Open Access Open Access  Restricted Access Subscription or Fee Access

A New Multimodel Representation of Fractional-Order Systems in Both Time and Frequency Domains

(*) Corresponding author

Authors' affiliations



In order to deal with some difficult problems in fractional-order systems, like high computational load of fractional-order operator, fractional-order transfer function is commonly approximated by an integer order model. However, the dimension of this model increases with its accuracy, which can make the design of a controller more difficult. In this paper, a new approach for modelling of fractional-order systems is investigated. Exploiting the multimodel technique, the suggested method replaces the unique fractional-order model by a set of simpler integer order models. The determination of the different models is based on an approximation of the fractional-order derivative operator sα. Then the global model is obtained through a fusion of the simple models weighted by their respective relevance degrees calculated by optimizing a constrained least squares problem. The resulting final model can represent adequately the fractional-order systems both in time and in frequency domains. Simulations and comparative studies carried out on academic examples indicate the interest, the clarity and the improvement in accuracy in both time and frequency domains of the proposed modelling method, compared to modelling by a single model based on the approximation of Oustaloup.
Copyright © 2020 Praise Worthy Prize - All rights reserved.


Fractional-Order Systems; Multimodel Approach; Fractional-Order Derivative

Full Text:



A. van der Ziel, Unified Presentation of 1/f Noise in Electronic Devices: Fundamental 1/f Noise Sources, Proceedings of the IEEE, Vol. 76(Issue 3):233-258, March 1988.

W. Willinger, M.S. Taqqu, W.E. Leland, and D.V. Wilson, Self-Similarity in High-Speed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements, Statistical Science, Vol. 10(Issue 1):67-85, February 1995.

A. Scotti, C. Meneveau, and S.G. Saddoughi, Fractal Dimension of Velocity Signals in High-Reynolds-Numbers Hydrodynamic Turbulence, Physical Review E, Vol. 51 (Issue 6):5594-5608, June 1995.

H.H. Sun, and B. Onaral, A Unified Approach to Represent Metal Electrode Polarization, IEEE Trans. Biomed. Eng., Vol. BME-30(Issue 7):399-406, July 1983.

C. Bonnet, and J.R. Partington, Coprime Factorizations and Stability of Fractional Differential Systems, Systems & Control Letters, Vol. 41(Issue 3):167-174, October 2000.

J. Sabatier, M. Moze, and C. Farges, LMI Stability Conditions for Fractional Order Systems, Computers and Mathematics with Applications, Vol. 59(Issue 5):1594-1609, March 2010.

S. Zheng, B. Liang, F. Liu, Z. Yang, and Y. Xie, Robust stability of fractional order system with polynomial uncertainties based on sum-of-squares approach, Journal of the Franklin Institute, Vol. 357(Issue 12): 8035–8058, 2020.

B. B. Alagoz, Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations, ISA Transactions, Vol. 70: 7-15, 2017.

B. K. Lenka, Fractional comparison method and asymptotic stability results for multivariable fractional order systems, Communications in Nonlinear Science and Numerical Simulation, Vol.69: pp.398-415, April 2019.

X. Zhang, and Z. Wang, Stability and robust stabilization of uncertain switched fractional order systems, ISA Transactions, Vol. 103, pp. 1-9, August 2020.

J. Yude, D. Mingxing and G. Yanping, Stabilization of Non-Linear Fractional-Order Uncertain Systems, Asian Journal of Control, Vol. 20 (Issue 2): 669-677, March 2018.

L. Chong, C. Bing, S. Peng, and Y. Jin-Peng, Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems, Systems & Control Letters, Vol. 112, pp. 31-35, February 2018.

W. Yiheng, W. Jiachang, L. Tianyu, and W. Yong, Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state, Journal of the Franklin Institute, Vol. 356 (Issue 4):1975-1990, March 2019.

F. Tian, W. Baowei, L. Lili, W. Yue-E, Finite-Time Stability and Stabilization of Fractional-Order Switched Singular Continuous-Time Systems, Circuits, Systems, and Signal Processing, Vol. 38 (Issue 12): 5528-554, December 2019.

S. Zhou, J. Cao, and Y. Chen, Genetic Algorithm-Based Identification of Fractional-Order Systems, Entropy, Vol. 15(Issue 5):1624-1642, May 2013.

W. Du, Q. Miao, L. Tong, and Y. Tang, Identification of fractional-order systems with unknown initial values and structure, Physics Letters A, Vol. 381(Issue23): 1943-1949, June 2017.

Y. Lu, Y. Tang, X Zhang, and S. Wang, Parameter identification of fractional order systems with nonzero initial conditions based on block pulse functions, Measurement, Vol. 158, July 2020.

Z. Gao, Modulating function-based system identification for a fractional-order system with a time delay involving measurement noise using least-squares method, International Journal of Systems Science, Vol. 48(Issue 7): 1460-1471, 2017.

