. 363-367, Mar. 2002.

M. A. Al-Alaui, "Novel digital integrator and differentiator", Electron. Lett., vol. 29, no. 4, pp. 376-378, 1993.

Y. Q. Chen and B. M. Vinagre, "A new IIR-type digital fractional-order differentiator", Signal Process., vol. 83, pp. 2359-2365, 2003.

A. Khodabakhshian, V. J. Gosbell and F. Coowar, "Discretization of power system transfer functions", IEEE Trans. Power Syst., vol. 9, no. 1, pp. 255-261, Feb. 1994.

H.-M. Cheng and T.-C. Chiu, "Wordlength estimation of digital controller synthesis for inkjet printer mechanism", J. Comput., vol. 3, no. 4, pp. 50-57, Apr. 2008.

M. J. Newmann and D. G. Holmes, "Delta operator digital filters for high performance inverter applications", IEEE Trans. Power Electron., vol. 18, no. 1, pp. 447-454, Jan. 2003.

G. C. Goodwin, R. H. Middleton and V. Poor, "High-speed digital signal processing and control",” Proc. IEEE, vol. 80, no. 2, pp. 240-259, Feb. 1992.

R. H. Middleton and G. C. Goodwin, Digital Control and Estimation. A Unified Approach. Englewood Cliffs, NJ: Prentice-Hall, 1990.

R. H. Middleton and G. C. Goodwin, "Improved finite word length characteristics in digital control using delta operators", IEEE Trans. Autom. Control, vol. AC-31, no. 11, pp. 1015-1021, Nov. 1986.

G. Maione, "High-Speed Digital Realizations of Fractional Operators in the Delta Domain", IEEE Trans. Autom. Control, vol. 56, no. 3, March 2011.

Y. Zhao and D. Zhang, "H∞ fault detection for uncertain delta operator systems with packet dropout and limited communication", In Proceedings of the American Control Conference (ACC), 2017, pp. 4772-4777.

O. Lamrabet, E. H. Tissir, and F. E. L. Haoussi, " Controller design for delta operator time-delay systems subject to actuator saturation", In Proceedings of the International Conference on Intelligent Systems and Computer Vision (ISCV 2020), Jun. 2020, pp. 1-6.

S. K. Dolai, A. Mondal and P. Sarkar, "Discretization of Fractional Order Operator in Delta Domai",.GU J. Sci., Part A, vol. 9, no. 4, pp. 401-420, 2022.

S. Ganguli, G. Kaur and P. Sarkar, "Global heuristic methods for reduced-order modelling of fractional-order systems in the delta domain: a unified approach", Ricerche di Matematica, Aug. 2021.

J. Gao, S. Chai, M. Shuai, B. Zhang and L. Cui, " Detecting False Data Injection Attack on Cyber-Physical System Based on Delta Operator", In Proceedings of the 37th Chinese Control Conference (CCC), 2018. pp. 5961–5966.

J. P. Mishra and X. Yu, "Delta-Operator-Based Reaching Laws for Sliding Mode Control Design", IEEE Trans. Circ. Syst. II: Express Briefs, vol. 69, no. 4, pp. 2136-2140, April 2022.

Y. Xue, J. Han, Z. Tu and X. Chen, "Stability analysis and design of cooperative control for linear delta operator system", AIMS Mathematics, vol. 8, no. 6, pp.12671-12693, 2023.

B. Zheng, Y. Wu, H. Li and Z. Chen, "Adaptive Sliding Mode Attitude Control of Quadrotor UAVs Based on the Delta Operator Framework", Symmetry, vol. 14, no. 3, p. 498, 2022.

X. Zhang, F. Ding, L. Xu and E. Yang, "Highly computationally efficient state filter based on the delta operator", Int. J. Adapt. Control Signal Process., vol. 6, pp. 875-889, 2019.

H. Rachid, L. Ouarda and T. El Houssaine, "Stabilization of Delta Operator Systems with Actuator Saturation via an Anti-Windup Compensator", Symmetry, vol. 11, no. 9, p. 1084, 2019.

J. Leo Amalraj, M. Maria Susai Manuel, M. Meganathan and M. Syed Ali, "The Generalized Fractional Proportional Delta Operator and New Generalized Transforms in Discrete Fractional Calculus", Math. Prob. Eng., Hindawi, vol. 2022, p. 4849312, 2022.

A. Biswas, A. Mondal and P. Sarkar," Design and implementation of digital controller in delta domain for buck converter", FU Elec. Energ., vol. 36, no. 1, pp. 103-119, 2023.

F. Yamin and Z. Duanjin,"Robust fault detection for delta operator switched fuzzy systems with bilateral packet losses", J. Syst. Eng. Electron., vol. 34, no. 1, pp. 214-223, Feb. 2023.

N. Sanjay and K. Kumar, "New close form approximations of ln(1 + x)", Teach. Math., vol. 12, no. 1, pp. 7-14, 2009.

S. K. Dolai, A. Mondal, and P. A. Sarkar, "New Approach for Direct Discretization of Fractional order Operator in Delta Domain", FU Elec. Energ., vol. 35, no. 3, pp. 313-331, 2022.

Y. Chen, B. M. Vinagre and I. Podlubny, "Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review", Nonlinear Dyn., vol. 38, no. 1, pp. 155-170, Dec. 2004.


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