A unified procedure for the identification of reduced-order fractional models based on the process reaction curve

Juan J. Gude; Oscar Camacho; Antonio Di Teodoro; Pablo García Bringas; Juan J. Gude; Oscar Camacho; Antonio Di Teodoro; Pablo García Bringas

doi:10.3934/math.2026653

AIMS Mathematics

2026, Volume 11, Issue 6: 15851-15886. doi: 10.3934/math.2026653

Previous Article Next Article

Research article Special Issues

A unified procedure for the identification of reduced-order fractional models based on the process reaction curve

1.
University of Deusto, Faculty of Engineering, Avda. de las Universidades, 24, Bilbao 48007, Bizkaia, Spain
2.
Universidad San Francisco de Quito, Colegio de Ciencias e Ingenierias, Avenida Diego de Robles y Vía Interoceánica, Quito 170157, Ecuador

Received: 31 January 2026 Revised: 19 April 2026 Accepted: 30 April 2026 Published: 05 June 2026
MSC : 26A33, 93B30, 93B40, 93C05, 93C15

This paper introduces a unified analytical method for the identification of fractional reduced-order models, specifically the fractional first-order plus dead time (FFOPDT) and fractional dual-pole plus dead time (FDPPDT) structures, using only three points from the open-loop step response. The method provides explicit parameter-estimation formulas that eliminate the need for iterative optimization, reducing computational effort while preserving the simplicity of traditional reaction-curve techniques. Numerical simulations demonstrate superior accuracy and robustness compared to existing analytical and hybrid techniques, especially for overdamped and S-shaped responses typical of thermal and chemical processes. The method is validated for fractional orders within the range $ \alpha \in [0.5, 1.0] $, covering the most relevant dynamics observed in practice. Laboratory experiments on a thermal system confirm the model's applicability under real-world conditions, including measurement noise, limited sensor resolution, and hardware constraints. Because the workflow aligns with standard industrial identification practices and does not require specialized knowledge of fractional calculus, it provides a practical means to incorporate fractional-order modeling into proportional-integral-derivative (PID)-based process control.
- fractional-order systems,
- process identification,
- reduced-order models,
- reaction curve,
- fractional PID control
Citation: Juan J. Gude, Oscar Camacho, Antonio Di Teodoro, Pablo García Bringas. A unified procedure for the identification of reduced-order fractional models based on the process reaction curve[J]. AIMS Mathematics, 2026, 11(6): 15851-15886. doi: 10.3934/math.2026653

Related Papers:

Abstract

This paper introduces a unified analytical method for the identification of fractional reduced-order models, specifically the fractional first-order plus dead time (FFOPDT) and fractional dual-pole plus dead time (FDPPDT) structures, using only three points from the open-loop step response. The method provides explicit parameter-estimation formulas that eliminate the need for iterative optimization, reducing computational effort while preserving the simplicity of traditional reaction-curve techniques. Numerical simulations demonstrate superior accuracy and robustness compared to existing analytical and hybrid techniques, especially for overdamped and S-shaped responses typical of thermal and chemical processes. The method is validated for fractional orders within the range $ \alpha \in [0.5, 1.0] $, covering the most relevant dynamics observed in practice. Laboratory experiments on a thermal system confirm the model's applicability under real-world conditions, including measurement noise, limited sensor resolution, and hardware constraints. Because the workflow aligns with standard industrial identification practices and does not require specialized knowledge of fractional calculus, it provides a practical means to incorporate fractional-order modeling into proportional-integral-derivative (PID)-based process control.

