Abstract
This paper presents a stable reduced order approximation method of fractional-order systems based on the grey wolf optimizer hybridized with cuckoo search algorithm (GWO-CS). The proposed method is applied to factional order transfer functions (FOTF) in order to obtain a low-order model exhibiting good fits to the Bode’s ideal transfer function (TF) of the original system, taking into account the stability criteria of the reduced model. In the first stage, the Oustaloup approximation of a given FOTF is applied to obtain a high integer-order TF that matches with the original FOTF. In the second stage, The GWO-CS method is compared with the frequency-limited balanced truncation method resulting from Oustaloup approximation of the original FOTF in a specified frequency range and other existing meta-heuristic based model order reduction approaches. Simulation of different numerical examples confirms and validates the effectiveness of the proposed approach.
Similar content being viewed by others
Data availability
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
References
R.K. Appiah, Linear model reduction using Hurwitz polynomial approximation. Int. J. Control 28(3), 477–488 (1978). https://doi.org/10.1080/00207177808922472
P. Benner, P. Kürschner, J. Saak, Frequency-limited balanced truncation with low-rank approximations. SIAM J. Sci. Comput. 38, A471–A499 (2016)
B. Bourouba, S. Ladaci, A. Chaabi, Reduced-order model approximation of fractional-order systems using differential evolution algorithm. J. Control Autom. Electr. Syst. 29, 32–43 (2018). https://doi.org/10.1007/s40313-017-0356-5
S. Chakraborty, S.S. Kandala, C.P. Vyasarayani, Reduced ordered modelling of time delay systems using Galerkin approximations and eigenvalue decomposition. Int. J. Dyn. Control 7, 1065–1083 (2019). https://doi.org/10.1007/s40435-019-00510-3
A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992). https://doi.org/10.1109/9.159595
B. Du, Y. Wei, S. Liang, Y. Wang, Rational approximation of fractional order systems by vector fitting method. Int. J. Control Autom. Syst. 15, 186–195 (2017)
Z. Erlangung, Model order reduction of linear control systems: comparison of balance truncation and singular perturbation approximation with application to optimal control, thesis (2016)
T.J. Freeborn, B. Maundy, A.S. Elwakil, Fractional order models of super capacitors, batteries and fuel cells: a survey. Mater. Renew. Sustain. Energy 4, 1–7 (2015)
R.W. Freund, Model reduction methods based on Krylov subspaces. Acta Numerica 12, 267–319 (2003)
M. Gonzalez-Lee, H. Vazquez-Leal, J.F. Gomez-Aguilar, L.J. Morales-Mendoza, V.M. Jimenez-Fernandez, J.R. Laguna-Camacho, C.M. Calderon-Ramon, Exploring the cross-correlation as a means for detecting digital watermarks and its reformulation into the fractional calculus framework. IEEE Access 6(6), 71699–71718 (2018). https://doi.org/10.1109/ACCESS.2018.2882405
S. Gugercin, A. Antoulas, A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2004)
A.K. Gupta, D. Kumar, P. Samuel, A meta-heuristic cuckoo search and eigen permutation approach for model order reduction. Sadhana 43, 65 (2018)
J. Jerabek, R. Sotner, J. Dvorak, J. Polak, D. Kubanek, N. Herencsar, J. Koton, Reconfigurable fractional-order filter with electronically controllable slope of attenuation, pole frequency and type of approximation. J. Circ. Syst. Comput. 26(10), 1750157 (2017)
W. Krajewski, U. Viaro, A method for the integer-order approximation of fractional-order systems. J. Frankl. Inst. 351, 555–564 (2014)