Y. Tang, N. Li, M. Liu, Y. Lu, and W. Wang, Identification of fractional-order systems with time delays using block pulse functions, Mechanical Systems and Signal Processing, Vol. 91: 382-394, July 2017.

S.M. Fahim, S. Ahmed, and S.A. Imtiaz, Fractional order model identification using the sinusoidal input, ISA Transactions, Vol. 83: 35-41, December 2018.

G. Liu, L. Wang, W.L. Luo, J.K. Liu, and Z.R. Lu, Parameter Identification of Fractional Order System using Enhanced Response Sensitivity Approach, Communications in Nonlinear Science and Numerical Simulation, Vol. 67: pp. 492-505, February 2019.

T.T. Hartley, and C.F. Lorenzo, Dynamics and Control of Initialized Fractional Order Systems, Nonlinear Dynamics, Vol. 29(Issue 1-4): 201-233, July 2002.

A. Rhouma, F. Bouani, B. Bouzouita, and M. Ksouri, Model Predictive Control of Fractional Order Systems, Journal of Computational and Nonlinear Dynamics, Vol. 9(Issue 3):310111-310117, February 2014.

A. J. Muñoz-Vázquez, J. D. Sánchez-Torres, and M. Defoort, Second-order predefined-time sliding-mode control of fractional-order systems, Asian Journal of Control, 1-9, 2020.

J. Wang, C. Shao, and Y-Q. Chen, Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance, Mechatronics, Vol. 53, pp. 8-19, August 2018.

A.J. Muñoz‐Vázquez, V. Parra‐Vega, A. Sánchez‐Orta, and F. Martínez‐Reyes, Robust Mittag‐Leffler stabilisation of fractional‐order systems, Asian Journal of Control, pp. 1-9, July 2019.

D. Xue, C. Zhao, and Y. Chen, A modified approximation method of fractional order system, International Conference on Mechatronics and Automation, pp 1043-1048, Luoyang, June 2006.

Y. Wei, J. Wang T. Liu, and Y. Wang, Fixed pole based modeling and simulation schemes for fractional order systems, ISA Transactions, Vol. 84: 43-54, 2019.

Gepreel, K., Nofal, T., Approximate Solutions for Nonlinear Partial Fractional Differential Equations, (2013) International Review of Physics (IREPHY), 7 (4), pp. 323-330.

B. Du, Y. Wei, S. Liang, and Y. Wang, Rational approximation of fractional order systems by vector fitting method, International Journal of Control, Automation and Systems, Vol. 15(Issue 1): 186-195, 2017.

M.D. Ortigueira, and J.A.T. Machado, New discrete-time fractional derivatives based on the bilinear transformation: definitions and properties, Journal of Advanced Research, Vol. 25, pp: 1-10, September 2020.

G. Maione, and M.P. Lazarevié, On the Symmetric Distribution of Interlaced Zero-Pole Pairs approximating the Discrete Fractional Tustin Operator, IEEE International Conference on Systems, Man and Cybernetics (SMC), pp. 2578-2583,Bari, Italy, October 2019.

D. Casagrande, W. Krajewski, and U. Viaro, The Integer-Order Approximation of Fractional-Order Systems in The Loewner Framework, International Federation of Automatic Control, 15th IFAC Symposium on Large Scale Complex Systems LSS, 26-28 May 2019, Delft, Netherlands, Vol. 52 (Issue 3): 43-48, 2019.

D.K. Robin, I.M. Cristina, and M.I. Clara, An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions, ISA Transactions, Vol.74, pp. 229-238, March 2018.

S. Mahata, R. Kar, and D. Mandal, Optimal approximation of fractional-order systems with model validation using CFOA, IET Signal Processing, Vol. 13 (Issue 9): 787-797, Decembre 2019.

M. Shibendu, S. Suman Kumar, K. Rajib, and M. Durbadal, Accurate integer-order rational approximation of fractional-order low-pass Butterworth filter using a metaheuristic optimisation approach, IET Signal Processing, Vol. 13 (Issue 5): 581-589, July 2018.

W. Krajewski, and U. Viaro, A Method for the Integer-Order Approximation of Fractional-Order Systems, Journal of Franklin Institute, Vol. 351(Issue 1): 555-564, January 2014.

M. Rydel, New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian, Advances in Computational Mathematics, Vol. 45 (Issue 2): 631-653, 2019.

M. Rydel, R. Stanisławski, and K. Latawiec, Krzysztof, Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems, Symmetry, Vol.11 (Issue 2), February 2019.

R. Stanisławski, M. Rydel, and K. Latawiec, Modeling of discrete-time fractional-order state space systems using the balanced truncation method, Journal of the Franklin Institute, Vol. 354 (Issue 7): 3008-3020, May 2017.