References

[1]	K. J. Åström, T. Hägglund, Advanced PID control, IEEE Contr. Syst., 26 (2006), 98–101. https://doi.org/10.1109/MCS.2006.1580160
[2]	A. O'dwyer, Handbook of PI and PID controller tuning rules, World Scientific, 2009. https://doi.org/10.1142/p575
[3]	H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[4]	I. Petráš, Volume 6 applications in control, In: Handbook of fractional calculus with applications, Boston: De Gruyter, 2019. https://doi.org/10.1515/9783110571745
[5]	C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, V. Feliu, Fractional-order systems and controls: fundamentals and applications, London: Springer, 2010. https://doi.org/10.1007/978-1-84996-335-0
[6]	I. Podlubny, Fractional differential equations, Academic Press, 1999.
[7]	L. Sadek, S. A. ldris, F. Jarad, The general caputo-katugampola fractional derivative and numerical approach for solving the fractional differential equations, Alex. Eng. J., 121 (2025), 539–557. https://doi.org/10.1016/j.aej.2025.02.065 doi: 10.1016/j.aej.2025.02.065
[8]	L. Sadek, A. Algefary, On quantum trigonometric fractional calculus, Alex. Eng. J., 120 (2025), 371–377. https://doi.org/10.1016/j.aej.2025.02.005 doi: 10.1016/j.aej.2025.02.005
[9]	K. Kothari, U. V. Mehta, R. Prasad, Fractional-order system modeling and its applications, J. Eng. Sci. Technol. Rev., 12 (2019), 6. https://doi.org/10.25103/jestr.126.01 doi: 10.25103/jestr.126.01
[10]	R. Almeida, N. R. Bastos, M. T. T. Monteiro, Modeling some real phenomena by fractional differential equations, Math. Method. Appl. Sci., 39 (2016), 4846–4855. https://doi.org/10.1002/mma.3818 doi: 10.1002/mma.3818
[11]	J. Sabatier, P. Lanusse, P. Melchior, A. Oustaloup, Fractional order differentiation and robust control design: CRONE, H-infinity and motion control, Springer, 2015. https://doi.org/10.1007/978-94-017-9807-5
[12]	M. Tavakoli-Kakhki, M. Haeri, M. S. Tavazoei, Simple fractional order model structures and their applications in control system design, Eur. J. Control, 16 (2010), 680–694. https://doi.org/10.3166/ejc.16.680-694 doi: 10.3166/ejc.16.680-694
[13]	Z. Nie, Q. Wang, R. Liu, Y. Lan, Identification and pid control for a class of delay fractional-order systems, IEEE/CAA J. Automatic., 3 (2016), 463–476. https://doi.org/10.1109/JAS.2016.7510103 doi: 10.1109/JAS.2016.7510103
[14]	J. J. Gude, P. García Bringas, Proposal of a general identification method for fractional-order processes based on the process reaction curve, Fractal Fract., 6 (2022), 526. https://doi.org/10.3390/fractalfract6090526 doi: 10.3390/fractalfract6090526
[15]	J. J. Gude, F. B. Baraldi, I. Oleagordia, P. García Bringas, Analytical fractional reduced-order model identification method for processes with overdamped and underdamped response, IFAC-PapersOnLine, 58 (2024), 191–196. https://doi.org/10.1016/j.ifacol.2024.08.188 doi: 10.1016/j.ifacol.2024.08.188
[16]	M. W. Campos, F. A. Ayres Jr, I. V. de Bessa, R. L. de Medeiros, P. R. Martins, E. kaminski Lenzi, et al., Fractional-order identification system based on sundaresan's technique, Chaos Soliton. Fract., 185 (2024), 115132. https://doi.org/10.1016/j.chaos.2024.115132 doi: 10.1016/j.chaos.2024.115132
[17]	A. Deb, A. Dasgupta, G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems, J. Franklin I., 343 (2006), 1–26. https://doi.org/10.1016/j.jfranklin.2005.06.005 doi: 10.1016/j.jfranklin.2005.06.005
[18]	S. K. Damarla, M. Kundu, A unified framework using orthogonal hybrid functions for solving linear and nonlinear fractional differential systems, AppliedMath, 5 (2025), 153. https://doi.org/10.3390/appliedmath5040153 doi: 10.3390/appliedmath5040153