P. Kürschner, Balanced truncation model order reduction in limited time intervals for large systems. Adv. Comput. Math. 44, 1821–1844 (2018)
J. Lam, Model reduction of delay systems using Padé approximants. Int. J. Control 57(2), 377–39 (1993)
J.E. Lavin-Delgado, J.E. Solis-Perez, J.F. Gomez-Aguilar, R.F. Escobar-Jimenez, Robust optical flow estimation involving exponential fractional-order derivatives. Optik (2020). https://doi.org/10.1016/j.ijleo.2019.163642
J.E. Lavin-Delgado, J.E. Solis-Perez, J.F. Gomez-Aguilar et al., A new fractional-order mask for image edge detection based on Caputo–Fabrizio fractional-order derivative without singular kernel. Circ. Syst. Signal Process. 39, 1419–1448 (2020). https://doi.org/10.1007/s00034-019-01200-3
J.E. Lavin-Delgado, J.E. Solis-Perez, J.F. Gomez-Aguilar et al., Fractional speeded up robust features detector with the Caputo–Fabrizio derivative. Multimed. Tools Appl. 79, 32957–32972 (2020). https://doi.org/10.1007/s11042-020-09547-5
N. Liu, S. Cao, J. Fei, Fractional-order PID controller for active power filter using active disturbance rejection control. Math. Probl. Eng. 2019, 10 (2019). https://doi.org/10.1155/2019/6907570
W. Michiels, G. Hilhorst, G. Pipeleers, T. Vyhlidal, J. Swevers, Reduced modelling and fixed order control of delay systems applied to a heat exchanger. IET Control Theory Appl. 11(18), 3341–3352 (2017)
S. Mirjalili, S.M. Mirjalili, A. Lewis, Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)
C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional Order Systems and Controls. Fundamentals and Applications, Ser. Advances in Industrial Control (Springer, London, 2010)
K. Oldham, J. Spanier, The fractional calculus; theory and applications of differentiation and integration to arbitrary order, in Mathematics in Science and Engineering 5 (Academic Press, New York, 1974)
A. Oustaloup, La Dérivation Non Entière: Théorie, Synthèse et Applications (Hermes, Paris, 1995)
A. Oustaloup, F. Levron, B. Mathieu, Frequency band complex non integer differentiator? Characterization and synthesis. IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(1), 25–39 (2000)
A.D. Pano-Azucena, B. Ovilla-Martinez, E. Tlelo-Cuautle, J.M. Munoz-Pacheco, L.G. de la Fraga, FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grunwald–Letnikov method. Commun. Nonlinear Sci. Numer. Simul. 72, 516–527 (2019). https://doi.org/10.1016/j.cnsns.2019.01.014
I. Petras, Tuning and implementation methods for fractional order controllers. Fract. Calc. Appl. Anal. 15(2), 282–303 (2012)
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Ser. Mathematics in Science and Engineering (Elsevier, Amsterdam, 1998), pp. 1–340
M. Rydel, R. Stanisławski, K.J. Latawiec, Balanced truncation model order reduction in limited frequency and time intervals for discrete-time commensurate fractional order systems. Symmetry 11(2), 258 (2019). https://doi.org/10.3390/sym11020258
Y. Shamash, Model reduction using the Routh stability criterion and the Pade approximation technique. Int. J. Control 21(3), 475–484 (1975). https://doi.org/10.1080/00207177508922004
A. Sikander, P. Thakur, Reduced order modelling of linear time-invariant system using modified cuckoo search algorithm. Soft Comput. 22, 3449–3459 (2018)
A. Silva-Juarez, E. Tlelo-Cuautle, L.G. de la Fraga, R. Li, FPAA-based implementation of fractional-order chaotic oscillators using first-order active filter blocks. J. Adv. Res. 25, 77–85 (2020). https://doi.org/10.1016/j.jare.2020.05.014
J.E. Solis-Perez, J.F. Gomez-Aguilar, R.F. Escobar-Jimenez, J. Reyes-Reyes, Blood vessel detection based on fractional Hessian matrix with non-singular Mittag–Leffler Gaussian kernel. Biomed. Signal Process. Control (2019). https://doi.org/10.1016/j.bspc.2019.101584