B .Bourouba, S. Ladaci, and A. Chaabi, Reduced-Order Model Approximation of Fractional-Order Systems Using Differential Evolution Algorithm, Journal of Control, Automation and Electrical Systems, Vol.29, pp. 32-43, February 2018.

S. Talmoudi, and M. Lahmari, The Multi-model Approach for Fractional-Order Systems Modelling, Transactions of the Institute of Measurement and Control, Vol. 40(Issue 1):331-340, January 2018.

N. Hassene, Contribution to analysis of fractional order systems, Ph.D. dissertation, Elect. Dept., University of Constantine 1, Algeria, 2013.

I. Podlubny, Fractional Differential Equations (Academic Press, 1999).

K.S. Miller, and B. Ross, An introduction to the fractional calculus and fractional differential equations (A Wiley-Interscience Publication, 1993).

M. Aoun, R. Malti, F. Levron, and A. Oustaloup, Numerical simulations of fractional systems, Biennial Conference on Mechanical Vibration and Noise, Vol. 5, pp. 745-752, Chicago, Illinois, September 2003.

K. B. Oldham, and J. Spanier, The Fractional Calculus, (Academic press, 1974).

S. Talmoudi, K. Abderrahim, R. Ben Abdennour, and M. Ksouri, Multimodel Approach Using Neural Networks for Complex Systems Modeling and Identification, Nonlinear Dynamics and Systems Theory, Vol. 8(Issue 3):299-316, 2008.

N. Elfelly, J.Y. Dieulot, M. Benrejeb, and P. Borne, A Multimodel Approach for Complex Systems Modeling Based on Classification Algorithms, Int. J. of Computers, Communications and Control, Vol. 7(Issue 4): 645-660, November 2012.

H. Bennasr, and F. M’Sahli, Multimodel representation of complex nonlinear systems: a multifaceted approach for real-time application, Mathematical Problems in Engineering, Vol. 2018, 1-15, 2018.

M. Ksouri, Contributions to multimodel control of complex processes, Ph.D. dissertation, Lille University of Science and Technology, France, 1999.

A.M. Nagy Kiss, B. Marx, J. Ragot, and G. Schutz, Systematic multimodeling methodology applied to an activated sludge reactor model, Industrial & Engineering Chemistry Research, Vol. 49(Issue 6):2790-2799, February 2010.

Laabidi, K., Ksouri, M., Multimodel Approach in the Frequency Domain, (2017) International Journal on Information Technology (IREIT), 5 (6), pp. 156-163.

F. Delmotte, Multimodel Analysis, Ph.D. dissertation, Lille University of Science and Technology, France, 1997.

S. Talmoudi, K. Abderrahim, R. Ben Abdennour, and M. Ksouri, A New Technique of Validities' Computation for Models' Base, WSEAS Transactions on Circuits and Systems, Vol. 2(Issue 4):680-685, October 2003.

R. Ben mohamed, H. Ben Nasr, and F. M’sahli, A Multimodel Approach for a Nonlinear System based on Neural Network Validity, International Journal of Intelligent Computing and Cybernetics, Vol. 4(Issue 3):331-352, August 2011.

A. Ben Messaoud, S. Talmoudi Ben Aoun, and M. Lahmari Ksouri, A New Strategy of Validities’ Computation for Multimodel Approach: Experimental Validation, International Journal of Advanced Computer Science and Applications (IJACSA), Vol. 8(Issue 7), January 2017.

A. Ben Messaoud, S. Talmoudi Ben Aoun, and M. Lahmari Ksouri, Multimodel Approach for Modelling of Nonlinear Systems: Validities’ Optimal Computation, International Journal for Computation and Mathematics in Electrical and Electronic Engineering (COMPEL), Vol. 37(Issue 1):153-175, 2018.

S.A. Emami, and A. Roudbari, Multi-Model ELM Based Identification of an Aircraft Dynamics in the Entire Flight Envelope, IEEE Transactions on Aerospace and Electronic Systems, Vol. 55 (Issue 5): 2181-2194, October 2019.

R. Malti, S. Victor, A. Oustaloup, and H. Garnier, An optimal instrumental variable method for continuous-time fractional model identification, IFAC Proceedings, Vol. 41, pp. 14379-14384, Seoul, Korea, July 2008.

H. Sheng, Y.Q. Chen, and T.S. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, (Springer London Dordrecht Heidelberg, 2012).

T. J. Freeborn, B. Maundy, and A. S. Elwakil, Fractional-Order Models of Supercapacitors, Batteries and Fuel Cells: a Survey, Mater. Renew. Sustain. Energy, Vol. 4(Issue 9), 2015.

J.J. Quintana, A. Ramos, I. Nuez, Identification of the fractional impedance of ultracapacitors, IFAC Proceedings Volumes, Vol.39 (Issues 11):432-436, January 2006.


  • There are currently no refbacks.

Please send any question about this web site to
Copyright © 2005-2023 Praise Worthy Prize