[19]	Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216 (2010), 2276–2285. https://doi.org/10.1016/j.amc.2010.03.063 doi: 10.1016/j.amc.2010.03.063
[20]	Y. Tang, H. Liu, W. Wang, Q. Lian, X. Guan, Parameter identification of fractional order systems using block pulse functions, Signal Process., 107 (2015), 272–281. https://doi.org/10.1016/j.sigpro.2014.04.011 doi: 10.1016/j.sigpro.2014.04.011
[21]	Y. Li, N. Sun, Numerical solution of fractional differential equations using the generalized block pulse operational matrix, Comput. Math. Appl., 62 (2011), 1046–1054. https://doi.org/10.1016/j.camwa.2011.03.032 doi: 10.1016/j.camwa.2011.03.032
[22]	E. Guevara, H. Meneses, O. Arrieta, R. Vilanova, A. Visioli, F. Padula, Fractional order model identification: Computational optimization, In: 2015 IEEE 20th Conference on Emerging Technologies and Factory Automation (ETFA), 2015. https://doi.org/10.1109/ETFA.2015.7301630
[23]	B. B. Alagoz, A. Tepljakov, A. Ates, E. Petlenkov, C. Yeroglu, Time-domain identification of one noninteger order plus time delay models from step response measurements, Int. J. Model. Simul. Sc., 10 (2019), 1941011. https://doi.org/10.1142/S1793962319410113 doi: 10.1142/S1793962319410113
[24]	I. Fidalgo Astorquia, N. Gómez-Larrakoetxea, J. J. Gude, I. Pastor, Fractional-order system identification: Efficient reduced-order modeling with particle swarm optimization and ai-based algorithms for edge computing applications, Mathematics, 13 (2025), 1308. https://doi.org/10.3390/math13081308 doi: 10.3390/math13081308
[25]	D. Zamora-Arranz, P. Garcia-Bringas, J. J. Gude, J. Del Ser, Leveraging programmable logic controllers for machine learning applications in industrial setups, Results Eng., 2026, 110194.
[26]	J. J. Gude, P. García Bringas, M. Herrera, L. Rincón, A. Di Teodoro, O. Camacho, Fractional-order model identification based on the process reaction curve: A unified framework for chemical processes, Results Eng., 21 (2024), 101757. https://doi.org/10.1016/j.rineng.2024.101757 doi: 10.1016/j.rineng.2024.101757
[27]	J. J. Gude, O. Camacho, A. Di Teodoro, P. García Bringas, Analytical method for the identification of higher-order fractional systems using fractional dual-pole plus dead-time models, Results Eng., 26 (2025), 105574. https://doi.org/10.1016/j.rineng.2025.105574 doi: 10.1016/j.rineng.2025.105574
[28]	V. M. Alfaro, R. Vilanova, Control of high-order processes: Repeated-pole plus dead-time models' identification, Int. J. Control, 97 (2024), 141–151. https://doi.org/10.1080/00207179.2021.1954240 doi: 10.1080/00207179.2021.1954240
[29]	H. Meneses, O. Arrieta, F. Padula, A. Visioli, R. Vilanova, Fopi/fopid tuning rule based on a fractional order model for the process, Fractal Fract., 6 (2022), 478. https://doi.org/10.3390/fractalfract6090478 doi: 10.3390/fractalfract6090478
[30]	A. Di Teodoro, J. J. Gude, S. Vega, R. Chalco, R. Montaluisa, O. Camacho, Fractional-order hybrid 2-dof pid control with a complex basis: A novel framework for mimo systems, Asian J. Control, 28 (2026), 57–75. https://doi.org/10.1002/asjc.3781 doi: 10.1002/asjc.3781
[31]	A. Di Teodoro, M. Herrera, L. Rincon, J. J. Gude, O. Camacho, A hybrid control framework for chemical processes with long time delay: Theory and experiments, ACS omega, 9 (2024), 32469–32480. https://doi.org/10.1021/acsomega.3c10514 doi: 10.1021/acsomega.3c10514
