### A NEW APPROACH FOR DIRECT DISCRETIZATION OF FRACTIONAL ORDER OPERATOR IN DELTA DOMAIN

**DOI Number**

**First page**

**Last page**

#### Abstract

The fractional order system (FOS)** **comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s^{±}^{m}, 0 < m < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s^{±}^{m}is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain.

#### Keywords

#### Full Text:

PDF#### References

K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Dover Books on Mathematics, 2006.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993.

R. Caponetto, G. Dongola, L. Fortuna and I. Petras, Fractional Order Systems, Modeling and Control Applications, Singapore: World Scientific, 2010, pp. 43–54.

H. H. Sun, B. Onaral and Y. Tsao, "Application of the positive reality principle to metal electrode linear polarization phenomena", IEEE Trans. Biomed., Eng. vol. 31, pp. 664–674, 1984.

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

S. B. Skaar, A. N. Michel and R. K. Miller, "Stability of viscoelastic control systems", IEEE Trans. Auto. Control., vol. 33, no. 4, pp. 348–357, 1988.

N. Engheta, "Fractional calculus and fractional paradigm in electromagnetic theory", In Proceedings of the VIIth International Conference on Mathemathical Methods in Electromagnetic Theory, 1998, pp. 879–880.

I. Podlubny, "Fractional-order systems and PIλDμ controllers", IEEE Trans. Automatic Contro.l, vol. 44, pp. 208–214, 1999.

N. Nakagava and K. Sorimachi,"Basic characteristics of a fractance device" IEICE Trans. Fundamentals, vol. E75-A, pp.1814–1819, 1992.

B. M. Vinagre, I Podlubny, A Hernández and V. Feliu, "Some approximation of fractional-order operator used in control theory and application", J. Fractional Calc. and Appl., vol. 3, pp. 231–248, 2000.

D. Xue, C. Zhao and Y. Chen, "A modified approximation method of fractional order system", In Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA), Luoyang, 2006, pp. 1043–1048.

M. Khanra, J. Pal and K. Biswas, "Rational approximation of fractional operator –a comparative study", In Proceedings of the IEEE international conference on power, Control and Embedded Systems (ICPCES), Allahabad, 2010, pp. 1–5.

A. Oustaloup, La Derivation Non Entiere: Theorie, Synthese et Applications. Hermes, Paris, 1995.

B. T. Krishna, "Studies on fractional-order differentiators and integrators: A survey", Signal Processing, vol. 91, pp. 386–426, 2011.

G. Maione, "High-speed digital realizations of fractional operator in the delta-domain", IEEE Trans. Automat. Control, vol. 56, pp. 697–702, 2011.

R. D. Keyser and C. I. Muresan, "Analysis of a new continuous-to-discrete-time operator for the approximation of fractional order system", In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC), Budapest, 2016, pp. 003211–003216.

B. T. Krishna, "Design of fractional-order differ integrators using reduced order s to z transforms", In Proceeding of the 10th IEEE International Conference on Industrial and Information Systems, Peradeniya, 2015, pp. 469–473.

Y. Q. Chen and K. L. Moore, "Discretization schemes for fractional-order differentiators and Integrators", IEEE Trans. Circuits System., vol. 49, no. 3, pp. 363–367, 2000.

Y. Chen, I. Petráš and D. Xue, "Fractional Order Control-A Tutorial", In Proceeding of the American Control Conference, 2009, pp. 1397–1411.

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

R. H. Middleton and G. C. Goodwin, "Improved finite word length characteristics in digital control using delta operator", IEEE Trans. Automatic Cont., vol. 31, pp. 1015–1021, 1986.

J. Cortes-Romero, A. luviana-Juarez and H. Sira-Ramirez, "A Delta Operator Approach for the Discrete-Time Active Disturbance Rejection Control on Induction Motors", Math. Probl. Eng., vol. 2013, pp. 1–9, 2013.

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, Seattle, 2017, pp. 4772–4777.

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 Proceeding of the 37th Chinese Control Conference, Wuhan, 2018, pp. 5961–5966.

J. Swarnakar, P. Sarkar, M. Dey and L. Joyprakash Singh, "A unified approach for reduced order modelling of fractional order system in delta domain- a unified approach", In Proceeding of the IEEE region 10 Humanitarian Technology Conference (R-10 HTC), 2017, pp. 144–150.

P. Sarkar, R. R. Shekh and A. Iqbal, "A unified approach for reduced order modelling of fractional order system in delta domain", In Proceeding of the IEEE International Automatic Control Conference (CACS), Taichung, 2016, pp. 257–262.

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, 2021.

L. A. Quezada-Téllez, L. Franco-Pérez and G. Fernandez-Anaya, "Controlling Chaos for a Fractional-Order Discrete System", IEEE Open J. Circuits Syst., vol. 1, pp. 263–269, 2020.

O. Lamrabet, E. H. Tissir and F. E. 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, pp. 1–6.

I. Pan and S. Das, Intelligent Fractional Order Systems and Control, Studies in computational intelligence, Springer-Verlag, Berlin Heidelberg, 2013.

B. M. Vinagre, Y. Q. Chen and I. Petrášc, "Two direct Tustin discretization methods for fractional-order differentiator/integrator", J. Franklin Institute, vol. 340, pp. 349–362, 2003.

S. K. Khattri, "New close form approximation of ln(1+x)", The Teaching of Mathematics, vol. 12, pp. 7–14, 2009.

J. Baranowski, W. Bauer, M. Zagorowska, T. Dziwinski and P. Piatek, "Time-domain oustaloup approximation",” In Proceedings of the 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 2015, pp. 116–120.

### Refbacks

- There are currently no refbacks.

ISSN: 0353-3670 (Print)

ISSN: 2217-5997 (Online)

COBISS.SR-ID 12826626