H.N. Soloklo, N. Bigdeli, Direct approximation of fractional order systems as a reduced integer/fractional-order model by genetic algorithm. Sadhana 45, 277 (2020). https://doi.org/10.1007/s12046-020-01503-1
R. Stanisławski, M. Rydel, K.J. Latawiec, Modelling of discrete-time fractional-order state space systems using the balanced truncation method. J. Frankl. Inst. 354, 3008–3020 (2017)
Y. Tang, H. Liu, W. Wang, X. Guan, Parameter identification of fractional order systems using block pulse functions. Signal Process. 107, 272–281 (2015)
M. Tavakoli-Kakhki, M. Haeri, Model reduction in commensurate fractional-order linear systems. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 223, 493–505 (2009)
G. Tsirimokou, C. Psychalinos, A.S. Elwakil, K.N. Salama, Electronically tunable fully integrated fractional-order resonator. IEEE Trans. Circ. Syst. II Express Br. Print (2017). https://doi.org/10.1109/TCSII.2017.2684710
C. Vastarouchas, G. Tsirimokou, T.J. Freeborn, C. Psychalinos, Emulation of an electrical-analogue of a fractional-order human respiratory mechanical impedance model using OTA topologies. AEU Int. J. Electron. Commun. 78, 201–208 (2017). https://doi.org/10.1016/j.aeue.2017.03.021
B.W. Wan, Linear model reduction using Mihailov criterion and Pade approximation technique. Int. J. Control 33(6), 1073–1089 (1981). https://doi.org/10.1080/00207178108922977
J. Wiora, A. Wiora, Influence of methods approximating fractional-order differentiation on the output signal illustrated by three variants of oustaloup filter. Symmetry 12(11), 1898 (2020). https://doi.org/10.3390/sym12111898
W. Wyss, The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)
H. Xu, X. Liu, J. Su, An improved grey wolf optimizer algorithm integrated with Cuckoo Search, in 9th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems Technology(IDAACS), pp. 490–493 (2017). https://doi.org/10.1109/IDAACS.2017.8095129
D. Xue, Fractional-Order Control Systems: Fundamentals and Numerical Implementations (De Gruyter, Berlin, 2017)
X.S. Yang, Nature-Inspired Metaheuristic Algorithms (Luniver Press, Apache, 2008)
X. Yang, S. Deb, Cuckoo Search via Levy flights, in World Congress on Nature & Biologically Inspired Computing (NaBIC), pp. 210–214(2009). https://doi.org/10.1109/NABIC.2009.5393690
C. Zou, L. Zhang, X. Hu, Z. Wang, T. Wik, M. Pecht, A review of fractional-order techniques applied to lithium-ion batteries, lead-acid batteries, and supercapacitors. J. Power Sour. 390, 286–296 (2018)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A: Singular Values Decomposition of the Studied Examples
Appendix A: Singular Values Decomposition of the Studied Examples
Taking into account the HSVD depicted in Table 15, the reduced order can be calculated as follows:
-
(1)
Example 1: \(\sum _{i>2}^{n} \sigma _{i}= 0.0396< \sigma _{2}=0.7695\)
-
(2)
Example 2: \(\sum _{i>2}^{n} \sigma _{i}= 0.0772< \sigma _{2}=2.1219\)
-
(3)
Example 3: \(\sum _{i>2}^{n} \sigma _{i}= 0.01< \sigma _{2}=0.2420\)
-
(4)
Example 4: \(\sum _{i>2}^{n} \sigma _{i}= 0.0375< \sigma _{2}=0.0687\)
Therefore, according to (16) the suitable reduced order for the studied examples is r = 3.
Rights and permissions
About this article
Cite this article
Mouhou, A., Badri, A. Low Integer-Order Approximation of Fractional-Order Systems Using Grey Wolf Optimizer-Based Cuckoo Search Algorithm. Circuits Syst Signal Process 41, 1869–1894 (2022). https://doi.org/10.1007/s00034-021-01872-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-021-01872-w