[32]	J. J. Gude, A. Di Teodoro, D. Agudelo, M. Herrera, L. Rincón, O. Camacho, Sliding mode control design using a generalized reduced-order fractional model for chemical processes, Results Eng., 24 (2024), 103032. https://doi.org/10.1016/j.rineng.2024.103032 doi: 10.1016/j.rineng.2024.103032
[33]	T. Hägglund, Process control in practice, Boston: De Gruyter, 2023. https://doi.org/10.1515/9783111104959
[34]	A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, elsevier, 2006.
[35]	S. Samko, A. Kilbas, O. Marichev, Fractional integrals and derivatives: Theory and applications, 1993.
[36]	S. Rogosin, The role of the mittag-leffler function in fractional modeling, Mathematics, 3 (2015), 368–381. https://doi.org/10.3390/math3020368 doi: 10.3390/math3020368
[37]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Berlin: Springer, 2020. https://doi.org/10.1007/978-3-662-61550-8
[38]	H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-leffler functions and their applications, J. Appl. Math., 3 (2015), 298628. https://doi.org/10.1155/2011/298628 doi: 10.1155/2011/298628
[39]	J. J. Gude, A. Di Teodoro, O. Camacho, P. García Bringas, A new fractional reduced-order model-inspired system identification method for dynamical systems, IEEE Access, 11 (2023), 103214–103231. https://doi.org/10.1109/ACCESS.2023.3317230 doi: 10.1109/ACCESS.2023.3317230
[40]	J. Ceballos, N. Coloma, A. Di Teodoro, D. Ochoa-Tocachi, Generalized fractional cauchy-riemann operator associated with the fractional cauchy-riemann operator, Adv. Appl. Clifford Algebras, 30 (2020), 70. https://doi.org/10.1007/s00006-020-01096-2 doi: 10.1007/s00006-020-01096-2
[41]	N. Coloma, A. Di Teodoro, D. Ochoa-Tocachi, F. Ponce, Fractional elementary bicomplex functions in the riemann-liouville sense, Adv. Appl. Clifford Algebras, 31 (2021), 63. https://doi.org/10.1007/s00006-021-01165-0 doi: 10.1007/s00006-021-01165-0
[42]	K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
[43]	H. Aboukheir, J. Romero, A. Di Teodoro, An approach for fractional commensurate order youla parametrization using q-weighted operator, In: Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024), 2024,706–713. https://doi.org/10.5220/0013058700003822
[44]	J. J. Gude, P. García Bringas, Influence of the selection of reaction curve's representative points on the accuracy of the identified fractional-order model, J. Math., 265 (2002), 7185131. https://doi.org/10.1155/2022/7185131 doi: 10.1155/2022/7185131
[45]	J. J. Gude, P. García Bringas, Improving a reaction curve-based analytical identification technique for fractional models, Int. J. Dynam. Control, 13 (2025), 106. https://doi.org/10.1007/s40435-025-01604-x doi: 10.1007/s40435-025-01604-x
[46]	V. Romero Segovia, T. Hägglund, K. J. Åström, Measurement noise filtering for pid controllers, J. Process Contr., 24 (2014), 299–313. https://doi.org/10.1016/j.jprocont.2014.01.017 doi: 10.1016/j.jprocont.2014.01.017
[47]	J. J. Gude, P. García Bringas, A novel control hardware architecture for implementation of fractional-order identification and control algorithms applied to a temperature prototype, Mathematics, 11 (2023), 143. https://doi.org/10.3390/math11010143 doi: 10.3390/math11010143
[48]	K. Oprzedkiewicz, W. Mitkowski, E. Gawin, Application of fractional order transfer functions to modeling of high-order systems, In: 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 2015, 1169–1174. https://doi.org/10.1109/MMAR.2015.7284044
[49]	T. Hägglund, Signal filtering in pid control, IFAC Proceedings Volumes, 45 (2012), 1–10, 2012. https://doi.org/10.3182/20120328-3-IT-3014.00002 doi: 10.3182/20120328-3-IT-3014.00002

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)